= √2

**Table 1.** Reconfiguration parameters. **Table 1.** Reconfiguration parameters.


Since all the current demands in the stator frame are synchronous with the rotor motion, they can be expressed into a rotating frame by applying two new Clarke–Park transformations [9,55] (Figure 8): Since all the current demands in the stator frame are synchronous with the rotor motion, they can be expressed into a rotating frame by applying two new Clarke–Park transformations [9,55] (Figure 8):


**Figure 8.** Reference frame transformation: (**a**) normal Clarke–Park transformation, (**b**) new transformation with phase *a* isolated. **Figure 8.** Reference frame transformation: (**a**) normal Clarke–Park transformation, (**b**) new transformation with phase *a* isolated.

$$
\begin{bmatrix}
\dot{i}\_{df}^{\#} \\
\dot{i}\_{df}^{\#} \\
\dot{i}\_{zf}^{\#}
\end{bmatrix} = \mathbf{R}\_{\text{PC}f}(m)\mathbf{T}\_{\text{PC}af}(\theta\_{\varepsilon}) \begin{bmatrix}
\dot{i}\_{xf}^{\#} \\
\dot{i}\_{yf}^{\#} \\
\dot{i}\_{\eta}^{\#}
\end{bmatrix},
\tag{37}
$$

where () is a rotation matrix that generalizes the transformation matrix () related to the case of isolation of phase *a* [9]: where *RCP f*(*m*) is a rotation matrix that generalizes the transformation matrix *TCPa f*(*θe*) related to the case of isolation of phase *a* [9]:

$$\mathbf{T\_{CPf}}(\theta\_{\varepsilon}) = \begin{bmatrix} k\_2 \mathbf{s}(\theta\_{\varepsilon}) - k\_1 \mathbf{c}(\theta\_{\varepsilon}) & -k\_2 \mathbf{s}(\theta\_{\varepsilon}) - k\_1 \mathbf{c}(\theta\_{\varepsilon}) & k\_3 \mathbf{c}(\theta\_{\varepsilon}) \\ k\_1 \mathbf{s}(\theta\_{\varepsilon}) + k\_2 \mathbf{c}(\theta\_{\varepsilon}) & k\_1 \mathbf{s}(\theta\_{\varepsilon}) - k\_2 \mathbf{c}(\theta\_{\varepsilon}) & -k\_3 \mathbf{s}(\theta\_{\varepsilon}) \\ 0 & 0 & -1/\sqrt{3} \end{bmatrix},\tag{38}$$

$$\mathbf{R\_{CPf}}(m) = \begin{bmatrix} \mathbf{c}(m2\pi/3) & -\mathbf{s}(m2\pi/3) & 0 \\ \mathbf{s}(m2\pi/3) & \mathbf{c}(m2\pi/3) & 0 \\ 0 & 0 & 1 \end{bmatrix},$$

$$k\_1 = \sqrt{6}/\left(6 + 4\sqrt{3}\right), \quad k\_2 = 1/\sqrt{2}, \quad k\_3 = 2/\sqrt{3},\tag{39}$$

The RCFTC accommodation is finally completed by inverting the direct transfor-

mation matrix to calculate the reference voltages for the converter:

#

#

#

The RCFTC accommodation is finally completed by inverting the direct transformation matrix to calculate the reference voltages for the converter: � # # � = � ()� −1 � ()� T � # # �, (40)

$$
\begin{bmatrix} V\_{\rm xnf}^{\#} \\ V\_{\rm ynf}^{\#} \\ V\_{\rm nof}^{\#} \end{bmatrix} = \left( T\_{\rm CPaf}(\theta\_{\varepsilon}) \right)^{-1} \left( \mathbf{R}\_{\rm PCf}(m) \right)^{\mathrm{T}} \begin{bmatrix} V\_{df}^{\#} \\ V\_{df}^{\#} \\ V\_{zf}^{\#} \end{bmatrix} . \tag{40}
$$

22

⎥

where

where

$$\left(T\_{\mathbb{C}Paf}(\theta\_{\varepsilon})\right)^{-1} = \begin{bmatrix} \frac{\mathbf{s}(\theta\_{\varepsilon})}{2k\_{2}} - \frac{\mathbf{c}(\theta\_{\varepsilon})}{2k\_{1}} & \frac{\mathbf{s}(\theta\_{\varepsilon})}{2k\_{1}} + \frac{\mathbf{c}(\theta\_{\varepsilon})}{2k\_{2}} & -1\\ -\frac{\mathbf{s}(\theta\_{\varepsilon})}{2k\_{2}} - \frac{\mathbf{c}(\theta\_{\varepsilon})}{2k\_{1}} & \frac{\mathbf{s}(\theta\_{\varepsilon})}{2k\_{1}} - \frac{\mathbf{c}(\theta\_{\varepsilon})}{2k\_{2}} & -1\\ 0 & 0 & -\sqrt{3} \end{bmatrix} \tag{41}$$

21

The integration of the proposed FTC strategy within the motor closed-loop system is schematically depicted in Figure 9. The integration of the proposed FTC strategy within the motor closed-loop system is schematically depicted in Figure 9.

21

−1

⎢

22

**Figure 9.** PMSM closed-loop architecture with FTC strategy. **Figure 9.** PMSM closed-loop architecture with FTC strategy.

When the system is in normal condition, the conventional Clarke–Parke transformations are employed ( , −1) and the central point of the Y-connection is isolated ( <sup>1</sup> and <sup>2</sup> are not used). Once an ITSC is detected and isolated, the reference frame transformations of the RCFTC are employed ( , −1 ), and the reconfigured voltage references are sent to the four-leg converter to signal the central point ( <sup>1</sup> and <sup>2</sup>). When the system is in normal condition, the conventional Clarke–Parke transformations are employed (*TCP*, *T* −1 *CP* ) and the central point of the Y-connection is isolated (*Tn*<sup>1</sup> and *Tn*<sup>2</sup> are not used). Once an ITSC is detected and isolated, the reference frame transformations of the RCFTC are employed (*TCP<sup>f</sup>* , *T* −1 *CP<sup>f</sup>* ), and the reconfigured voltage references are sent to the four-leg converter to signal the central point (*Tn*<sup>1</sup> and *Tn*2).

#### **4. Results and Discussion 4. Results and Discussion**

### *4.1. Failure Transient Characterization 4.1. Failure Transient Characterization*

The effectiveness of the presented FTC strategy has been tested by using the nonlinear model of the propulsion system. The model is entirely developed in the MATLAB/Simulink environment, and its numerical solution is obtained via the fourth order Runge–Kutta method, using a 10−6 s integration step. It is worth noting that the choice of a fixed-step solver is not strictly related to the objectives of this work (in which the model is used for "off-line" simulations testing the FTC), but it has been selected for the next step of the project, when the FTC system will be implemented in the ECU boards via the automatic MATLAB compiler and executed in "real-time". The effectiveness of the presented FTC strategy has been tested by using the nonlinear model of the propulsion system. The model is entirely developed in the MATLAB/Simulink environment, and its numerical solution is obtained via the fourth order Runge–Kutta method, using a 10−<sup>6</sup> s integration step. It is worth noting that the choice of a fixed-step solver is not strictly related to the objectives of this work (in which the model is used for "off-line" simulations testing the FTC), but it has been selected for the next step of the project, when the FTC system will be implemented in the ECU boards via the automatic MATLAB compiler and executed in "real-time".

The closed-loop control is executed at a 20 kHz sampling rate and the maximum allowable fault latency has been set to 50 ms (for details, see Section 4.2). All the simulations started (*t* = 0 *s*) with a healthy PMSM, driving the propeller at 5800 rpm (UAV in straightand-level flight at sea level Table B1). The closed-loop control is executed at a 20 kHz sampling rate and the maximum allowable fault latency has been set to 50 ms (for details, see Section 4.2). All the simulations started (*t* = 0 s) with a healthy PMSM, driving the propeller at 5800 rpm (UAV in straightand-level flight at sea level Table A1).

The FTC strategy has been assessed by simulating the occurrence of an ITSC fault with *µ* = 0.5 on phase *a* at *t* = 150 ms. The failure transient is characterized by applying or not the proposed FTC and by comparing the responses with those in healthy conditions. As shown by Figure 10, though its relevant extension, the ITSC fault implies minor impacts on the propeller speed response during stationary operations (Figure 10a), even if the FTC application assures a faster recovery of the pre-fault speed value. On the other hand, the failure transient during unsteady operations is much more limited with FTC, even if smallamplitude ripples (at approximately 100 Hz) appear immediately after the accommodation (Figure 10b). These responses highlight the importance of applying the FTC for ITSC faults: since the fault effects are minor during stationary operations, its detection is very difficult, but the ITSC is typically unstable, and it progressively spreads along the phase windings if the coil is not isolated. The FTC strategy has been assessed by simulating the occurrence of an ITSC fault with ൌ 0.5 on phase *a* at *t* = 150 ms. The failure transient is characterized by applying or not the proposed FTC and by comparing the responses with those in healthy condi‐ tions. As shown by Figure 10, though its relevant extension, the ITSC fault implies minor impacts on the propeller speed response during stationary operations (Figure 10a), even if the FTC application assures a faster recovery of the pre‐fault speed value. On the other hand, the failure transient during unsteady operations is much more limited with FTC, even if small‐amplitude ripples (at approximately 100 Hz) appear immediately after the accommodation (Figure 10b). These responses highlight the importance of applying the FTC for ITSC faults: since the fault effects are minor during stationary operations, its de‐ tection is very difficult, but the ITSC is typically unstable, and it progressively spreads along the phase windings if the coil is not isolated. The FTC strategy has been assessed by simulating the occurrence of an ITSC fault with ൌ 0.5 on phase *a* at *t* = 150 ms. The failure transient is characterized by applying or not the proposed FTC and by comparing the responses with those in healthy condi‐ tions. As shown by Figure 10, though its relevant extension, the ITSC fault implies minor impacts on the propeller speed response during stationary operations (Figure 10a), even if the FTC application assures a faster recovery of the pre‐fault speed value. On the other hand, the failure transient during unsteady operations is much more limited with FTC, even if small‐amplitude ripples (at approximately 100 Hz) appear immediately after the accommodation (Figure 10b). These responses highlight the importance of applying the FTC for ITSC faults: since the fault effects are minor during stationary operations, its de‐ tection is very difficult, but the ITSC is typically unstable, and it progressively spreads along the phase windings if the coil is not isolated.

**Figure 10.** Propeller speed tracking with ITSC ( ൌ 0.5) on phase *a* at *t* = 150 ms: (**a**) stationary and (**b**) unsteady operations. **Figure 10.** Propeller speed tracking with ITSC (*µ* = 0.5) on phase *a* at *t* = 150 ms: (**a**) stationary and (**b**) unsteady operations. **Figure 10.** Propeller speed tracking with ITSC ( ൌ 0.5) on phase *a* at *t* = 150 ms: (**a**) stationary and (**b**) unsteady operations.

The failure transient in terms of motor torque is then reported in Figure 11. It can be noted that, if the FTC is not applied, the post‐fault behaviour is characterized, during both stationary and unsteady operations, by relevant high‐frequency ripples (at approximately 1 kHz, i.e., twice the electrical frequency of the motor). In particular, during stationary operations, the FTC permits to rapidly restore the pre‐fault torque level, by eliminating high‐frequency loads that would inevitably cause damages at mechanical and electrical parts (Figure 11a). The failure transients in terms of phase currents are then reported in Figure 12, when the FTC is applied. The fault generates a short circuit current (, Figure 12) causing the loss of symmetry of the three‐phase system. Thanks to the RCFTC accom‐ modation, the phase *a* is disengaged and the fourth leg of the converter is activated: this action stops the short circuit current and opens a current path through the central point (, Figure 12). The failure transient in terms of motor torque is then reported in Figure 11. It can be noted that, if the FTC is not applied, the post-fault behaviour is characterized, during both stationary and unsteady operations, by relevant high-frequency ripples (at approximately 1 kHz, i.e., twice the electrical frequency of the motor). In particular, during stationary operations, the FTC permits to rapidly restore the pre-fault torque level, by eliminating high-frequency loads that would inevitably cause damages at mechanical and electrical parts (Figure 11a). The failure transients in terms of phase currents are then reported in Figure 12, when the FTC is applied. The fault generates a short circuit current (*i f* , Figure 12) causing the loss of symmetry of the three-phase system. Thanks to the RCFTC accommodation, the phase *a* is disengaged and the fourth leg of the converter is activated: this action stops the short circuit current and opens a current path through the central point (*in*, Figure 12). The failure transient in terms of motor torque is then reported in Figure 11. It can be noted that, if the FTC is not applied, the post‐fault behaviour is characterized, during both stationary and unsteady operations, by relevant high‐frequency ripples (at approximately 1 kHz, i.e., twice the electrical frequency of the motor). In particular, during stationary operations, the FTC permits to rapidly restore the pre‐fault torque level, by eliminating high‐frequency loads that would inevitably cause damages at mechanical and electrical parts (Figure 11a). The failure transients in terms of phase currents are then reported in Figure 12, when the FTC is applied. The fault generates a short circuit current (, Figure 12) causing the loss of symmetry of the three‐phase system. Thanks to the RCFTC accom‐ modation, the phase *a* is disengaged and the fourth leg of the converter is activated: this action stops the short circuit current and opens a current path through the central point (, Figure 12).

**Figure 11.** Normalized motor torque with ITSC ( ൌ 0.5) on phase *a* at *t* = 150 ms: (**a**) stationary (ೌೣ ൎ 12 ) and (**b**) unsteady operations (ೌೣ ൎ 15 ). **Figure 11.** Normalized motor torque with ITSC ( ൌ 0.5) on phase *a* at *t* = 150 ms: (**a**) stationary (ೌೣ ൎ 12 ) and (**b**) unsteady operations (ೌೣ ൎ 15 ). **Figure 11.** Normalized motor torque with ITSC (*µ* = 0.5) on phase *a* at *t* = 150 ms: (**a**) stationary (*Qmmax* ≈ 12 Nm) and (**b**) unsteady operations (*Qmmax* ≈ 15 Nm).

*Aerospace* **2022**, *8*, x FOR PEER REVIEW 16 of 25

**Figure 12.** Normalized phase currents with ITSC ( ൌ 0.5) on phase *a* at *t* = 150 ms (௦௧ ൌ 80): (**a**) stationary (**b**) unsteady operations. **Figure 12.** Normalized phase currents with ITSC (*µ* = 0.5) on phase *a* at *t* = 150 ms (*Isat* = 80 A): (**a**) stationary (**b**) unsteady operations. **Figure 12.** Normalized phase currents with ITSC ( ൌ 0.5) on phase *a* at *t* = 150 ms (௦௧ ൌ 80): (**a**) stationary (**b**) unsteady operations. **Figure 12.** Normalized phase currents with ITSC ( ൌ 0.5) on phase *a* at *t* = 150 ms (௦௧ ൌ 80): (**a**) (a) (b)

The current phasor trajectories in the Clarke space are shown in Figure 13. It can be noted that, when an ITSC fault is injected in phase *a*, the trajectories are not strictly ellip‐ tical. This depends on the fact that the faulty currents are not perfectly sinusoidal, but they also contain higher harmonic contents. The phenomenon is caused by the phase voltages saturation. Despite the presence of these higher harmonic contents, the ellipse fitting tech‐ nique successfully operates by demonstrating the relevant robustness of the FDI algo‐ rithm. On the other hand, when the fault is accommodated, the trajectory involves the neutral axis too, in such a way that, its projection on the , plane overlaps the healthy circular trajectory. Finally, the geometrical parameters of the current phasor elliptical tra‐ jectory (semi‐axes lengths and major axis inclination) are plotted in Figure 14. It can be noted that, when the FTC intervenes, the projection of the current trajectory on the , plane becomes circular (ெ⁄ ൌ 1) and the ellipse inclination returns to zero. The current phasor trajectories in the Clarke space are shown in Figure 13. It can be noted that, when an ITSC fault is injected in phase *a*, the trajectories are not strictly elliptical. This depends on the fact that the faulty currents are not perfectly sinusoidal, but they also contain higher harmonic contents. The phenomenon is caused by the phase voltages saturation. Despite the presence of these higher harmonic contents, the ellipse fitting technique successfully operates by demonstrating the relevant robustness of the FDI algorithm. On the other hand, when the fault is accommodated, the trajectory involves the neutral axis too, in such a way that, its projection on the *α*, *β* plane overlaps the healthy circular trajectory. Finally, the geometrical parameters of the current phasor elliptical trajectory (semi-axes lengths and major axis inclination) are plotted in Figure 14. It can be noted that, when the FTC intervenes, the projection of the current trajectory on the *α*, *β* plane becomes circular (*sM*/*s<sup>m</sup>* = 1) and the ellipse inclination returns to zero. The current phasor trajectories in the Clarke space are shown in Figure 13. It can be noted that, when an ITSC fault is injectedin phase *a*, the trajectories are not strictly ellip‐ tical. This depends on the fact that the faulty currents are not perfectly sinusoidal, but they also contain higher harmonic contents. The phenomenon is caused by the phase voltages saturation. Despite the presence of these higher harmonic contents, the ellipse fitting tech‐ nique successfully operates by demonstrating the relevant robustness of the FDI algo‐ rithm. On the other hand, when the fault is accommodated, the trajectory involves the neutral axis too, in such a way that, its projection on the , plane overlaps the healthy circular trajectory. Finally, the geometrical parameters of the current phasor elliptical tra‐ jectory (semi‐axes lengths and major axis inclination) are plotted in Figure 14. It can be noted that, when the FTC intervenes, the projection of the current trajectory on the , plane becomes circular (ெ⁄ ൌ 1) and the ellipse inclination returns to zero. stationary (**b**) unsteady operations. The current phasor trajectories in the Clarke space are shown in Figure 13. It can be noted that, when an ITSC fault is injected in phase *a*, the trajectories are not strictly ellip‐ tical. This depends on the fact that the faulty currents are not perfectly sinusoidal, but they also contain higher harmonic contents. The phenomenon is caused by the phase voltages saturation. Despite the presence of these higher harmonic contents, the ellipse fitting tech‐ nique successfully operates by demonstrating the relevant robustness of the FDI algo‐ rithm. On the other hand, when the fault is accommodated, the trajectory involves the neutral axis too, in such a way that, its projection on the , plane overlaps the healthy circular trajectory. Finally, the geometrical parameters of the current phasor elliptical tra‐ jectory (semi‐axes lengths and major axis inclination) are plotted in Figure 14. It can be noted that, when the FTC intervenes, the projection of the current trajectory on the ,

**Figure 13.** Current phasor trajectory in Clarke space with ITSC ( ൌ 0.5) on phase *a* (௦௧ ൌ 80): (**a**) stationary (**b**) unsteady operations. **Figure 13.** Current phasor trajectory in Clarke space with ITSC ( ൌ 0.5) on phase *a* (௦௧ ൌ 80): (**a**) stationary (**b**) unsteady operations. **Figure 13.** Current phasor trajectory in Clarke space with ITSC (*µ* = 0.5) on phase *a* (*Isat* = 80 A): (**a**) stationary (**b**) unsteady operations. **Figure 13.** Current phasor trajectory in Clarke space with ITSC ( ൌ 0.5) on phase *a* (௦௧ ൌ 80): (**a**) stationary (**b**) unsteady operations.

operations. **Figure 14.** Ellipse parameters with ITSC ( ൌ 0.5) on phase *a* at *t* = 40 ms: (**a**) stationary (**b**) unsteady operations. **Figure 14.** Ellipse parameters with ITSC ( ൌ 0.5) on phase *a* at *t* = 40 ms: (**a**) stationary (**b**) unsteady operations. **Figure 14.** Ellipse parameters with ITSC (*µ* = 0.5) on phase *a* at *t* = 40 ms: (**a**) stationary (**b**) unsteady operations.

### *4.2. FDI Parameters Definition 4.2. FDI Parameters Definition* As described in Section 3.2, the design of the FDI algorithm requires the definition of

As described in Section 3.2, the design of the FDI algorithm requires the definition of four parameters, i.e., *εdth*, *εith*, *nth*, and *n*. Concerning the ellipse fitting samples *n*, this parameter has been defined by pursuing a good balance between the trajectory reconstruction accuracy and the FDI latency, both increasing when *n* increases. Considering that the propeller speed tracking bandwidth is approximately 15 Hz, the maximum allowable FDI latency has been set to 50 ms, which could be reasonably targeted by an equivalent monitoring frequency of 500 Hz. Being the sampling rate of the sensor system at 20 kHz, *n* = 40 has been imposed. The values of the remaining parameters (*εdth*, *εith*, *nth*) have been instead defined via time domain simulations, aiming to obtain correct ITSC FDI with very limited extension (*µ* = 0.1). Two FDI design simulations have been performed, Figure 15: four parameters, i.e., ௗ௧, ௧, ௧, and . Concerning the ellipse fitting samples , this parameter has been defined by pursuing a good balance between the trajectory recon‐ struction accuracy and the FDI latency, both increasing when increases. Considering that the propeller speed tracking bandwidth is approximately 15 Hz, the maximum al‐ lowable FDI latency has been set to 50 ms, which could be reasonably targeted by an equiv‐ alent monitoring frequency of 500 Hz. Being the sampling rate of the sensor system at 20 kHz, ൌ 40 has been imposed. The values of the remaining parameters (ௗ௧, ௧, ௧) have been instead defined via time domain simulations, aiming to obtain correct ITSC FDI with very limited extension ( ൌ 0.1). Two FDI design simulations have been performed, Figure 15:

• Simulation 1: cruise speed hold, Simulation 1: cruise speed hold,

• Simulation 2: maximum speed ramp demand. Simulation 2: maximum speed ramp demand.

*Aerospace* **2022**, *8*, x FOR PEER REVIEW 17 of 25

**Figure 15.** Propeller speed demands imposed for FDI design simulations (ITSC with ൌ 0.1 at *t* = 150 ms). **Figure 15.** Propeller speed demands imposed for FDI design simulations (ITSC with *µ* = 0.1 at *t* = 150 ms). *Aerospace* **2022**, *8*, x FOR PEER REVIEW 18 of 25

**Figure 16.** FDI signals during FDI design simulations (ITSC with ൌ 0.1 at *t* = 150 ms). **Figure 16.** FDI signals during FDI design simulations (ITSC with *µ* = 0.1 at *t* = 150 ms).

*4.3. Critical Comparison with Other ITSC FDI Methods* A comparative analysis of the proposed FDI method with the most relevant ones de‐ veloped in the literature (Table 2) has been carried out by using the list of capabilities given in Table 3. The results are reported in Table 4. **Table 2.** Relevant FDI methods for ITSC faults. The FDI parameter plotted in Figure 16 shows that low values of *εdth* would clearly imply a too-sensitive algorithm, with high counts *n<sup>c</sup>* during unsteady operations and in turn high values of *nth* to avoid false alarms. On the other hand, high values of *εdth* will make the algorithm too robust to false alarms but it will reduce the effectiveness in detecting ITSC at an early stage. In the proposed case study, a good balance between the robustness against false alarms and the effectiveness in diagnosing ITSC higher than *µ* = 0.1 can be

**Acronym Method Reference**

M3 PVA [31] M4 CNN [39] M5 APVA Present work

(*latency time*).

**Method**

**Capability M1 M2 M3 M4 M5** C1 Yes No No Yes Yes C2 No No Yes No Yes C3 No Yes No Not provided Yes

**Acronym Method**

**Table 4.** Comparison of FDI methods for ITSC faults.

C1 Able to detect the faulty phase. C2 Insensitive to operating loads. C3 Robust against speed changes. C4 Robust against current waveform. C5 Minimum number of detected shorted turns.

C6 Electrical periods for FDI

C7 Real‐time computation. C8 Tuning simplicity.

**Table 3.** FDI capabilities for ITSC faults.

obtained by selected *εdth* = 0.6 A and *nth* = 20, which guarantees an FDI latency lower than the 50 ms among the two cases reported in Figure 15.

## *4.3. Critical Comparison with Other ITSC FDI Methods*

A comparative analysis of the proposed FDI method with the most relevant ones developed in the literature (Table 2) has been carried out by using the list of capabilities given in Table 3. The results are reported in Table 4.



**Table 3.** FDI capabilities for ITSC faults.


**Table 4.** Comparison of FDI methods for ITSC faults.


The FFT methods are robust against noise, but they are not applicable in unsteady operations. They can be enhanced by applying HHT transforms, but the capability to detect the faulty phase is lost. Similarly, the main disadvantage of the PVA methods is the inability to locate the fault. The CNN method is competitive in terms of the ITSC location, but their robustness in unsteady operations has not been proved; furthermore, the design of the neural detector requires a complex design and training process. On the other hand, the proposed APVA method demonstrates excellent capabilities in unsteady operations, it succeeds in fault location and detects ITSC faults with a very limited extension (10%, less than four turns) independently from the load level. All the methods considered are capable

of detecting incipient faults, but many of them lack enough information for a comparison in terms of detection and isolation latency.

## **5. Conclusions**

A novel FTC strategy for a high-speed PMSM with a four-leg converter employed for UAV propulsion is developed and characterized in terms of FDI and fault accommodation capabilities. The FTC performances are assessed via dynamic simulation by using a detailed nonlinear model of the electric propulsion system, which includes a physically based modelling of ITSC faults. For the proposed FTC, an original FDI algorithm is developed and applied, based on an innovative current signature technique, which uses as fault symptoms the geometrical parameters of the elliptical trajectory of the currents phasor in the Clarke plane (e.g., major and minor axes lengths, and major axis inclination). In addition, a theoretical analysis is carried out to support the FDI algorithms, by demonstrating that the major axis inclination can be used as a symptom for the faulty phase identification.

A comparative analysis with other ITSC FDI methods from the literature is also carried out. If compared with neural networks methods, which exhibit the best sensibility to incipient faults (a shorted turn is isolated within 10 electrical periods), the proposed technique behaves slightly worse (four shorted turns, corresponding to a 10% extension along the coil, and are isolated within 20 electrical periods). Nevertheless, the neural methods are more complicated to be tuned (due to the training process), while the proposed method only requires tuning four parameters. Finally, the paper demonstrates that a fault accommodation based on the RCFTC technique succeeds in minimizing the failure transient and eliminates the high-frequency torque ripples induced by the fault.

**Author Contributions:** Conceptualization, methodology and investigation, A.S. and G.D.R.; software, data curation and writing—original draft preparation, A.S.; validation, formal analysis and writing review and editing, G.D.R.; resources, supervision and visualization, G.D.R. and R.G.; project administration and funding acquisition, R.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was co-funded by the Italian Government (*Ministero Italiano dello Sviluppo Economico*, MISE) and by the Tuscany Regional Government, in the context of the R&D project "*Tecnologie Elettriche e Radar per SAPR Autonomi* (TERSA)", Grant number: F/130088/01-05/X38.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors wish to thank Luca Sani, from the University of Pisa (*Dipartimento di Ingegneria dell'Energia, dei Sistemi, del Territorio e delle Costruzioni*), for the support in the definition of the PMSM model parameters, and Francesco Schettini, from Sky Eye Systems (Italy), for the support in the definition of the UAV propeller loads.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Appendix A.**

The general ellipse expression is defined by an implicit second-order polynomial with specific constraints on coefficients.

$$\begin{cases} A\alpha^2 + B\alpha\beta + C\beta^2 + D\alpha + E\beta + F = 0\\ B^2 - 4AC < 0 \end{cases} \tag{A1}$$

in which *A*, *B*, *C*, *D*, *E*, and *F* are the ellipse coefficients, while *α* and *β* are the Cartesian coordinates of the ellipse points. The related vectorial conic definition is:

$$
\Gamma \cdot \gamma = 0,\tag{A2}
$$

where **Γ** = - *α* 2 , *αβ*, *β* 2 , *α*, *β*, 1 and *γ* = [*A*, *B*, *C*, *D*, *E*, *F*] T .

The ellipse fitting to a set of coordinate points (*α<sup>i</sup>* , *βi*), coming from *i* monitoring measurements (where *i* = 1, . . . , *n*, and *n* is greater than the number of conic coefficients, i.e., *n* > 6) is an over-determined problem, which can be approached by minimizing the sum of distances of the points (*α<sup>i</sup>* , *βi*) to the conic represented by coefficients *γ*:

$$\begin{cases} \min\_{a} \sum\_{i=1}^{n} \left(\Gamma\_{i} \cdot \gamma\right)^{2} \\ B^{2} - 4AC < 0 \end{cases} \tag{A3}$$

Due to the constraint, the problem cannot be solved directly with a conventional least-square approach. However, Fitzgibbon [53] showed that under a proper scaling, the inequality in Equation (A3) can be changed into an equality constraint as,

$$\begin{cases} \min\_{a} \sum\_{i=1}^{n} (\Gamma\_{i} \cdot \gamma)^{2} \\ 4AC - B^{2} = 1 \end{cases} \tag{A4}$$

This minimization problem can be conveniently formulated as:

$$\begin{cases} \min \left\| \mathbb{D} \gamma \right\|^2\\ \gamma^{\mathrm{T}} \mathbb{C} \gamma = 1 \end{cases} \tag{A5}$$

where D and C are known as the design and constraint matrices, respectively, defined as

$$\mathbb{D} = \begin{bmatrix} a\_1^2 & a\_1\beta\_1 & \beta\_1^2 & a\_1 & \beta\_1 & 1\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ a\_n^2 & a\_n\beta\_n & \beta\_n^2 & a\_n & \beta\_n & 1 \end{bmatrix}, \quad \mathbb{C} = \begin{bmatrix} 0 & 0 & 2 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \tag{A6}$$

and it can be solved as a quadratically-constrained least-squares minimization by applying Lagrange multipliers, as:

$$\begin{cases} \mathcal{S}\gamma = \lambda \mathcal{C}\gamma\\ \gamma^\mathrm{T}\mathcal{C}\gamma = 1 \end{cases} \tag{A7}$$

where S is the *scatter matrix*, defined as:

$$\mathbb{S} = \mathbb{D}^{\mathsf{T}} \mathbb{D},\tag{A8}$$

The optimal solution of Equation (A7) is the eigenvector corresponding to the minimum positive eigenvalue *λ<sup>k</sup>* **.** It is worth noting that the matrix C is singular, and S is also singular if all data points lie exactly on an ellipse. Because of that, the computation of eigenvalues is numerically unstable and it can produce wrong results (as infinite or complex numbers). To overcome the drawback, Halìˇr [52] suggested to partition the C and S matrices. The constraint matrix is defined as:

$$\mathbb{C} = \begin{bmatrix} \mathbb{C}\_1 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \tag{A9}$$

and

$$\mathbb{C}\_1 = \begin{bmatrix} 0 & 0 & 2 \\ 0 & -1 & 0 \\ 2 & 0 & 0 \end{bmatrix} \, \tag{A10}$$

The partition of matrix S is obtained by splitting the matrix D into its quadratic and linear parts:

$$\mathbb{D} = \{ \mathbb{D}\_1 \: \mathbb{D}\_2 \},\tag{A11}$$

where

$$\mathbb{D}\_1 = \begin{bmatrix} \alpha\_1^2 & \alpha\_1 \beta\_1 & \beta\_1^2 \\ \vdots & \vdots & \vdots \\ \alpha\_n^2 & \alpha\_n \beta\_n & \beta\_n^2 \end{bmatrix}, \quad \mathbb{D}\_2 = \begin{bmatrix} \alpha\_1 & \beta\_1 & 1 \\ \vdots & \vdots & \vdots \\ \alpha\_n & \beta\_n & 1 \end{bmatrix}. \tag{A12}$$

Then, the scatter matrix is constructed as:

$$\mathbb{S} = \begin{bmatrix} \mathbb{S}\_1 & \mathbb{S}\_2 \\ \mathbb{S}\_2^T & \mathbb{S}\_3 \end{bmatrix}' \tag{A13}$$

in which

$$\mathbb{S}\_1 = \mathbb{D}\_1^\mathrm{T} \mathbb{D}\_1, \quad \mathbb{S}\_2 = \mathbb{D}\_1^\mathrm{T} \mathbb{D}\_2, \quad \mathbb{S}\_3 = \mathbb{D}\_2^\mathrm{T} \mathbb{D}\_2. \tag{A14}$$

Similarly, the coefficients vector is partitioned as:

$$\boldsymbol{\gamma} = \begin{bmatrix} \gamma \ \boldsymbol{\gamma} \end{bmatrix}^{\mathrm{T}} \text{.}\tag{A15}$$

where

$$
\gamma\_1 = \begin{bmatrix} A & B & C \end{bmatrix}^\mathrm{T}, \quad \gamma\_2 = \begin{bmatrix} D & E & F \end{bmatrix}^\mathrm{T}. \tag{A16}
$$

Based on this decomposition, Equation (A7) can be written as:

$$\begin{cases} \mathbb{S}\_1 \gamma\_1 + \mathbb{S}\_2 \gamma\_2 = \lambda \mathbb{C}\_1 a\_1 \\ \mathbb{S}\_2^\mathrm{T} \gamma\_1 + \mathbb{S}\_3 \gamma\_2 = \mathbf{0} \\ \gamma\_1^\mathrm{T} \mathbb{C}\_1 \gamma\_1 = 1 \end{cases} \,, \tag{A17}$$

Considering that matrix S<sup>3</sup> is singular only if all the points lie on a line [52], the second of equation in Equation (A17) can be solved to obtain *γ*2. By substituting in Equation (A17), and by considering that C<sup>1</sup> is not singular, we have:

$$\begin{cases} \mathbb{M}\gamma\_1 = \lambda\gamma\_1\\ \gamma\_1^\mathrm{T}\mathbb{C}\_1\gamma\_1 = 1\\ \gamma\_2 = -\mathbb{S}\_3^{-1}\mathbb{S}\_2^\mathrm{T}\gamma\_1\\ \gamma = \left(\gamma\_1\ \gamma\_2\right)^\mathrm{T} \end{cases} \tag{A18}$$

in which M is the reduced scatter matrix:

$$\mathbb{C}\mathbb{M} = \mathbb{C}\_1^{-1} \left( \mathbb{S}\_1 - \mathbb{S}\_2 \mathbb{S}\_3^{-1} \mathbb{S}\_2^{\mathrm{T}} \right). \tag{A19}$$

The optimal solution corresponds to the eigenvector *γ* that yields a minimal nonnegative eigenvalue *λ*. Once obtained *γ*, the lengths of major and minor semi-axes *s<sup>M</sup>* and *s<sup>m</sup>* are [54]:

$$s\_{M,m} = \frac{\sqrt{\frac{2\left(A\mathcal{E}^2 + \mathcal{C}\mathcal{D}^2 - \mathcal{B}\mathcal{D}\mathcal{E} + (\mathcal{B}^2 - 4\mathcal{A}\mathcal{C})F\right)}{\left(A + \mathcal{C} \pm \sqrt{\left(A - \mathcal{C}\right)^2 + \mathcal{B}^2}\right)^{-1}}}{4A\mathcal{C} - \mathcal{B}^2}.\tag{A20}$$

while the major axis inclination *ϕel* is [54]:

$$\varphi\_{el} = \begin{cases} 0 & \text{for B = 0, } A < \mathbb{C} \\ \pi/2 & \text{for B = 0, } A > \mathbb{C} \\ 1/2\cot^{-1}((A - \mathbb{C})/\mathcal{B}) & \text{for B \neq 0, } A < \mathbb{C} \\ \pi/2 + 1/2\cot^{-1}((A - \mathbb{C})/\mathcal{B}) & \text{for B \neq 0, } A > \mathbb{C} \end{cases} \tag{A21}$$

## **Appendix B.**

This section contains tables reporting the parameters of the UAV propeller (Table A1), the simulation model of the propulsion system (Table A2), and the design parameters of the FTC system (Table A3).


**Table A2.** System model parameters.


**Table A3.** FTC Algorithm parameters.


## **References**


## *Article* **Application of Deep Reinforcement Learning in Reconfiguration Control of Aircraft Anti-Skid Braking System**

**Shuchang Liu <sup>1</sup> , Zhong Yang <sup>1</sup> , Zhao Zhang 2,\* , Runqiang Jiang <sup>2</sup> , Tongyang Ren <sup>2</sup> , Yuan Jiang <sup>2</sup> , Shuang Chen <sup>3</sup> and Xiaokai Zhang <sup>3</sup>**


**Abstract:** The aircraft anti-skid braking system (AABS) plays an important role in aircraft taking off, taxiing, and safe landing. In addition to the disturbances from the complex runway environment, potential component faults, such as actuators faults, can also reduce the safety and reliability of AABS. To meet the increasing performance requirements of AABS under fault and disturbance conditions, a novel reconfiguration controller based on linear active disturbance rejection control combined with deep reinforcement learning was proposed in this paper. The proposed controller treated component faults, external perturbations, and measurement noise as the total disturbances. The twin delayed deep deterministic policy gradient algorithm (TD3) was introduced to realize the parameter self-adjustments of both the extended state observer and the state error feedback law. The action space, state space, reward function, and network structure for the algorithm training were properly designed, so that the total disturbances could be estimated and compensated for more accurately. The simulation results validated the environmental adaptability and robustness of the proposed reconfiguration controller.

**Keywords:** aircraft anti-skid braking system; actuator faults; reconfiguration control; linear activedisturbance rejection control; deep reinforcement learning; twin delayed deep deterministic policy gradient algorithm

## **1. Introduction**

The aircraft anti-skid braking system (AABS) is an essential airborne utilities system to ensure the safe and smooth landing of aircraft [1]. With the development of aircraft towards high speed and large tonnage, the performance requirements of AABS are increasing. Moreover, AABS is a complex system with strong nonlinearity, strong coupling, and timevarying parameters, and is sensitive to the runway environment [2]. These characteristics make AABS controller design an interesting and challenging topic.

The most widely used control method in practice is PID + PBM, which is a speed differential control law. However, it suffers from low-speed slipping and underutilization of ground bonding forces, making it difficult to meet high performance requirements. To this end, researchers have proposed many advanced control methods to improve the AABS performance, such as mixed slip deceleration PID control [3], model predictive control [4], extremum-seeking control [5], sliding mode control [6], reinforcement Q-learning control [7], and so on. Zhang et al. [8] proposed a feedback linearization controller with a prescribed performance function to ensure the transient and steady-state braking performance. Qiu et al. [9] combined backstepping dynamic surface control with an asymmetric barrier Lyapunov function to obtain a robust tracking response in the presence of disturbance

**Citation:** Liu, S.; Yang, Z.; Zhang, Z.; Jiang, R.; Ren, T.; Jiang, Y.; Chen, S.; Zhang, X. Application of Deep Reinforcement Learning in Reconfiguration Control of Aircraft Anti-Skid Braking System. *Aerospace* **2022**, *9*, 555. https://doi.org/ 10.3390/aerospace9100555

Academic Editor: Gianpietro Di Rito

Received: 11 July 2022 Accepted: 19 September 2022 Published: 26 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and runway surface transitions. Mirzaei et al. [10] developed a fuzzy braking controller optimized by a genetic algorithm and introduced an error-based global optimization approach for fast convergence near the optimum point. The above-mentioned works provide an in-depth study on AABS control; however, the adverse effects caused by typical component faults such as actuator faults are neglected. Since most AABS are designed based on hydraulic control systems, the long hydraulic pipes create an enormous risk of air mixing with oil, and internal leakage. Without regular maintenance, it is easy to cause functional degradation or even failure, which raises many security concerns [11,12]. How to ensure the stability and the acceptable braking performance of AABS after actuator faults becomes a key issue.

In order to actually improve the safety and reliability of AABS, the fault probability can be reduced by reliability design and redundant technology on the one hand [13]. However, due to the production factors (cost/weight/technological level), the redundancy of aircraft components is so limited that the system reliability is hard to increase. On the other hand, fault-tolerant control (FTC) technology can be introduced into the AABS controller design, which is the future development direction of AABS and the key technology that needs urgent attention [14]. Reconfiguration control is a popular branch of FTC that has been widely used in many safety-critical systems, especially in aerospace engineering [15,16]. The essence of reconfiguration control is to consider the possible faults of the plant in the controller design process. When component faults occur, the fault system information is used to reconfigure the controller structure or parameters automatically [17]. In this way, the adverse effects caused by faults can be restrained or eliminated, thus realizing an asymptotically stable and acceptable performance of the closed-loop system. A number of common reconfiguration control methods can be classified as follows: adaptive control [18,19], multi-model switching control [20], sliding mode control [21], fuzzy control [22], other robust control [23], etc. In addition, the characteristics of AABS increase the difficulty of accurate modeling, and many nonlinear reconfiguration control methods are complex and relatively hard to apply in engineering. Therefore, it is crucial to design a reconfiguration controller with a clear structure, and which is model-independent, strong fault-perturbation resistant, and easy to implement.

Han retained the essence of PID control and proposed an active disturbance rejection control (ADRC) technique that requires low model accuracy and shows good control performance [24]. ADRC can estimate disturbances in internal and external systems and compensate for them [25]. Furthermore, ADRC has been widely used in FTC system design because of its obvious advantages in solving control problems of nonlinear models with uncertainty and strong disturbances [26–28]. Although the structure is not difficult to implement with modern digital computer technology, ADRC needs to tune a bunch of parameters which makes it hard to use in practice [29]. To overcome the difficulty, Gao proposed linear active disturbance rejection control (LADRC), which is based on linear extended state observer (LESO) and linear state error feedback (LSEF) [30,31]. The bandwidth tuning method greatly reduced the number of LADRC parameters. LADRC has been applied to solve various control problems [32–34].

However, it is well known that a controller with fixed parameters may not be able to maintain the acceptable (rated or degraded) performance of a fault system. For this reason, some advanced algorithms with parameter adaptive capabilities have been introduced by researchers that further improve the robustness and environmental adaptability of ADRC, such as neural networks [35,36], fuzzy logic [37,38], and the sliding mode [39,40]. With the development of artificial intelligence techniques, reinforcement learning has been applied to control science and engineering [41,42], and good results have been achieved. Yuan et al. proposed a novel online control algorithm for a thickener which is based on reinforcement learning [43]. Pang et al. studied the infinite-horizon adaptive optimal control of continuous-time linear periodic systems, using reinforcement learning techniques [44]. A Q-learning-based adaptive method for ADRC parameters was proposed by Chen et al. and has been applied to the ship course control [45].

Motivated by the above observations, in this paper, a reconfiguration control scheme via LADRC combined with deep reinforcement learning was developed for AABS which is subject to various fault perturbations. The proposed reconfiguration control method is a remarkable control strategy compared to previous methods for three reasons:

(1) AABS is extended with a new state variable, which is the sum of all unknown dynamics and disturbances not noticed in the fault-free system description. This state variable can be estimated using LESO. It indirectly simplifies the AABS modeling;

(2) Artificial intelligence technology is introduced and combined with the traditional control method to solve special control problems. By combining LADRC with the deep reinforcement learning TD3 algorithm, the selection of controller parameters is equivalent to the choice of agent actions. The parameter adaptive capabilities of LESO and LSEF are endowed through the continuous interaction between the agent and the environment, which not only eliminates the tedious manual tuning of the parameters, but also results in more accurate estimation and compensation for the adverse effects of fault perturbations;

(3) It is a data-driven robust control strategy that does not require any additional fault detection or identification (FDI) module, while the controller parameters are adaptive. Therefore, the proposed method corresponds to a novel combination of active reconfiguration control and FDI-free reconfiguration control, which makes it an interesting solution under unknown fault conditions.

The paper is organized as follows. Section 2 describes AABS dynamics with an actuator fault factor. The reconfiguration controller is presented in Section 3. The simulation results are presented to demonstrate the merits of the proposed method in Section 4, and conclusions are drawn in Section 5.

## **2. AABS Modeling**

The AABS mainly consists of the following components: aircraft fuselage, landing gear, wheels, a hydraulic servo system, a braking device, and an anti-skid braking controller. The subsystems are strongly coupled and exhibit strong nonlinearity and complexity.

Based on the actual process and objective facts of anti-skid braking, the following reasonable assumptions can be made [46]:


## *2.1. Aircraft Fuselage Dynamics*

The force diagram of the aircraft fuselage is shown in Figure 1 and the specific parameters described in the diagram are shown in Table 1. *Aerospace* **2022**, *9*, x FOR PEER REVIEW 4 of 27

**Figure 1.** Force diagram of aircraft fuselage. **Figure 1.** Force diagram of aircraft fuselage.

**Table 1.** Parameters of aircraft fuselage dynamics.

*V* Aircraft speed *T*<sup>0</sup> Engine force *Fx* Aerodynamic drag *Fy* Aerodynamic lift *Fs* Parachute drag

*y* Center of gravity height variation

<sup>1</sup>*<sup>f</sup>* Braking friction force between main

<sup>2</sup>*<sup>f</sup>* Braking friction force between front

*<sup>t</sup> <sup>h</sup>* Distance between engine force line

*<sup>s</sup> <sup>h</sup>* Distance between parachute drag line

*a* Distance between main wheel and center

*<sup>b</sup>* Distance between front wheel and center

*T*0

ρ

*N*<sup>1</sup> Main wheel support force *N*<sup>2</sup> Front wheel support force

wheel and ground

wheel and ground

*m* Mass of the aircraft 1761 kg *g* Gravitational acceleration 9.8 m/s2

*I* Fuselage inertia 4000 kg·s2·m *S* Wing aera 50.88 m2 *<sup>s</sup> S* Parachute area 20 m2 *Cx* Aerodynamic drag coefficient 0.1027 *Cy* Aerodynamic lift coefficient 0.6 *Cxs* Parachute drag coefficient 0.75

′ Intimal engine force 426 kg *Kv* Velocity coefficient of engine 1 kg·s/m

Air density 4000kg·s2/m4

and center of gravity 0.1 m

and center of gravity 0.67 m

of gravity 1.076 m

of gravity 6.727 m


**Table 1.** Parameters of aircraft fuselage dynamics.

The aircraft force and torque equilibrium equations are:

$$\begin{cases} m\dot{V} + F\_{\text{x}} + F\_{\text{s}} + f\_1 + f\_2 - T\_0 = 0 \\ F\_{\text{y}} + N\_1 + N\_2 - mg = 0 \\ N\_2\dot{b} + F\_{\text{s}}h\_{\text{s}} - N\_1a - T\_0h\_{\text{f}} - f\_1H - f\_2H = 0 \end{cases} \tag{1}$$

According to the influence of aerodynamic characteristics, we can obtain [46]:

$$\begin{cases} \begin{aligned} T\_0 &= T\_0' + K\_v V\\ F\_x &= \frac{1}{2} \rho \mathbf{C}\_x S V^2\\ F\_y &= \frac{1}{2} \rho \mathbf{C}\_y S V^2\\ F\_s &= \frac{1}{2} \rho \mathbf{C}\_{xs} S\_s V^2\\ f\_1 &= \mu\_1 N\_1\\ f\_2 &= \mu\_2 N\_2 \end{aligned} \end{cases} \tag{2}$$

## *2.2. Landing Gear Dynamics*

The main function of the landing gear is to support and buffer the aircraft, thus improving the longitudinal and vertical forces. In addition to the wheel and braking device, the struts, buffers, and torque arm are also the main components of the landing gear. In this paper, it is assumed that the stiffness of the torque arm is large enough, and the torsional freedom of the wheel with respect to the strut and the buffer is ignored, so the torque arm is not considered.

The buffer can be reasonably simplified as a mass-spring-damping system [46], and the force acting on the aircraft fuselage by the buffer can be described as:

$$\begin{cases} \text{ N}\_1 = \text{K}\_1 \text{X}\_1 + \text{C}\_1 \dot{\text{X}}\_1^2 \\ \text{ N}\_2 = \text{K}\_2 \text{X}\_2 + \text{C}\_2 \dot{\text{X}}\_2^2 \end{cases} \tag{3}$$

$$\begin{cases} X\_1 = a + y \\ X\_2 = -b + y \end{cases} \tag{4}$$

whose parameters are shown in Table 2.

**Table 2.** Parameters of the buffer.


Due to the non-rigid connection between the landing gear and the aircraft fuselage, horizontal and angular displacements are generated under the action of braking forces. However, the struts are cantilever beams, and their angular displacements are very small and negligible. Therefore, the lateral stiffness model can be expressed by the following equivalent second-order equation: and negligible. Therefore, the lateral stiffness model can be expressed by the following equivalent second-order equation: 1 0 2 1 2 <sup>1</sup> *a f <sup>K</sup> <sup>d</sup> s s* ξ <sup>−</sup> <sup>=</sup> + +

2

$$\begin{cases} \begin{aligned} d\_d &= \frac{-f\_1}{K\_0} \\ \frac{1}{\mathcal{W}\_n^2} s^2 + \frac{2\xi}{\mathcal{W}\_n} s + 1 \\ d\_V &= \frac{d}{dt} (d\_d) \end{aligned} \tag{5} \end{cases} \tag{5}$$

(5)

(6)

whose parameters are shown in Table 3. **Name Description Value** 

**Table 3.** Parameters of the landing gear lateral stiffness model. *<sup>a</sup> d* Navigation vibration displacement Please see Equation (5)

**Table 3.** Parameters of the landing gear lateral stiffness model.


It can be seen that during the taxiing, the main wheel is subjected to a combined effect

*j s zx*

σ

*M M V J R*

*g*

lateral stiffness, there is a longitudinal axle velocity *Vzx* along the fuselage, which is superimposed by the aircraft velocity *V* and the navigation vibration velocity *Vd* . The dy-

> *w g zx V*

ω

*V R V Vd R R Nk M NR n*

 − = +

*j g*

μ

*g*

ω

 <sup>=</sup> = + = − =

*2.3. Wheel Dynamics 2.3. Wheel Dynamics* 

The force diagram of the main wheel brake is shown in Figure 2. The force diagram of the main wheel brake is shown in Figure 2.

**Figure 2.** Force diagram of the main wheel. **Figure 2.** Force diagram of the main wheel.

namics equation of the main wheel is [46]:

whose parameters are shown in Table 4.

It can be seen that during the taxiing, the main wheel is subjected to a combined effect of the braking torque *M<sup>s</sup>* and the ground friction torque *M<sup>j</sup>* . Due to the effect of the lateral stiffness, there is a longitudinal axle velocity *Vzx* along the fuselage, which is superimposed by the aircraft velocity *V* and the navigation vibration velocity *dV*. The dynamics equation of the main wheel is [46]:

$$\begin{cases} \begin{aligned} \dot{\omega} &= \frac{M\_{\dot{j}} - M\_{\text{s}}}{f} + \frac{V\_{\text{rx}}}{R\_{\text{g}}} \\ V\_{\text{w}} &= \omega R\_{\text{g}} \\ V\_{\text{z}x} &= V + d\_{V} \\ R\_{\text{g}} &= R - N k\_{\sigma} \\ M\_{\dot{j}} &= \mu N R\_{\text{g}} n \end{aligned} \tag{6}$$

whose parameters are shown in Table 4.



During the braking, the tires are subjected to the braking torque that keeps the aircraft speed always greater than the wheel speed, that is *V* > *Vw*. Thus, the slip ratio *λ* is defined to represent the slip motion ratio of the wheels relative to the runway. For the main wheel, using *Vzx* instead of *V* to calculate *λ* can avoid false brake release due to landing gear deformation, thus effectively reducing the landing gear walk situation [46]. The following equation is used to calculate the slip rate in this paper:

$$
\lambda = \frac{V\_{\rm zx} - V\_{\rm w}}{V\_{\rm zx}} \tag{7}
$$

The tire–runway combination coefficient is related to many factors, including real-time runway conditions, aircraft speed, slip rate, and so on. A simple empirical formula called 'magic formula' developed by Pacejka [47] is widely used to calculate and can be expressed as follows:

$$
\mu(\lambda, \tau\_{\flat}) = \tau\_1 \sin(\tau\_2 \text{arctg}(\tau\_3 \lambda)) \tag{8}
$$

where *τj*(*j* = 1, 2, 3), *τ*1, *τ*2, *τ*<sup>3</sup> are peak factor, stiffness factor, and curve shape factor, respectively. Table 5 lists the specific parameters for several different runway statuses [48].

**Table 5.** Parameters of the runway status.


## *2.4. Hydraulic Servo System and Braking Device Modeling*

Due to the complex structure of the hydraulic servo system, in this paper, some simplifications have been made so that only electro-hydraulic servo valves and pipes are considered. Their transfer functions are given as follows:

$$\begin{cases} M(s) = \frac{K\_{sv}}{\frac{s^2}{\omega\_{sv}^2} + \frac{2\zeta\_{sv}s}{\omega\_{sv}} + 1} \\\ L(s) = \frac{K\_p}{T\_p s + 1} \end{cases} \tag{9}$$

whose parameters are shown in Table 6.

It should be noted that the anti-skid braking controller should realize both braking control and anti-skid control. To this end, there is an approximately linear relationship between the brake pressure *P* and the control current *Ic*, which can be described as follows:

$$P = -I\_{\mathbb{C}}M(s)L(s) + P\_0 \tag{10}$$

where *<sup>P</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>×</sup> <sup>10</sup><sup>7</sup> Pa.

The braking device serves to convert the brake pressure into brake torque, which is calculated as follows:

$$M\_{\rm s} = \mu\_{\rm mc} N\_{\rm mc} P R\_{\rm mc} \tag{11}$$

whose parameters are shown in Table 6.


**Table 6.** Parameters of the hydraulic servo system.

The hydraulic servo system, as the actuator of AABS, is inevitably subject to some potential faults. Problems such as hydraulic oil mixing with air, internal leakage, and vibration seriously affect the efficiency of the hydraulic servo system [49]. Therefore, in this paper, the loss of efficiency (LOE) is introduced to represent a typical AABS actuator fault, which is characterized by a decrease in the actuator gain from its nominal value [26]. In the case of an actuator LOE fault, the brake pressure generated by the hydraulic servo system deviates from the commanded output expected by the controller. In other words, one instead has:

$$P\_{fault} = k\_{LOE} \text{P} \tag{12}$$

where *Pf ault* represents the actuator actual output, and *kLOE* ∈ (0, 1] refers the LOE fault factor.

**Remark 1.** *n*% *LOE is equivalent to the LOE fault gain kLOE* = 1 − *n*/100, *kLOE* = 1 *indicates that the actuator is fault-free.*

**Remark 2.** *Note that if the components do not always have the same characteristics as those of fault-free, it is necessary to establish the fault model. This not only provides an accurate model for the next reconfiguration on controller design, but also ensures that the adverse effects caused by fault perturbation can be effectively observed and compensated for.*

Thus, Equation (11) can be rewritten as follows:

$$\mathbf{M\_s}' = \mu\_{mc} \mathbf{N\_{mc}} P\_{fault} \mathbf{R\_{mc}} \tag{13}$$

where *M<sup>s</sup>* 0 is the actual brake torque.

**Remark 3.** *As can be seen from the entire modeling process described above, AABS is nonlinear and highly coupled. The actuator fault leads to a sudden jump in the model parameters with greater internal perturbation compared to the fault-free case. Meanwhile, external disturbances such as the runway environment cannot be ignored.*

## **3. Reconfiguration Controller Design**

*3.1. Problem Description*

Despite the aircraft having three degrees of freedom, only longitudinal taxiing is focused on in AABS. In this paper, AABS adopted the slip speed control type [48], that is, the braked wheel speed *V<sup>ω</sup>* was used as the reference input, and the aircraft speed *V* was dynamically adjusted by the AABS controller to achieve anti-skid braking. According to Section 2, the AABS longitudinal dynamics model can be rewritten as follows:

$$
\ddot{V} = f(V, \dot{V}, \mathcal{O}\_{\text{out}}, \mathcal{O}\_f) + b\_{\text{v}}u \tag{14}
$$

where *f*(·) is the controlled plant dynamics, *vout* represents the external disturbance, *v<sup>f</sup>* is an uncertain term including component faults, *b<sup>v</sup>* is the control gain, and *u* is the system input. . .

Let *x*<sup>1</sup> = *V*, *x*<sup>2</sup> = *V*. Set *f*(*V*, *V*, *vout*, *v<sup>f</sup>* ) as the system generalized total perturbation and extend it to a new system state variable, i.e., *x*<sup>3</sup> = *f*(*V*, . *V*, *vout*, *v<sup>f</sup>* ). Then the state equation of System (14) can be obtained:

$$\begin{cases}
\dot{\mathbf{x}}\_1 = \mathbf{x}\_2 \\
\dot{\mathbf{x}}\_2 = \mathbf{x}\_3 + b\_v u \\
\dot{\mathbf{x}}\_3 = h(V, \dot{V}, \mathcal{a}\_{out} \boldsymbol{\omega}\_f)
\end{cases} \tag{15}$$

where *x*1, *x*2, *x*<sup>3</sup> are system state variables, and *h*(*V*, . *V*, *vout*, *v<sup>f</sup>* ) = . *f*(*V*, . *V*, *vout*, *v<sup>f</sup>* ).

**Assumption 1.** *Both the system generalized total perturbation f*(*V*, . *V*, *vout*, *v<sup>f</sup>* ) *and its differential h*(*V*, . *V*, *vout*, *v<sup>f</sup>* ) *are bounded, i.e., <sup>f</sup>*(*V*, . *V*, *vout*, *v<sup>f</sup>* ) <sup>≤</sup> *<sup>σ</sup>*<sup>1</sup> *h*(*V*, . *V*, *vout*, *v<sup>f</sup>* ) <sup>≤</sup> *<sup>σ</sup>*<sup>2</sup> , *where σ*1, *σ*<sup>2</sup> *are two*

*positive numbers.*

For System (14), affected by the total perturbation, a LADRC reconfiguration controller was designed next to restrain or eliminate the adverse effects, thus realizing the asymptotic stability and acceptable performance of the closed-loop system.

### *3.2. LADRC Controller Design Aerospace* **2022**, *9*, x FOR PEER REVIEW 10 of 27

The control schematic of the LADRC is shown in Figure 3.

Firstly, the following tracking differentiator (TD) was designed:

 = 

() () ()

*<sup>r</sup> ek v k v k*

1 12

where *<sup>r</sup> v* is the desired input, <sup>1</sup> *v* is the transition process of *<sup>r</sup> v* , <sup>2</sup> *v* is the derivative of <sup>1</sup> *v* , and *r* and *h* are adjusted accordingly as filter coefficients. The function fhan ( )⋅

*x x rh <sup>a</sup> r ad*

1 2 11 1 2 3 21 1 3 31 1

vation of the variables in System (14) [50], i.e., 1 1 *z v* → , 2 2 *z v* → , <sup>3</sup> (,, , ) *out f z fVV* <sup>→</sup>

β

β

*z z zv*

*z zv*

 =− − =− − +

β

*u*

ϖ ϖ

Further, the bandwidth method [50] was used and we could obtain:

then the system (15) can be simplified to a double integral series structure:

=− −

fh fhan , , ,

*v k v k hv k vk vk h*

1

= −

( ) ()

2 2

( )

fhan , , , ,

1 2

 

We established the following form, LESO:

Selecting the suitable observer gains <sup>123</sup> (, , )

When <sup>3</sup> *z* can estimate (,, , ) *out f fVV*

1

 += + += +

( ) () ()

1 fh

( ) () ()

( ) <sup>0</sup>

sgn ,

<sup>&</sup>gt; = −

*d*

( ) ( )

( )

βββ

0 3 *v u z*

*b*

111 222 0 11 2 2

*evz evz u ke k e*

= +

3 00 ((,, , ) ) *V fVV z u u* = − +≈ ϖ ϖ

1

 = = =

β

β

β

2

3

2

3

ω

ω

3 3 *o o o*

ω

 = − = −

*z z z v bu*

<sup>≤</sup>

*r aa d*

0

*v*

, LESO then enabled real-time obser-

<sup>−</sup> <sup>=</sup> (19)

without error, let LSEF be:

*out f* (21)

(16)

(17)

(18)

(20)

(22)

ϖ ϖ.

*ek v k rh*

2

**Figure 3.** Control schematic of LADRC. **Figure 3.** Control schematic of LADRC.

is defined as follows:

Set

Firstly, the following tracking differentiator (TD) was designed:

$$\begin{cases} e(k) = v\_1(k) - v\_r(k) \\ \text{fh} = \text{fhan}(e(k), v\_2(k), r, h) \\ v\_1(k+1) = v\_1(k) + hv\_2(k) \\ v\_2(k+1) = v\_2(k) + h\text{fh} \end{cases} \tag{16}$$

where *v<sup>r</sup>* is the desired input, *v*<sup>1</sup> is the transition process of *v<sup>r</sup>* , *v*<sup>2</sup> is the derivative of *v*1, and *r* and *h* are adjusted accordingly as filter coefficients. The function fhan(·) is defined as follows:

$$\text{fhan}(\mathbf{x}\_1, \mathbf{x}\_2, r, h) = -\left\{ \begin{array}{l} r \text{sgn}(a), |a| > d\_0 \\ r\_{d'}^{a} |a| \le d\_0 \end{array} \right. \tag{17}$$

We established the following form, LESO:

$$\begin{cases} \dot{z}\_1 = z\_2 - \beta\_1 (z\_1 - v\_1) \\ \dot{z}\_2 = z\_3 - \beta\_2 (z\_1 - v\_1) + b\_v u \\ \dot{z}\_3 = -\beta\_3 (z\_1 - v\_1) \end{cases} \tag{18}$$

Selecting the suitable observer gains (*β*1, *β*2, *β*3), LESO then enabled real-time observation of the variables in System (14) [50], i.e., *z*<sup>1</sup> → *v*<sup>1</sup> , *z*<sup>2</sup> → *v*<sup>2</sup> , *z*<sup>3</sup> → *f*(*V*, . *V*, *vout*, *v<sup>f</sup>* ).

$$\text{Set}$$

$$
\mu = \frac{\mu\_0 - z\_3}{b\_v} \tag{19}
$$

When *z*<sup>3</sup> can estimate *f*(*V*, . *V*, *vout*, *v<sup>f</sup>* ) without error, let LSEF be:

$$\begin{cases} \ e\_1 = v\_1 - z\_1 \\ \ e\_2 = v\_2 - z\_2 \\ \ u\_0 = k\_1 e\_1 + k\_2 e\_2 \end{cases} \tag{20}$$

then the system (15) can be simplified to a double integral series structure:

$$\ddot{V} = \left( f(V, \dot{V}, \mathfrak{a}\_{\text{out}}, \mathfrak{a}\_f) - z\_3 \right) + \mathfrak{u}\_0 \approx \mathfrak{u}\_0 \tag{21}$$

Further, the bandwidth method [50] was used and we could obtain:

$$\begin{cases} \begin{aligned} \beta\_1 &= 3\omega\_o \\ \beta\_2 &= 3\omega\_o^2 \\ \beta\_3 &= \omega\_o^3 \end{aligned} \tag{22}$$

where *ω<sup>o</sup>* is the observer bandwidth. The larger *ω<sup>o</sup>* is, the smaller LESO observation errors are. However, the sensitivity of the system to noise may be increased, so the *ω<sup>o</sup>* selection requires comprehensive consideration.

Similarly, according to the parameterization method and engineering experience [32], the LSEF parameters can be chosen as:

$$\begin{cases} \begin{array}{c} k\_1 = \omega\_c^2 \\ k\_2 = 2\mathfrak{J}\omega\_c \end{array} \end{cases} \tag{23}$$

where *ω<sup>c</sup>* is the controller bandwidth, *ξ* is the damping ratio, and in this paper *ξ* = 1. Therefore, the parameter tuning problem of LADRC controller was simplified to the observer bandwidth *ω<sup>o</sup>* and controller bandwidth *ω<sup>c</sup>* configuration.

#### *3.3. TD3 Algorithm 3.3. TD3 Algorithm*

server bandwidth

ω

where

where

ω

TD3 algorithm is an offline RL algorithm based on DDPG proposed in 2015 [51]. This approach adopted a similar method implemented in Double-DQN [52] to reduce the overestimation in function approximation, delaying the update frequency in the actor– network, and adding noises to target the actor–network to release the sensitivity and instability in DDPG. The structure of TD3 is shown in Figure 4. TD3 algorithm is an offline RL algorithm based on DDPG proposed in 2015 [51]. This approach adopted a similar method implemented in Double-DQN [52] to reduce the overestimation in function approximation, delaying the update frequency in the actor–network, and adding noises to target the actor–network to release the sensitivity and instability in DDPG. The structure of TD3 is shown in Figure 4.

ω

Similarly, according to the parameterization method and engineering experience

2

ξω

ω

ω

1 <sup>2</sup> 2 *c c*

Therefore, the parameter tuning problem of LADRC controller was simplified to the ob-

<sup>=</sup>

*k k*

ξ

*o* and controller bandwidth

rors are. However, the sensitivity of the system to noise may be increased, so the

*o* is, the smaller LESO observation er-

<sup>=</sup> (23)

is the damping ratio, and in this paper

*<sup>c</sup>* configuration.

ω*o* se-

ξ=1.

*Aerospace* **2022**, *9*, x FOR PEER REVIEW 11 of 27

*o* is the observer bandwidth. The larger

lection requires comprehensive consideration.

*c* is the controller bandwidth,

ω

[32], the LSEF parameters can be chosen as:

**Figure 4.** Structure of TD3. **Figure 4.** Structure of TD3.

action is defined as:

Updating the parameters of critic networks by minimizing loss: Updating the parameters of critic networks by minimizing loss:

$$L = N^{-1} \sum \left( y - Q\_{\theta\_i}(s, a) \right)^2 \tag{24}$$

where *s* is the current state, *a* is the current action, and ( ) , *<sup>i</sup> Q sa* θ stands for the parameterized state-action value function *Q* with parameter where *s* is the current state, *a* is the current action, and *Qθ<sup>i</sup>* (*s*, *a*) stands for the parameterized state-action value function *Q* with parameter *θ<sup>i</sup>* .

$$y = r + \gamma \min\_{i=1,2} Q\_{\theta\_i'}(s', \hat{a}) \tag{25}$$

= is the target value of the function *Q sa* ( ) , θ , γ ∈[0,1] is the discount factor, and the target is the target value of the function *Q<sup>θ</sup>* (*s*, *a*), *γ* ∈ [0, 1] is the discount factor, and the target action is defined as:

'

θ

*i i*

$$
\widetilde{a} = \pi\_{\phi'}(s) + \varepsilon'\tag{26}
$$

*a s*( ) πφ′ = + ′ є (26) where noise *e* 0 follows a clipped normal distribution clip [N (0, *σ*), −*c*, *c*], *c* > 0. This implies that *e* 0 is a random variable with N (0, *σ*) and belongs to the interval [−*c*, *c*].

The inputs of the actor network are both *Q<sup>θ</sup>* (*s*, *a*) from the critic network and the minibatch form the memory, and the output is the action given by:

$$a\_t = \pi\_\Phi(s\_t) + \varepsilon \tag{27}$$

where *φ* is the parameter of the actor network, and *π<sup>φ</sup>* is the output form the actor network, which is a deterministic and continuous value. Noise *e* follows the normal distribution N (0, *σ*), and is added for exploration.

Updating the parameters of the actor–network based on deterministic gradient strategy:

$$\nabla\_{\Phi} I(\phi) = N^{-1} \sum \nabla\_{a} Q\_{\theta\_{1}}(s, a)|\_{a = \pi\_{\Phi}(s)} \nabla\_{\Phi} \pi\_{\Phi}(s) \tag{28}$$

TD3 updates the actor–network and all three target networks every d steps periodically in order to avoid a too fast convergence. The parameters of the critic target networks and the actor–target network are updated according to:

$$\begin{cases} \ \theta\_i' \leftarrow \tau \theta\_i + (1 - \tau)\theta\_i' \\\ \phi' \leftarrow \tau \phi + (1 - \tau)\phi' \end{cases} \tag{29}$$

The pseudocode of the proposed approach is given in Algorithm 1.

## **Algorithm 1**. TD3

	- <sup>12</sup> Update critics *<sup>θ</sup><sup>i</sup>* <sup>←</sup> min*θ<sup>i</sup> <sup>N</sup>*−<sup>1</sup> <sup>∑</sup> *y* − *Qθ<sup>i</sup>* (*s*, *a*) 2 ;
	- 13 Every *d* steps:
	- 14 Update *φ* by the deterministic policy gradient:
	- <sup>15</sup> <sup>∇</sup>*<sup>φ</sup> <sup>J</sup>*(*φ*) = *<sup>N</sup>*−<sup>1</sup> <sup>∑</sup> <sup>∇</sup>*<sup>a</sup> <sup>Q</sup>θ*<sup>1</sup> (*s*, *a*) *<sup>a</sup>*=*πφ*(*s*)∇*φπφ*(*s*); **17** ( ) 1 θ τθ τ θ*ii i* ′ ′ ← +−
	- 16 Update target network: **18** φ′ ′ ← +− τφ ( ) 1 τ φ;
	- 17 *θ* 0 *<sup>i</sup>* ← *τθ<sup>i</sup>* + (1 − *τ*)*θ* 0 *i*
	- 18 *φ* <sup>0</sup> ← *τφ* + (1 − *τ*)*φ* 0 ; 0 ; **19** *s* ← *s*′ ;
	- 19 *s* ← *s*

do:

(1) The aircraft speed *V* < 2 ;

20 **Until** *s* reaches terminal state *sT*. **20 Until** *s* reaches terminal state *Ts* .

### *3.4. TD3-LADRC Reconfiguration Controller Design 3.4. TD3-LADRC Reconfiguration Controller Design*

Lack of environment adaptability, poor control performance, and weak robustness are the main shortcomings of parameter-fixed controllers [36]. When a fault occurs, it may not be possible to maintain the acceptable (rated or degraded) performance of the damaged system. Motivated by the above analysis, a reconfiguration controller called TD3-LADRC is proposed in this paper, and its control schematic is shown in Figure 5. Lack of environment adaptability, poor control performance, and weak robustness are the main shortcomings of parameter-fixed controllers [36]. When a fault occurs, it may not be possible to maintain the acceptable (rated or degraded) performance of the damaged system. Motivated by the above analysis, a reconfiguration controller called TD3- LADRC is proposed in this paper, and its control schematic is shown in Figure 5.

The deep reinforcement learning algorithm TD3 is introduced to realize the LADRC parameters adaption. The details of each part have been described above. The selection of control parameters is treated as the agent's action *<sup>t</sup> a* , and the response result of the control system *<sup>t</sup> s* is considered as the state, i.e., as follows: The deep reinforcement learning algorithm TD3 is introduced to realize the LADRC parameters adaption. The details of each part have been described above. The selection of control parameters is treated as the agent's action *a<sup>t</sup>* , and the response result of the control system *s<sup>t</sup>* is considered as the state, i.e., as follows:

[ ]<sup>T</sup>

*s s eeV V*

[ ]

The reward function plays a crucial role in the reinforcement learning algorithm. The appropriateness of the reward function design directly affects the training effect of the reinforcement learning, which in turn affects the effectiveness of the whole reconfiguration controller. According to the working characteristics of AABS, the following reward

The stop conditions for each training episode are as follows, and one of the three will

0, 4 100, 200

[ ]

*r V V V Ve <sup>t</sup>* = − ≤ ≤− + >− <− − < > 1 6 4 0 4 6 100 2 20 ( )( ) ( ) (32)

,

*t oc*

ω ω

*t obs*

*c o*

 ∈ ∈

ω

ω

function is selected after several attempts to ensure stable and smooth braking:

(2) The error between main wheel speed and aircraft speed *e* > 20 ;

*a*

The range of each controller parameter is selected as follows:

=

T

= = (30)

(31)

,, ,

where *<sup>w</sup> eVV* = − , and *obs s* is the agent observations vector.

$$\begin{cases} \boldsymbol{a\_{t}} = \begin{bmatrix} \boldsymbol{\omega\_{0}}, \boldsymbol{\omega\_{c}} \end{bmatrix}^{\mathrm{T}} \\\\ \boldsymbol{s\_{t}} = \boldsymbol{s\_{obs}} = \begin{bmatrix} \boldsymbol{e\_{r}} \dot{\boldsymbol{e}}\_{r} \boldsymbol{V\_{r}} \dot{\boldsymbol{V}} \end{bmatrix}^{\mathrm{T}} \end{cases} \tag{30}$$

where *e* = *V* − *Vw*, and *sobs* is the agent observations vector.

The range of each controller parameter is selected as follows:

$$\begin{cases} \ \omega\_{\mathcal{C}} \in [0, 4] \\ \ \omega\_{\mathcal{O}} \in [100, 200] \end{cases} \tag{31}$$

The reward function plays a crucial role in the reinforcement learning algorithm. The appropriateness of the reward function design directly affects the training effect of the reinforcement learning, which in turn affects the effectiveness of the whole reconfiguration controller. According to the working characteristics of AABS, the following reward function is selected after several attempts to ensure stable and smooth braking:

$$\tau\_{\rm I} = 1 \left( -6 \le \dot{V} \le -4 \right) + 0 \left( \dot{V} > -4 \parallel \dot{V} < -6 \right) - 100 \left( V < 2 \parallel e > 20 \right) \tag{32}$$

The stop conditions for each training episode are as follows, and one of the three will do:


**Remark 4.** *TD3, TD, LESO, and LSEF together constitute the TD3-LADRC controller. Compared to normal LADRC, TD3-LADRC realizes the parameter adaption that makes the controller reconfigurable. The robustness and immunity are greatly improved. It can effectively compensate the adverse effects caused by the total perturbations including faults.*

## *3.5. TD3-LESO Estimation Capability Analysis*

In order to prove the stability of the whole closed-loop system, the convergence of TD3-LESO is first analyzed in conjunction with Assumption 1 [53]. Let the estimation errors of TD3-LESO be *<sup>x</sup>*e*<sup>i</sup>* <sup>=</sup> *<sup>x</sup><sup>i</sup>* <sup>−</sup> *<sup>z</sup><sup>i</sup>* , *i* = 1, 2, 3, and the estimation error equation of the observer can be obtained as: .

$$\begin{cases} \dot{\tilde{\boldsymbol{x}}}\_{1} = \tilde{\boldsymbol{x}}\_{2} - 3\omega\_{o}\tilde{\boldsymbol{x}}\_{1} \\ \dot{\tilde{\boldsymbol{x}}}\_{2} = \tilde{\boldsymbol{x}}\_{3} - 3\omega\_{o}^{2}\tilde{\boldsymbol{x}}\_{1} \\ \dot{\tilde{\boldsymbol{x}}}\_{3} = h(\boldsymbol{V}, \dot{\boldsymbol{V}}, \boldsymbol{\mathcal{w}}\_{\text{out}}, \boldsymbol{\mathcal{o}}\_{f}) - \omega\_{o}^{3}\tilde{\boldsymbol{x}}\_{1} \end{cases} \tag{33}$$

Let *ε<sup>i</sup>* = *x*e*i ω i*−1 *o* , *i* = 1, 2, 3, then Equation (33) can be rewritten as:

$$\dot{\varepsilon} = \omega\_o A\_3 \varepsilon + B \frac{h(V, \dot{V}, \mathfrak{a}\_{out}, \mathfrak{a}\_f)}{\omega\_o^2} \tag{34}$$

where *A*<sup>3</sup> = −3 1 0 −3 0 1 −1 0 0 , *B* = [0 0 1] T .

Based on Assumption 1 and Theorem 2 in Reference [54], the following theorem can be obtained:

.

**Theorem 1.** *Under the condition that h*(*V*, *V*, *vout*, *v<sup>f</sup>* ) *is bounded, the TD3-LESO estimation errors are bounded and their upper bound decrease monotonically with the increase of the observer bandwidth ωo*.

The proof is given in the Appendix A. Thus, it is clear that there are three positive numbers *υ<sup>i</sup>* , *<sup>i</sup>* <sup>=</sup> 1, 2, 3, such that the state estimation error <sup>|</sup>*x*e*<sup>i</sup>* |≤ *υ<sup>i</sup>* holds, i.e., the TD3-LESO estimation errors are bounded, which can effectively estimate the states of the controlled plant and the total perturbation.

## *3.6. Stability Analysis of Closed-loop System*

The closed-loop system consisted of the control laws (19) and (20), and the controlled object (21) is: ..

$$V = f - z\_3 + k\_1 e\_1 + k\_2 e\_2 \tag{35}$$

If we defined the tracking errors as *ε<sup>i</sup>* = *v<sup>i</sup>* − *x<sup>i</sup>* , *i* = 1, 2, then we could attain:

$$\begin{cases}
\dot{\varepsilon}\_1 = \dot{r}\_1 - \dot{\mathbf{x}}\_1 = r\_2 - \mathbf{x}\_2 = \tilde{\mathbf{e}}\_2 \\
\dot{\varepsilon}\_2 = \dot{r}\_2 - \dot{\mathbf{x}}\_2 = r\_3 - \ddot{V} \\
= -k\_1 \varepsilon\_1 - k\_1 \tilde{\mathbf{x}}\_1 - k\_2 \varepsilon\_2 - k\_2 \tilde{\mathbf{x}}\_2 - \tilde{\mathbf{x}}\_3
\end{cases} \tag{36}$$

Let *ε* = [*ε*1,*ε*2] *T* , *<sup>x</sup>*<sup>e</sup> <sup>=</sup> [*x*e1, *<sup>x</sup>*e2, *<sup>x</sup>*e3] *T* , then:

$$
\dot{\varepsilon}(t) = A\_{\varepsilon}\varepsilon(t) + A\_{\tilde{\mathbf{x}}}\tilde{\mathbf{x}}(t) \tag{37}
$$

where *A<sup>e</sup>* = 0 1 −*k*<sup>1</sup> −*k*<sup>2</sup> , *<sup>A</sup>x*<sup>e</sup> <sup>=</sup> 0 0 0 −*k*<sup>1</sup> −*k*<sup>2</sup> −1 By solving Equation (37):

$$
\varepsilon(t) = e^{A\_{\xi}t}\varepsilon(0) + \int\_0^t e^{A\_{\xi}(t-\tau)} A\_{\tilde{\varkappa}} \tilde{\chi}(\tau)d\tau \tag{38}
$$

.

Combining Assumption 1, Theorem 1, Theorem 3, and Theorem 4 in the literature [54], the following theorem was proposed to analyze the stability of the closed-loop system:

**Theorem 2.** *Under the condition that the TD3-LESO estimation errors are bounded, there exists a controller bandwidth ωc*, *such that the tracking error of the closed-loop system is bounded. Thus, for a bounded input, the output of the closed-loop system is bounded, i.e., the closed-loop system is BIBO-stable.*

See the Appendix A for proof.

## **4. Simulation Results**

In order to verify the reconfiguration capability and disturbance rejection capabilities of the proposed method, the corresponding simulations are carried out in this section and compared with conventional PID + PBM and LADRC.

The initial states of the aircraft are set as follows:


To prevent deep wheel slippage as well as tire blowout, the wheel speed was kept following the aircraft speed quickly at first, and the brake pressure was applied only after 1.5 s. The anti-skid brake control was considered to be over when *V* was less than 2 m/s.

In the experiment, both the critic networks and the actor networks were realized by a fully connected neural network with three hidden layers. The number of neurons in the hidden layer was (50,25,25). The activation function of the hidden layer was selected as the ReLU function, and the activation function of the output layer of the actor network was selected as the tanh function. In addition, the parameters of the actor network and the critic network were tuned by an Adam optimizer. The remaining parameters of TD3-LADRC are shown in Table 7.


**Table 7.** Parameters of TD3-LADRC.

**Remark 5.** *It is noted that the braking time t and braking distance x are selected as the criteria for braking efficiency, and the system stability is observed by slip rate λ*. *Aerospace* **2022**, *9*, x FOR PEER REVIEW 16 of 27

The model simulation was carried out in MATLAB 2022a, and the TD3 algorithm was realized through the reinforcement learning toolbox. The simulation time was 20 s, the sampling time was 0.001 s. The training stopped when the average reward reached 12,000. The training took about 6 h to complete. The learning curves of the reward obtained by the agent for each interaction with the environment during the training process are shown in Figure 6. The model simulation was carried out in MATLAB 2022a, and the TD3 algorithm was realized through the reinforcement learning toolbox. The simulation time was 20 s, the sampling time was 0.001 s. The training stopped when the average reward reached 12,000. The training took about 6 h to complete. The learning curves of the reward obtained by the agent for each interaction with the environment during the training process are shown in Figure 6.

**Figure 6.** Learning curves. **Figure 6.** Learning curves.

*that A*ε

*braking results.*

*stants the agent considers* 0

**Table 8.** AABS performance index.

It can be seen that at the beginning of the training, the agent was in the exploration phase and the reward obtained was relatively low. Later, the reward gradually increased, and after 40 episodes, the reward was steadily maintained at a high level and the algorithm gradually converges. It can be seen that at the beginning of the training, the agent was in the exploration phase and the reward obtained was relatively low. Later, the reward gradually increased, and after 40 episodes, the reward was steadily maintained at a high level and the algorithm gradually converges.

### *4.1. Case 1: Fault-Free and External Disturbance-Free in Dry Runway Condition 4.1. Case 1: Fault-Free and External Disturbance-Free in Dry Runway Condition*

**Remark 6**. *During the braking process, it is observed that in some instants* 0

*is Hurwitz (see Proof of Theorem 2 for details). On the other hand,* 

*changed by the agent through a continuous interaction with the environment, and in these in-*

Braking time(s) 20.48 16.73 14.79 Braking distance(m) 811.9 595.46 571.18

The simulation results of the dynamic braking process for different control schemes are shown in Figures 7 and 8 and Table 8. The simulation results of the dynamic braking process for different control schemes are shown in Figures 7 and 8 and Table 8.

As can be seen from Figure 7, PID + PBM leads to numerous skids during braking, which may cause serious loss to the tires. In contrast, LADRC and TD3-LADRC not only skid less frequently, but also have shorter braking time and braking distance. Moreover, the control effect of TD3-LADRC is better than LADRC. Figure 8 shows that TD3-LADRC can dynamically tune the controller parameters to accurately observe and compensate for the total disturbances, and thus improve the AABS performance. As can be seen from Figure 7, PID + PBM leads to numerous skids during braking, which may cause serious loss to the tires. In contrast, LADRC and TD3-LADRC not only skid less frequently, but also have shorter braking time and braking distance. Moreover, the control effect of TD3-LADRC is better than LADRC. Figure 8 shows that TD3-LADRC can dynamically tune the controller parameters to accurately observe and compensate for the total disturbances, and thus improve the AABS performance.

ω

ω

ω

*<sup>c</sup>* = *as optimal, i.e., no anti-skid braking control leads to better* 

*<sup>c</sup>* = *. It may not* 

*<sup>c</sup> does not change the fact* 

*<sup>c</sup> is constantly* 

ω

*affect the stability of the whole system. On the one hand, the value of* 

input.

**Figure 7.** (**a**) Aircraft velocity and wheel velocity; (**b**) breaking distance; (**c**) slip ratio; (**d**) control **Figure 7.** (**a**) Aircraft velocity and wheel velocity; (**b**) breaking distance; (**c**) slip ratio; (**d**) control input.

ω*o* .

**Figure 8.** (**a**) Extended state of TD3-LADRC; (**b**) controller bandwidth ω*<sup>c</sup>* ; (**c**) observer bandwidth **Figure 8.** (**a**) Extended state of TD3-LADRC; (**b**) controller bandwidth *ωc*; (**c**) controller bandwidth *ωo*.

*4.2. Case 2: Actuator LOE Fault in Dry Runway Condition*  **Table 8.** AABS performance index.


ing much less efficient and risks dragging and flat tires. In addition, LADRC cannot brake the aircraft to a stop which is not allowed in practice. Figure 9c shows that there is a high frequency of wheel slip in the low-speed phase of the aircraft. In contrast, TD3-LADRC retains the experience gained from the agent's prior training and continuously adjusts the controller parameters online based on the plant states, which ultimately allows the aircraft to brake smoothly. From Figure 10a, it can be seen that the total fault perturbations are **Remark 6.** *During the braking process, it is observed that in some instants ω<sup>c</sup>* = 0. *It may not affect the stability of the whole system. On the one hand, the value of ω<sup>c</sup> does not change the fact that A<sup>ε</sup>* is Hurwitz (see Proof of Theorem 2 for details). On the other hand, *ω<sup>c</sup> is constantly changed by the agent through a continuous interaction with the environment, and in these instants the agent considers ω<sup>c</sup>* = 0 *as optimal, i.e., no anti-skid braking control leads to better braking results*.

#### improves the robustness and immunity of the controller in fault-perturbed conditions, but *4.2. Case 2: Actuator LOE Fault in Dry Runway Condition*

estimated fast and accurately based on the adaptive LESO. Overall, TD3-LADRC not only

also greatly significantly improves the safety and reliability of AABS. **Table 9.** AABS performance index. The fault considered here assumed a 20% actuator LOE at 5 s and escalated to 40% LOE at 10 s. The simulation results are shown in Figures 9 and 10 and Table 9.

**Performance Index PID + PBM LADRC TD3-LADRC**  Braking time(s) 23.48 -- 17.70

input.

**Figure 9.** (**a**) Aircraft velocity and wheel velocity; (**b**) breaking distance; (**c**) slip ratio; (**d**) control **Figure 9.** (**a**) Aircraft velocity and wheel velocity; (**b**) breaking distance; (**c**) slip ratio; (**d**) control input.

ω*o* .

proved.

**Figure 10.** (**a**) Extended state of TD3-LADRC; (**b**) controller bandwidth ω*<sup>c</sup>* ; (**c**) observer bandwidth **Figure 10.** (**a**) Extended state of TD3-LADRC; (**b**) controller bandwidth *ωc*; (**c**) controller bandwidth *ωo*.

*4.3. Case 3: Actuator LOE Fault in Mixed Runway Condition*  **Table 9.** AABS performance index.


The deterioration of the runway conditions has resulted in a very poor tire–ground bond. It can be seen from Figure 11 that both braking time and braking distance have increased compared to the dry runway. Figure 12 shows that TD3-LADRC is still able to achieve controller parameters adaption, accurately observe the total fault perturbations, and effectively compensate for the adverse effects. The whole reconfiguration control system adapts well to runway changes. The environmental adaptability of AABS is im-**Table 10.** AABS performance index. **Performance Index PID + PBM LADRC TD3-LADRC**  Braking time (s) 49.14 29.82 24.19 As can be seen in Figure 9, PID + PBM continuously performed a large braking and releasing operation under the combined effect of fault and disturbance. This makes braking much less efficient and risks dragging and flat tires. In addition, LADRC cannot brake the aircraft to a stop which is not allowed in practice. Figure 9c shows that there is a high frequency of wheel slip in the low-speed phase of the aircraft. In contrast, TD3-LADRC retains the experience gained from the agent's prior training and continuously adjusts the controller parameters online based on the plant states, which ultimately allows the aircraft to brake smoothly. From Figure 10a, it can be seen that the total fault perturbations are estimated fast and accurately based on the adaptive LESO. Overall, TD3-LADRC not only improves the robustness and immunity of the controller in fault-perturbed conditions, but also greatly significantly improves the safety and reliability of AABS.

Braking distance (m) 1228.71 739.99 672.03

input.

## *4.3. Case 3: Actuator LOE Fault in Mixed Runway Condition*

The mixed runway structure is as follows: dry runway in the interval of 0–10 s, wet runway in the interval of 10–20 s, and snow runway after 20 s. The fault considered here assumed a 10% actuator LOE at 10 s. The simulation results are shown in Figures 11 and 12 and Table 10. *Aerospace* **2022**, *9*, x FOR PEER REVIEW 21 of 27

**Figure 11.** (**a**) Aircraft velocity and wheel velocity; (**b**) breaking distance; (**c**) slip ratio; (**d**) control **Figure 11.** (**a**) Aircraft velocity and wheel velocity; (**b**) breaking distance; (**c**) slip ratio; (**d**) control input.

**Figure 12.** (**a**) Extended state of TD3-LADRC; (**b**) controller bandwidth ω*<sup>c</sup>* ; (**c**) observer bandwidth **Figure 12.** (**a**) Extended state of TD3-LADRC; (**b**) controller bandwidth *ωc*; (**c**) controller bandwidth *ωo*.



A TD3-LADRC reconfiguration controller was developed, and the parameters of LSEF and LESO were adjusted online using the TD3 algorithm. The simulation results under different conditions verified that the designed controller can effectively improve the antiskid braking performance even under faults and perturbations, as well as different runway environments. It successfully strengthened the robustness, immunity, and environmental adaptability of the AABS, thereby improving the safety and reliability of the aircraft. However, TD3-LADRC is so complex that its control effectiveness was verified only The deterioration of the runway conditions has resulted in a very poor tire–ground bond. It can be seen from Figure 11 that both braking time and braking distance have increased compared to the dry runway. Figure 12 shows that TD3-LADRC is still able to achieve controller parameters adaption, accurately observe the total fault perturbations, and effectively compensate for the adverse effects. The whole reconfiguration control system adapts well to runway changes. The environmental adaptability of AABS is improved.

#### by simulations in this paper. The combined effect caused by various uncertainties in prac-**5. Conclusions**

ω*o* .

tical applications on the robustness of the controller cannot be completely considered. Therefore, in future work, an aircraft braking hardware-in-loop experimental platform is necessary to build, consisting of the host PC, the target CPU, the anti-skid braking controller, the actuators, and the aircraft wheel. The host PC and the target CPU are the soft-A linear active disturbance rejection reconfiguration control scheme based on deep reinforcement learning was proposed to meet the higher performance requirements of AABS under fault-perturbed conditions. According to the composition structure and working principle, AABS mathematical model with an actuator fault factor is established.

ware simulation part, while the other four parts are the hardware part.

A TD3-LADRC reconfiguration controller was developed, and the parameters of LSEF and LESO were adjusted online using the TD3 algorithm. The simulation results under different conditions verified that the designed controller can effectively improve the anti-skid braking performance even under faults and perturbations, as well as different runway environments. It successfully strengthened the robustness, immunity, and environmental adaptability of the AABS, thereby improving the safety and reliability of the aircraft. However, TD3- LADRC is so complex that its control effectiveness was verified only by simulations in this paper. The combined effect caused by various uncertainties in practical applications on the robustness of the controller cannot be completely considered. Therefore, in future work, an aircraft braking hardware-in-loop experimental platform is necessary to build, consisting of the host PC, the target CPU, the anti-skid braking controller, the actuators, and the aircraft wheel. The host PC and the target CPU are the software simulation part, while the other four parts are the hardware part.

**Author Contributions:** Conceptualization, S.L., Z.Y. and Z.Z.; methodology, S.L., Z.Y. and Z.Z.; software, S.L. and Z.Z.; validation, S.L., Z.Y., Z.Z., R.J., T.R., Y.J., S.C. and X.Z.; formal analysis, S.L., Z.Y. and Z.Z.; investigation, S.L., Z.Y., Z.Z., R.J., T.R., Y.J., S.C. and X.Z.; resources, Z.Y., Z.Z., R.J., T.R., Y.J., S.C. and X.Z.; data curation, S.L., Z.Y., Z.Z., R.J., T.R., Y.J., S.C. and X.Z.; writing—original draft, S.L. and Z.Z.; writing—review and editing, S.L. and Z.Z.; visualization, S.L. and Z.Z.; supervision, S.L., Z.Y., Z.Z., R.J., T.R., Y.J., S.C. and X.Z.; project administration, S.L., Z.Y. and Z.Z.; funding acquisition, Z.Y., Z.Z., R.J., T.R., Y.J., S.C. and X.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Key Laboratory Projects of Aeronautical Science Foundation of China, grant numbers 201928052006 and 20162852031, and Postgraduate Research & Practice Innovation Program of NUAA, grant number xcxjh20210332.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Appendix A. Proof of Theorems**

**Proof of Theorem 1.** By solving Equation (34) we can attain:

$$\varepsilon(t) = e^{\omega\_o A\_3 t} \varepsilon(0) + \int\_0^t e^{\omega\_o A\_3 (t - \tau)} B \frac{h(V(\tau), \dot{V}(\tau), \mathcal{O}\_{\text{out}, \omega} \mathcal{O}\_f)}{\omega\_o^2} d\tau \tag{A1}$$

Define *ζ*(*t*) as follows:

$$\mathcal{L}(t) = \int\_0^t e^{\omega\_o A\_3(t-\tau)} B \frac{h(V(\tau), \dot{V}(\tau), \mathcal{o}\_{\text{out}}, \mathcal{o}\_f)}{\omega\_o^2} d\tau \tag{A2}$$

From the fact that *h*(*V*(*τ*), . *V*(*τ*), *vout*, *v<sup>f</sup>* ) is bounded, we have:

$$|\zeta\_i(t)| \le \frac{\sigma}{\omega\_o^3} \left[ \left| \left( A\_3^{-1} B \right)\_i \right| + \left| \left( A\_3^{-1} e^{\omega\_o A\_3 t} B \right)\_i \right| \right] \tag{A3}$$

$$\text{Because } A\_3^{-1} = \begin{bmatrix} 0 & 0 & -1 \\ 1 & 0 & -3 \\ 0 & 1 & -3 \end{bmatrix} \text{, we can attain: }$$

$$\left| \begin{pmatrix} A\_3^{-1}B \end{pmatrix}\_i \right| \le 3 \tag{A4}$$

Considering that *A*<sup>3</sup> is Hurwitz, there is thus a finite time *T*<sup>1</sup> so that for any *t* ≥ *T*1, *i*, *j* = 1, 2, 3, the following formula holds [54]:

$$\left| \left[ e^{\omega\_o A\_3 t} \right]\_{ij} \right| \le \frac{1}{\omega\_o^3} \tag{A5}$$

Therefore, the following formula is satisfied:

$$\left| \left[ e^{\omega\_0 A\_3 t} \mathcal{B} \right]\_i \right| \le \frac{1}{\omega\_o^3} \tag{A6}$$

Finally, we can attain:

$$\left| \left( A\_3^{-1} e^{\omega\_0 A\_3 t} B \right)\_i \right| \le \frac{4}{\omega\_o^3} \tag{A7}$$

From Equations (A3), (A4), and (A7) we can attain:

$$|\zeta\_i(t)| \le \frac{3\sigma}{\omega\_o^3} + \frac{4\sigma}{\omega\_o^6} \tag{A8}$$

Let *ε*sum(0) = |*ε*1(0)| + |*ε*2(0)| + |*ε*3(0)|, for all *t* ≥ *T*1, the following formula holds:

$$\left| \left[ e^{\omega\_0 A\_3 t} \varepsilon(\mathbf{0}) \right]\_i \right| \le \frac{\varepsilon\_{sum}(\mathbf{0})}{\omega\_o^3} \tag{A9}$$

Form Equation (A1) we can attain:

$$|\varepsilon\_i(t)| \le \left| \left[ e^{\omega\_0 A\_3 t} \varepsilon(0) \right]\_i \right| + |\zeta\_i(t)| \tag{A10}$$

Let *<sup>x</sup>*e*sum*(0) = <sup>|</sup>*x*e1(0)<sup>|</sup> <sup>+</sup> <sup>|</sup>*x*e2(0)<sup>|</sup> <sup>+</sup> <sup>|</sup>*x*e3(0)|, from *<sup>ε</sup><sup>i</sup>* <sup>=</sup> *x*e*i ω i*−1 *o* and formulas (A8)–(A10), we can attain:

$$\text{ иачал.}$$

$$|\widetilde{\mathfrak{x}}\_{i}(t)| \le \left| \frac{\widetilde{\mathfrak{x}}\_{sum}(0)}{\omega\_{o}^{3}} \right| + \frac{3\sigma}{\omega\_{o}^{4-i}} + \frac{4\sigma}{\omega\_{o}^{7-i}} = \upsilon\_{i} \tag{A11}$$

For all *t* ≥ *T*1, *i* = 1, 2, 3, the above formula holds.

**Proof of Theorem 2.** According to Equation (37) and Theorem 1, we can attain:

$$\begin{cases} \left[ A\_{\widetilde{\mathcal{X}}} \widetilde{\mathfrak{x}}(\tau) \right]\_1 = 0\\ \left| \left[ A\_{\widetilde{\mathcal{X}}} \widetilde{\mathfrak{x}}(\tau) \right]\_2 \right| \le k\_{sum} \upsilon\_{\widetilde{\mathfrak{x}}} = \gamma\_{I\prime} \,\forall t \ge T\_1 \end{cases} \tag{A12}$$

where *ksum* = 1 + *k*<sup>1</sup> + *k*2, bringing in the controller bandwidth *ksum* = 1 + *ω*<sup>2</sup> *<sup>c</sup>* + 2*ωc*, and taking the parameters in this way ensures that *A<sup>ε</sup>* is Hurwitz [54].

Define Θ = [0 *γ<sup>l</sup> T* , let *ϑ*(*t*) = R *<sup>t</sup>* 0 *e <sup>A</sup>ε*(*t*−*τ*)*Ax*e*x*e(*τ*)*dτ*, then we can attain:

$$\left|\vartheta\_{i}(t)\right| \le \left|\left(A\_{\varepsilon}^{-1}\Theta\right)\_{i}\right| + \left|\left(A\_{\varepsilon}^{-1}e^{A\_{\varepsilon}t}\Theta\right)\_{i}\right|, i = 1, 2\tag{A13}$$

$$\begin{cases} \left| \left( A\_{\varepsilon}^{-1} \Theta \right)\_{1} \right| = \frac{\gamma\_{l}}{k\_{1}} = \frac{\gamma\_{l}}{\omega\_{\varepsilon}^{2}} \\ \left| \left( A\_{\varepsilon}^{-1} \Theta \right)\_{2} \right| = 0 \end{cases} \tag{A14}$$

Consider that *A<sup>ε</sup>* is Hurwitz; thus, there is a finite time *T*<sup>2</sup> so that for any *t* ≥ *T*2, *i*, *j* = 1, 2, 3, the following formula holds [54]:

$$\left| \left[ e^{A\_{\ell}t} \right]\_{ij} \right| \le \frac{1}{\omega\_{\mathcal{C}}^3} \tag{A15}$$

Let *T*<sup>3</sup> = max{*T*1, *T*2}, for any *t* ≥ *T*3, *i* = 1, 2, we can attain:

$$\left| \left( e^{A\_{\mathbb{E}}t} \Theta \right)\_{\mathbb{i}} \right| \leq \frac{\gamma\_{\mathbb{I}}}{\omega\_{\mathbb{C}}^3} \tag{A16}$$

Then we can attain:

$$\left| \left( A\_{\varepsilon}^{-1} e^{A\_{\varepsilon}t} \Theta \right)\_{i} \right| \leq \begin{cases} \frac{1 + k\_2}{\omega\_{\varepsilon}^2} \frac{\gamma\_l}{\omega\_{\varepsilon}^3}, & i = 1 \\\frac{\gamma\_l}{\omega\_{\varepsilon}^3}, & i = 2 \end{cases} \tag{A17}$$

From Equations (A13), (A14), and (A17) we can attain that for any *t* ≥ *T*3:

$$|\vartheta\_{i}(t)| \le \begin{cases} \frac{\gamma\_{l}}{\omega\_{\cir}^{2}} + \frac{(1+k\_{2})\gamma\_{l}}{\omega\_{\cir}^{5}}, & i=1\\ \frac{\gamma\_{l}}{\omega\_{\cir}^{3}}, & i=2 \end{cases} \tag{A18}$$

Let *εs*(0) = |*ε*1(0)| + |*ε*2(0)|, then for any *t* ≥ *T*3:

$$\left| \left[ e^{A\_t t} \varepsilon(0) \right]\_i \right| \le \frac{\varepsilon\_s(0)}{\omega\_c^3} \tag{A19}$$

From Equation (A12), we can attain:

$$|\varepsilon\_{i}(t)| \le \left| \left[ e^{A\_{\varepsilon}t} \varepsilon(0) \right]\_{i} \right| + |\vartheta\_{i}(t)| \tag{A20}$$

From Equations (A12), (A18)–(A20), we can attain that for any *t* ≥ *T*3, *i* = 1, 2:

$$|\varepsilon\_i(t)| \le \begin{cases} \frac{\varepsilon\_s(0)}{\omega\_c^3} + \frac{k\_{sum}\upsilon\_i}{\omega\_c^2} + \frac{(1+k\_2)k\_{sum}\upsilon\_i}{\omega\_c^5}, & i=1\\\frac{k\_{sum}\upsilon\_i + \varepsilon\_s(0)}{\omega\_c^3}, & i=2 \end{cases} \tag{A21}$$

## **References**


## *Article* **Multi-Mode Shape Control of Active Compliant Aerospace Structures Using Anisotropic Piezocomposite Materials in Antisymmetric Bimorph Configuration**

**Xiaoming Wang 1,\* , Xinhan Hu <sup>2</sup> , Chengbin Huang <sup>2</sup> and Wenya Zhou <sup>2</sup>**


huxinhan@mail.dlut.edu.cn (X.H.); 904368073@mail.dlut.edu.cn (C.H.); zwy@dlut.edu.cn (W.Z.) **\*** Correspondence: wangxm@gzhu.edu.cn

**Abstract:** The mission performance of future advanced aerospace structures can be synthetically improved via active shape control utilizing piezoelectric materials. Multiple work modes are required. Bending/twisting mode control receives special attention for many classic aerospace structures, such as active reflector systems, active blades, and compliant morphing wings. Piezoelectric fiber composite (Piezocomposite) material features in-plane anisotropic actuation, which is very suitable for multiple work modes. In this study, two identical macro-fiber composite (MFC) actuators of the F1 type were bonded to the base plate structure in an "antisymmetric angle-ply bimorph configuration" in order to achieve independent bending/twisting shape control. In terms of the finite element model and homogenization strategy, the locations of bimorph MFCs were determined by considering the effect of trade-off control capabilities on the bending and twisting shapes. The modal characteristics were investigated via both experimental and theoretical approaches. The experimental tests implied that the shape control accuracy was heavily reduced due to various uncertainties and nonlinearities, including hysteresis and the creep effect of the actuators, model errors, and external disturbances. A multi-mode feedback control law was designed and the experimental tests indicated that synthetic (independent and coupled) bending/twisting deformations were achieved with improved shape accuracy. This study provides a feasible multi-mode shape control approach with high surface accuracy, especially by employing piezocomposite materials.

**Keywords:** shape control; macro-fiber composites; bending; twisting; experimental validation; control system

## **1. Introduction**

Owing to the increasingly stringent requirements for industrial equipment, particularly for the aerospace fields, smart devices and structures are receiving increasing attention to improve the synthetic or specific performances of systems [1–3]. Amongst these devices and structures, piezoelectric materials are the most widely used actuators or sensors for various applications, including shape control, vibration suppression, and health monitoring [4,5]. Active structures integrated with piezoelectric materials can change their shape or profile to enhance the accuracy and adaptability of the system via an active control approach. Some classic instances are active reflector systems, active blades, and compliant morphing wings [6–8].

Structural shape control utilizing piezoelectric materials is mainly implemented for two purposes: (1) To correct the shape error of the structure due to manufacturing error or external disturbance; and (2) to provide active shape control for morphing applications with a specific purpose, such as mechanically reconfigurable reflectors (MRRs) and compliant morphing wings. For reflector systems, particularly the large deployable antennas

**Citation:** Wang, X.; Hu, X.; Huang, C.; Zhou, W. Multi-Mode Shape Control of Active Compliant Aerospace Structures Using Anisotropic Piezocomposite Materials in Antisymmetric Bimorph Configuration. *Aerospace* **2022**, *9*, 195. https://doi.org/10.3390/ aerospace9040195

Academic Editor: Gianpietro Di Rito

Received: 21 February 2022 Accepted: 2 April 2022 Published: 6 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

that will be used in the future, high surface precision is required to maintain position accuracy; however, surface errors are induced by many factors, including assembly error, deployment accuracy, environmental loads, or mechanical creep [9]. Active shape control has been proven as a feasible approach to correct reflector surface errors and ensure high shape accuracy. Bradford [6] investigated active reflector systems actuated by macro-fiber composite (MFC) arrays to correct thermally induced deformations and manufacturing errors. Hill [10] investigated the feasibility of using distributed polyvinylidene fluoride actuators in the active control of a large-scale reflector under thermal load. Wang [11] used PZT actuators to adjust the surfaces of flexible cable net antennas using quadratic criteria. Song [12] presented an experimental validation of a PZT-actuated CFRP reflector using a closed-loop iterative shape control method based on the influence coefficient matrix. In addition to modifying the shape error, piezoelectric actuators can also be used for mechanically reconfigurable reflectors that can actively reshape its surface according to the purpose, such as modifying the service coverage. Tanaka and Hiraku [13,14] developed MRR prototypes with six spherical piezoelectric actuators. Shao [7] designed a mechanically reconfigurable reflector using 30 piezoelectric inchworm actuators and presented a distributed time-sharing control strategy to minimize the size and power of the MRR system. In the aviation field, piezoelectric materials are often used for the active shape control of both fixed and rotating wings. Monner [15,16] developed "active twist blades" and used MFCs for twist actuation. Li [17] presented an experimental validation of the feasibility of using piezoelectric actuators to improve rolling power at all dynamic pressures via elastic wing twist. In the last decade, the use of piezocomposite materials, especially MFCs, in compliant morphing wing designs has been broadly investigated. Bilgen [8,18,19] devoted many studies towards the design, optimization, and wind-tunnel testing of MFC-actuated compliant morphing wings, and implemented the flight control of micro-air-vehicles (MAVs). LaCroix [20] used MFCs to deform the surface of the forward-swept thin, compliant composite wings of MAVs. Molinari [21] designed a threedimensional adaptive compliant wing with embedded MFCs and presented aero-structural optimization. These prior works demonstrated that MFCs produce a large actuation effect to deform compliant structures. Compared with conventional hinged, discrete-control surfaces, MFC-actuated morphing wings perform with lower drag and offer more efficient production of control forces and moments.

For future smart structures, multiple work modes, such as bending, twisting, and expansion, will need to be controlled in many circumstances. Concerning some common structures, such as beams or plates, bending deformations are mostly considered due to their larger deformation magnitude and lower inherent frequencies. Twisting deformation and corresponding control issues receive specific attention in many fields, particularly for flexile wing structures. Twist morphing is one of the most popular categories of morphing wings and has resulted in a large number of wind-tunnel and flight tests in aircraft [1]. Some other instances are rotating blades, solar panels, and robot arms, whose behaviors also consist of both bending and twisting modes.

In a piezo-actuation context, in-plane polarized, anisotropic piezocomposite materials are the natural choices for multiple work modes including bending/twisting shape control. Conventional piezoceramics, such as lead zirconium titanate (PZT), are typically capable of large actuation forces; however, they have some limitations, such as small strains and low flexibility [22]. In addition, traditional actuators with through-the-thickness poling possess transverse isotropy in the plane and cannot supply sufficient twisting actuation moment [23]. Piezocomposite materials have emerged as the new class of hybrid materials; they consist of piezoelectric fiber reinforcements embedded in the epoxy matrix and interdigitated electrodes, so they can provide a wide range of effective material properties, good conformability, and strength integrity [24,25]. Piezocomposite materials can utilize the *d*<sup>33</sup> piezoelectric effect in the direction of PZT fibers, which is larger than the *d*<sup>31</sup> piezoelectric effect of conventional piezoelectric actuators [26]. In particular, piezocomposite actuator patches feature anisotropic actuation effects, which makes it possible to expand their work

modes by designing specific PZF fiber orientations [15,23,27–29]. For the MFCs used in this study, three work modes (expansion, bending, and torsion), were realized and, of course, the appropriate actuator types and configurations were chosen for specific modes. Smart Material Corporation [30] provides two types of standard MFC actuator patch that utilize the *d*<sup>33</sup> effect: P1 types with 0◦ fiber orientation and F1 types with 45◦ fiber orientation. The P1-type actuators are mainly used for the bending control of structures and the F1-type actuators are used for twisting control. In addition to unimorph-configuration actuators, piezocomposite actuators in bimorph configuration are commonly adopted to enhance control authority [21,31–34]. Bilgen [31] designed a lightweight high-voltage electronic circuit for MFC bimorphs and remedied the situations in which the MFCs had an asymmetry range in the positive and negative voltage directions. In the authors' previous studies, piezocomposite materials in "antisymmetric angle-ply bimorph configuration" were presented for the bending/twisting shape control of plate-like flexible wing structures [35,36]. The actuator optimizations (in unimorph or bimorph configuration) [37,38] and structure/actuatorintegrated designs [39] for such piezocomposite-actuated structures were also presented. In addition to theoretical investigations, experimental validations are also presented in this paper to demonstrate the feasibility of bimorph MFCs for bending/twisting mode control and real-time control systems.

The primary aim of this paper is to present both theoretical and experimental investigations of the control performance of anisotropic piezocomposite actuators in synthetic (independent and coupled) bending/twisting shape control. In this study, two MFCs of the F1 type in bimorph configuration were used for the shape control of a cantilever aluminum plate. The two identical actuators were orientated at ±45◦ from the front and backside surfaces, respectively. Thus, in ideal situations, the same potential would induce pure twisting, whereas the opposite potential would cause pure bending. The finite element method was applied to model the system and optimize the actuators' locations. An experimental setup was built for the MFC-actuated flexible plate, whose deformation was measured using two laser displacement sensors. A feedback closed-loop control law was designed to improve the shape control accuracy when it was subjected to uncertainties and nonlinearities. Finally, the multi-mode control scheme was experimentally verified using pure bending, twisting, and coupled bending/twisting shape control.

## **2. Model Formulation**

## *2.1. MFC-Actuated Plate Structures*

The active structure in this study is characterized by means of a cantilever aluminum plate, as shown in Figure 1. Two identical MFC patches of M8557-F1 type, which were produced by Smart Material Corp., Sarasota, FL, US, were symmetrically glued to the front and backside surfaces of the base plate, respectively. This type of F1 MFC actuator features a 45◦ fiber orientation with respect to its length direction. Practically, due to the opposite surfaces, the actual fiber orientations for the two actuators were −45◦ and 45◦ with respect to the global *x*-axis, respectively. Note that the electrodes of the actuator always remained perpendicular to the fiber orientation. That is to say, the actuators were in the so-called "antisymmetric angle-ply bimorph configuration", which offers several unique advantages for active shape control. First and foremost, independent bending or twisting deformations could be produced using the opposite or the same potential, respectively; the detailed descriptions are given in the subsequent subsections. Compared with unimorph configuration, larger actuation ability and control authority could be produced. Furthermore, the elastic axis of the base plate is unchanged under bimorph configuration, which may be beneficial for flexible wings [35,36]. Table 1 lists the geometric and material properties (theoretical value) of the base plate and MFC actuators.

**Figure 1.** Schematics of the MFC-actuated compliant plate structure. **Figure 1.** Schematics of the MFC-actuated compliant plate structure.


Because the laser displacement sensors were used to measure the elastic deformation, two measurement points, which were symmetrical relative to the midline, as shown in the **Table 1.** The geometric and material properties (theoretical value) of plant.

Measurement point locations 105 mm from the tip Piezoelectric constants (m/V) 400 × 10 −12 , −170 × 10 Fiber orientations (deg) 45 Electrode spacing 0.5 mm Because the laser displacement sensors were used to measure the elastic deformation, two measurement points, which were symmetrical relative to the midline, as shown in the figure, were chosen to represent the bending and twisting deformations. The optimization of the locations of the MFC actuators is described in the following sections.

### *2.2. Finite Element Model 2.2. Finite Element Model*

A mathematical model is commonly needed to predict the behaviors of systems and can also be used for model-based control system design. The finite element method was used in this study to model the piezocomposite-actuated plate. Figure 2 depicts the finite element model. Quadrilateral plate elements were adopted to discretize the structure. The key issue was to model the MFC actuator, which is a hybrid, layered material that consists of a rectangular cross-section, unidirectional piezoceramic fibers, the epoxy matrix, Kapton, and interdigitated electrodes [40]. Due to the complexity of MFCs, a homogenization strategy was adopted; thus, the actuator could be modeled in the form of homogenized orthotropic materials with arbitrary PZT fiber orientations, as well as composite materials [41,42]. Moreover, the local mass and stiffness change induced by the bonded patch was determined through composite laminate theory. Detailed descriptions of the FEM approach have been presented in [38,39]. For the sake of simplicity, the final governing equations are obtained and written as A mathematical model is commonly needed to predict the behaviors of systems and can also be used for model-based control system design. The finite element method was used in this study to model the piezocomposite-actuated plate. Figure 2 depicts the finite element model. Quadrilateral plate elements were adopted to discretize the structure. The key issue was to model the MFC actuator, which is a hybrid, layered material that consists of a rectangular cross-section, unidirectional piezoceramic fibers, the epoxy matrix, Kapton, and interdigitated electrodes [40]. Due to the complexity of MFCs, a homogenization strategy was adopted; thus, the actuator could be modeled in the form of homogenized orthotropic materials with arbitrary PZT fiber orientations, as well as composite materials [41,42]. Moreover, the local mass and stiffness change induced by the bonded patch was determined through composite laminate theory. Detailed descriptions of the FEM approach have been presented in [38,39]. For the sake of simplicity, the final governing equations are obtained and written as

$$\mathbf{M}\ddot{\mathbf{x}} + \mathbf{K}\mathbf{x} = \mathbf{B}\_{\mathsf{ul}}\boldsymbol{\mu} \tag{1}$$

*M*

*K*

 *M*

 p

*Mx Kx B u* + =

−12

(1)

and

denote the

mass matrix and stiffness matrix, which are assembled as *M M*= <sup>+</sup>bwhere *x* is the vector of the nodal displacements. The matrices *M* and *K* denote the mass matrix and stiffness matrix, which are assembled as *M* = *M*<sup>b</sup> + *M*<sup>p</sup> and *K* = *K*<sup>b</sup> + *K*p, respectively, where subscripts b and p denote the contribution of the base plate layer and piezocomposite layers, respectively. *u* = - *u*<sup>1</sup> *u*<sup>1</sup> T is the vector of the applied voltages. *B<sup>u</sup>* is the coefficient matrix, which depends on the actuator locations.

where

*x*

*K K K*

= <sup>+</sup>b p

plied voltages.

*B u*

plate layer and piezocomposite layers, respectively.

**Figure 2.** Finite element model of the piezo-actuated plate: (**a**) Quadrilateral plate element; (**b**) crosssections of the elements; (**c**) meshes of the structure. **Figure 2.** Finite element model of the piezo-actuated plate: (**a**) Quadrilateral plate element; (**b**) crosssections of the elements; (**c**) meshes of the structure.

, respectively, where subscripts b and p denote the contribution of the base

*u* = *u u*

is the coefficient matrix, which depends on the actuator locations.

1 1

T

is the vector of the ap-

The deflections of the two measurement points P1and P2are given byThe deflections of the two measurement points P<sup>1</sup> and P<sup>2</sup> are given by

$$\begin{aligned} w\_{\mathbb{P}\_1} &= \mathbb{C}\_{\mathbb{P}\_1} \mathfrak{x} \\ w\_{\mathbb{P}\_2} &= \mathbb{C}\_{\mathbb{P}\_2} \mathfrak{x} \end{aligned} \tag{2}$$

where P1 *C* and P2 *C* are the output matrices, which depend on the measurement locations of the laser sensors. where *C*P<sup>1</sup> and *C*P<sup>2</sup> are the output matrices, which depend on the measurement locations of the laser sensors.

 2  P

2

P

## *2.3. Actuator Position Optimization*

*2.3. Actuator Position Optimization* It is an important issue to determine how to efficiently use piezoelectric capabilities to their fullest extent in active shape control. Accordingly, the locations of the actuators must be optimized before performing experiments. Extensive research on the optimal placement of piezoelectric actuators for structural control has also been carried out. Some detailed literature reviews on this topic can be found in [43–45]. The authors have also presented the optimization approach for the anisotropic piezocomposite actuators by considering the PZT fiber orientations as well as the distributed positions [37–39]. However, due to the fixed size and fiber orientation of M8557-F1-type MFC actuators, only the position of the actuators in the length direction can be designed. Thus, the best position can be chosen via traversal simulations, which is not time-consuming. Moreover, the influence of the actuator's position on shape control ability can also be reflected in this way. There-It is an important issue to determine how to efficiently use piezoelectric capabilities to their fullest extent in active shape control. Accordingly, the locations of the actuators must be optimized before performing experiments. Extensive research on the optimal placement of piezoelectric actuators for structural control has also been carried out. Some detailed literature reviews on this topic can be found in [43–45]. The authors have also presented the optimization approach for the anisotropic piezocomposite actuators by considering the PZT fiber orientations as well as the distributed positions [37–39]. However, due to the fixed size and fiber orientation of M8557-F1-type MFC actuators, only the position of theactuators in the length direction can be designed. Thus, the best position can be chosen via traversal simulations, which is not time-consuming. Moreover, the influence of theactuator's position on shape control ability can also be reflected in this way. Therefore, only one design variable is concerned and given as

$$l\_d \in [\mathbf{0}, L - l] \tag{3}$$

(3)

(4)

where *a l* is the distance from the plate root to the left edge of the active area of the MFC, as shown in Figure 1. Note that, to facilitate the theoretical simulation, only the active area of MFC is considered. The values *L* and *l* denote the length of the base plate and actuator, where *l<sup>a</sup>* is the distance from the plate root to the left edge of the active area of the MFC, as shown in Figure 1. Note that, to facilitate the theoretical simulation, only the active area of MFC is considered. The values *<sup>L</sup>* and *<sup>l</sup>* denote the length of the base plate andactuator, respectively.

*a*

1

,

 P

 *w* =

=

respectively. Because both bending and twisting are concerned, two corresponding criteria are Because both bending and twisting are concerned, two corresponding criteria are used to evaluate the shape control capabilities and given by

> =

=

*u*

 −

$$J\_1 = \begin{bmatrix} w\_{\text{P}1} \end{bmatrix}, \text{ } \mathfrak{u} = \begin{bmatrix} 500, -500 \end{bmatrix}^T \tag{4}$$

$$J\_2 = |a|, \ \mathfrak{u} = [500, 500]^\mathrm{T} \tag{5}$$

$$\alpha = \arcsin \frac{w\_{\mathbb{P}\_1} - w\_{\mathbb{P}\_2}}{|\mathbb{P}\_1 \mathbb{P}\_2|} \approx \frac{w\_{\mathbb{P}\_1} - w\_{\mathbb{P}\_2}}{|\mathbb{P}\_1 \mathbb{P}\_2|} \tag{6}$$

where |P1P2| denotes the distance between two measurement points.

1

The variations of *J*<sup>1</sup> and *J*<sup>2</sup> with *l<sup>a</sup>* are shown in Figure 3a, respectively. It was found that the bending deformation decreased monotonically with the length position of the actuator, i.e., the best position was the root area, as depicted in Figure 3(b.1). However, it was preferable to place the actuator in the area of 0.1 m~0.35 m so as to obtain enhanced twisting control capability, while the largest twisting deformation occurred in *la*= 0.22 m, as depicted in Figure 3(b.2). Hence, the area of the shaded part in Figure 3a is a kind of Pareto optimal area, in which any point could be viewed as an acceptable location. Therefore, after considering the trade-off in control authority between the bending and torsional modes, we chose a final position of *la*= 0.1 m. The above discussions explain why we placed the MFC actuators in this position. it was preferable to place the actuator in the area of 0.1 m~0.35 m so as to obtain enhanced twisting control capability, while the largest twisting deformation occurred in as depicted in Figure 3(b.2). Hence, the area of the shaded part in Figure 3a is a kind of Pareto optimal area, in which any point could be viewed as an acceptable location. Therefore, after considering the trade-off in control authority between the bending and torsional modes, we chose a final position of *a l* = 0.1 m. The above discussions explain why we placed the MFC actuators in this position.

1

and

*J*

*Aerospace* **2022**, *9*, x 6 of 16

=

2

*J*

P P

*w w*

denotes the distance between two measurement points.

*l*

*a*

arcsin

with

1

1 2

found that the bending deformation decreased monotonically with the length position of the actuator, i.e., the best position was the root area, as depicted in Figure 3(b.1). However,

−

P P  2

   P P

 *w w*

> 1

> > P P

 1 2

 −

> 2

are shown in Figure 3a, respectively. It was

(6)

*a l*

.

= 0.22 m,

**Figure 3.** Variations in the bending and twisting deformations with the position of the actuator: (**a**) The variations in the bending and twisting deformations with actuator position; (**b.1**) the best position for bending control; (**b.2**) the best position for twisting control; (**b.3**) the chosen position for multi-mode shape control. **Figure 3.** Variations in the bending and twisting deformations with the position of the actuator: (**a**) The variations in the bending and twisting deformations with actuator position; (**b.1**) the best position for bending control; (**b.2**) the best position for twisting control; (**b.3**) the chosen position for multi-mode shape control.

#### *2.4. Theoretical Bending/Twisting Shape Control 2.4. Theoretical Bending/Twisting Shape Control*

where

PP1 2

The variations of

As previously mentioned, the primary aim of this study was to achieve synthetic (independent and coupled) bending and twisting shape control. To theoretically demonstrate the bending/twisting control principle, the deformation configuration of the plate under the actuation of a single MFC is presented in Figure 4. The results demonstrate that both bending and twisting deformations were produced under the actuation of a single F1-type MFC. For the MFC1 that was bonded on the front side of the plate, the deflection amplitude of the lower measurement point P .MFC1 2 *w* was larger than the upper measurement P ,MFC1 1 *w* . By contrast, the deformation trend produced by the MFC2 alone was opposite in not only the direction but also in the relations between *w*and *w*, i.e.,As previously mentioned, the primary aim of this study was to achieve synthetic (independent and coupled) bending and twisting shape control. To theoretically demonstrate the bending/twisting control principle, the deformation configuration of the plate under the actuation of a single MFC is presented in Figure 4. The results demonstrate that both bending and twisting deformations were produced under the actuation of a single F1-type MFC. For the MFC1 that was bonded on the front side of the plate, the deflection amplitude of the lower measurement point was larger than the upper measurement *w*P<sup>1</sup> , MFC1. By contrast, the deformation trend produced by the MFC2 alone was opposite in not only the direction but also in the relations between *w*P<sup>1</sup> and *w*P<sup>2</sup> , i.e.,

$$\begin{vmatrix} w\_{\text{P}\_1,\text{MFC1}} \\ w\_{\text{P}\_1,\text{MFC2}} \end{vmatrix} < \begin{vmatrix} w\_{\text{P}\_2,\text{MFC1}} \\ w\_{\text{P}\_2,\text{MFC2}} \end{vmatrix} \tag{7}$$

1 2

P1

P2

(7)

In ideal situations, the following relation exists:

 

$$\begin{vmatrix} w\_{\text{P}\_1, \text{MFC1}} \\ w\_{\text{P}\_1, \text{MFC2}} \end{vmatrix} = \begin{vmatrix} w\_{\text{P}\_2, \text{MFC2}} \\ w\_{\text{P}\_2, \text{MFC1}} \end{vmatrix} \tag{8}$$

Consequently, employing this bimorph configuration, the two actuators could be polarized in the same direction for twisting deformation and polarized in opposite directions for bending deformation. The voltage input applied for the front and backside actuators are defined as *u*<sup>1</sup> and *u*2, respectively. By applying the same voltages (i.e., *u*<sup>1</sup> = *u*2), pure moments of torque are generated, while the bending moments are canceled out. On the other hand, pure bending deformation can be achieved by using the opposite voltages (*u*<sup>1</sup> = −*u*2). Of course, a combination of bending and twisting deformations can be performed by designing the two voltages. In a practical context, for any two voltages, the decomposition is given as

$$\begin{aligned} \mu\_1 &= \mu\_\text{s} + \mathfrak{u}\_\text{o} \\ \mathfrak{u}\_2 &= \mathfrak{u}\_\text{s} - \mathfrak{u}\_\text{o} \end{aligned} \tag{9}$$

where *u<sup>s</sup>* = *u*1+*u*<sup>2</sup> 2 , *u<sup>o</sup>* = *u*1−*u*<sup>2</sup> 2 are the voltage components for twisting and bending deformation, respectively. The subscripts "*s*" and "*o*" correspond to the same and the opposite components. Figure 5 visualizes the voltage distribution for the two MFC actuators and Figure 6 shows the bending and twisting deformation obtained from the simulation by applying different voltages. *Aerospace* **2022**, *9*, x 7 of 16

**Figure 4.** Deformation configuration of the substrate plate actuated by the single MFC. (**a**) Deformation configuration of the substrate plate actuated by the MFC1; (**b**) deformation configuration of the substrate plate actuated by the MFC2. **Figure 4.** Deformation configuration of the substrate plate actuated by the single MFC. (**a**) Deformation configuration of the substrate plate actuated by the MFC1; (**b**) deformation configuration of the substrate plate actuated by the MFC2. *Aerospace* **2022**, *9*, x 8 of 16 *Aerospace* **2022**, *9*, x 8 of 16

(8)

), pure

(9)

1 2

*u u* =

larized in the same direction for twisting deformation and polarized in opposite directions **Figure 5.** Voltage distributions for the two MFC actuators. **Figure 5.** Voltage distributions for the two MFC actuators. **Figure 5.** Voltage distributions for the two MFC actuators.

*u u*

−

  *u*=**Figure 6.** Bending and twisting shape control effect using different voltages for the actuators. **Figure 6.** Bending and twisting shape control effect using different voltages for the actuators. **Figure 6.** Bending (**a**) and twisting (**b**) shape control effect using different voltages for the actuators.

*s o*

 −  *u*

are the voltage components for twisting and bending de-

2

*u*

#### where 1 2 2 *s u u u* + = , 1 2 2*o u* = **3. Experiment Implementation 3. Experiment Implementation**

by applying different voltages.

back closed-loop control laws.

back closed-loop control laws.

#### formation, respectively. The subscripts "*s*" and "*o*" correspond to the same and the oppo-**3. Experiment Implementation** *3.1.* Setup *3.1. Setup*

site components. Figure 5 visualizes the voltage distribution for the two MFC actuators and Figure 6 shows the bending and twisting deformation obtained from the simulation *3.1.* Setup An experimental setup was designed for validation according to the previous theoretical investigation, as shown in Figure 7. The base structure was a cantilever aluminum An experimental setup was designed for validation according to the previous theoretical investigation, as shown in Figure 7. The base structure was a cantilever aluminum An experimental setup was designed for validation according to the previous theoretical investigation, as shown in Figure 7. The base structure was a cantilever aluminum

plate, whose dimensional and material properties are listed in Table 1. Two identical MFC

plate, whose dimensional and material properties are listed in Table 1. Two identical MFC patches of type M8557-F1, which were produced by Smart Material Corp., Sarasota, FL,

actuators were located 0.1 m from the plate root, as determined previously. A PCI-1721 DAQ card, which was produced by Advantech Co., Ltd., Kunshan, China, was used to convert the digital signals from the computer to the analog signals. Since the operational voltage of the MFCs ranged from −500 to 1500 V, a HVA 1500-2 high-voltage amplifier, which was produced by Physical Instruments Corp., Berlin, Germany, was used to supply the high voltage for the MFC. The elastic deformation of the plate was measured by two OPTEX-CDX-30A-type laser displacement sensors, which was produced by Guangzhou Optex Industrial Automation Control Equipment Co., Ltd. Guangzhou, China; thus, the twisting angle could be computed using Equation (6). The measured data obtained from the laser sensors were conducted using an ADAM-USB card, which was produced by Advantech Co., Ltd., Kunshan, China; thus the signals could be received by the computer. A computer integrated with MATLAB (which was produced by MathWorks Corp, Natick, MA, USA) and LabVIEW (which was produced by NI Corp, Austin, TX, USA) code was used to generate the voltage signals, receive the sensor signals, and implement the feed-

actuators were located 0.1 m from the plate root, as determined previously. A PCI-1721 DAQ card, which was produced by Advantech Co., Ltd., Kunshan, China, was used to convert the digital signals from the computer to the analog signals. Since the operational voltage of the MFCs ranged from −500 to 1500 V, a HVA 1500-2 high-voltage amplifier, which was produced by Physical Instruments Corp., Berlin, Germany, was used to supply the high voltage for the MFC. The elastic deformation of the plate was measured by two OPTEX-CDX-30A-type laser displacement sensors, which was produced by Guangzhou Optex Industrial Automation Control Equipment Co., Ltd. Guangzhou, China; thus, the twisting angle could be computed using Equation (6). The measured data obtained from the laser sensors were conducted using an ADAM-USB card, which was produced by Advantech Co., Ltd., Kunshan, China; thus the signals could be received by the computer. A computer integrated with MATLAB (which was produced by MathWorks Corp, Natick, MA, USA) and LabVIEW (which was produced by NI Corp, Austin, TX, USA) code was used to generate the voltage signals, receive the sensor signals, and implement the feed-

plate, whose dimensional and material properties are listed in Table 1. Two identical MFC patches of type M8557-F1, which were produced by Smart Material Corp., Sarasota, FL, US [30], were glued to the front and backside surfaces of the base plate, respectively. The actuators were located 0.1 m from the plate root, as determined previously. A PCI-1721 DAQ card, which was produced by Advantech Co., Ltd., Kunshan, China, was used to convert the digital signals from the computer to the analog signals. Since the operational voltage of the MFCs ranged from −500 to 1500 V, a HVA 1500-2 high-voltage amplifier, which was produced by Physical Instruments Corp., Berlin, Germany, was used to supply the high voltage for the MFC. The elastic deformation of the plate was measured by two OPTEX-CDX-30A-type laser displacement sensors, which was produced by Guangzhou Optex Industrial Automation Control Equipment Co., Ltd. Guangzhou, China; thus, the twisting angle could be computed using Equation (6). The measured data obtained from the laser sensors were conducted using an ADAM-USB card, which was produced by Advantech Co., Ltd., Kunshan, China; thus the signals could be received by the computer. A computer integrated with MATLAB (which was produced by MathWorks Corp, Natick, MA, USA) and LabVIEW (which was produced by NI Corp, Austin, TX, USA) code was used to generate the voltage signals, receive the sensor signals, and implement the feedback closed-loop control laws. *Aerospace* **2022**, *9*, x 9 of 16

**Figure 7.** Experiment setup. **Figure 7.** Experiment setup.

### *3.2. Modal Analysis 3.2. Modal Analysis*

Before the active shape control tests, modal analysis was implemented first, for two purposes: (1) To investigate the dynamic characteristics of the compliant structure; and (2) to evaluate the influences of the bonded MFC actuators on the system. The modal frequencies and shapes can be determined by solving an eigenvalue problem of the FE model as follows: Before the active shape control tests, modal analysis was implemented first, for two purposes: (1) To investigate the dynamic characteristics of the compliant structure; and (2) to evaluate the influences of the bonded MFC actuators on the system. The modal frequencies and shapes can be determined by solving an eigenvalue problem of the FE model as follows:

$$\left[\mathbf{K} - \omega^2 \mathbf{M}\right] \Phi = 0\tag{10}$$

**Figure 8.** Structural modal shapes. (**a**) Structural modal shapes without MFC actuators; (**b**) struc-

The frequencies were recognized in terms of the free vibration data of the plate using fast Fourier transform. Table 2 lists the natural frequencies for the first three bending modes and the first torsional mode with and without the MFC actuators. In general, the theoretical values were in good agreement with the experimental results. The results showed that, after adding MFC actuators, the frequencies of the first bending and first torsional modes increased, while the frequencies of the second and third bending modes decreased. Note that the influences of piezoelectric actuators depend on many issues, including the size, position, and fiber orientation of anisotropic piezocomposite materials.

(10)

where *ω* is the natural frequencies and *Φ* denotes the corresponding modal shapes. Figure 8 shows the structural modal shapes of the first six modes, including four bending modes and two torsional modes (third and sixth modes). It was observed that the bonded MFCs had relatively little influence on the modal shapes of the structure. where *ω* is the natural frequencies and **Φ** denotes the corresponding modal shapes. Figure 8 shows the structural modal shapes of the first six modes, including four bending modes and two torsional modes (third and sixth modes). It was observed that the bonded MFCs had relatively little influence on the modal shapes of the structure.

The frequencies were recognized in terms of the free vibration data of the plate using fast Fourier transform. Table 2 lists the natural frequencies for the first three bending modes and the first torsional mode with and without the MFC actuators. In general, the theoretical values were in good agreement with the experimental results. The results showed that, after adding MFC actuators, the frequencies of the first bending and first torsional modes increased, while the frequencies of the second and third bending modes decreased. Note that the influences of piezoelectric actuators depend on many issues, including the size, position, and fiber orientation of anisotropic piezocomposite materials.

is the natural frequencies and

Before the active shape control tests, modal analysis was implemented first, for two purposes: (1) To investigate the dynamic characteristics of the compliant structure; and (2) to evaluate the influences of the bonded MFC actuators on the system. The modal frequencies and shapes can be determined by solving an eigenvalue problem of the FE model

2

*Φ*

Figure 8 shows the structural modal shapes of the first six modes, including four bending modes and two torsional modes (third and sixth modes). It was observed that the bonded

−

 

 

 =

P1

P2

) under the sin-

*w*

*w*,

, so the corresponding voltage of the MFC2 is

denotes the corresponding modal shapes.

(10)

*K ω M Φ 0*

MFCs had relatively little influence on the modal shapes of the structure.

**Figure 8.** Structural modal shapes. (**a**) Structural modal shapes without MFC actuators; (**b**) structural modal shapes with MFC actuators. **Figure 8.** Structural modal shapes. (**a**) Structural modal shapes without MFC actuators; (**b**) structural modal shapes with MFC actuators.


The frequencies were recognized in terms of the free vibration data of the plate using **Table 2.** Natural frequencies of the structure with and without MFCs (Hz). **Table 2.** Natural frequencies of the structure with and without MFCs (Hz).

*Aerospace* **2022**, *9*, x 10 of 16

**Figure 7.** Experiment setup.

*3.2. Modal Analysis*

as follows:

where

*ω*

### *3.3. Control Ability 3.3. Control Ability* In this study, the control ability of the MFCs in unimorph and bimorph was tested

In this study, the control ability of the MFCs in unimorph and bimorph was tested experimentally through static deformation analysis. To this end, the experimental steadystate deflection was obtained by directly applying a certain constant voltage for the MFC and recorded after a relative long time. experimentally through static deformation analysis. To this end, the experimental steadystate deflection was obtained by directly applying a certain constant voltage for the MFC and recorded after a relative long time. (a) Using single MFC

#### (a) Using single MFC The elastic deflections of the two measurement points (i.e.,

The elastic deflections of the two measurement points (i.e., *w*P<sup>1</sup> , *w*P<sup>2</sup> ) under the single MFC are given in Figure 9. It can be observed that using a single MFC, both bending and twisting deformation were produced due to the off-line PZT fiber orientation. Moreover, as previously noted, the deflection amplitude of the lower-point P<sup>2</sup> produced by the MFC1 actuator was larger than the upper point P1, i.e., *w*P<sup>1</sup> ,MFC1  < *w*P2.MFC1 for both positive and negative voltages. Conversely, the deflections produced by the MFC2 showed the opposite trend, i.e., *w*P<sup>1</sup> ,MFC2  > *w*P<sup>2</sup> ,MFC2 . That is to say, the remaker given by Equation (7) and Figure 4 was verified by experiments. However, it can be observed that the actuation abilities of the two actuators were not identical, since the deflection amplitude generated by the MFC1 is larger than the MFC2. This deviation was induced by a variety of uncertainties and errors that are discussed in Section 3.4. That is to say, the theoretical remaker given by Equation (8) was not strictly met in the experiments. gle MFC are given in Figure 9. It can be observed that using a single MFC, both bending and twisting deformation were produced due to the off-line PZT fiber orientation. Moreover, as previously noted, the deflection amplitude of the lower-point P<sup>2</sup> produced by the MFC1 actuator was larger than the upper point P1, i.e., P ,MFC1 P .MFC1 1 2 *w w* for both positive and negative voltages. Conversely, the deflections produced by the MFC2 showed the opposite trend, i.e., P ,MFC2 P ,MFC2 1 2 *w w* . That is to say, the remaker given by Equation (7) and Figure 4 was verified by experiments. However, it can be observed that the actuation abilities of the two actuators were not identical, since the deflection amplitude generated by the MFC1 is larger than the MFC2. This deviation was induced by a variety of uncertainties and errors that are discussed in Section 3.4. That is to say, the theoretical remaker given by Equation (8) was not strictly met in the experiments.

(b) Using bimorph MFCs

2

*u*

 1

 *u* =−

labeled by the voltage of the MFC1, i.e.,

**Figure 9.** Deflection of the measurement points using a single MFC actuator. were generally equal; the deflection directions are opposite in Figure 10a (i.e., twisting **Figure 9.** Deflection of the measurement points using a single MFC actuator.

1 *u*

the operation voltage range is −500~1500 V for the same sign; however, it is −500~500 V for the opposite sign. It can be observed that the deflection amplitudes of the two points

. Because of the asymmetrical operation voltages (i.e., −500~1500 V) for the MFCs,

## (b) Using bimorph MFCs

The deflections of the plate under the actuation of the bimorph MFCs using the same and opposite voltages are shown in Figure 10, respectively. The *x*-axis in Figure 10b is labeled by the voltage of the MFC1, i.e., *u*1, so the corresponding voltage of the MFC2 is *u*<sup>2</sup> = −*u*1. Because of the asymmetrical operation voltages (i.e., −500~1500 V) for the MFCs, the operation voltage range is −500~1500 V for the same sign; however, it is −500~500 V for the opposite sign. It can be observed that the deflection amplitudes of the two points were generally equal; the deflection directions are opposite in Figure 10a (i.e., twisting deformation) and the same in Figure 10b (i.e., bending deformation), respectively. Moreover, the deflection amplitudes *w*P<sup>1</sup>  and *w*P<sup>2</sup> , which should be identical in terms of FE analysis, were still not the same in the two experimental cases. The above experiments imply that bending and twisting shape control can be qualitatively realized as theoretically predicted; however, they cannot be implemented with high control accuracy through an open-loop control approach alone. *Aerospace* **2022**, *9*, x 11 of 16 deformation) and the same in Figure 10b (i.e., bending deformation), respectively. Moreover, the deflection amplitudes P1 *w* and P2 *w* , which should be identical in terms of FE analysis, were still not the same in the two experimental cases. The above experiments imply that bending and twisting shape control can be qualitatively realized as theoretically predicted; however, they cannot be implemented with high control accuracy through an open-loop control approach alone.

**Figure 10.** Deflection of the measurement points using bimorph MFC actuators. (**a**) Deflection of the measurement points using bimorph MFC actuators under the same voltages; (**b**) deflection of the measurement points using bimorph MFC actuators under the opposite voltages. **Figure 10.** Deflection of the measurement points using bimorph MFC actuators. (**a**) Deflection of the measurement points using bimorph MFC actuators under the same voltages; (**b**) deflection of the measurement points using bimorph MFC actuators under the opposite voltages.

### *3.4. Uncertainty Analysis 3.4. Uncertainty Analysis*

on.

The above experimental results demonstrate the independent bending/twisting shape control resulting from the use of MFCs in an antisymmetric angle-ply bimorph configuration. However, the open shape control accuracy is generally low due to a variety of The above experimental results demonstrate the independent bending/twisting shape control resulting from the use of MFCs in an antisymmetric angle-ply bimorph configuration. However, the open shape control accuracy is generally low due to a variety of uncertainties and nonlinearities.

uncertainties and nonlinearities. Firstly, the shape control accuracy is also influenced by the non-uniform distribution of the substrate material of the plate, which is assumed as an isotropic aluminum plate in FE modeling. This non-uniform distribution induces changes in both the bending and torsional stiffness properties. Secondly, the MFC layer and the substrate plate are glued by using epoxy. Thus, the thickness and distribution of the epoxy layer also affect the actuation effect of the MFCs, especially the synchronization of the two actuators. Furthermore, the actuation performance is heavily affected by the nonlinearities of the MFCs, including hysteresis, creep, and varied piezoelectric coefficients [30]. Hysteresis and creep effects are the intrinsic nonlinear characteristics of piezoelectric actuators and can significantly affect the control performance [46], as shown in Figure 11. The hysteresis exhibited at a given time depends on the present input and the operational history of the system. Creep is related to the drift effect of output displacement with a constant applied voltage over Firstly, the shape control accuracy is also influenced by the non-uniform distribution of the substrate material of the plate, which is assumed as an isotropic aluminum plate in FE modeling. This non-uniform distribution induces changes in both the bending and torsional stiffness properties. Secondly, the MFC layer and the substrate plate are glued by using epoxy. Thus, the thickness and distribution of the epoxy layer also affect the actuation effect of the MFCs, especially the synchronization of the two actuators. Furthermore, the actuation performance is heavily affected by the nonlinearities of the MFCs, including hysteresis, creep, and varied piezoelectric coefficients [30]. Hysteresis and creep effects are the intrinsic nonlinear characteristics of piezoelectric actuators and can significantly affect the control performance [46], as shown in Figure 11. The hysteresis exhibited at a given time depends on the present input and the operational history of the system. Creep is related to the drift effect of output displacement with a constant applied voltage over extended periods. Furthermore, the piezoelectric coefficients of MFCs are also varied, depending on the situation; an example of this is the d<sup>33</sup> and d<sup>31</sup> piezoelectric constants in

extended periods. Furthermore, the piezoelectric coefficients of MFCs are also varied, depending on the situation; an example of this is the d<sup>33</sup> and d<sup>31</sup> piezoelectric constants in

*Aerospace* **2022**, *9*, x 12 of 16

high electric field intensity compared with low field intensity [30]. Furthermore, the shape control accuracy is also affected by the placement of wires, external disturbance, and so on.

**Figure 11.** Hysteresis and creep effects of the MFC actuator. (**a**) Hysteresis effect of the MFC actuator; (**b**) creep effect of the MFC actuator. **Figure 11.** Hysteresis and creep effects of the MFC actuator. (**a**) Hysteresis effect of the MFC actuator; (**b**) creep effect of the MFC actuator.

Because of the above uncertainties and nonlinearities, accurate bending/twisting shape control cannot be ensured in experiments by using the voltage values estimated by the linear FE simulation. Some feedforward control schemes can be implemented to reduce or cancel out unwanted issues. For example, a feedforward inverse compensation control can be designed to cancel out the hysteresis and creep nonlinearities of MFCs in terms of a phenomenon-based model and experimental data [32]. However, it is still a difficult task to perform open-loop control given all the undesired effects. Therefore, a feedback control scheme is necessary to realize accurate bending/twisting shape control. It is easy to design a shape control law for a single-input–single-output (SISO) system, such as the bending shape control of a plate using M8528-P1-type MFCs in our previous study [47]. However, the plant in this study constitutes a two-input–two-output system, and the key point is how to design the control law to achieve both independent and coupled bending/twisting shape control. Because of the above uncertainties and nonlinearities, accurate bending/twisting shape control cannot be ensured in experiments by using the voltage values estimated by the linear FE simulation. Some feedforward control schemes can be implemented to reduce or cancel out unwanted issues. For example, a feedforward inverse compensation control can be designed to cancel out the hysteresis and creep nonlinearities of MFCs in terms of a phenomenon-based model and experimental data [32]. However, it is still a difficult task to perform open-loop control given all the undesired effects. Therefore, a feedback control scheme is necessary to realize accurate bending/twisting shape control. It is easy to design a shape control law for a single-input–single-output (SISO) system, such as the bending shape control of a plate using M8528-P1-type MFCs in our previous study [47]. However, the plant in this study constitutes a two-input–two-output system, and the key point is how to design the control law to achieve both independent and coupled bending/twisting shape control.

## **4. Closed-Loop Multi-Mode Shape Control System**

### **4. Closed-Loop Multi-Mode Shape Control System** *4.1. Feedback Control Law*

components, i.e.,

mations are organized as follows:

P1

 *B T*

= +,

*w*

*4.1. Feedback Control Law* To achieve the synthetic control of bending and twisting shape, the structural defor-To achieve the synthetic control of bending and twisting shape, the structural deformations are organized as follows:

$$\begin{array}{c} B = \frac{w\_{\text{P}\_1} + w\_{\text{P}\_2}}{2} \\ T = \frac{w\_{\text{P}\_1} - w\_{\text{P}\_2}}{2} \end{array} \tag{11}$$

1 2 P P 2 *w w T* − = (11) where *B* and *T* denote the bending and twisting components in the deformation, respectively. On the other hand, the deflections of the two points can also be represented by the components, i.e.,*w*P<sup>1</sup> = *B* + *T*, *w*P<sup>2</sup> = *B* − *T*.

2

where *B* and *T* denote the bending and twisting components in the deformation, respec-Hence, the shape control error can be given as

P2

*w*

$$\begin{aligned} \Delta B &= B - B\_{\rm d} = \frac{\Lambda w p\_{\rm i} + \Lambda w p\_{\rm p}}{2} \\ \Delta T &= T - T\_{\rm d} = \frac{\Lambda w p\_{\rm i} - \Lambda w p\_{\rm p}}{2} \end{aligned} \tag{12}$$

1 2 P P d 2 *w w B B B* + = − = (12) where *B*<sup>d</sup> and *T*<sup>d</sup> denote the command bending and twisting requirements, respectively. To adjust the shape control error, a feedback control law is designed as

$$\begin{aligned} \Delta u\_s &= \mathcal{K}\_s \Delta T + \mathcal{K}\_{sB} \Delta B \\ \Delta u\_p &= \mathcal{K}\_p \Delta B + \mathcal{K}\_{pT} \Delta T \end{aligned} \tag{13}$$

P P

*w w*

−

where *B*d and *T*d denote the command bending and twisting requirements, respectively. To adjust the shape control error, a feedback control law is designed as where ∆*u<sup>s</sup>* and ∆*u<sup>p</sup>* are the incremental voltage values for the same and the opposite components of the actuators in each time step, respectively. The values *K<sup>s</sup>* and *K<sup>p</sup>* denote the primary control gains, which are designed according to ideal situations. The values *KsB* and *KpT* are used to compensate for the shape error associated with the uncertainties. Subsequently, the voltages for the two MFC actuators can be easily determined according to Equation (9). The multi-mode shape control was directly implemented based on the experiment by using *B c* and *T c* as the control requirements. According to Equation (11), the arbitrary deformation of the plate can be represented by a combination of bending and twisting components. The pure twisting shape control results obtained by using 0.8*T*and

*Aerospace* **2022**, *9*, x 13 of 16

*s*

*u K T K B*

 = + 

 = +

components of the actuators in each time step, respectively. The values

*u K B K T*

*p*

 *s*

> *p*

note the primary control gains, which are designed according to ideal situations. The val-

tainties. Subsequently, the voltages for the two MFC actuators can be easily determined

 *sB*

> *pT*

 

are the incremental voltage values for the same and the opposite

are used to compensate for the shape error associated with the uncer-

(13)

de-

*K*

*c*

=

*s*

and

*K*

*p*

0 as

=

*B*

*c*

#### *4.2. Multi-Mode Shape Control Results* the objectives are shown in Figure 12. Figure 12a gives the deflection histories of the two

where

ues

*KsB*

*s u*

and

according to Equation (9).

and

*KpT*

*4.2. Multi-Mode Shape Control Results*

*p u*

The multi-mode shape control was directly implemented based on the experiment by using *B<sup>c</sup>* and *T<sup>c</sup>* as the control requirements. According to Equation (11), the arbitrary deformation of the plate can be represented by a combination of bending and twisting components. measurement points, i.e., P1 *w* and P2 *w* , which demonstrate that the same deformation amplitudes with opposite directions were achieved. The time histories of *B* and *T* shown in Figure 12b demonstrate that the pure twisting deformation of the plate was achieved,

The pure twisting shape control results obtained by using *T<sup>c</sup>* = 0.8 and *B<sup>c</sup>* = 0 as the objectives are shown in Figure 12. Figure 12a gives the deflection histories of the two measurement points, i.e., *w*P<sup>1</sup> and *w*P<sup>2</sup> , which demonstrate that the same deformation amplitudes with opposite directions were achieved. The time histories of *B* and *T* shown in Figure 12b demonstrate that the pure twisting deformation of the plate was achieved, while the bending deformation was maintained at zero. It can be observed that the shape control accuracy was greatly improved, especially compared with the previous open-loop results. Figure 12c shows the time histories of the voltages for the two MFC actuators. It can be observed that the two voltages did not converge to the same value in terms of the ideal situation. Due to the influences of the creep effect, the voltages slowly varied with time. Furthermore, the deformations converged to the desired values. Such voltage profiles are difficult to determine by feedforward control due to complex uncertainties. The results again prove the necessity of closed-loop control in high-precision shape control. while the bending deformation was maintained at zero. It can be observed that the shape control accuracy was greatly improved, especially compared with the previous open-loop results. Figure 12c shows the time histories of the voltages for the two MFC actuators. It can be observed that the two voltages did not converge to the same value in terms of the ideal situation. Due to the influences of the creep effect, the voltages slowly varied with time. Furthermore, the deformations converged to the desired values. Such voltage profiles are difficult to determine by feedforward control due to complex uncertainties. The results again prove the necessity of closed-loop control in high-precision shape control. Similarly, Figure 13 shows the pure bending shape control results obtained by using 0 *T c* = and 0.8 *B c* = as the objectives. The displacement of the two measuring points was consistent, without producing twisting deformation. The voltage profiles of the MFC1 and MFC2 were generally opposite to each other, as theoretically predicted; however, they still varied with time to resist the uncertainties and nonlinearities.

**Figure 12.** Pure twisting shape control performance. (**a**) The deflection histories of the two measurement points; (**b**) the time histories of *B* and *T*; (**c**) the time histories of the voltages for the two MFC actuators. **Figure 12.** Pure twisting shape control performance. (**a**) The deflection histories of the two measurement points; (**b**) the time histories of *B* and *T*; (**c**) the time histories of the voltages for the two MFC actuators.

Similarly, Figure 13 shows the pure bending shape control results obtained by using *T<sup>c</sup>* = 0 and *B<sup>c</sup>* = 0.8 as the objectives. The displacement of the two measuring points was consistent, without producing twisting deformation. The voltage profiles of the MFC1 and MFC2 were generally opposite to each other, as theoretically predicted; however, they still varied with time to resist the uncertainties and nonlinearities. *Aerospace* **2022**, *9*, x 14 of 16

**Figure 13.** Pure bending shape control performance. (**a**) The deflection histories of the two measurement points; (**b**) the time histories of *B* and *T*; (**c**) the time histories of the voltages for the two MFC actuators. **Figure 13.** Pure bending shape control performance. (**a**) The deflection histories of the two measurement points; (**b**) the time histories of *B* and *T*; (**c**) the time histories of the voltages for the two MFC actuators.

*T*

*c*

=

achieved by employing the multi-mode feedback control approach.

0.3 and

Finally, the coupled bending/twisting shape control (or arbitrary shape control) re-

=

*B*

14. It can be observed that both the bending and twisting deformations reached the com-

according to the feedback control law to achieve arbitrary deformation. The trend in the voltage profiles can also be explained by Figures 12 and 13. The above results imply that independent bending and twisting shape control with improved shape accuracy was

**Figure 14.** Arbitrary shape control performance. (**a**) The deflection histories of the two measurement points; (**b**) the time histories of *B* and *T*; (**c**) the time histories of the voltages for the two MFC

In this paper, the multi-mode shape control of piezo-actuated compliant morphing structures was achieved in both theoretical and experimental ways. Independent and coupled bending/twisting shape control of a plate structure was achieved by using F1-type MFCs in an antisymmetric angle-ply bimorph configuration. The optimal locations of the MFC actuators were determined by comprehensively considering the control of the bending and twisting deformations. The experimental tests implied that the shape control accuracy was heavily reduced due to various uncertainties and nonlinearities, including hysteresis and the creep effect of the actuators, model errors, and external disturbances. A multi-mode feedback control law was designed to cancel out the shape error. The experimental results implied that synthetic (independent and coupled) bending and twisting shape control with improved shape accuracy was achieved by employing the multi-

**Author Contributions:** Conceptualization, X.W.; Data curation, X.H.; Formal analysis, C.H.; Funding acquisition, X.W.; Investigation, X.H.; Methodology, X.W. and W.Z.; Software, X.H. and C.H.; Supervision, W.Z.; Writing—original draft, X.W.; Writing—review & editing, W.Z. All authors have

*c*

0.8 as the objectives, as shown in Figure

actuators.

**5. Conclusions**

mode feedback control approach.

read and agreed to the published version of the manuscript.

sults are presented by using

Finally, the coupled bending/twisting shape control (or arbitrary shape control) results are presented by using *T<sup>c</sup>* = 0.3 and *B<sup>c</sup>* = 0.8 as the objectives, as shown in Figure 14. It can be observed that both the bending and twisting deformations reached the command values with high shape accuracy. The two voltage values were adjusted in time according to the feedback control law to achieve arbitrary deformation. The trend in the voltage profiles can also be explained by Figures 12 and 13. The above results imply that independent bending and twisting shape control with improved shape accuracy was achieved by employing the multi-mode feedback control approach. Finally, the coupled bending/twisting shape control (or arbitrary shape control) results are presented by using 0.3 *T c* = and 0.8 *B c* = as the objectives, as shown in Figure 14. It can be observed that both the bending and twisting deformations reached the command values with high shape accuracy. The two voltage values were adjusted in time according to the feedback control law to achieve arbitrary deformation. The trend in the voltage profiles can also be explained by Figures 12 and 13. The above results imply that independent bending and twisting shape control with improved shape accuracy was achieved by employing the multi-mode feedback control approach.

MFC actuators.

**Figure 13.** Pure bending shape control performance. (**a**) The deflection histories of the two measurement points; (**b**) the time histories of *B* and *T*; (**c**) the time histories of the voltages for the two

*Aerospace* **2022**, *9*, x 14 of 16

**Figure 14.** Arbitrary shape control performance. (**a**) The deflection histories of the two measurement points; (**b**) the time histories of *B* and *T*; (**c**) the time histories of the voltages for the two MFC **Figure 14.** Arbitrary shape control performance. (**a**) The deflection histories of the two measurement points; (**b**) the time histories of *B* and *T*; (**c**) the time histories of the voltages for the two MFC actuators.

## **5. Conclusions**

actuators.

**5. Conclusions** In this paper, the multi-mode shape control of piezo-actuated compliant morphing structures was achieved in both theoretical and experimental ways. Independent and coupled bending/twisting shape control of a plate structure was achieved by using F1-type MFCs in an antisymmetric angle-ply bimorph configuration. The optimal locations of the MFC actuators were determined by comprehensively considering the control of the bending and twisting deformations. The experimental tests implied that the shape control accuracy was heavily reduced due to various uncertainties and nonlinearities, including hysteresis and the creep effect of the actuators, model errors, and external disturbances. A multi-mode feedback control law was designed to cancel out the shape error. The experimental results implied that synthetic (independent and coupled) bending and twisting shape control with improved shape accuracy was achieved by employing the multimode feedback control approach. In this paper, the multi-mode shape control of piezo-actuated compliant morphing structures was achieved in both theoretical and experimental ways. Independent and coupled bending/twisting shape control of a plate structure was achieved by using F1-type MFCs in an antisymmetric angle-ply bimorph configuration. The optimal locations of the MFC actuators were determined by comprehensively considering the control of the bending and twisting deformations. The experimental tests implied that the shape control accuracy was heavily reduced due to various uncertainties and nonlinearities, including hysteresis and the creep effect of the actuators, model errors, and external disturbances. A multi-mode feedback control law was designed to cancel out the shape error. The experimental results implied that synthetic (independent and coupled) bending and twisting shape control with improved shape accuracy was achieved by employing the multi-mode feedback control approach.

**Author Contributions:** Conceptualization, X.W.; Data curation, X.H.; Formal analysis, C.H.; Funding acquisition, X.W.; Investigation, X.H.; Methodology, X.W. and W.Z.; Software, X.H. and C.H.; Supervision, W.Z.; Writing—original draft, X.W.; Writing—review & editing, W.Z. All authors have read and agreed to the published version of the manuscript. **Author Contributions:** Conceptualization, X.W.; Data curation, X.H.; Formal analysis, C.H.; Funding acquisition, X.W.; Investigation, X.H.; Methodology, X.W. and W.Z.; Software, X.H. and C.H.; Supervision, W.Z.; Writing—original draft, X.W.; Writing—review & editing, W.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (11872381, 12102096) and the Guangdong Basic and Applied Basic Research Foundation (2019A1515010859).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The author would like to thank the reviewers and the editors for their valuable comments and constructive suggestions that helped to improve the paper significantly.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


## *Article* **Finite Element Method-Based Optimisation of Magnetic Coupler Design for Safe Operation of Hybrid UAVs**

**Sami Arslan 1,\* , Ires Iskender <sup>2</sup> and Tu˘gba Selcen Navruz <sup>3</sup>**


**\*** Correspondence: sami.arslan1@gazi.edu.tr; Tel.: +90-5300407511

**Abstract:** The integration of compact concepts and advances in permanent-magnet technology improve the safety, usability, endurance, and simplicity of unmanned aerial vehicles (UAVs) while also providing long-term operation without maintenance and larger air gap use. These developments have revealed the demand for the use of magnetic couplers to magnetically isolate aircraft engines and starter-generator shafts, allowing contactless torque transmission. This paper explores the design aspects of an active cylindrical-type magnetic coupler based on finite element analyses to achieve an optimum model for hybrid UAVs using a piston engine. The novel model is parameterised in Ansys Maxwell for optimetric solutions, including magnetostatics and transients. The criteria of material selection, coupler types, and topologies are discussed. The Torque-Speed bench is set up for dynamic and static tests. The highest torque density is obtained in the 10-pole configuration with an embrace of 0.98. In addition, the loss of synchronisation caused by the piston engine shaft locking and misalignment in the case of bearing problems is also examined. The magnetic coupler efficiency is above 94% at the maximum speed. The error margin of the numerical simulations is 8% for the Maxwell 2D and 4.5% for 3D. Correction coefficients of 1.2 for the Maxwell 2D and 1.1 for 3D are proposed.

**Keywords:** active cylindrical coupler; correction coefficient; finite element method; hybrid UAV; magnetic coupler; magnetic coupling; noncontact torque transmission

## **1. Introduction**

Newly increased environmental apprehension, consciousness, and continuous development to improve the safety and reliability of all aircraft are some of the biggest challenges to be addressed in aviation. Such impressive and challenging issues require the development of more efficient and innovative hybrid systems, as shown in Figure 1.

In conventional systems, the propulsion system of small UAVs is provided only by fuel engines, usually piston engines (PEs). In hybrid systems, there is a high-speed, direct-drive, and highly efficient electric machine called starter/generator (S/G) [1] that provides the initial starting mechanism of the PE and charges the system battery group in generator mode while cruising or contributes to the propulsion in motor mode during climbing.

The modernisation of unmanned aerial vehicles (UAVs) under the concept of More Electric Aircraft (MEA) has been on the agenda. However, the challenge of isolating the shafts of the aircraft engine and the electrically driven system, typically the S/G, for more functional and stealthy operations [2] imposes a critical function on magnetic couplers (MCs). MCs provide both contactless torque transmission and hermetic separation using static seals or containment shrouds, which are essential for hybrid UAVs.

**Citation:** Arslan, S.; Iskender, I.; Navruz, T.S. Finite Element Method-Based Optimisation of Magnetic Coupler Design for Safe Operation of Hybrid UAVs. *Aerospace* **2023**, *10*, 140. https://doi.org/ 10.3390/aerospace10020140

Academic Editor: Gianpietro Di Rito

Received: 12 December 2022 Revised: 27 January 2023 Accepted: 30 January 2023 Published: 2 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

*Aerospace* **2023**, *10*, x FOR PEER REVIEW 2 of 23

**Figure 1.** Simple hybrid UAV illustration with the Sullivan S676 Starter/Generator. **Figure 1.** Simple hybrid UAV illustration with the Sullivan S676 Starter/Generator. **Figure 1.** Simple hybrid UAV illustration with the Sullivan S676 Starter/Generator.

MCs have significant advantages such as overload protection, reduced maintenance, simple design, and highly tolerant shaft misalignment, vibration, and noise absorption. An MC consists of permanent magnets (PMs), rotor yokes, a protective cover to protect PMs from high speeds, a containment shroud for sealing, and shafts. Typically, it consists of two rotating parts, an inner and outer rotor, and is grouped as shown in Figure 2 [3]. MCs have significant advantages such as overload protection, reduced maintenance, simple design, and highly tolerant shaft misalignment, vibration, and noise absorption. An MC consists of permanent magnets (PMs), rotor yokes, a protective cover to protect PMs from high speeds, a containment shroud for sealing, and shafts. Typically, it consists of two rotating parts, an inner and outer rotor, and is grouped as shown in Figure 2 [3]. MCs have significant advantages such as overload protection, reduced maintenance, simple design, and highly tolerant shaft misalignment, vibration, and noise absorption. An MC consists of permanent magnets (PMs), rotor yokes, a protective cover to protect PMs from high speeds, a containment shroud for sealing, and shafts. Typically, it consists of two rotating parts, an inner and outer rotor, and is grouped as shown in Figure 2 [3].

**Figure 2.** Type of MCs: (**a**) Active/Reactive coupler; (**b**) Hysteresis coupler; (**c**) Eddy-current cou-**Figure 2.** Type of MCs: (**a**) Active/Reactive coupler; (**b**) Hysteresis coupler; (**c**) Eddy-current coupler. **Figure 2.** Type of MCs: (**a**) Active/Reactive coupler; (**b**) Hysteresis coupler; (**c**) Eddy-current coupler.

pler. Active and reactive couplers have PMs inserted into both rotors. In reactive couplers, the PMs are mounted on only one-half of the rotor, while the other half is steel in the form of PMs. The inner and outer rotors rotate at synchronous speed. The hysteresis and eddycurrent couplers have PMs on the half-rotor side, while the other side has hysteresis and Active and reactive couplers have PMs inserted into both rotors. In reactive couplers, the PMs are mounted on only one-half of the rotor, while the other half is steel in the form of PMs. The inner and outer rotors rotate at synchronous speed. The hysteresis and eddycurrent couplers have PMs on the half-rotor side, while the other side has hysteresis and Active and reactive couplers have PMs inserted into both rotors. In reactive couplers, the PMs are mounted on only one-half of the rotor, while the other half is steel in the form of PMs. The inner and outer rotors rotate at synchronous speed. The hysteresis and eddy-current couplers have PMs on the half-rotor side, while the other side has hysteresis and conductive materials, respectively. The inner and outer rotor speeds are not the same.

conductive materials, respectively. The inner and outer rotor speeds are not the same. Recently, substantial work has been concentrated on MCs using analytical and numerical approaches. Carpentier et al. [4] suggested implementing the analytical virtual work approach to the framework of the volume integral method to compute the magnetic forces. Li et al. [5] obtained 3D analytical torque equations with a closed form for an ideal radial MC. Ravaud et al. [6] performed a 3D semi-analytical study of the transmitted torque between uniformly magnetised PMs based on the Coulombian model. The theoconductive materials, respectively. The inner and outer rotor speeds are not the same. Recently, substantial work has been concentrated on MCs using analytical and numerical approaches. Carpentier et al. [4] suggested implementing the analytical virtual work approach to the framework of the volume integral method to compute the magnetic forces. Li et al. [5] obtained 3D analytical torque equations with a closed form for an ideal radial MC. Ravaud et al. [6] performed a 3D semi-analytical study of the transmitted torque between uniformly magnetised PMs based on the Coulombian model. The theoretical aspects of these studies are predominant. Apart from this, the coupler parameters Recently, substantial work has been concentrated on MCs using analytical and numerical approaches. Carpentier et al. [4] suggested implementing the analytical virtual work approach to the framework of the volume integral method to compute the magnetic forces. Li et al. [5] obtained 3D analytical torque equations with a closed form for an ideal radial MC. Ravaud et al. [6] performed a 3D semi-analytical study of the transmitted torque between uniformly magnetised PMs based on the Coulombian model. The theoretical aspects of these studies are predominant. Apart from this, the coupler parameters affecting the transferable torque have not been comparatively studied.

retical aspects of these studies are predominant. Apart from this, the coupler parameters affecting the transferable torque have not been comparatively studied. On the other hand, studies have accelerated with the development of powerful numerical analysers [7] using finite element methods (FEMs) [8]. Ziolkowski et al. [9] compared transient, quasi-static, and fast-quasi-static modelling techniques to calculate force profiles using FEMs. Ose-Zala et al. [10] investigated the influence of basic design parameters on the mechanical torque for cylindrical MCs with rounded PMs using QuickField software based on 2D FEMs. Kang et al. [11] showed the torque calculation and parametric analysis of synchronous PM couplers. The analytical results are compared with 2D FEMs. affecting the transferable torque have not been comparatively studied. On the other hand, studies have accelerated with the development of powerful numerical analysers [7] using finite element methods (FEMs) [8]. Ziolkowski et al. [9] compared transient, quasi-static, and fast-quasi-static modelling techniques to calculate force profiles using FEMs. Ose-Zala et al. [10] investigated the influence of basic design parameters on the mechanical torque for cylindrical MCs with rounded PMs using QuickField software based on 2D FEMs. Kang et al. [11] showed the torque calculation and parametric analysis of synchronous PM couplers. The analytical results are compared with 2D FEMs. On the other hand, studies have accelerated with the development of powerful numerical analysers [7] using finite element methods (FEMs) [8]. Ziolkowski et al. [9] compared transient, quasi-static, and fast-quasi-static modelling techniques to calculate force profiles using FEMs. Ose-Zala et al. [10] investigated the influence of basic design parameters on the mechanical torque for cylindrical MCs with rounded PMs using QuickField software based on 2D FEMs. Kang et al. [11] showed the torque calculation and parametric analysis of synchronous PM couplers. The analytical results are compared with 2D FEMs. Nevertheless, different temperature and grade conditions of PMs have not been studied. Torque variations against different rotor materials have not yet been investigated.

Baiba Ose et al. [12] examined the influence of the PM width and the number of pole pairs on the mechanical torque of MCs. Meng et al. [13] performed transient magnetic field calculations for MCs by using Ansys Maxwell 3D software. However, the studies do not simultaneously examine multi-objective design parameters that affect each other. field calculations for MCs by using Ansys Maxwell 3D software. However, the studies do not simultaneously examine multi-objective design parameters that affect each other. In addition, different MC topologies [14] have been the subject of comparison. Kang et al. [15] compared the torque of synchronous PM couplers with parallel and Halbach-

Nevertheless, different temperature and grade conditions of PMs have not been studied.

Baiba Ose et al. [12] examined the influence of the PM width and the number of pole pairs on the mechanical torque of MCs. Meng et al. [13] performed transient magnetic

Torque variations against different rotor materials have not yet been investigated.

*Aerospace* **2023**, *10*, x FOR PEER REVIEW 3 of 23

In addition, different MC topologies [14] have been the subject of comparison. Kang et al. [15] compared the torque of synchronous PM couplers with parallel and Halbachmagnetised magnets by using field calculations. Recently, studies of an axially magnetised MC [16] have been reported. At the same time, magnetic gear concepts [17–19] inspired by mechanical gearboxes were studied. Structures combining magnetic gears and electrical machines [20,21] have begun to be developed. Moreover, hybrid coupler studies [22] have become widespread. Loss calculations [23,24] for MC efficiency studies are shown. However, static tests in response to torque angle variations have not been investigated. magnetised magnets by using field calculations. Recently, studies of an axially magnetised MC [16] have been reported. At the same time, magnetic gear concepts [17–19] inspired by mechanical gearboxes were studied. Structures combining magnetic gears and electrical machines [20,21] have begun to be developed. Moreover, hybrid coupler studies [22] have become widespread. Loss calculations [23,24] for MC efficiency studies are shown. However, static tests in response to torque angle variations have not been investigated. MCs are safely used in many areas [25–27], such as automotive, marine, pump, and compressor applications. One of the practical benefits of MCs is to prevent mechanical

MCs are safely used in many areas [25–27], such as automotive, marine, pump, and compressor applications. One of the practical benefits of MCs is to prevent mechanical faults [28,29] due to torque overloads in some critical applications with the help of slipping when excessive torque is applied. MCs are also impactful for use in hazardous or corrosive environments while transmitting torque through a containment shroud [30]. faults [28,29] due to torque overloads in some critical applications with the help of slipping when excessive torque is applied. MCs are also impactful for use in hazardous or corrosive environments while transmitting torque through a containment shroud [30]. Optimisation studies [31,32] have been performed to achieve the optimum design. Furthermore, it has been investigated whether magnetic bearings [33] could be used in-

Optimisation studies [31,32] have been performed to achieve the optimum design. Furthermore, it has been investigated whether magnetic bearings [33] could be used instead of mechanical bearings to reduce maintenance and operating costs. stead of mechanical bearings to reduce maintenance and operating costs. Although not in large numbers, MCs have started to be used in the aviation industry [34,35]. Benarous et al. [36] summarised all the findings and revealed test data from a

Although not in large numbers, MCs have started to be used in the aviation industry [34,35]. Benarous et al. [36] summarised all the findings and revealed test data from a magnetic gear coupler designed for an aerospace application. Finally, coreless design [37], which is demanded chiefly in aviation applications, has also been mentioned. magnetic gear coupler designed for an aerospace application. Finally, coreless design [37], which is demanded chiefly in aviation applications, has also been mentioned. Since most systems traditionally have design limitations that directly affect output characteristics, the system-specific design of MCs is required where performance investi-

Since most systems traditionally have design limitations that directly affect output characteristics, the system-specific design of MCs is required where performance investigations against correlative system parameters are considered. gations against correlative system parameters are considered. This paper clarifies the design aspects and implications of active cylindrical MCs, particularly for small-sized UAVs, to achieve the optimum design. The use of proposed

This paper clarifies the design aspects and implications of active cylindrical MCs, particularly for small-sized UAVs, to achieve the optimum design. The use of proposed MCs in hybrid UAVs comprising PE and S/G units is important because they provide a significant advantage in protecting the UAV, especially under severe conditions such as excessive loading and shaft lock-up [38,39]. In such catastrophic situations, the S/G is operated in motor mode, allowing the UAV to continue its cruise mission or land safely with the help of the loss of synchronisation between the inner and outer rotors of the MC, as shown in Figure 3. Although this loss of synchronisation may seem like a problem in ordinary machine designs, the use of MCs provides a great advantage in terms of protection against breakage in hybrid UAVs. MCs in hybrid UAVs comprising PE and S/G units is important because they provide a significant advantage in protecting the UAV, especially under severe conditions such as excessive loading and shaft lock-up [38,39]. In such catastrophic situations, the S/G is operated in motor mode, allowing the UAV to continue its cruise mission or land safely with the help of the loss of synchronisation between the inner and outer rotors of the MC, as shown in Figure 3. Although this loss of synchronisation may seem like a problem in ordinary machine designs, the use of MCs provides a great advantage in terms of protection against breakage in hybrid UAVs.

**Figure 3.** (**a**) Block diagram of the hybrid UAV; (**b**) 3D drawing of the proposed model. **Figure 3.** (**a**) Block diagram of the hybrid UAV; (**b**) 3D drawing of the proposed model.

The novel MC is part of the customised Bearcat F85F model aircraft, which is a 60% hybrid by replacing the conventional 3W-140i PE [40] with the 3W-55XI PE and S/G of 6 kW and 4500 rpm. The MCs are prototyped, and the given dimensions are verified.

The main contribution of this paper is to explore the effects of the design parameters of the MC by applying a multi-objective optimisation approach. Supporting the numerical analyses with experiments and systematically collecting the results under a unique study paves the way for researchers to facilitate the design process and validate the proof of concept. This work is distinguished from other studies with the following novelties:


In addition, exploring the effect of PM temperature changes, orientations, and grades on pullout torque increases the novelty of the article. In previous studies, the multi-objective optimisation of the MC design was considered analytically [10–14]. However, nonlinear parameters affecting the performance, such as leakage flux, core losses, and end effects, are ignored in analytical methods to avoid complex and time-consuming calculations. In addition, the eddy-current losses induced in the PMs due to the continuously varying torque angle depending on the natural vibration of the piston engine are very difficult to handle analytically. Therefore, it requires more precise FEM analysis.

For this purpose, the design parameters are considered as a whole, and accordingly, the system is numerically optimised. Thus, the leakage flux, core losses, and end effects are evaluated with the FEM model. The efficiency of the optimised MC is considered an important performance indicator and is analysed together with the nonlinear effects of the materials. Furthermore, the experimental verification of an optimised FEM model in accordance with PE output parameters for hybrid UAVs also makes this study interesting for researchers. The experimental results are in agreement with the FEM outputs.

This study consists of four main frameworks. Section 2 covers design considerations such as dimension criteria, constraints, and rotor topologies. Analytical pre-dimensioning and FEMs by Ansys Maxwell are included in Section 3. The MC is dynamically modelled to improve the simulation accuracy. The effects of the air gap clearance, model length, pole numbers, PM thickness, and thickness of the rotor yokes are investigated in magnetostatics and transients. The torque ripple of the MC is explored. Section 4 comparatively presents and discusses the performance test results of the MCs with different design parameters carried out on the dynamic test bench. Locked-rotor and dynamic tests are performed with steps, full loads, and overloads [41]. Finally, the findings are reviewed in Section 5.

## **2. Design Considerations**

The block diagrams of the hybrid UAV system and the proposed model are shown in Figure 3a,b, respectively. Numerous criteria are used for UAV classification [42], such as the mean take-off weight (MTOW), size, operating conditions, and capabilities.

The modernised Bearcat F85F Warbirds 1/4.2 scale aircraft with 22 kg, 256 cm wingspan, 204 cm length, and 150 m ceiling altitude is in the Open Category A3 (small size) based on European Union Aviation Safety Agency (EASA) regulations [43,44].

The maximum torque of the replaced 3W-55XI PE and, therefore, the minimum torque to be transmitted by the MC is 4.4 N·m. However, considering the load variations due to sudden manoeuvres, a safety factor of 1.2 is determined. In addition, the correction coefficient of 1.3 is initially chosen at the beginning of the design to account for the simulation errors and high starting kickback torque of the PE. Accordingly, in light of the MTOW, including PE and S/G, the allowable weight and length for the MC are set by the manufacturer at 375 g and 15 mm, respectively. The optimisation parameters of the MC design sought in the design reviews and given by the UAV manufacturer are summarised in Table 1.


torque within the manufacturable size and weight limitations to compensate for the un-

Minimum torque density, required 18.4 N.m/kg

manufacturer at 375 g and 15 mm, respectively. The optimisation parameters of the MC design sought in the design reviews and given by the UAV manufacturer are summarised

Pullout torque, with safety factor and correction coefficient 6.9 N.m

**Parameters Value** 

Rated speed 4500 rpm

In the design of active couplers, an objective function such as torque per magnet volume, torque per coupler volume, or cost per weight should be considered to obtain the final design. The minimum weight that meets the requirements is often preferred for hybrid UAVs. However, the optimum design study is based on the achievable maximum torque within the manufacturable size and weight limitations to compensate for the unpredictable high kickback torque experienced during the initial start-up of the PE. The optimal design parameters are identified in Section 3.5 by comparing different topologies. MCs are also classified by the shape of PMs [45], such as star-type, cylindrical, ring-type, rectangular or sector shape, and toothed surface. In terms of practical use, the cylindrical type is more popular. Further classification can be performed according to the magnetisation direction of PMs as radial, axial, and linear orientations. The active cylindrical type is intended for synchronous speed and radial motion requirements. predictable high kickback torque experienced during the initial start-up of the PE. The optimal design parameters are identified in Section 3.5 by comparing different topologies. MCs are also classified by the shape of PMs [45], such as star-type, cylindrical, ring-type, rectangular or sector shape, and toothed surface. In terms of practical use, the cylindrical type is more popular. Further classification can be performed according to the magnetisation direction of PMs as radial, axial, and linear orientations. The active cylindrical type is intended for synchronous speed and radial motion requirements. *2.1. Determination of the Minimum Outer Diameter of the Inner Rotor*  The inner rotor of the MC is directly connected to the flange of the PE, as marked in red in Figure 4a, thus providing magnetic separation [46] between the shafts of the PE and S/G to improve the safe operation [47] of the hybrid UAV.

### *2.1. Determination of the Minimum Outer Diameter of the Inner Rotor* The design of the MC should start from the inner rotor to the outer rotor, as opposed

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The inner rotor of the MC is directly connected to the flange of the PE, as marked in red in Figure 4a, thus providing magnetic separation [46] between the shafts of the PE and S/G to improve the safe operation [47] of the hybrid UAV. to the conventional method, due to the diameter limitations of the inner flange in the 3W-55XI PE, as shown in Figure 4b. It is ensured that the outer rotor of the MC is also the rotor of the S/G to take advantage of this design limitation.

**Figure 4.** Mounting illustration of the (**a**) Inner rotor; (**b**) Inner flange. **Figure 4.** Mounting illustration of the (**a**) Inner rotor; (**b**) Inner flange.

The design of the MC should start from the inner rotor to the outer rotor, as opposed to the conventional method, due to the diameter limitations of the inner flange in the 3W-55XI PE, as shown in Figure 4b. It is ensured that the outer rotor of the MC is also the rotor of the S/G to take advantage of this design limitation.

## *2.2. Selection of Rotor Topology*

**Table 1.** Design parameters.

**Table 1.** Design parameters.

in Table 1.

Figure 5a–f illustrate the conventional rotor topologies selected depending on the objective function and the application area. Some disruptive topologies have also been applied, such as Halbach arrays to increase the field strength of PMs, as shown in Figure 5g [15], and enhanced hybrid couplers to increase the torque density, as shown in Figure 5h [48]. However, arc surface PMs are preferred due to the ease of fabrication and access.

**Figure 5.** Rotor topologies: (**a**) Arc surface-mounted; (**b**) Rectangular surface-mounted; (**c**) Ringtype; (**d**) Buried arc type; (**e**) Buried type; (**f**) Inset type; (**g**) Halbach arrays; (**h**) Enhanced hybrid. **Figure 5.** Rotor topologies: (**a**) Arc surface-mounted; (**b**) Rectangular surface-mounted; (**c**) Ring-type; (**d**) Buried arc type; (**e**) Buried type; (**f**) Inset type; (**g**) Halbach arrays; (**h**) Enhanced hybrid.

Figure 5a–f illustrate the conventional rotor topologies selected depending on the objective function and the application area. Some disruptive topologies have also been applied, such as Halbach arrays to increase the field strength of PMs, as shown in Figure 5g [15], and enhanced hybrid couplers to increase the torque density, as shown in Figure 5h [48]. However, arc surface PMs are preferred due to the ease of fabrication and access.

#### *2.3. Materials Overview 2.3. Materials Overview*

*2.2. Selection of Rotor Topology* 

Electric steel, carbon steel, and metals are used as MC rotor materials. NdFeB and SmCo stand out among ferrite, ceramic, and alnico magnets due to their high energy density. Epoxy, the most common type of coating for aerospace applications, has been preferred among coating types such as zinc, gold, plastic, nickel, and Teflon. The temperature assignment of PMs is made at 80 degrees Celsius, which is the most likely to be encountered in the system. In some special cases, a protective sleeve made of stainless steel, fibres, or plastics is used to prevent the PMs from leaving the rotor surface. Electric steel, carbon steel, and metals are used as MC rotor materials. NdFeB and SmCo stand out among ferrite, ceramic, and alnico magnets due to their high energy density. Epoxy, the most common type of coating for aerospace applications, has been preferred among coating types such as zinc, gold, plastic, nickel, and Teflon. The temperature assignment of PMs is made at 80 degrees Celsius, which is the most likely to be encountered in the system. In some special cases, a protective sleeve made of stainless steel, fibres, or plastics is used to prevent the PMs from leaving the rotor surface.

The containment shroud for sealing fixed to the stationary part of the MC hermetically separates the inner and outer rotors. There are several materials, such as nonferrous stainless steel, Hastelloy, carbon fibre peek, oxide ceramics, and nonmetallics. The containment shroud for sealing fixed to the stationary part of the MC hermetically separates the inner and outer rotors. There are several materials, such as nonferrous stainless steel, Hastelloy, carbon fibre peek, oxide ceramics, and nonmetallics.

### **3. Design Studies 3. Design Studies**

Analytical approaches [49,50] are simple and fast methods for estimating preliminary design dimensions. The margins of error are high because the calculations are made under the assumptions that the magnetisation of PMs is homogeneous, the model length and the average air gap radius are very large compared to the PM thickness and air gap length, and the rotor materials are not saturated and have high permeability [51]. However, analytical calculations involving these effects are laborious and complicated. Analytical approaches [49,50] are simple and fast methods for estimating preliminary design dimensions. The margins of error are high because the calculations are made under the assumptions that the magnetisation of PMs is homogeneous, the model length and the average air gap radius are very large compared to the PM thickness and air gap length, and the rotor materials are not saturated and have high permeability [51]. However, analytical calculations involving these effects are laborious and complicated.

## *3.1. Analytical Preliminary Sizing*

*3.1. Analytical Preliminary Sizing*  The analytical subdomain method based on the Maxwell stress tensor and virtual work approach are accurate methods for the analytical calculations of the transmitted torque of MCs. The analytical subdomain method uses Laplace's and Poisson's equations [7] for the air gap and PM regions to find the flux density distribution by using the derivative of the vector potential equation in the air gap, as in Equations (1) and (2). Then, the The analytical subdomain method based on the Maxwell stress tensor and virtual work approach are accurate methods for the analytical calculations of the transmitted torque of MCs. The analytical subdomain method uses Laplace's and Poisson's equations [7] for the air gap and PM regions to find the flux density distribution by using the derivative of the vector potential equation in the air gap, as in Equations (1) and (2). Then, the transmitted torque is calculated from the Maxwell stress tensor method as in Equation (3).

transmitted torque is calculated from the Maxwell stress tensor method as in Equation (3).

$$B\_{IIr}(r, \Theta) = \frac{1}{r} \frac{\partial A\_{II}}{\partial \Theta} \tag{1}$$

$$B\_{II\Theta}(r,\theta) = \frac{1}{r}\frac{\partial A\_{II}}{\partial \theta} \tag{2}$$

$$T = \frac{l\_s r\_{mean}}{\mu\_0} \int\_0^{2\pi} [B\_{IIr}(r\_{mean}, \Theta) B\_{II\Theta}(r\_{mean}, \Theta)] d\Theta \tag{3}$$

µ where ୍୍୰ሺ, Ɵሻ and ୍୍Ɵሺ, Ɵሻ are the air gap flux density distributions depending on the radial distance (*r*) and angle (Ɵ) according to polar coordinate adoption, respectively. where *BI Ir*(*r*, θ) and *BI I*θ(*r*, θ) are the air gap flux density distributions depending on the radial distance (*r*) and angle (θ) according to polar coordinate adoption, respectively.*∂AI I* and *l<sup>s</sup>* are the vector potential in the air gap and the total model active length, respectively. *rmean* is the radius of the middle of the air gap, as shown in Figure 6a. roi and rio are the outer radius of the inner rotor and inner diameter of the outer rotor, respectively.

∂୍୍ and ௦ are the vector potential in the air gap and the total model active length, respectively. is the radius of the middle of the air gap, as shown in Figure 6a. roi and rio are the outer radius of the inner rotor and inner diameter of the outer rotor, respectively. However, the virtual work method [52] is a practical approach for the fast calculation of the air gap volume depending on the angular displacement between the PMs based on

Ɵ ൌ ሺଵሻଶሺሻ

where , Ɵ, Ɵ, ଵ, , 2 are the change in stored energy in the air gap in joules, mechanical angle of a pole in radians, angular displacement between poles depending on the load, fundamental component of the flux density in the middle of the air gap in Tesla, air gap volume in m3, and the number of poles, respectively. ௧௧ is the total torque exerted on the middle of the air gap used to estimate the torque on the rotors with regard to the total number of poles. The active couplers work without any slip until the pullout torque is exceeded. The pullout torque is expressed as the maximum torque that the MC can

The model of the proposed cylindrical MC comprises an inner and outer rotor (1,7), weight reduction holes (2,9), mounting holes (3,8), PM housings (4,10), and PMs (5,6), as

The torque angle (θ) is the mechanical angle between the d-axes of the inner rotor PMs and the outer rotor PMs when the MC is loaded, as shown in Figure 6a. The angle at the maximum torque is called the critical angle [3] and is calculated as in Equation (5),

Mdeg represents the mechanical angle. The critical angle is 18 Mdeg and 0.314 radians for 10-poles. Preliminary sizing calculations are performed using Equation (4) and summarised in Table 2 for the 10-pole configuration at the critical angle in which (Ɵ-dƟ)

The respective air gap volume is calculated as 1289.5 mm3. The bore diameter of the inner rotor is 29 mm due to the mounting hole diameters on the flange, as shown in Figure 4b. Similarly, the outer diameter of the inner rotor is to be a minimum of 43 mm for the model. Considering the thickness of the rotor yokes and PMs, rmean is initially chosen to be 27.5 mm. With the initial assumption of an air gap length of 1.5 mm, the outer diameter of the inner rotor and the inner diameter of the outer rotor are found to be 26.75 mm and 28.25 mm, respectively. Thus, the corresponding model length is found to be 5 mm.

is directly equal to dƟ. The Ɵ is 36 Mdeg and 0.628 radians for 10-poles.

Ɵ௧ ൌ <sup>360</sup>°

2ሺƟ െ Ɵሻ ∗ ሺ2ሻ (4)

ሺ2ሻ ∗ 2 , (5)

the energy change in the air gap and is given as follows.

handle.

illustrated in Figure 6b.

equivalent to 90 electrical degrees.

௧௧ ൌ

**Figure 6. Figure 6.** Model of the MCs ( Model of the MCs ( **a**) Parameter definitions for the VM method; ( **a**) Parameter definitions for the VM method; ( **b b** ) Material definitions. ) Material definitions.

However, the virtual work method [52] is a practical approach for the fast calculation of the air gap volume depending on the angular displacement between the PMs based on the energy change in the air gap and is given as follows.

$$T\_{\text{total}} = \frac{d\mathcal{W}}{d\Theta} = \frac{(B\_{\text{g1}})^2 (V\_{\text{ag}})}{2\mu\_0 (\Theta - d\Theta)} \* (2p) \tag{4}$$

where *dW*, θ, *d*θ, *Bg*1, *Vag*, 2*p* are the change in stored energy in the air gap in joules, mechanical angle of a pole in radians, angular displacement between poles depending on the load, fundamental component of the flux density in the middle of the air gap in Tesla, air gap volume in m<sup>3</sup> , and the number of poles, respectively. *Ttotal* is the total torque exerted on the middle of the air gap used to estimate the torque on the rotors with regard to the total number of poles. The active couplers work without any slip until the pullout torque is exceeded. The pullout torque is expressed as the maximum torque that the MC can handle.

The model of the proposed cylindrical MC comprises an inner and outer rotor (1,7), weight reduction holes (2,9), mounting holes (3,8), PM housings (4,10), and PMs (5,6), as illustrated in Figure 6b.

The torque angle (θ) is the mechanical angle between the d-axes of the inner rotor PMs and the outer rotor PMs when the MC is loaded, as shown in Figure 6a. The angle at the maximum torque is called the critical angle [3] and is calculated as in Equation (5), equivalent to 90 electrical degrees.

Mdeg represents the mechanical angle. The critical angle is 18 Mdeg and 0.314 radians for 10-poles. Preliminary sizing calculations are performed using Equation (4) and summarised in Table 2 for the 10-pole configuration at the critical angle in which (θ-dθ) is directly equal to dθ. The θ is 36 Mdeg and 0.628 radians for 10-poles.

$$
\theta\_{critical} = \frac{360^{\circ}}{(2p) \ast 2}, \quad \text{Mdeg} \tag{5}
$$

The respective air gap volume is calculated as 1289.5 mm<sup>3</sup> . The bore diameter of the inner rotor is 29 mm due to the mounting hole diameters on the flange, as shown in Figure 4b. Similarly, the outer diameter of the inner rotor is to be a minimum of 43 mm for the model. Considering the thickness of the rotor yokes and PMs, rmean is initially chosen to be 27.5 mm. With the initial assumption of an air gap length of 1.5 mm, the outer diameter of the inner rotor and the inner diameter of the outer rotor are found to be 26.75 mm and 28.25 mm, respectively. Thus, the corresponding model length is found to be 5 mm.


**Table 2.** Analytical design calculations by virtual work approach for preliminary sizing.

## *3.2. Maxwell 2D Static Analyses*

The increased ability to use all processor cores and the symmetry properties tend to directly use FEM-based software in the design and optimisation of MCs [53], resulting in a tangible increase in simulation accuracy. The FEM accurately calculates the air gap flux density and transmitted torque by considering the material nonlinearities, leakage fluxes, core losses, PM magnetisation directions and temperature changes, induced eddy-current losses on PMs, and dynamic effects on the MC.

Ansys Maxwell multi-functional analyses are performed in a magnetostatic environment for static simulations and in a transient environment for dynamic simulations [54].

## 3.2.1. Correlation of Effective Air Gap Diameter and Model Length

The analyses are started with a 10-pole configuration to determine the effective air gap diameter and the model length for the required torque and torque density. Considering the air gap and yoke thicknesses, the analysis started with a minimum effective air gap diameter of 45 mm. The effective air gap is defined as the middle of the air gap.

Figure 7 investigates the pullout torque for the range of effective air gap diameters and model lengths. Figure 7a shows that the required pullout torque of 6.9 N·m specified in Table 1 is met with a minimum effective air gap diameter of 47 mm for a model length of 10 mm or a minimum model length of 6 mm for an effective air gap length of 67 mm. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 9 of 23

**Figure 7.** Pullout torque versus (**a**) Effective air gap length in model length; (**b**) Model length. **Figure 7.** Pullout torque versus (**a**) Effective air gap length in model length; (**b**) Model length.

3.2.2. Investigation of Optimum Pole Number In magnetic systems, such as MCs, the pole number configuration of the rotors significantly affects the transmitted torque and, thus, the torque density. The pole number of 10 for the inner and outer rotors is chosen as the optimum point because the highest pullout torque is provided, as shown in Figure 8a. The relationship between the effective air gap diameter and the length of the MC directly affects the inertia and transferrable torque of the hybrid UAV. The choice of an effective air gap diameter as high as possible will increase the system inertia and allow modular construction for higher torque transmissions. Therefore, the effective air gap diameter is chosen to be 57.5 mm, approaching the maximum limit of the range.

On the other hand, Figure 8b examines viable or impracticable cases of the torque density for different numbers of inner and outer rotor poles. The torque density expresses the pullout torque per unit weight in N.m/kg. The torque density decreases dramatically On the other hand, the linear increase in the model length corresponds to an almost linear increase in the pullout torque. The required torque of 6.9 N·m is achieved with a minimum coupler length of 7.4 mm, as shown in Figure 7b. However, for easy manufac-

in the case of different inner and outer rotor pole number configurations. However, a different number of poles on both rotors is possible with the appropriate design of the mod-

**Figure 8.** (**a**) Torque vs same rotor pole number; (**b**) Torque density vs different rotor pole number.

Air gap clearance has a direct effect on the torque since it affects the total reluctance. Reducing the air gap will increase the torque, as seen in Figure 9, but it will also increase production costs and cause the rotors to rub against each other in the case of imbalance. Considering the fabrication issues and PE vibration, the air gap is set to 1.5 mm, as marked in green in Figure 9a. A twofold increase in the air gap length does not mean a twofold reduction in the torque but instead a reduction of 13%, as shown in Figure 9b.

(**a**) (**b**)

3.2.3. Effect of Air Gap Clearance on Pullout Torque

turability and higher tolerance to disturbance, the length is chosen to be 10 mm, in which case the pullout torque is 9.4 N·m, hereafter referred to as the updated torque requirement. pullout torque is provided, as shown in Figure 8a. On the other hand, Figure 8b examines viable or impracticable cases of the torque density for different numbers of inner and outer rotor poles. The torque density expresses

In magnetic systems, such as MCs, the pole number configuration of the rotors significantly affects the transmitted torque and, thus, the torque density. The pole number of 10 for the inner and outer rotors is chosen as the optimum point because the highest

**Figure 7.** Pullout torque versus (**a**) Effective air gap length in model length; (**b**) Model length.

### 3.2.2. Investigation of Optimum Pole Number the pullout torque per unit weight in N.m/kg. The torque density decreases dramatically

3.2.2. Investigation of Optimum Pole Number

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(**a**) (**b**)

In magnetic systems, such as MCs, the pole number configuration of the rotors significantly affects the transmitted torque and, thus, the torque density. The pole number of 10 for the inner and outer rotors is chosen as the optimum point because the highest pullout torque is provided, as shown in Figure 8a. in the case of different inner and outer rotor pole number configurations. However, a different number of poles on both rotors is possible with the appropriate design of the modulator in the air gap. Thus, the magnetic gear concept [19] is formed.

**Figure 8.** (**a**) Torque vs same rotor pole number; (**b**) Torque density vs different rotor pole number. **Figure 8.** (**a**) Torque vs same rotor pole number; (**b**) Torque density vs different rotor pole number.

3.2.3. Effect of Air Gap Clearance on Pullout Torque Air gap clearance has a direct effect on the torque since it affects the total reluctance. Reducing the air gap will increase the torque, as seen in Figure 9, but it will also increase production costs and cause the rotors to rub against each other in the case of imbalance. Considering the fabrication issues and PE vibration, the air gap is set to 1.5 mm, as marked in green in Figure 9a. A twofold increase in the air gap length does not mean a On the other hand, Figure 8b examines viable or impracticable cases of the torque density for different numbers of inner and outer rotor poles. The torque density expresses the pullout torque per unit weight in N·m/kg. The torque density decreases dramatically in the case of different inner and outer rotor pole number configurations. However, a different number of poles on both rotors is possible with the appropriate design of the modulator in the air gap. Thus, the magnetic gear concept [19] is formed.

### twofold reduction in the torque but instead a reduction of 13%, as shown in Figure 9b. 3.2.3. Effect of Air Gap Clearance on Pullout Torque

Air gap clearance has a direct effect on the torque since it affects the total reluctance. Reducing the air gap will increase the torque, as seen in Figure 9, but it will also increase production costs and cause the rotors to rub against each other in the case of imbalance. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 10 of 23

**Figure 9.** Pullout torque investigation versus: (**a**) Pole number in air gap length; (**b**) Air gap length. **Figure 9.** Pullout torque investigation versus: (**a**) Pole number in air gap length; (**b**) Air gap length.

3.2.4. Determination of PM Thickness The PM thickness changes the average flux density in the air gap and, hence, the transmitted torque [55]. The thickness of the inner rotor PMs is set at 4 mm because the Considering the fabrication issues and PE vibration, the air gap is set to 1.5 mm, as marked in green in Figure 9a. A twofold increase in the air gap length does not mean a twofold reduction in the torque but instead a reduction of 13%, as shown in Figure 9b.

### maximum increase in the torque density is met, as shown in Figure 10a. On the other hand, the thickness of the outer rotor PMs of 4 mm is chosen because the maximum weight of 3.2.4. Determination of PM Thickness

375 gr is reached, as illustrated in red in Figure 10b. The PM thickness changes the average flux density in the air gap and, hence, the transmitted torque [55]. The thickness of the inner rotor PMs is set at 4 mm because the maximum increase in the torque density is met, as shown in Figure 10a. On the other hand,

**Figure 10.** Inner and outer rotor PM thickness versus: **(a)** Torque density; **(b)** Coupler weight.

The thickness of the rotor yokes should be selected carefully, as it changes the total reluctance and, hence, the air gap magnetic flux density. The minimum thickness of the rotor yokes that meet the updated torque requirement results in the minimum coupler

For this purpose, the inner and outer rotor yoke thicknesses are determined to be 14 mm and 8 mm, as shown in Figure 11a and 11b, respectively. The choice of the values is evaluated based on a fraction-free approach for ease of production and in light of the minimum wall thickness necessary to eliminate material deformation during fabrication.

(**a**) (**b**)

3.2.5. Determination of the Thickness of Rotor Yokes

weight.

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(**a**) (**b**)

3.2.4. Determination of PM Thickness

the thickness of the outer rotor PMs of 4 mm is chosen because the maximum weight of 375 gr is reached, as illustrated in red in Figure 10b. 375 gr is reached, as illustrated in red in Figure 10b.

**Figure 9.** Pullout torque investigation versus: (**a**) Pole number in air gap length; (**b**) Air gap length.

The PM thickness changes the average flux density in the air gap and, hence, the transmitted torque [55]. The thickness of the inner rotor PMs is set at 4 mm because the maximum increase in the torque density is met, as shown in Figure 10a. On the other hand, the thickness of the outer rotor PMs of 4 mm is chosen because the maximum weight of

**Figure 10.** Inner and outer rotor PM thickness versus: **(a)** Torque density; **(b)** Coupler weight. **Figure 10.** Inner and outer rotor PM thickness versus: (**a**) Torque density; (**b**) Coupler weight.

3.2.5. Determination of the Thickness of Rotor Yokes 3.2.5. Determination of the Thickness of Rotor Yokes

The thickness of the rotor yokes should be selected carefully, as it changes the total reluctance and, hence, the air gap magnetic flux density. The minimum thickness of the rotor yokes that meet the updated torque requirement results in the minimum coupler weight. The thickness of the rotor yokes should be selected carefully, as it changes the total reluctance and, hence, the air gap magnetic flux density. The minimum thickness of the rotor yokes that meet the updated torque requirement results in the minimum coupler weight.

For this purpose, the inner and outer rotor yoke thicknesses are determined to be 14 mm and 8 mm, as shown in Figure 11a and 11b, respectively. The choice of the values is evaluated based on a fraction-free approach for ease of production and in light of the minimum wall thickness necessary to eliminate material deformation during fabrication. For this purpose, the inner and outer rotor yoke thicknesses are determined to be 14 mm and 8 mm, as shown in Figures 11a and 11b, respectively. The choice of the values is evaluated based on a fraction-free approach for ease of production and in light of the minimum wall thickness necessary to eliminate material deformation during fabrication. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 11 of 23

**Figure 11.** Pullout torque versus: (**a**) Thickness of inner yoke; (**b**) Thickness of outer yoke. **Figure 11.** Pullout torque versus: (**a**) Thickness of inner yoke; (**b**) Thickness of outer yoke.

3.2.6. Investigation of the PM Embrace and Offset Effect 3.2.6. Investigation of the PM Embrace and Offset Effect

Pole embrace represents the ratio of the pole arc to the pole pitch. The arched PMs may not be concentric with the rotor. In the absence of a uniform air gap, the offset between the centre of the bottom and top of the PM arc is called the pole arc offset. The embrace has a more significant effect on the pullout torque, while the offset has a limited Pole embrace represents the ratio of the pole arc to the pole pitch. The arched PMs may not be concentric with the rotor. In the absence of a uniform air gap, the offset between the centre of the bottom and top of the PM arc is called the pole arc offset. The embrace has a more significant effect on the pullout torque, while the offset has a limited effect.

effect. The embrace of 0.98 offers a higher torque density, as marked in red in Figure 12a. The pole arc offset may not be preferred because it negatively affects the output torque, as seen in Figure 12b, except for mandatory situations such as cogging torque. Thus, the The embrace of 0.98 offers a higher torque density, as marked in red in Figure 12a. The pole arc offset may not be preferred because it negatively affects the output torque, as seen in Figure 12b, except for mandatory situations such as cogging torque. Thus, the maximum pullout torque is achieved with the maximum embrace and minimum offset.

maximum pullout torque is achieved with the maximum embrace and minimum offset.

Transient analyses allow performance outputs corresponding to the design parameters to be dynamically realised under no-load, rated-load, and overload conditions. Simulations are carried out in which the moment of inertia, the mechanical losses in terms of damping factors, and the load type acting are modelled. Thus, the moment of inertia of the inner and outer rotors is calculated as 0.42 kg-cm2 and 1.95 kg-cm2, respectively. Mechanical losses, i.e., wind and friction losses, ventilator losses, and bearing losses, are practically accepted at 3.5% of the output power [56]. The load type is considered such that

The pole types applied to the PM machines can also be employed in the MCs. Surfacemounted PMs are more production viable than internal PMs and can be divided into three parts, as shown in Figure 13a–c. Figure 13d shows that the effect of the pole type on the torque density is minimal. However, type-1 is preferred due to its ease of installation.

the load varies nonlinearly with the square of the speed, such as fan load [54].

(**a**) (**b**) **Figure 12.** Pullout torque versus: (**a**) PM embrace; (**b**) PM offset.

*3.3. Maxwell 2D Transient Analyses* 

3.3.1. Comparison of Pole Types

(**a**) (**b**)

3.2.6. Investigation of the PM Embrace and Offset Effect

**Figure 11.** Pullout torque versus: (**a**) Thickness of inner yoke; (**b**) Thickness of outer yoke.

Pole embrace represents the ratio of the pole arc to the pole pitch. The arched PMs may not be concentric with the rotor. In the absence of a uniform air gap, the offset between the centre of the bottom and top of the PM arc is called the pole arc offset. The embrace has a more significant effect on the pullout torque, while the offset has a limited

The embrace of 0.98 offers a higher torque density, as marked in red in Figure 12a. The pole arc offset may not be preferred because it negatively affects the output torque, as seen in Figure 12b, except for mandatory situations such as cogging torque. Thus, the maximum pullout torque is achieved with the maximum embrace and minimum offset.

**Figure 12.** Pullout torque versus: (**a**) PM embrace; (**b**) PM offset. **Figure 12.** Pullout torque versus: (**a**) PM embrace; (**b**) PM offset.

### *3.3. Maxwell 2D Transient Analyses 3.3. Maxwell 2D Transient Analyses*

effect.

Transient analyses allow performance outputs corresponding to the design parameters to be dynamically realised under no-load, rated-load, and overload conditions. Simulations are carried out in which the moment of inertia, the mechanical losses in terms of damping factors, and the load type acting are modelled. Thus, the moment of inertia of the inner and outer rotors is calculated as 0.42 kg-cm2 and 1.95 kg-cm2, respectively. Mechanical losses, i.e., wind and friction losses, ventilator losses, and bearing losses, are practically accepted at 3.5% of the output power [56]. The load type is considered such that the load varies nonlinearly with the square of the speed, such as fan load [54]. Transient analyses allow performance outputs corresponding to the design parameters to be dynamically realised under no-load, rated-load, and overload conditions. Simulations are carried out in which the moment of inertia, the mechanical losses in terms of damping factors, and the load type acting are modelled. Thus, the moment of inertia of the inner and outer rotors is calculated as 0.42 kg-cm<sup>2</sup> and 1.95 kg-cm<sup>2</sup> , respectively. Mechanical losses, i.e., wind and friction losses, ventilator losses, and bearing losses, are practically accepted at 3.5% of the output power [56]. The load type is considered such that the load varies nonlinearly with the square of the speed, such as fan load [54].

### 3.3.1. Comparison of Pole Types 3.3.1. Comparison of Pole Types

The pole types applied to the PM machines can also be employed in the MCs. Surfacemounted PMs are more production viable than internal PMs and can be divided into three parts, as shown in Figure 13a–c. Figure 13d shows that the effect of the pole type on the torque density is minimal. However, type-1 is preferred due to its ease of installation. The pole types applied to the PM machines can also be employed in the MCs. Surfacemounted PMs are more production viable than internal PMs and can be divided into three parts, as shown in Figure 13a–c. Figure 13d shows that the effect of the pole type on the torque density is minimal. However, type-1 is preferred due to its ease of installation. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 12 of 23

**Figure 13.** Pole types versus: (**a**) Type-1; (**b**) Type-2; (**c**) Type-3; (**d**) Torque density in rev,M. **Figure 13.** Pole types versus: (**a**) Type-1; (**b**) Type-2; (**c**) Type-3; (**d**) Torque density in rev,M.

#### 3.3.2. Effect of PM Type, Grade, and Temperature on Pullout Torque 3.3.2. Effect of PM Type, Grade, and Temperature on Pullout Torque

The grade, type, orientation, and operating temperature of PMs have a significant role in MC design. Figure 14 examines all implications for the 10-pole configuration at a critical angle. The pullout torque increases with the increasing PM grade, as shown in Figure 14a. On the other hand, SmCo magnets can operate at higher temperatures and in harsher conditions. Nevertheless, their energy density is lower than that of NdFeB, resulting in a lower pullout torque, as shown in Figure 14b for the same thickness of PMs. The grade, type, orientation, and operating temperature of PMs have a significant role in MC design. Figure 14 examines all implications for the 10-pole configuration at a critical angle. The pullout torque increases with the increasing PM grade, as shown in Figure 14a. On the other hand, SmCo magnets can operate at higher temperatures and in harsher conditions. Nevertheless, their energy density is lower than that of NdFeB, resulting in a lower pullout torque, as shown in Figure 14b for the same thickness of PMs.

In addition, the magnetisation direction has little effect on the pullout torque, as shown in Figure 14c. However, the operating temperature of the PMs significantly affects the torque, as shown in Figure 14d. Radially oriented N48H is preferred for accessibility.

 (**a**) (**b**)

(**c**) (**d**)

3.3.3. Rotor Flux Density and Mesh Distribution

**Figure 14.** Effect of PM: (**a**) Grade; (**b**) Type; (**c**) Magnetic orientation; (**d**) Operating temperature.

The flux density of the rotor yokes should be close to, but not reach, the saturation point, which is the knee point on the BH curve to achieve the maximum torque density, as shown in Figure 15a. However, the minimum wall thickness required to prevent material deformation during manufacturing and dynamic effects limits the design of the yoke thickness close to the saturation point. The yoke design is based on adjusting the yoke thicknesses as close as possible to the saturation point, as shown in Figure 15a. In this case, the outer rotor yoke thicknesses (tyo2 and tyo1) can be a minimum of 2 mm to prevent

(**a**) (**b**) (**c**) (**d**)

**Figure 14.** Effect of PM: (**a**) Grade; (**b**) Type; (**c**) Magnetic orientation; (**d**) Operating temperature. **Figure 14.** Effect of PM: (**a**) Grade; (**b**) Type; (**c**) Magnetic orientation; (**d**) Operating temperature.

**Figure 13.** Pole types versus: (**a**) Type-1; (**b**) Type-2; (**c**) Type-3; (**d**) Torque density in rev,M.

The grade, type, orientation, and operating temperature of PMs have a significant role in MC design. Figure 14 examines all implications for the 10-pole configuration at a critical angle. The pullout torque increases with the increasing PM grade, as shown in Figure 14a. On the other hand, SmCo magnets can operate at higher temperatures and in harsher conditions. Nevertheless, their energy density is lower than that of NdFeB, resulting in a lower pullout torque, as shown in Figure 14b for the same thickness of PMs.

In addition, the magnetisation direction has little effect on the pullout torque, as shown in Figure 14c. However, the operating temperature of the PMs significantly affects the torque, as shown in Figure 14d. Radially oriented N48H is preferred for accessibility.

3.3.2. Effect of PM Type, Grade, and Temperature on Pullout Torque

3.3.3. Rotor Flux Density and Mesh Distribution The flux density of the rotor yokes should be close to, but not reach, the saturation point, which is the knee point on the BH curve to achieve the maximum torque density, In addition, the magnetisation direction has little effect on the pullout torque, as shown in Figure 14c. However, the operating temperature of the PMs significantly affects the torque, as shown in Figure 14d. Radially oriented N48H is preferred for accessibility. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 13 of 23

### as shown in Figure 15a. However, the minimum wall thickness required to prevent material deformation during manufacturing and dynamic effects limits the design of the yoke 3.3.3. Rotor Flux Density and Mesh Distribution

thickness close to the saturation point. The yoke design is based on adjusting the yoke thicknesses as close as possible to the saturation point, as shown in Figure 15a. In this case, the outer rotor yoke thicknesses (tyo2 and tyo1) can be a minimum of 2 mm to prevent The flux density of the rotor yokes should be close to, but not reach, the saturation point, which is the knee point on the BH curve to achieve the maximum torque density, as shown in Figure 15a. However, the minimum wall thickness required to prevent material deformation during manufacturing and dynamic effects limits the design of the yoke thickness close to the saturation point. The yoke design is based on adjusting the yoke thicknesses as close as possible to the saturation point, as shown in Figure 15a. In this case, the outer rotor yoke thicknesses (tyo2 and tyo1) can be a minimum of 2 mm to prevent fabrication deformation. The inner rotor inner yoke thickness (tyi1) is set to a minimum of 2.5 mm to avoid reducing the mechanical strength and flywheel effect, and the inner rotor outer yoke thickness (tyi2) is set to 5.24 mm to ensure the selected effective air gap diameter. fabrication deformation. The inner rotor inner yoke thickness (tyi1) is set to a minimum of 2.5 mm to avoid reducing the mechanical strength and flywheel effect, and the inner rotor outer yoke thickness (tyi2) is set to 5.24 mm to ensure the selected effective air gap diameter. Figure 15b exhibits the mesh distribution of the model. The critical regions, such as the air gap and PMs, are subjected to dense meshing. The total number of mesh elements is 9440. The mesh method is TAU. Thus, the solution accuracy is increased. The parts marked in pink are the mounting holes. The diameter of the weight reduction holes, marked in red, is chosen to not reduce the mechanical strength. Thus, the total weight is reduced by 4 gr.

**Figure 15.** (**a**) Flux density distributions; (**b**) Mesh distributions. **Figure 15.** (**a**) Flux density distributions; (**b**) Mesh distributions.

3.3.4. Investigation of Negative Torque at Loss of Synchronisation Loss of synchronisation (LoS) refers to the situation where the synchronisation between the rotors is disrupted by exceeding the torque limit and critical torque angle as a Figure 15b exhibits the mesh distribution of the model. The critical regions, such as the air gap and PMs, are subjected to dense meshing. The total number of mesh elements is 9440. The mesh method is TAU. Thus, the solution accuracy is increased. The parts marked

In the 2D simulation, the LoS torque is analysed by setting the inner rotor speed to 0 rpm and rotating the outer rotor at different speeds. In the test system, the LoS torque is measured by locking the inner rotor so that it cannot rotate and gradually rotating the outer rotor at different speeds by a speed source. As shown in Figure 16, the LoS torque at the maximum speed is −0.61 N.m for the 2D simulation and −0.6 N.m for the test results. The deviations in the results are due to the sensitivity of the sensors on the test bench and the higher moment of inertia of the test bench compared to the UAV dynamic model in

result of a shaft malfunction on the PE shaft. However, in the case of LoS, while the drive system is sustained by S/G, the negative torque acting from the inner rotor needs to be

**Figure 16.** Negative torque at loss of synchronisation depending on operating speed.

the simulations.

ter.

reduced by 4 gr.

in pink are the mounting holes. The diameter of the weight reduction holes, marked in red, is chosen to not reduce the mechanical strength. Thus, the total weight is reduced by 4 gr. **Figure 15.** (**a**) Flux density distributions; (**b**) Mesh distributions.

(**a**) (**b**)

fabrication deformation. The inner rotor inner yoke thickness (tyi1) is set to a minimum of 2.5 mm to avoid reducing the mechanical strength and flywheel effect, and the inner rotor outer yoke thickness (tyi2) is set to 5.24 mm to ensure the selected effective air gap diame-

Figure 15b exhibits the mesh distribution of the model. The critical regions, such as the air gap and PMs, are subjected to dense meshing. The total number of mesh elements is 9440. The mesh method is TAU. Thus, the solution accuracy is increased. The parts marked in pink are the mounting holes. The diameter of the weight reduction holes, marked in red, is chosen to not reduce the mechanical strength. Thus, the total weight is

3.3.4. Investigation of Negative Torque at Loss of Synchronisation 3.3.4. Investigation of Negative Torque at Loss of Synchronisation Loss of synchronisation (LoS) refers to the situation where the synchronisation be-

*Aerospace* **2023**, *10*, x FOR PEER REVIEW 13 of 23

Loss of synchronisation (LoS) refers to the situation where the synchronisation between the rotors is disrupted by exceeding the torque limit and critical torque angle as a result of a shaft malfunction on the PE shaft. However, in the case of LoS, while the drive system is sustained by S/G, the negative torque acting from the inner rotor needs to be analysed and accounted for in the safety factor to identify the power limits of the S/G. tween the rotors is disrupted by exceeding the torque limit and critical torque angle as a result of a shaft malfunction on the PE shaft. However, in the case of LoS, while the drive system is sustained by S/G, the negative torque acting from the inner rotor needs to be analysed and accounted for in the safety factor to identify the power limits of the S/G. In the 2D simulation, the LoS torque is analysed by setting the inner rotor speed to 0

In the 2D simulation, the LoS torque is analysed by setting the inner rotor speed to 0 rpm and rotating the outer rotor at different speeds. In the test system, the LoS torque is measured by locking the inner rotor so that it cannot rotate and gradually rotating the outer rotor at different speeds by a speed source. As shown in Figure 16, the LoS torque at the maximum speed is −0.61 N·m for the 2D simulation and −0.6 N·m for the test results. The deviations in the results are due to the sensitivity of the sensors on the test bench and the higher moment of inertia of the test bench compared to the UAV dynamic model in the simulations. rpm and rotating the outer rotor at different speeds. In the test system, the LoS torque is measured by locking the inner rotor so that it cannot rotate and gradually rotating the outer rotor at different speeds by a speed source. As shown in Figure 16, the LoS torque at the maximum speed is −0.61 N.m for the 2D simulation and −0.6 N.m for the test results. The deviations in the results are due to the sensitivity of the sensors on the test bench and the higher moment of inertia of the test bench compared to the UAV dynamic model in the simulations.

**Figure 16.** Negative torque at loss of synchronisation depending on operating speed. **Figure 16.** Negative torque at loss of synchronisation depending on operating speed.

## *3.4. Maxwell 3D Static and Transient Optimetric Analyses*

Static simulations examine the MC behaviour in the steady state, i.e., when the inner or outer rotor shaft is locked. Therefore, model losses are not considered in the static analyses. On the other hand, transient analysis is more accurate because it considers losses, coupling effects, end effects, eddy-current losses on PMs, and material wall thickness.

## 3.4.1. Static Locked-Rotor Torque and Transient Torque Ripple Analyses

Locked-rotor or static torque refers to the torque capability of the coupler. It can be examined in different pole numbers depending on the torque angle, as shown in Figure 17a. The maximum static torque is provided as 9.4 N·m in the 10-pole configuration.

In the dynamic state, torque transmission in response to instantaneous load variations causes torque ripple in MCs due to the different moments of inertia of the rotors and the flywheel effect. It is 20 mN·m for the proposed model, as shown in Figure 17b. The rotors hold each other until the critical angle is exceeded, resulting in minimal torque ripple in the active MCs.

ple in the active MCs.

*3.4. Maxwell 3D Static and Transient Optimetric Analyses* 

3.4.1. Static Locked-Rotor Torque and Transient Torque Ripple Analyses

**Figure 17.** (**a**) Locked-rotor torque versus torque angle; (**b**) Torque ripple in mechanical revolution. **Figure 17.** (**a**) Locked-rotor torque versus torque angle; (**b**) Torque ripple in mechanical revolution.

Static simulations examine the MC behaviour in the steady state, i.e., when the inner or outer rotor shaft is locked. Therefore, model losses are not considered in the static analyses. On the other hand, transient analysis is more accurate because it considers losses, coupling effects, end effects, eddy-current losses on PMs, and material wall thickness.

Locked-rotor or static torque refers to the torque capability of the coupler. It can be examined in different pole numbers depending on the torque angle, as shown in Figure 17a. The maximum static torque is provided as 9.4 N.m in the 10-pole configuration.

In the dynamic state, torque transmission in response to instantaneous load variations causes torque ripple in MCs due to the different moments of inertia of the rotors and the flywheel effect. It is 20 mN.m for the proposed model, as shown in Figure 17b. The rotors hold each other until the critical angle is exceeded, resulting in minimal torque rip-

3.4.2. Investigation of Different Rotor Materials and Air Gap Flux Density 3.4.2. Investigation of Different Rotor Materials and Air Gap Flux Density

Carbon steel, electric steel, and stainless steel can be used as rotor materials. Although electrical steel has lower core losses for synchronous MCs, it does not provide an advantage in the proposed model due to its low model volume. Moreover, the design results in a higher yoke thickness due to the lower saturation point if electrical steel is Carbon steel, electric steel, and stainless steel can be used as rotor materials. Although electrical steel has lower core losses for synchronous MCs, it does not provide an advantage in the proposed model due to its low model volume. Moreover, the design results in a higher yoke thickness due to the lower saturation point if electrical steel is preferred.

preferred. However, some exceptional cases, such as military applications, require the use of nonmagnetic materials, such as Steel-316, called yokeless design. In such cases, it is inevitable to increase the PM thickness to avoid a drastic drop in the torque of approximately 60%, as shown in Figure 18a. Steel-1020 is used as the rotor material in the production of However, some exceptional cases, such as military applications, require the use of nonmagnetic materials, such as Steel-316, called yokeless design. In such cases, it is inevitable to increase the PM thickness to avoid a drastic drop in the torque of approximately 60%, as shown in Figure 18a. Steel-1020 is used as the rotor material in the production of MCs. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 15 of 23

**Figure 18.** Investigation of: (**a**) Effect of different rotor materials on torque; (**b**) Air gap flux density. **Figure 18.** Investigation of: (**a**) Effect of different rotor materials on torque; (**b**) Air gap flux density.

3.4.3. Study of Pullout Torque depending on Misalignment Length As part of the worst-case scenario analysis, it is essential to examine the reduction in the pullout torque due to the misalignment length caused by the propeller pulling the On the other hand, the air gap flux density in the air gap is examined in Figure 18b when magnetic Steel-1020 is used as the rotor material. It is 0.5 T at the critical angle of 18 Mdeg for the proposed design, which is far from the demagnetisation point of the PMs.

### system forwards until it is unable to generate thrust in the event of any extreme bearing failure. 3.4.3. Study of Pullout Torque Depending on Misalignment Length

Figure 19a examines the pullout torque depending on the misalignment. Figure 19b exhibits the misalignment. The torque decreases as the misalignment increases. Although the test results and the simulations agree with each other, the differences in the results are due to the difficulty in the precision adjustment of the misalignment length in the test As part of the worst-case scenario analysis, it is essential to examine the reduction in the pullout torque due to the misalignment length caused by the propeller pulling the system forwards until it is unable to generate thrust in the event of any extreme bearing failure.

system. Figure 19a examines the pullout torque depending on the misalignment. Figure 19b exhibits the misalignment. The torque decreases as the misalignment increases. Although the test results and the simulations agree with each other, the differences in the results are due to the difficulty in the precision adjustment of the misalignment length in the test system.

**Figure 19.** (**a**) Pullout torque depending on misalignment length; (**b**) Exhibition of misalignment.

Magnetic coupler losses are composed of rotor core losses, induced eddy-current losses on PMs, and mechanical losses. The mechanical losses, estimated at 3.5% of the output power, are thus calculated as 0.00051 W/(rad/sec)2. On the other hand, vibration due to the natural operation of the PE, load disturbances due to different UAV operating zones, and torque fluctuations during load changes cause the torque angle to change continuously. As a result, high eddy-current losses are induced on the PMs, which increase the temperature of the PMs and reduce the transmitted torque by reducing their residual flux density, and their impact on the system should be investigated. The proposed MC design is realised in light of all these effects and the design requirements are provided.

As seen in Figure 20a, the eddy-current losses are simulated as 237 W. The efficiency of the MC is 94.3% at the maximum speed of 6500 rpm and 95% at the minimum speed of 2500 rpm, as shown in Figure 20b at the critical angle. However, to measure the efficiency on the test bench, a second transmitter is required to measure the input power in addition to the torque/speed transmitter measuring the output power. Due to the difficulty in the setup, the efficiency cannot be measured on the test bench. However, using the proposed correction coefficient, the actual efficiency can be estimated from the numerical efficiency

3.4.4. Magnetic Coupler Efficiency and Induced Eddy-Current Losses on PMs

graph.

failure.

system.

(**a**) (**b**)

3.4.3. Study of Pullout Torque depending on Misalignment Length

**Figure 18.** Investigation of: (**a**) Effect of different rotor materials on torque; (**b**) Air gap flux density.

As part of the worst-case scenario analysis, it is essential to examine the reduction in the pullout torque due to the misalignment length caused by the propeller pulling the system forwards until it is unable to generate thrust in the event of any extreme bearing

Figure 19a examines the pullout torque depending on the misalignment. Figure 19b exhibits the misalignment. The torque decreases as the misalignment increases. Although the test results and the simulations agree with each other, the differences in the results are due to the difficulty in the precision adjustment of the misalignment length in the test

**Figure 19.** (**a**) Pullout torque depending on misalignment length; (**b**) Exhibition of misalignment. **Figure 19.** (**a**) Pullout torque depending on misalignment length; (**b**) Exhibition of misalignment.

3.4.4. Magnetic Coupler Efficiency and Induced Eddy-Current Losses on PMs 3.4.4. Magnetic Coupler Efficiency and Induced Eddy-Current Losses on PMs

Magnetic coupler losses are composed of rotor core losses, induced eddy-current losses on PMs, and mechanical losses. The mechanical losses, estimated at 3.5% of the output power, are thus calculated as 0.00051 W/(rad/sec)2. On the other hand, vibration due to the natural operation of the PE, load disturbances due to different UAV operating zones, and torque fluctuations during load changes cause the torque angle to change continuously. As a result, high eddy-current losses are induced on the PMs, which increase the temperature of the PMs and reduce the transmitted torque by reducing their residual flux density, and their impact on the system should be investigated. The proposed MC design is realised in light of all these effects and the design requirements are provided. Magnetic coupler losses are composed of rotor core losses, induced eddy-current losses on PMs, and mechanical losses. The mechanical losses, estimated at 3.5% of the output power, are thus calculated as 0.00051 W/(rad/s)<sup>2</sup> . On the other hand, vibration due to the natural operation of the PE, load disturbances due to different UAV operating zones, and torque fluctuations during load changes cause the torque angle to change continuously. As a result, high eddy-current losses are induced on the PMs, which increase the temperature of the PMs and reduce the transmitted torque by reducing their residual flux density, and their impact on the system should be investigated. The proposed MC design is realised in light of all these effects and the design requirements are provided.

As seen in Figure 20a, the eddy-current losses are simulated as 237 W. The efficiency of the MC is 94.3% at the maximum speed of 6500 rpm and 95% at the minimum speed of 2500 rpm, as shown in Figure 20b at the critical angle. However, to measure the efficiency on the test bench, a second transmitter is required to measure the input power in addition to the torque/speed transmitter measuring the output power. Due to the difficulty in the setup, the efficiency cannot be measured on the test bench. However, using the proposed correction coefficient, the actual efficiency can be estimated from the numerical efficiency graph. As seen in Figure 20a, the eddy-current losses are simulated as 237 W. The efficiency of the MC is 94.3% at the maximum speed of 6500 rpm and 95% at the minimum speed of 2500 rpm, as shown in Figure 20b at the critical angle. However, to measure the efficiency on the test bench, a second transmitter is required to measure the input power in addition to the torque/speed transmitter measuring the output power. Due to the difficulty in the setup, the efficiency cannot be measured on the test bench. However, using the proposed correction coefficient, the actual efficiency can be estimated from the numerical efficiency graph. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 16 of 23

**Figure 20**. (**a**) Induced eddy-current losses on PMs; (**b**) Efficiency of the MC at maximum speed. **Figure 20.** (**a**) Induced eddy-current losses on PMs; (**b**) Efficiency of the MC at maximum speed.

### *3.5. Summary List of Various MC Designs 3.5. Summary List of Various MC Designs*

UAVs.

\* Optimum values.

sults.

**4. Results and Discussion** 

The various MC designs obtained from the simulations are summarised in Table 3. The optimum design is realised within the system constraints, and the optimum parameters are indicated with an asterisk. The PM thickness is included in the measurement of the inner rotor outer diameter and outer rotor inner diameter. Validation of the multiobjective simulations and the tests sheds light on the safe usability of MCs in hybrid The various MC designs obtained from the simulations are summarised in Table 3. The optimum design is realised within the system constraints, and the optimum parameters are indicated with an asterisk. The PM thickness is included in the measurement of the inner rotor outer diameter and outer rotor inner diameter. Validation of the multi-objective simulations and the tests sheds light on the safe usability of MCs in hybrid UAVs.

Outer diameter of outer rotor 79 mm 83 mm 83 mm \* Inner diameter of outer rotor 57 mm 59 mm 59 mm \* Outer diameter of inner rotor 54 mm 56 mm 56 mm \* Inner diameter of inner rotor 20 mm 20 mm 20 mm \*

Air gap length 1.5 mm 1.5 mm 1.5 mm \* Effective air gap diameter 55.5 mm 57.5 mm 57.5 mm \* Model length 10 mm 10 mm 10 mm \*

Pullout torque (N.m), dynamic 3.9 6.9 7.2 7.5 8.1 **8.7**  Torque density (N.m/kg) 12.2 19.7 20.5 20.3 23 **23.45**

outer and inner rotors of the MC, respectively.

**Table 3.** Dimension and performance list of various MCs in summary. **Pole Number 8-Pole 10-Pole \*** 

**Grade of PM N48H N45H N48H N48H N48H \*** 

Total weight (gr) 320 350 351 370 352 **371** 

This section reveals and discusses the test results and compares them with the 2D and 3D simulation data. In an optimisation process, the accuracy of the analyses is determined, and a correction coefficient is proposed based on the correlation between the re-

Various MCs with different design parameters listed in Table 3 have been produced, and some of them are illustrated in Figure 21. The upper and lower parts illustrate the


**Table 3.** Dimension and performance list of various MCs in summary.

## **4. Results and Discussion**

This section reveals and discusses the test results and compares them with the 2D and 3D simulation data. In an optimisation process, the accuracy of the analyses is determined, and a correction coefficient is proposed based on the correlation between the results.

Various MCs with different design parameters listed in Table 3 have been produced, and some of them are illustrated in Figure 21. The upper and lower parts illustrate the outer and inner rotors of the MC, respectively. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 17 of 23

**Figure 21.** Various MC productions of: (**a**) 8-pole/0.8-embrace/3 mm-PM thickness/10 mm length; (**b**) 8-pole/0.98-embrace/4 mm-PM thickness/10 mm length; (**c**) 10-pole/0.8-embrace/4 mm-PM thickness/10 mm length; (**d**) 10-pole/0.98-embrace/4 mm-PM thickness/10 mm length; (**e**) 10-pole/0.98 embrace/4 mm-PM thickness/20 mm length. **Figure 21.** Various MC productions of: (**a**) 8-pole/0.8-embrace/3 mm-PM thickness/10 mm length; (**b**) 8-pole/0.98-embrace/4 mm-PM thickness/10 mm length; (**c**) 10-pole/0.8-embrace/4 mm-PM thickness/10 mm length; (**d**) 10-pole/0.98-embrace/4 mm-PM thickness/10 mm length; (**e**) 10-pole/0.98-embrace/4 mm-PM thickness/20 mm length.

Due to the MCs being included in the group of noncontact electrical machines [57], their performance examinations are carried out in a similar way to special rotating machines, using direct or indirect test methods [58]. On the other hand, the propulsion platform, including the PE, MC, and propeller, is set up to conduct force tests on the system. Due to the MCs being included in the group of noncontact electrical machines [57], their performance examinations are carried out in a similar way to special rotating machines, using direct or indirect test methods [58]. On the other hand, the propulsion platform, including the PE, MC, and propeller, is set up to conduct force tests on the system.

The direct test method provides more accurate results because it dynamically measures MC parameters such as the output torque and output speed with a torque/speed transmitter. Figure 22a demonstrates the installation of the MC for the direct dynamic test system, while Figure 22b shows its installation on the PE and propeller. The direct test method provides more accurate results because it dynamically measures MC parameters such as the output torque and output speed with a torque/speed transmitter. Figure 22a demonstrates the installation of the MC for the direct dynamic test system, while Figure 22b shows its installation on the PE and propeller.

**Figure 22.** Assembly of (**a**) MC for dynamic tests; (**b**) MC with piston engine and propeller.

The operation principle of the direct dynamic test system shown in Figure 23a is as follows: The MC (3) device under test (D.U.T) is driven by a geared induction motor (4) with a high torque capacity. Dynamic tests are performed by gradually or directly loading the hysteresis brake for 1 h for each set of tests (1), depending on the type of test, such as no-load, rated-load, or overload. At this stage, as the load changes, the output torque and speed are measured by the torque/speed transmitter (2) and recorded by the panel.

In the thrust test system shown in Figure 23b, the load cells and sensors are used to measure the force, thrust, and temperature while the PE is operated at idle speed, cruising speed, and overspeed. The sensors and transducers in the test systems are calibrated by

(**a**) (**b**)

system, while Figure 22b shows its installation on the PE and propeller.

(**a**) (**b**) (**c**) (**d**) (**e**)

embrace/4 mm-PM thickness/20 mm length.

**Figure 21.** Various MC productions of: (**a**) 8-pole/0.8-embrace/3 mm-PM thickness/10 mm length; (**b**) 8-pole/0.98-embrace/4 mm-PM thickness/10 mm length; (**c**) 10-pole/0.8-embrace/4 mm-PM thickness/10 mm length; (**d**) 10-pole/0.98-embrace/4 mm-PM thickness/10 mm length; (**e**) 10-pole/0.98-

Due to the MCs being included in the group of noncontact electrical machines [57], their performance examinations are carried out in a similar way to special rotating machines, using direct or indirect test methods [58]. On the other hand, the propulsion platform, including the PE, MC, and propeller, is set up to conduct force tests on the system. The direct test method provides more accurate results because it dynamically measures MC parameters such as the output torque and output speed with a torque/speed transmitter. Figure 22a demonstrates the installation of the MC for the direct dynamic test

**Figure 22.** Assembly of (**a**) MC for dynamic tests; (**b**) MC with piston engine and propeller. **Figure 22.** Assembly of (**a**) MC for dynamic tests; (**b**) MC with piston engine and propeller.

The operation principle of the direct dynamic test system shown in Figure 23a is as follows: The MC (3) device under test (D.U.T) is driven by a geared induction motor (4) with a high torque capacity. Dynamic tests are performed by gradually or directly loading the hysteresis brake for 1 h for each set of tests (1), depending on the type of test, such as no-load, rated-load, or overload. At this stage, as the load changes, the output torque and speed are measured by the torque/speed transmitter (2) and recorded by the panel. The operation principle of the direct dynamic test system shown in Figure 23a is as follows: The MC (3) device under test (D.U.T) is driven by a geared induction motor (4) with a high torque capacity. Dynamic tests are performed by gradually or directly loading the hysteresis brake for 1 h for each set of tests (1), depending on the type of test, such as no-load, rated-load, or overload. At this stage, as the load changes, the output torque and speed are measured by the torque/speed transmitter (2) and recorded by the panel. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 18 of 23 an organisation with an international accreditation certificate [59]. In addition, the tests are performed three times in total at different times, and the average results are used.

**Figure 23.** Test system of: (**a**) Direct dynamic method; (**b**) Thrust measurement method. **Figure 23.** Test system of: (**a**) Direct dynamic method; (**b**) Thrust measurement method.

*4.1. Locked-Rotor Test Results in Summary*  In the locked-rotor or static test, the shaft is heavily loaded by the hysteresis brake so that it cannot be rotated. Depending on the torque angle, torque measurement is performed by gradually adjusting the load, and the data are recorded. Figure 24a shows the results of the locked-rotor test at a critical torque angle with a 10 mm coupler length for In the thrust test system shown in Figure 23b, the load cells and sensors are used to measure the force, thrust, and temperature while the PE is operated at idle speed, cruising speed, and overspeed. The sensors and transducers in the test systems are calibrated by an organisation with an international accreditation certificate [59]. In addition, the tests are performed three times in total at different times, and the average results are used.

### configurations with a higher torque density, while Figure 24b exhibits the results for a *4.1. Locked-Rotor Test Results in Summary*

lower torque density. The maximum locked-rotor torque is achieved at the configuration of 10-pole and 0.98-embrace, as marked in black in Figure 24a. However, the maximum locked-rotor torque for 8 poles with the same configuration as the 10-pole structure results in a reduction of approximately 14%, as marked in blue in Figure 24a. Furthermore, if the effect of the model length is examined, a twofold increase in the length increases the locked-rotor torque by almost a factor of two. In the locked-rotor or static test, the shaft is heavily loaded by the hysteresis brake so that it cannot be rotated. Depending on the torque angle, torque measurement is performed by gradually adjusting the load, and the data are recorded. Figure 24a shows the results of the locked-rotor test at a critical torque angle with a 10 mm coupler length for configurations with a higher torque density, while Figure 24b exhibits the results for a lower torque density. The maximum locked-rotor torque is achieved at the configuration of 10-pole and 0.98-embrace, as marked in black in Figure 24a.

(**b**)

*4.2. Investigation of Pullout Torque in Transient and Static Torque versus Torque Angle*  The locked-rotor and dynamic test results are consistent with each other. Therefore, the test results are plotted only for the optimum design with a configuration of 10-pole, 0.98-embrace, and 10 mm length to avoid visual pollution. However, the maximum locked-rotor torque for 8 poles with the same configuration as the 10-pole structure results in a reduction of approximately 14%, as marked in blue in Figure 24a. Furthermore, if the effect of the model length is examined, a twofold increase in the length increases the locked-rotor torque by almost a factor of two.

### Figure 25a compares the results obtained from the Maxwell 2D and 3D simulations and dynamic tests depending on time. The average dynamic pullout torque obtained from *4.2. Investigation of Pullout Torque in Transient and Static Torque versus Torque Angle*

the Maxwell 2D and 3D simulations and dynamic tests are 9.35 N.m, 9.15 N.m, and 8.7 N.m, respectively. Figure 25b plots the static torque results for the different torque angles. The maximum static torque obtained from the Maxwell 2D and 3D simulations and static The locked-rotor and dynamic test results are consistent with each other. Therefore, the test results are plotted only for the optimum design with a configuration of 10-pole, 0.98-embrace, and 10 mm length to avoid visual pollution.

tests at the critical angle of 18 Mdeg are 9.32 N.m, 9.11 N.m, and 8.72 N.m, respectively. Figure 25a compares the results obtained from the Maxwell 2D and 3D simulations and dynamic tests depending on time. The average dynamic pullout torque obtained from the Maxwell 2D and 3D simulations and dynamic tests are 9.35 N·m, 9.15 N·m, and 8.7 N·m, respectively. Figure 25b plots the static torque results for the different torque angles. The maximum static torque obtained from the Maxwell 2D and 3D simulations and static tests at the critical angle of 18 Mdeg are 9.32 N·m, 9.11 N·m, and 8.72 N·m, respectively.

The Maxwell simulations are in close agreement with the test results. The numerical simulations have an acceptable margin of error compared to the test results, which is 8% for the Maxwell 2D and 4.5% for the Maxwell 3D, where the safety factor is not included. A more effective design is achieved if a relevant difference or margin of error, called a correction coefficient, is provided in the first step of the numerical design.

**Figure 25.** Test results of the optimum MC in the 10-pole, 0.98-embrace, and 10 mm length configuration: (**a**) Pullout torque in transients; (**b**) Static torque versus torque angle in magnetostatics. **Figure 25.** Test results of the optimum MC in the 10-pole, 0.98-embrace, and 10 mm length configuration: (**a**) Pullout torque in transients; (**b**) Static torque versus torque angle in magnetostatics.

The Maxwell simulations are in close agreement with the test results. The numerical simulations have an acceptable margin of error compared to the test results, which is 8% for the Maxwell 2D and 4.5% for the Maxwell 3D, where the safety factor is not included. A more effective design is achieved if a relevant difference or margin of error, called a In this sense, taking into account the safety factor [60], the Maxwell 2D and 3D simulations are, on average, 14% and 9% higher than the direct dynamic test results for the proposed model, respectively. As a result, a correction coefficient of 1.2 for the Maxwell 2D and 1.1 for the Maxwell 3D is proposed for the use of MCs in hybrid UAVs.

### correction coefficient, is provided in the first step of the numerical design. **5. Conclusions**

In this sense, taking into account the safety factor [60], the Maxwell 2D and 3D simulations are, on average, 14% and 9% higher than the direct dynamic test results for the proposed model, respectively. As a result, a correction coefficient of 1.2 for the Maxwell 2D and 1.1 for the Maxwell 3D is proposed for the use of MCs in hybrid UAVs. **5. Conclusions**  This article contributes to exploring all design parameter effects of active cylindrical MCs with multi-objective simulations based on FEMs. The magnetic design was optimised in Ansys Maxwell using optimetric and tuning tools. Increasing the number of poles results in a maximum torque density only up to a certain point. The highest pullout torque was achieved with the configuration of the 10-pole and 0.98-embrace, offering an 18% higher torque than the 8-pole configuration. Increasing the embrace provides more output torque. A 20% increase in the embrace results in a 7.5% increase in the torque density. Reducing the embrace to less than 0.6 almost halves the output torque. Increasing the offset reduces the transmitted torque by a maximum of 10%. Using PMs with a lower residual flux density reduces the pullout torque. The MCs using Nd-Fe-B PMs provide a higher torque density than the couplers using Sm-Co. A double increase in the air gap length reduces the pullout torque by 13%. The reduction in the PM thickness and yoke thickness significantly reduces the torque density. Operating the MC as close to the saturation point as possible ensures the minimum weight of the system. A direct-type dynamic test system was set up for the transient and locked-rotor tests. A thrust test system was also installed for the force tests of the MC on the PE. Exceeding the critical torque angle causes synchronisation loss between the inner and outer rotors. The torque fluctuation at load changes is approximately 0.25%. The loss of synchronisation torque at the maximum speed is −0.61 N.m. The magnetic coupler efficiency is above 94% at the maximum speed. The Maxwell 2D FEM results are higher than the 3D and dynamic tests, but the results agree with a reasonable margin of error. The test results differ by 8% with the This article contributes to exploring all design parameter effects of active cylindrical MCs with multi-objective simulations based on FEMs. The magnetic design was optimised in Ansys Maxwell using optimetric and tuning tools. Increasing the number of poles results in a maximum torque density only up to a certain point. The highest pullout torque was achieved with the configuration of the 10-pole and 0.98-embrace, offering an 18% higher torque than the 8-pole configuration. Increasing the embrace provides more output torque. A 20% increase in the embrace results in a 7.5% increase in the torque density. Reducing the embrace to less than 0.6 almost halves the output torque. Increasing the offset reduces the transmitted torque by a maximum of 10%. Using PMs with a lower residual flux density reduces the pullout torque. The MCs using Nd-Fe-B PMs provide a higher torque density than the couplers using Sm-Co. A double increase in the air gap length reduces the pullout torque by 13%. The reduction in the PM thickness and yoke thickness significantly reduces the torque density. Operating the MC as close to the saturation point as possible ensures the minimum weight of the system. A direct-type dynamic test system was set up for the transient and locked-rotor tests. A thrust test system was also installed for the force tests of the MC on the PE. Exceeding the critical torque angle causes synchronisation loss between the inner and outer rotors. The torque fluctuation at load changes is approximately 0.25%. The loss of synchronisation torque at the maximum speed is −0.61 N·m. The magnetic coupler efficiency is above 94% at the maximum speed. The Maxwell 2D FEM results are higher than the 3D and dynamic tests, but the results agree with a reasonable margin of error. The test results differ by 8% with the Maxwell 2D results and 4.5% with the Maxwell 3D results. The difference is due to the density differences in the adaptive meshes, the inclusion of end-leakage effects in the 3D FEM, and temperature assignments. As a result, a correction coefficient of 1.2 for the Maxwell 2D and 1.1 for the Maxwell 3D is proposed. A comprehensive examination of the active cylindrical MCs contributes to the use of MCs for other applications such as robotics, hydraulics, automotive, medical, and pumps.

Maxwell 2D results and 4.5% with the Maxwell 3D results. The difference is due to the density differences in the adaptive meshes, the inclusion of end-leakage effects in the 3D FEM, and temperature assignments. As a result, a correction coefficient of 1.2 for the Maxwell 2D and 1.1 for the Maxwell 3D is proposed. A comprehensive examination of the **Author Contributions:** Conceptualisation, S.A. and I.I.; methodology, S.A.; software, S.A.; validation, I.I. and T.S.N.; formal analysis, T.S.N.; investigation, S.A.; resources, I.I.; data curation, S.A.; writing—original draft preparation, S.A.; writing—review and editing, I.I.; visualisation, T.S.N.; supervision, I.I.; project administration, S.A.; funding acquisition, S.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work in part is supported by The Scientific and Technological Research Council of TURKIYE (TUBITAK) 1501 support program with project number 3192296.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** FEM is conducted with the licenced use of Ansys Electronics v2020R1. The 3D models are held with the licenced use of Solidworks software.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


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## *Article* **Fault-Tolerant Control of a Dual-Stator PMSM for the Full-Electric Propulsion of a Lightweight Fixed-Wing UAV**

**Aleksander Suti \* , Gianpietro Di Rito and Roberto Galatolo**

Department of Civil and Industrial Engineering, University of Pisa, Largo Lucio Lazzarino 2, 56122 Pisa, Italy; gianpietro.di.rito@unipi.it (G.D.R.); roberto.galatolo@unipi.it (R.G.) **\*** Correspondence: aleksander.suti@dici.unipi.it; Tel.: +39-0502217211

**Abstract:** The reliability enhancement of electrical machines is one of the key enabling factors for spreading the full-electric propulsion to next-generation long-endurance UAVs. This paper deals with the fault-tolerant control design of a Full-Electric Propulsion System (FEPS) for a lightweight fixed-wing UAV, in which a dual-stator Permanent Magnet Synchronous Machine (PMSM) drives a twin-blade fixed-pitch propeller. The FEPS is designed to operate with both stators delivering power (active/active status) during climb, to maximize performances, while only one stator is used (active/stand-by status) in cruise and landing, to enhance reliability. To assess the faulttolerant capabilities of the system, as well as to evaluate the impacts of its failure transients on the UAV performances, a detailed model of the FEPS (including three-phase electrical systems, digital regulators, drivetrain compliance and propeller loads) is integrated with the model of the UAV longitudinal dynamics, and the system response is characterized by injecting a phase-to-ground fault in the motor during different flight manoeuvres. The results show that, even after a stator failure, the fault-tolerant control permits the UAV to hold altitude and speed during cruise, to keep on climbing (even with reduced performances), and to safely manage the flight termination (requiring to stop and align the propeller blades with the UAV wing), by avoiding potentially dangerous torque ripples and structural vibrations.

**Keywords:** fixed-wing UAV; full-electric propulsion system; axial-flux PMSMS; fault-tolerant control; phase-to-ground short circuit; failure transient analysis

## **1. Introduction**

The global market size of Unmanned Aerial Vehicles (UAVs) was 27.4 billion USD in 2021 and, despite the negative impact of the COVID-19 pandemic, it is expected to grow within 2026 up to 58.4 billion USD, at a Compound Annual Growth Rate (CAGR) of 16.4% [1]. Additionally, pushed by the wider objectives of the aerospace electrification, the design of next-generation long-endurance UAVs is undoubtedly moving toward the use of Full-Electric Propulsion Systems (FEPSs). Although immature nowadays in terms of reliability and energy density (e.g., lithium-ion battery packs typically range about 300 kJ/kg, which is 100 times lower than gasoline [2]), FEPSs are expected to obtain large investments in the forthcoming years, aiming to replace the conventional internal combustion motors, as well as to outclass the hybrid or hydrogen-based solutions [3]. Coherently, the global market size of electric motors is projected to grow within 2028 up to 181.9 billion USD, at a CAGR of 7.0% [4]. In particular, the segment of Permanent Magnet Synchronous Machines (PMSMs) is forecast to hold more and more significant markets, due to their advantages in terms of power density, efficiency, low torque ripple and dynamic performances. In this context, the Italian Government and the Tuscany Regional Government co-funded the project TERSA (*Tecnologie Elettriche e Radar per Sistemi aeromobili a pilotaggio remoto Autonomi*) [5], led by Sky Eye Systems (Italy) in collaboration with the University of Pisa and other Italian industries.

**Citation:** Suti, A.; Di Rito, G.; Galatolo, R. Fault-Tolerant Control of a Dual-Stator PMSM for the Full-Electric Propulsion of a Lightweight Fixed-Wing UAV. *Aerospace* **2022**, *9*, 337. https:// doi.org/10.3390/aerospace9070337

Academic Editor: Wim J. C. Verhagen

Received: 9 May 2022 Accepted: 22 June 2022 Published: 24 June 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The TERSA project aims to develop an Unmanned Aerial System (UAS) with fixedwing UAV, Figure 1, having the following main characteristics:

	- # Synthetic aperture radar, to support surveillance missions in adverse environmental conditions;
	- # Sense-and-avoid system, integrating a camera with a miniaturised radar, to support autonomous flight capabilities in emergency conditions.

**Figure 1.** Rendering of the TERSA UAV [5].

With particular reference to the activities related to the TERSA UAV propulsion system development (which this work refers to), special attention has been dedicated to the demonstration of its fault-tolerant capabilities. It is well-known that, compared with solutions based on internal combustion motors, FEPSs on UAVs would guarantee smaller CO<sup>2</sup> emissions, higher efficiency, lower noise, reduced thermal signature (crucial for military applications), higher service ceiling and simplified maintenance [6], but several reliability and safety issues are still open, especially for long-endurance flights in unsegregated airspaces. As relevant example, the failure rate of a simplex FEPS solution with a three-phase PMSM driven by three-leg converter typically ranges about 2.4 per thousand flight hours [7,8], which is far from the reliability and safety levels required for the airworthiness certification [9].

Provided that the weight and envelopes required by UAV applications impede the extensive use of hardware redundancy (e.g., redundant motors), the reliability enhancement of FEPSs can be achieved only through motor phase redundancy or by using unconventional converters. Different solutions are proposed in the literature, and they can be split in two categories: those applying conventional three-leg converters (using multiple phases [10,11] or multiple three-phase arrangements [12,13]), and those using four-leg converters [8,14,15]. In the latter solution, a couple of power switches are added as stand-by devices to the conventional three-leg bridge, enabling the control of the central point of the motor Yconnection. Although the four-leg solution permits to save weight, it requires an ad hoc design of the motor and its power electronics [8,16]. On the other hand, PMSMs with multiple three-phase arrangements are less compact, but they use conventional converters driven by standard techniques [13].

The failure rate of electric machines is essentially driven by faults on motor phases and converters (open-switch in a converter leg, open-phase, phase-to-ground fault, interturn short circuit, or capacitor short circuit [17]), that cover about 70% of the system fault modes [7]. Stator faults can initiate for different causes, such as dielectric breakdown, degradation of the winding insulation, thermal stress, overburden, or mechanical vibrations [18], and many research efforts have been carried out for their diagnosis and the compensation, especially for open-phase faults [8,19–21] and inter-turns short circuits [22–25], while the literature is poorer for phase-to-ground faults in electrical machines [26–28]. As discussed in [29], phase-to-ground faults fall into the short circuit faults category. Usually, a short circuit initiates as an inter-turn fault (very difficult to detect at an early stage), which typically evolves into a coil-to-coil, phase-to-phase, or into a phase-to-ground short circuit. Phase-to-ground faults are particularly dangerous, because they can cause irreversible damages to both windings and core. If the motor windings could be replaced, the core damage is irreversible and it requires the entire motor removal.

When addressing UAV applications, the basic consequence related to motor faults is the decrease or loss of thrust power, which essentially impacts on the altitude hold and/or climb capabilities of the vehicle. Together with other major UAV failures such as those affecting control actuators and sensors, the hazard mitigation requires the application of suitable fault-tolerant techniques. A comprehensive survey on methods for fault diagnosis and fault-tolerant control against UAV failures is provided by [30], and the reference highlights that the works addressing propulsion failures for fixed-wing UAVs is very limited. Most of the literature is actually focused on the effects of faults to control actuators and sensors for both single UAVs [31–34] and UAVs in formation flight [35–37], while the faults to propulsion systems are typically modelled with rough or very simplified approaches (e.g., total propulsion loss as in [38] or increase in the drivetrain friction as in [39]).

Together with reliability requirements, an FEPS for UAVs must have high compactness, high power-to-weight ratio, high torque density, and excellent efficiency. For these highperformance applications, Axial-Flux PMSMs (AFPMSMs) are preferred to conventional PMSMs with radial flux linkages [40]. In fact, although conventional PMSMs have higher technology readiness, AFPMSMs are superior in terms of weight (core material is reduced), torque-to-weight ratio (magnets are thinner), efficiency (rotor losses are minimized), and versatility (the axial air gaps are easily adjustable) [41–43]. The FEPS of the reference TERSA UAV is actually equipped with a dual-stator AFPMSM, capable of operating in both active/active and active/stand-by configurations to obtain fault-tolerant capabilities for the system [44].

This paper aims to contribute to the literature of FEPSs for fixed-wing UAVs by dealing with the fault-tolerant control design and the dynamic performance characterization of the TERSA UAV, particularly addressing the impacts of failure transients on both the motor and the vehicle performances in different flight phases (climb, cruise, flight termination/landing) if a phase-to-ground short circuit in a stator is simulated.

The basic objective of the investigation is, through a detailed fault modelling, to characterise both the fault symptoms (at both the motor level and UAV level) and the failure transients related to the application of fault-tolerant techniques. For this reason, the paper does not include the description of the health-monitoring algorithms, but it simply assumes that they exist and are capable of detecting the fault with a pre-defined latency; after that a compensation is applied (the failed stator is de-energized and the control on healthy one is activated or reconfigured).

The work is articulated as follows: the first part is dedicated to the system description and to the nonlinear FEPS model; successively, the main features of the fault-tolerant control design are presented. Finally, an excerpt of simulation results is proposed, by highlighting and discussing the effects of a phase-to-ground fault during different flight manoeuvres, and by demonstrating the effectiveness of the proposed design.

## **2. Materials and Methods**

## *2.1. FEPS Description*

The fault-tolerant FEPS of the TERSA UAV is basically composed of (Figure 2):

	- # Control/monitoring (CON/MON) module, for the implementation of the closed-loop control and health-monitoring functions;
	- # Conventional three-leg converter;
	- # Three Current Sensors (CSa, CSb, CSc), one per each motor phase;
	- # Angular Position Sensor (APS), measuring the motor rotation;
	- # Power Supply Unit (PSU), providing all ECU components with the required electrical supply;

**Figure 2.** Schematics of the FEPS architecture.

The fault-tolerant FEPS is designed to guarantee mission accomplishment even after the failure of one of the two AFPMSM stators. Different operations of the stators are thus defined by the CON/MON modules, so that each stator can


In addition, since the UAV flight termination and landing is obtained by deploying a parachute and by inflating airbags to be used as landing gears, the propeller blades must be aligned with the wing before opening the parachute to avoid interferences, and a specific control mode must be foreseen. As a consequence, four operation modes have been defined to control each stator of the AFPMSM:


As reported in Table 1 and represented in terms of a flow chart in Figure 3, depending on the MON fault flags (generated by the health-monitoring algorithms and communicated to the FCC) and on the mission phase (received from the FCC), the CON modules can be switched to FMM, FTM, HSB or CSB modes.


**Table 1.** FEPS operation modes as functions of mission phases and detected faults.

**Figure 3.** Flow chart defining FEPS operation modes.

*2.2. Mechanical Transmission and Propeller Loads Modelling*

The dynamics of the aero-mechanical section of the FEPS, providing the UAV with the thrust *Tp*, is schematically depicted in Figure 4a and is modelled by [8,12]:

$$\begin{cases} J\_p \ddot{\theta}\_p = -\mathcal{Q}\_p - \mathcal{C}\_{gb} \left( \dot{\theta}\_p - \dot{\theta}\_m \right) - \mathcal{K}\_{gb} \left( \theta\_p - \theta\_m \right) + \mathcal{Q}\_d \\\ J\_m \ddot{\theta}\_m = \mathcal{Q}\_m + \mathcal{C}\_{gb} \left( \dot{\theta}\_p - \dot{\theta}\_m \right) + \mathcal{K}\_{gb} \left( \theta\_p - \theta\_m \right) + \mathcal{Q}\_c \\\ \mathcal{Q}\_c = \mathcal{Q}\_{c\max} \sin(n\_h n\_d \theta\_m) \end{cases} \tag{1}$$

where *J<sup>p</sup>* and *θp*, *J<sup>m</sup>* and *θ<sup>m</sup>* are the inertias and angles of the propeller and the motor shafts, respectively, *Q<sup>p</sup>* is the propeller torque, *Q<sup>d</sup>* is a gust-induced disturbance torque, *Q<sup>m</sup>* is the motor torque, *Q<sup>c</sup>* is the cogging torque and *Qcmax* is its maximum amplitude, *n<sup>d</sup>* is the pole

pairs number, *n<sup>h</sup>* is the harmonic index of the cogging disturbance, while *Kgb* and *Cgb* are the stiffness and the damping of the mechanical coupling joint.

**Figure 4.** FEPS: (**a**) mechanical scheme; (**b**) equivalent three-phase PMSM scheme (one pole pair).

Concerning the aerodynamic torque, the FEPS is equipped with the twin-blade fixedpitch composite propeller APC22 × 10E, which is characterized by the following thrust and torque expressions:

$$T\_p = \mathbb{C}\_{T\_p} \left( \dot{\theta}\_{p\prime} AR \right) \rho D\_p^4 \dot{\theta}\_p^2 \,. \tag{2}$$

$$Q\_p = \mathbb{C}\_{Q\_p} \left( \dot{\theta}\_{p\prime} AR \right) \rho D\_p^5 \dot{\theta}\_{p\prime}^2 \tag{3}$$

$$AR = \mathcal{V}\_{\mathfrak{a}} / D\_p \dot{\theta}\_p \, \tag{4}$$

where *CT<sup>p</sup>* and *CQ<sup>p</sup>* are the nondimensional thrust and torque coefficients, *AR* is the propeller advance ratio, *D<sup>p</sup>* is the propeller diameter, *ρ* is the air density, and V*<sup>a</sup>* is the UAV forward speed.

It is worth noting that the manufacturer database provides the nondimensional coefficients *CT<sup>p</sup>* and *CQ<sup>p</sup>* only for *AR* < 0.65 [45]. This range adequately covers the FEPS operating conditions in FMM (Table 1), but it is not adequate for the FTM, where *AR* theoretically tends to infinite (because the propeller stops rotating), so that a loads model extension was carried out. This was carried out via Equation (5), by linearly extrapolating the coefficient trends at *AR*\* = 0.65 (Figure 5), with an approach that typically provides conservative estimates [46,47].

$$\mathbf{C}\_{X\_{\mathcal{P}}} = \begin{cases} \mathbf{C}\_{X\_{\mathcal{P}}}^{(DB)} \left( \dot{\theta}\_{p\prime} AR \right) & AR \leq AR^\* \\ \mathbf{C}\_{X\_{\mathcal{P}}}^{(DB)} \left( \dot{\theta}\_{p\prime} AR^\* \right) + \left. \frac{\partial \mathbf{C}\_{X\_{\mathcal{P}}}^{(DB)} \right|}{\dot{\theta}\_{p\prime} AR^\*} & AR > AR^\* \end{cases} \tag{5}$$

In Equation (5), *CX<sup>p</sup>* represents the thrust (*X* = *T*) or torque (*X* = *Q*) propeller coefficient, while *C* (*DB*) *Xp* is the related quantity given in the manufacturer database.

**Figure 5.** APC 22 × 10E propeller curves: thrust coefficient (**a**) and torque coefficient (**b**), thrust (**c**) and torque (**d**) at sea level.

## *2.3. Three-Phase PMSM Modelling*

Apart from the architectural dissimilarity from a conventional radial-flux PMSM, the mathematical modelling of a AFPMSM is essentially identical [48,49]. With reference to the schematics in Figure 4b, and under the following assumptions [13]:


$$\mathbf{V}\_{abc} = \mathbf{R}\dot{\mathbf{i}}\_{abc} + \mathbf{L}\frac{d}{dt}\mathbf{i}\_{abc} + \mathbf{e}\_{abc\prime} \tag{6}$$

$$e\_{\rm abc} = \lambda\_m n\_d \dot{\theta}\_m \left[ \sin(n\_d \theta\_m), \sin\left(n\_d \theta\_m - \frac{2}{3}\pi\right), \sin\left(n\_d \theta\_m + \frac{2}{3}\pi\right) \right]^T,\tag{7}$$

In Equations (5)–(6), *Vabc* = [*V<sup>a</sup>* − *Vn*, *V<sup>b</sup>* − *Vn*, *V<sup>c</sup>* − *Vn*] *T* is the applied voltages vector, *iabc* = [*ia*, *i<sup>b</sup>* , *ic*] *T* is the stator currents vector, *eabc* is the back-electromotive forces vector, *R* and *L* are the resistance and inductance of the phases, *λ<sup>m</sup>* is the magnet flux linkage, and *V<sup>n</sup>* is the neutral point voltage. The motor torque (*Qm*) is thus given by:

$$Q\_{m} = \lambda\_{m} n\_{d} \left[ i\_{d} \sin(n\_{d} \theta\_{m}) + i\_{b} \sin \left( n\_{d} \theta\_{m} - \frac{2}{3} \pi \right) + i\_{c} \sin \left( n\_{d} \theta\_{m} + \frac{2}{3} \pi \right) \right] \tag{8}$$

More conveniently, the motor torque can be expressed with reference to quantities in the rotating reference frame, by applying the Clarke–Park transformations [13], so that:

$$\dot{\mathbf{u}}\_{\text{a}\[\gamma\}} = \mathbf{T}\_{\text{C}} \dot{\mathbf{u}}\_{\text{abc}} = \sqrt{\frac{2}{3}} \begin{bmatrix} 1 & -1/2 & -1/2 \\ 0 & \sqrt{3}/2 & -\sqrt{3}/2 \\ \sqrt{2}/2 & \sqrt{2}/2 & \sqrt{2}/2 \end{bmatrix} \dot{\mathbf{u}}\_{\text{abc}}.\tag{9}$$

$$\dot{\mathbf{u}}\_{dqz} = \mathbf{T}\_{\mathbf{P}} \dot{\mathbf{u}}\_{d\boldsymbol{\beta}\gamma} = \begin{bmatrix} \cos(n\_d \theta\_m) & \sin(n\_d \theta\_m) & 0\\ -\sin(n\_d \theta\_m) & \cos(n\_d \theta\_m) & 0\\ 0 & 0 & 1 \end{bmatrix} \dot{\mathbf{u}}\_{d\boldsymbol{\beta}\gamma\prime} \tag{10}$$

$$\mathbf{i}\_{dqz} = \mathbf{T}\_\mathbf{P} \mathbf{T}\_\mathbf{C} \mathbf{i}\_{abc} = \sqrt{\frac{2}{3}} \begin{bmatrix} \cos(n\_d \theta\_m) & \cos(n\_d \theta\_m - \frac{2\pi}{3}) & \cos\left(n\_d \theta\_m + \frac{2\pi}{3}\right) \\ -\sin(n\_d \theta\_m) & -\sin\left(n\_d \theta\_m - \frac{2\pi}{3}\right) & -\sin\left(n\_d \theta\_m + \frac{2\pi}{3}\right) \\ \sqrt{2}/2 & \sqrt{2}/2 & \sqrt{2}/2 \end{bmatrix} \mathbf{i}\_{abc} \quad (11)$$

where *iαβγ* and *idqz* are the current vectors in the Clark and Clark–Park reference frames, respectively. By using Equation (7), we finally have:

$$Q\_m = \sqrt{\frac{3}{2}} \lambda\_m n\_d i\_q = k\_t i\_{q\prime} \tag{12}$$

in which *k<sup>t</sup>* is the motor torque constant.

## *2.4. Fault-Tolerant Control System Design*

The multi-mode closed-loop system of the FEPS, schematically depicted in Figure 6, has been entirely developed as a finite-state machine, by using the Matlab–Simulink– Stateflow tools, with mode switch signals that can be generated by the MON modules or overridden by the commands sent by the FCC. In FMM (Table 1), the CON modules receive the speed setpoint ( . *θ* # *<sup>m</sup>*) from the FCC, while in FTM the angle setpoint (*θ* # *<sup>m</sup>*) is constant and pre-defined.

**Figure 6.** FEPS closed-loop architecture.

In the CON modules, all the regulators implement digital signal processing and apply proportional/integral actions on tracking error signals, plus an anti-windup function with back-calculation technique to compensate for command saturation. In particular, the generic *j*-th digital regulator (with *j* = *C*, *S* and *R*, indicating the current, speed and rotation loops, respectively, Table A3) is governed by Equations (12)–(13):

$$y\_{PI}^{(j)} = k\_P^{(j)} \varepsilon^{(j)} + \frac{k\_I^{(j)} T\_s^{(j)}}{z - 1} \left[ \varepsilon^{(j)} + k\_{AW}^{(j)} \left( y^{(j)} - y\_{PI}^{(j)} \right) \right],\tag{13}$$

$$\begin{array}{l} \begin{array}{l} y^{(j)} = \begin{cases} y\_{PI}^{(j)} \\ y\_{sat}^{(j)} \text{sgn}\left(y\_{PI}^{(j)}\right) \end{array} \end{array} \Big| \begin{array}{l} \begin{aligned} y\_{PI}^{(j)} < y\_{sat}^{(j)} \\ \left| y\_{PI}^{(j)} \right| \ge y\_{sat}^{(j)} \end{array} \end{array} \tag{14}$$

where *z* is the discrete-time operator, *ε* (*j*) is the regulator input (tracking error), *y* (*j*) is the regulator output, *y* (*j*) *PI* is the saturator block input (proportional–integral with reference to error, if no saturation is present), while *k* (*j*) *P* and *k* (*j*) *I* are the proportional and integral gains, *k* (*j*) *AW* is the back-calculation anti-windup gain, *y* (*j*) *sat* is the saturation limit, and *T* (*j*) *s* is the sampling rate, Table A3.

In the MON modules, a set of monitoring algorithms are real-time executed at 10 kHz sampling rate, to detect and isolate the major FEPS faults (open-phase, shorted-phase, overheating, overcurrent, hardover, jamming, etc.), and to define the correct operation mode of the AFPMSM stators (Table 1). The maximum fault detection latency for all health-monitoring algorithms was set to 250 ms and the FEPS failure transients will be thus characterized in Section 3 with reference to this worst-case scenario.

## *2.5. UAV Longitudinal Dynamics Modelling*

The UAV dynamics is simulated via a reduced-order model, by taking into account the longitudinal phugoid behaviour only (Figure 7). By assuming that:


$$\begin{cases} m\_a \dot{\mathcal{V}}\_a = T\_p \cos(\mathfrak{a} - \mathfrak{a}\_0) - D - m\_d g \sin(\gamma) \\ m\_a \mathcal{V}\_a \dot{\gamma} = L - m\_d g \cos(\gamma) + T\_p \sin(\mathfrak{a} - \mathfrak{a}\_0) \\ \mathcal{M} = 0 \end{cases} \tag{15}$$

where

$$\begin{cases} L = 1/2\rho S \mathcal{V}^2 \mathbb{C}\_L = 1/2\rho S \mathcal{V}^2 (\mathbb{C}\_{La}\mathfrak{a} + \mathbb{C}\_{L\delta\_\varepsilon}\delta\_\varepsilon) \\ D = 1/2\rho S \mathcal{V}^2 \mathbb{C}\_D = 1/2\rho S \mathcal{V}^2 \left(\mathbb{C}\_{D0} + k\mathbb{C}\_L^2\right) \\\ M = 1/2\rho S \mathcal{V}^2 \overline{\mathfrak{c}} (\mathbb{C}\_{m0} + \mathbb{C}\_{ma}\mathfrak{a} + \mathbb{C}\_{m\delta\_\varepsilon}\delta\_\varepsilon) \end{cases} \tag{16}$$

In Equations (14)–(15), *ma*, V*a*, *γ*, and (*α*0) *α* are the UAV mass, forward speed, path angle, and (zero-lift) angle-of-attack; *L*, *D*, and *T<sup>p</sup>* are the UAV lift, drag and thrust (Figure a and Figure ); *M* is the total aerodynamic pitch moment; *S* is the wing area; *c* is the UAV mean aerodynamic chord; *Cm*<sup>0</sup> is the base pitch moment coefficient; *Cm<sup>α</sup>* and *Cmδ<sup>e</sup>* are the pitch moment–slope coefficients; and *CL<sup>α</sup>* and *CLδ<sup>e</sup>* are the lift–slope coefficients, while *CD*<sup>0</sup> and *k* are the zero-lift drag coefficient and the induced drag factor, respectively.

To evaluate the impacts of motor failures at the vehicle level, the system simulation also includes the closed-loop control on the UAV Rate-of-Climb (RoC, Equation (16)), as described by the scheme in Figure 8.

$$RoC = \mathcal{V}\_d \sin(\gamma) \tag{17}$$

**Figure 7.** Reference schematics for the UAV longitudinal phugoid dynamics.

**Figure 8.** Rate-of-climb closed-loop system.

The RoC regulator receives the setpoint (*RoC*# ) from the mission management functions and implements a proportional/integral action on the tracking error with back-calculation anti-windup (similarly to FEPS ones, the regulator is governed by Equations (12)–(13), in which *j* = RoC, Table A3). In addition, a stability augmentation system related to advance speed perturbation (loop with gain *kSAS* in Figure 8) is also applied.

## **3. Results**

## *3.1. Simulation Campaign Definition*

The performances of the fault-tolerant FEPS were assessed by using a MATLAB/Simulink model of the complete system and numerically solved via the fourth-order Runge–Kutta method, using a 10−<sup>6</sup> s integration step. It is worth noting that the choice of a fixed-step solver is not strictly related to the objectives of this work (in which the model is used for "off-line" simulations), but it has been selected for the next steps of the project, when the control system will be implemented in the ECU boards via automatic MATLAB compilers and executed in "real-time".

The fault-tolerant capabilities of the FEPS are tested by injecting a phase-to-ground fault in the phase A of the stator 2 (Figure 4b) in different flight manoeuvres, i.e.:

	- Start (*t* = 0 s): the FEPS works in normal operation (no faults) and drives the propeller at 5800 rpm with the UAV at 26 m/s in level flight at sea altitude; propeller at 5800 rpm with the UAV at 26 m/s in level flight at sea altitude; FEPS command (t = 1 s), i.e.,
	- FEPS command (*t* = 1 s), i.e., o For climb, the maximum RoC of 3.5 m/s is requested by the FCC;
		- # For climb, the maximum RoC of 3.5 m/s is requested by the FCC; o For cruise, the propeller speed setpoint is held;
		- # For cruise, the propeller speed setpoint is held; o For flight termination/landing, the propeller speed setpoint is decreased
	- # For flight termination/landing, the propeller speed setpoint is decreased from the cruise value at a <sup>−</sup>60 rad/s<sup>2</sup> rate; from the cruise value at a −60 rad/s<sup>2</sup> rate; Event 1 (E1, fault injection): a phase-to-ground fault on phase a of stator 2 is
	- Event 1 (E1, fault injection): a phase-to-ground fault on phase a of stator 2 is imposed; imposed; Event 2 (E2, fault detection and isolation): a CSB mode is set on the faulty stator;
	- Event 2 (E2, fault detection and isolation): a CSB mode is set on the faulty stator; Event 3 (E3, fault compensation):
	- Event 3 (E3, fault compensation): o For climb, the current demand for the healthy stator is doubled and the RoC
		- # For climb, the current demand for the healthy stator is doubled and the RoC setpoint is reduced to 1 m/s; setpoint is reduced to 1 m/s; o For cruise and flight termination/landing, the healthy stator is activated
		- # For cruise and flight termination/landing, the healthy stator is activated (250 ms delay is assumed to achieve the full electric supply) and controlled; (250 ms delay is assumed to achieve the full electric supply) and controlled; Event 4 (E4, only for flight termination/landing): the active stator is switched to
	- Event 4 (E4, only for flight termination/landing): the active stator is switched to operate from FMM to FTM. operate from FMM to FTM. To permit the evaluation of failure transient impacts on system performances, the

To permit the evaluation of failure transient impacts on system performances, the results of two simulations will be proposed in Sections 3.2–3.4 by applying or not the system health monitoring, so that a comparison between uncompensated and compensated behaviours is documented. results of two simulations will be proposed in Sections 3.2–4 by applying or not the system health monitoring, so that a comparison between uncompensated and compensated behaviours is documented.

### *3.2. Failure Transients in Climb* 3.2. Failure Transients in Climb

The simulations can be described with reference to Figure 9: firstly, the tracking performances of the closed-loop control on RoC are assessed, by requesting (*t* = 1 s) the UAV to achieve the maximum-climb rate (3.5 m/s) at 1.5*g* load factor; secondly, for both simulations, E1 is imposed while the UAV is performing a steady climb (*t* = 12 s). In one of the two simulations, the fault is detected and compensated (E2+E3, at *t* = 12.25 s), and the CON modules switch to operate from FMM/FMM to FMM/CSB. The simulations can be described with reference to Figure 9: firstly, the tracking performances of the closed-loop control on RoC are assessed, by requesting (t = 1 s) the UAV to achieve the maximum-climb rate (3.5 m/s) at 1.5g load factor; secondly, for both simulations, E1 is imposed while the UAV is performing a steady climb (t = 12 s). In one of the two simulations, the fault is detected and compensated (E2+E3, at t = 12.25 s), and the CON modules switch to operate from FMM/FMM to FMM/CSB.

Figure 9. UAV response in climb with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): RoC (a), airspeed (b). **Figure 9.** UAV response in climb with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): RoC (**a**), airspeed (**b**).

In the healthy condition, the RoC tracking is characterized by a rise time of about 5 s and negligible overshoot, Figure 9a. An undetected phase-to-ground fault drastically In the healthy condition, the RoC tracking is characterized by a rise time of about 5 s and negligible overshoot, Figure 9a. An undetected phase-to-ground fault drastically

impacts on performances: without detection, the RoC actually goes below zero, meaning that the UAV follows a descent motion. On the other hand, the fault compensation permits

the airspeed exhibits relevant oscillations (with about a 12 s period, close to the Lanchester's phugoid prediction, i.e., √2/, [51]), while it rapidly recovers the cruise

value if the health-monitoring is applied, Figure 9b.

impacts on performances: without detection, the RoC actually goes below zero, meaning that the UAV follows a descent motion. On the other hand, the fault compensation permits to hold the UAV in climb, even if with reduced performance. If the fault is not detected, the airspeed exhibits relevant oscillations (with about a 12 s period, close to the Lanchester's phugoid prediction, i.e., *π* √ 2V*a*/*g*, [51]), while it rapidly recovers the cruise value if the health-monitoring is applied, Figure 9b. Aerospace 2022, 9, x FOR PEER REVIEW 12 of 22 The propeller speed is plotted in Figure 10a. In case of undetected fault, the faulty

The propeller speed is plotted in Figure 10a. In case of undetected fault, the faulty stator brakes down the propeller, by reducing the speed up to about 1000 rpm below the cruise value (5800 rpm), thus resulting in negative RoC, Figure 9a. If the health-monitoring is applied, immediately after the compensation, the output of the SAS block (Figure 8) increases because of the airspeed reduction, Figure 9b, as well as for the diminishing output from the RoC regulator, thus causing an initial increase in the motor demand speed (up to about *t* = 16 s). During this transient period (from 12.25 s to 16 s), the healthy stator operates in saturation condition, as confirmed by the quadrature current output in Figure 10b. It can be also observed that the phase-to-ground fault introduces relevant ripples of current (hence torque) at about 800 Hz frequency, which is twice the motor electrical frequency . stator brakes down the propeller, by reducing the speed up to about 1000 rpm below the cruise value (5800 rpm), thus resulting in negative RoC, Figure 9a. If the healthmonitoring is applied, immediately after the compensation, the output of the SAS block (Figure 8) increases because of the airspeed reduction, Figure 9b, as well as for the diminishing output from the RoC regulator, thus causing an initial increase in the motor demand speed (up to about t = 16 s). During this transient period (from 12.25 s to 16 s), the healthy stator operates in saturation condition, as confirmed by the quadrature current output in Figure 10b. It can be also observed that the phase-to-ground fault introduces relevant ripples of current (hence torque) at about 800 Hz frequency, which is twice the

(*n<sup>d</sup> θm*). The current peaks reach five times the maximum value for continuous duty cycle operations (*Isat*), with a mean value of about −0.5 *Isat*, which produces a braking torque contribution, Figure 10b. motor electrical frequency (ௗ̇). The current peaks reach five times the maximum value for continuous duty cycle operations (௦௧), with a mean value of about −0.5 ௦௧, which produces a braking torque contribution, Figure 10b.

Figure 10. FEPS response in climb with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): propeller speed (a), quadrature current (b), with ௦௧ = 46 Arms. **Figure 10.** FEPS response in climb with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): propeller speed (**a**), quadrature current (**b**), with *Isat* = 46 Arms.

In Figure 11, the phase currents and voltages of the two stators for the compensated case are finally shown. The results are also proposed in the time range between the fault injection and compensation to emphasize the detailed dynamic behaviours: differently from the normal operation, the phase currents in the faulty stator (although still balanced, i.e., their sum is null) are not symmetric, Figure 11a. In fact, the current in phase c roughly push-pulls with reference to the one in phase a, while in the phase b the current progressively shifts to be roughly synchronous with it. The loss of current symmetry results from the voltage grounding on pin a (Figure 11b), which implies that the phase a voltage is driven by the neutral point voltage only. As a consequence of the symmetry loss, the Clarke—Park transform on stator 2 is no longer effective, and the direct and In Figure 11, the phase currents and voltages of the two stators for the compensated case are finally shown. The results are also proposed in the time range between the fault injection and compensation to emphasize the detailed dynamic behaviours: differently from the normal operation, the phase currents in the faulty stator (although still balanced, i.e., their sum is null) are not symmetric, Figure 11a. In fact, the current in phase *c* roughly push-pulls with reference to the one in phase *a*, while in the phase *b* the current progressively shifts to be roughly synchronous with it. The loss of current symmetry results from the voltage grounding on pin *a* (Figure 11b), which implies that the phase *a* voltage is driven by the neutral point voltage only. As a consequence of the symmetry loss, the Clarke—Park transform on stator 2 is no longer effective, and the direct and quadrature current demands (*id*2, *iq*2) become harmonic quantities, Figure 10b.

) become harmonic quantities, Figure 10b.

, ଶ

quadrature current demands (ௗଶ

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Figure 11. FEPS currents and voltages responses in climb with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): stator 2 currents (a) with ௦௧ = 46 Arms; stator 2 voltages (b) with = 36 V. **Figure 11.** FEPS currents and voltages responses in climb with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): stator 2 currents (**a**) with *Isat* = 46 Arms; stator 2 voltages (**b**) with *VDC* = 36 V. The simulations can be described with reference to Figure 12: for both simulations, E1 is imposed while the UAV is in steady cruise (t = 1 s); while in one of them, the fault is firstly detected (E2, at t = 1.25 s) and then compensated (E3, at t = 1.5 s), so that the CON

#### 3.3. Failure Transients in Cruise *3.3. Failure Transients in Cruise* modules are switched to operate from HSB/FMM to FMM/CSB. Since in cruise the FEPS operates with one stator only, the undetected fault drastically

The simulations can be described with reference to Figure 12: for both simulations, E1 is imposed while the UAV is in steady cruise (t = 1 s); while in one of them, the fault is firstly detected (E2, at t = 1.25 s) and then compensated (E3, at t = 1.5 s), so that the CON modules are switched to operate from HSB/FMM to FMM/CSB. Since in cruise the FEPS operates with one stator only, the undetected fault drastically The simulations can be described with reference to Figure 12: for both simulations, E1 is imposed while the UAV is in steady cruise (*t* = 1 s); while in one of them, the fault is firstly detected (E2, at *t* = 1.25 s) and then compensated (E3, at *t* = 1.5 s), so that the CON modules are switched to operate from HSB/FMM to FMM/CSB. impacts on UAV response, with the RoC that settles to about −2 m/s, Figure 12a. On the other hand, in the compensated case, the performances are fully restored. In both cases, the airspeed oscillates around the cruise value with the phugoid period, and a maximum deviation of about 1 m/s is observed, Figure 12b.

impacts on UAV response, with the RoC that settles to about −2 m/s, Figure 12a. On the

Figure 12. UAV response in cruise with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): RoC (a), airspeed (b). **Figure 12.** UAV response in cruise with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): RoC (**a**), airspeed (**b**).

E1

E3

Figure 12. UAV response in cruise with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): RoC (a), airspeed (b). Since in cruise the FEPS operates with one stator only, the undetected fault drastically impacts on UAV response, with the RoC that settles to about −2 m/s, Figure 12a. On the other hand, in the compensated case, the performances are fully restored. In both cases, the airspeed oscillates around the cruise value with the phugoid period, and a maximum deviation of about 1 m/s is observed, Figure 12b.

The propeller speed is plotted in Figure 13a. In the undetected case, the faulty stator brakes down the motor, reducing the speed up to about 1200 rpm below the cruise value (5800 rpm), justifying the negative RoC in Figure 12a. If the health-monitoring is applied, the delay time required for the full electric supply of the stand-by stator (from *t* = 1.25 s

Aerospace 2022, 9, x FOR PEER REVIEW 14 of 22

to *t* = 1.5 s) implies that resistive aerodynamic loads are applied to the propeller, and the speed rate diminishes, so that, immediately after the fault, the faulty stator acts as a brake. fault introduces high-frequency ripples, and, immediately after the fault, the mean value of the quadrature current is negative. On the other hand, in steady condition, it becomes positive (0.25 ௦௧), so that the faulty stator delivers power to the propeller.

The propeller speed is plotted in Figure 13a. In the undetected case, the faulty stator brakes down the motor, reducing the speed up to about 1200 rpm below the cruise value (5800 rpm), justifying the negative RoC in Figure 12a. If the health-monitoring is applied, the delay time required for the full electric supply of the stand-by stator (from t = 1.25 s to t = 1.5 s) implies that resistive aerodynamic loads are applied to the propeller, and the speed rate diminishes, so that, immediately after the fault, the faulty stator acts as a brake. The above discussion is further enforced by observing the response in terms of quadrature currents, Figure 13b. As also highlighted in Section 3.2, the phase-to-ground

Figure 13. FEPS response in cruise with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): propeller speed (a), quadrature current (b), with ௦௧ = 46 Arms. **Figure 13.** FEPS response in cruise with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): propeller speed (**a**), quadrature current (**b**), with *Isat* = 46 Arms.

Finally, in Figure 14, the phase currents and voltages of the two stators for the compensated case are shown. The results are also proposed in the time range between the fault injection and compensation to emphasize out the detailed dynamic behaviours. The full electric activation of the stator 1 (at t = 1.5 s) is characterized by relevant peaks of the phase currents (Figure 14a), up to reach about three times the maximum value in continuous duty cycle operations (௦௧). As discussed in Section 3.2, the phase-to-ground The above discussion is further enforced by observing the response in terms of quadrature currents, Figure 13b. As also highlighted in Section 3.2, the phase-to-ground fault introduces high-frequency ripples, and, immediately after the fault, the mean value of the quadrature current is negative. On the other hand, in steady condition, it becomes positive (0.25 *Isat*), so that the faulty stator delivers power to the propeller.

fault causes the loss of the currents symmetry while maintaining their balance, Figure 14b. Finally, in Figure 14, the phase currents and voltages of the two stators for the compensated case are shown. The results are also proposed in the time range between the fault injection and compensation to emphasize out the detailed dynamic behaviours. The full electric activation of the stator 1 (at *t* = 1.5 s) is characterized by relevant peaks of the phase currents (Figure 14a), up to reach about three times the maximum value in continuous duty cycle operations (*Isat*). As discussed in Section 3.2, the phase-to-ground fault causes the loss of the currents symmetry while maintaining their balance, Figure 14b. Aerospace 2022, 9, x FOR PEER REVIEW 15 of 22

Figure 14. FEPS currents and voltages responses in cruise with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): stator 2 currents (a) with ௦௧ = 46 Arms; stator 2 voltages (b) with = 36 V. **Figure 14.** FEPS currents and voltages responses in cruise with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): stator 2 currents (**a**) with *Isat* = 46 Arms; stator 2 voltages (**b**) with *VDC* = 36 V.

The simulations can be described with reference to Figure 15: for both simulations,

The UAV behaviour can be interpreted via Equation (12): since the elevator deflection maintains the pitch equilibrium, a reduction in the propeller speed implies a thrust reduction, which causes an oscillatory descendant trajectory, Figure 15a. On the other hand, the airspeed oscillates by following the phugoid behaviour, while keeping its mean

E1

E3

E2

(a) (b)

E4a

one of the two ones, the fault is firstly detected (E2, at t = 4.75 s) and then compensated (E3, at t = 5 s), so that the CON modules switch from HSB/FMM to FMM/CSB. Successively, when the speed is adequately small (<1 rad/s), the CON modules switch from HSB/FMM to HSB/FTM in the undetected case (E4a, at t = 10.4 s), and from FMM/CSB

3.4. Failure Transient and Transition from FMM to FTM in Flight Termination/Landing

value roughly to the one before the fault, Figure 15b.

E4a

E4b

E2

E3

to FTM/CSB if the health-monitoring is applied (E4b, at t = 10.7 s).

E1E4b

with = 36 V.

### *3.4. Failure Transient and Transition from FMM to FTM in Flight Termination/Landing* Successively, when the speed is adequately small (<1 rad/s), the CON modules switch

3.4. Failure Transient and Transition from FMM to FTM in Flight Termination/Landing

The simulations can be described with reference to Figure 15: for both simulations, E1 is imposed while the UAV is performing a steady descent motion (*t* = 4.5 s); while in one of the two ones, the fault is firstly detected (E2, at *t* = 4.75 s) and then compensated (E3, at *t* = 5 s), so that the CON modules switch from HSB/FMM to FMM/CSB. Successively, when the speed is adequately small (<1 rad/s), the CON modules switch from HSB/FMM to HSB/FTM in the undetected case (E4a, at *t* = 10.4 s), and from FMM/CSB to FTM/CSB if the health-monitoring is applied (E4b, at *t* = 10.7 s). from HSB/FMM to HSB/FTM in the undetected case (E4a, at t = 10.4 s), and from FMM/CSB to FTM/CSB if the health-monitoring is applied (E4b, at t = 10.7 s). The UAV behaviour can be interpreted via Equation (12): since the elevator deflection maintains the pitch equilibrium, a reduction in the propeller speed implies a thrust reduction, which causes an oscillatory descendant trajectory, Figure 15a. On the other hand, the airspeed oscillates by following the phugoid behaviour, while keeping its mean value roughly to the one before the fault, Figure 15b.

Figure 14. FEPS currents and voltages responses in cruise with phase-to-ground fault on stator 2 (E1, E2 and E3 defined in Section 3.1): stator 2 currents (a) with ௦௧ = 46 Arms; stator 2 voltages (b)

(a) (b)

The simulations can be described with reference to Figure 15: for both simulations, E1 is imposed while the UAV is performing a steady descent motion (t = 4.5 s); while in one of the two ones, the fault is firstly detected (E2, at t = 4.75 s) and then compensated (E3, at t = 5 s), so that the CON modules switch from HSB/FMM to FMM/CSB.

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**Figure 15.** UAV response in flight termination/landing with phase-to-ground fault on stator 2 (E1, E2, E3 and E4 defined in Section 3.1): RoC (**a**), airspeed (**b**). Aerospace 2022, 9, x FOR PEER REVIEW 16 of 22

The UAV behaviour can be interpreted via Equation (12): since the elevator deflection maintains the pitch equilibrium, a reduction in the propeller speed implies a thrust reduction, which causes an oscillatory descendant trajectory, Figure 15a. On the other hand, the airspeed oscillates by following the phugoid behaviour, while keeping its mean value roughly to the one before the fault, Figure 15b. Figure 15. UAV response in flight termination/landing with phase-to-ground fault on stator 2 (E1, E2, E3 and E4 defined in Section 3.1): RoC (a), airspeed (b).

The closed-loop tracking on propeller speed and position are reported in Figure 16. It is worth noting that the mode transition is executed when the speed is not zeroed yet, to anticipate the parachute opening, and that in both cases the position tracking to the predefined setpoint (180 deg) is correctly accomplished. The quadrature currents response (Figure 17a) also points out that, compared with the climb and cruise simulations, the failure transient for the undetected case impacts on the mechanical transmission too, because the electrical frequency sweeps down, up to equalling the drivetrain resonant frequency (located at 100 Hz), Figure 17b. The closed-loop tracking on propeller speed and position are reported in Figure 16. It is worth noting that the mode transition is executed when the speed is not zeroed yet, to anticipate the parachute opening, and that in both cases the position tracking to the predefined setpoint (180 deg) is correctly accomplished. The quadrature currents response (Figure 17a) also points out that, compared with the climb and cruise simulations, the failure transient for the undetected case impacts on the mechanical transmission too, because the electrical frequency sweeps down, up to equalling the drivetrain resonant frequency (located at 100 Hz), Figure 17b.

Figure 16. FEPS response in flight termination/landing with phase-to-ground fault on stator 2 (E1, E2, E3 and E4 defined in Section 3.1): propeller speed (a), propeller angle (b). **Figure 16.** FEPS response in flight termination/landing with phase-to-ground fault on stator 2 (E1, E2, E3 and E4 defined in Section 3.1): propeller speed (**a**), propeller angle (**b**).

Figure 17. FEPS response in flight termination/landing with phase-to-ground fault on stator 2 (E1, E2, E3 and E4 defined in Section 3.1): quadrature currents (a) with ௦௧ = 46 Arms, propeller speed

(a) (b)

In Figure 18, the phase currents and voltages of the two stators for the compensated case are finally reported. The full electric activation of the stator 1 (at t = 5 s) is characterized by relevant peaks of the phase currents (Figure 18a), reaching up to about seven times the maximum value for continuous duty cycle (௦௧ ), while the stator 1 currents operate in saturation (√3/2௦௧) until the propeller stops (at t = 10 s). Similarly to what was discussed in Sections 3.2–3.3, the phase-to-ground fault again causes the loss of the currents' symmetry while maintaining their balance (Figure 18a). Finally, it is worth noting that the reduction in voltage amplitudes is coherent with the reduction in the backelectromotive forces caused by the speed decrease (the homopolar voltage component is

(b).

also represented in Figure 18b).

(b).

E2, E3 and E4 defined in Section 3.1): propeller speed (a), propeller angle (b).

(a) (b)

Figure 17. FEPS response in flight termination/landing with phase-to-ground fault on stator 2 (E1, E2, E3 and E4 defined in Section 3.1): quadrature currents (a) with ௦௧ = 46 Arms, propeller speed **Figure 17.** FEPS response in flight termination/landing with phase-to-ground fault on stator 2 (E1, E2, E3 and E4 defined in Section 3.1): quadrature currents (**a**) with *Isat* = 46 Arms, propeller speed (**b**).

Figure 16. FEPS response in flight termination/landing with phase-to-ground fault on stator 2 (E1,

Figure 15. UAV response in flight termination/landing with phase-to-ground fault on stator 2 (E1,

The closed-loop tracking on propeller speed and position are reported in Figure 16. It is worth noting that the mode transition is executed when the speed is not zeroed yet, to anticipate the parachute opening, and that in both cases the position tracking to the predefined setpoint (180 deg) is correctly accomplished. The quadrature currents response (Figure 17a) also points out that, compared with the climb and cruise simulations, the failure transient for the undetected case impacts on the mechanical transmission too, because the electrical frequency sweeps down, up to equalling the drivetrain resonant

E2, E3 and E4 defined in Section 3.1): RoC (a), airspeed (b).

frequency (located at 100 Hz), Figure 17b.

In Figure 18, the phase currents and voltages of the two stators for the compensated case are finally reported. The full electric activation of the stator 1 (at t = 5 s) is characterized by relevant peaks of the phase currents (Figure 18a), reaching up to about seven times the maximum value for continuous duty cycle (௦௧ ), while the stator 1 currents operate in saturation (√3/2௦௧) until the propeller stops (at t = 10 s). Similarly to what was discussed in Sections 3.2–3.3, the phase-to-ground fault again causes the loss of the currents' symmetry while maintaining their balance (Figure 18a). Finally, it is worth noting that the reduction in voltage amplitudes is coherent with the reduction in the backelectromotive forces caused by the speed decrease (the homopolar voltage component is In Figure 18, the phase currents and voltages of the two stators for the compensated case are finally reported. The full electric activation of the stator 1 (at *t* = 5 s) is characterized by relevant peaks of the phase currents (Figure 18a), reaching up to about seven times the maximum value for continuous duty cycle (*Isat*), while the stator 1 currents operate in saturation (<sup>√</sup> 3/2*Isat*) until the propeller stops (at *t* = 10 s). Similarly to what was discussed in Sections 3.2 and 3.3, the phase-to-ground fault again causes the loss of the currents' symmetry while maintaining their balance (Figure 18a). Finally, it is worth noting that the reduction in voltage amplitudes is coherent with the reduction in the back-electromotive forces caused by the speed decrease (the homopolar voltage component is also represented in Figure 18b). Aerospace 2022, 9, x FOR PEER REVIEW 17 of 22

Figure 18. FEPS response in flight termination/landing with phase-to-ground fault on stator 2 (E1, E2, E3 and E4 defined in Section 3.1): stator 2 currents (a) with ௦௧ = 46 Arms; stator 2 voltages (b) with = 36 V. **Figure 18.** FEPS response in flight termination/landing with phase-to-ground fault on stator 2 (E1, E2, E3 and E4 defined in Section 3.1): stator 2 currents (**a**) with *Isat* = 46 Arms; stator 2 voltages (**b**) with *VDC* = 36 V.

temperature. Furthermore, another contribution to demagnetization derives from the flux weakening caused by the direct current increase. The direct current in the faulty stator is characterized by high-amplitude oscillations, so the flux linkage on the quadrature axis

Furthermore, the phase-to-ground fault can also impact the structural integrity of system drivetrain (e.g., bearings). In fact, the symmetry loss of the faulty stator currents implies that the magnetic fields generated via the FOC technique are not synchronous with the rotor magnet's motion. The ripple manifests at twice the electrical frequency and potential criticalities can arise if this frequency equals the structural resonance of the

The failure transient's analysis proposed in Section 3 highlights that the phase-toground fault can strongly impact performances at the UAV level too. If the fault is not detected, the propeller torque reduces to below the one required for level flight, thus causing the UAV to fall. Furthermore, if one compares the failure transient in climb with the one in cruise, it is worth noting that the larger the speed demand for the faulty stator, the lower its torque output. During climb, the demand after the fault settles at the maximum value (7500 rpm, Figure 10a) and the mean torque to a negative value (Figure 10b); while in cruise, when the speed setpoint is kept at 5800 rpm (Figure 13a), the mean torque settles to a positive value (Figure 13b). Concerning the flight termination/landing, it is demonstrated that, even without fault detection, the FEPS is capable of stopping and aligning the propeller blades with the wing, even if the electrical frequency sweep related

As confirmed by the results in Section 3, the phase-to-ground fault can determine damages to different parts of the electric machine. In fact, to compensate for the voltage supply lack on the faulty phase, the currents in other phases strongly increase, leading to extremely hot temperatures with consequent deterioration of the magnets. Partial or

( = + ௗ) can even overcome the magnet coercivity.

4. Discussion

drivetrain.

## **4. Discussion**

As confirmed by the results in Section 3, the phase-to-ground fault can determine damages to different parts of the electric machine. In fact, to compensate for the voltage supply lack on the faulty phase, the currents in other phases strongly increase, leading to extremely hot temperatures with consequent deterioration of the magnets. Partial or complete demagnetization may occur, since the magnet coercivity decreases with temperature. Furthermore, another contribution to demagnetization derives from the flux weakening caused by the direct current increase. The direct current in the faulty stator is characterized by high-amplitude oscillations, so the flux linkage on the quadrature axis (*λ<sup>q</sup>* = *λ<sup>m</sup>* + *Li<sup>d</sup>* ) can even overcome the magnet coercivity.

Furthermore, the phase-to-ground fault can also impact the structural integrity of system drivetrain (e.g., bearings). In fact, the symmetry loss of the faulty stator currents implies that the magnetic fields generated via the FOC technique are not synchronous with the rotor magnet's motion. The ripple manifests at twice the electrical frequency and potential criticalities can arise if this frequency equals the structural resonance of the drivetrain.

The failure transient's analysis proposed in Section 3 highlights that the phase-toground fault can strongly impact performances at the UAV level too. If the fault is not detected, the propeller torque reduces to below the one required for level flight, thus causing the UAV to fall. Furthermore, if one compares the failure transient in climb with the one in cruise, it is worth noting that the larger the speed demand for the faulty stator, the lower its torque output. During climb, the demand after the fault settles at the maximum value (7500 rpm, Figure 10a) and the mean torque to a negative value (Figure 10b); while in cruise, when the speed setpoint is kept at 5800 rpm (Figure 13a), the mean torque settles to a positive value (Figure 13b). Concerning the flight termination/landing, it is demonstrated that, even without fault detection, the FEPS is capable of stopping and aligning the propeller blades with the wing, even if the electrical frequency sweep related to the motor slowdown can generate potentially dangerous vibrations when the drivetrain resonant frequency is intercepted.

The future developments of the research will be focused on:

	- # AFPMSM model, via experimental testing with reference to normal operation (failure transient characterisation will be always simulated, but using updated parameters);
	- # Propeller loads model, via CFD simulations, with special focus on the region of *AR* > 0.65 (no data from manufacturer);
	- # Mechanical drivetrain model, via experimental testing;
	- # UAV longitudinal dynamics, via flight data.

## **5. Conclusions**

The fault-tolerant control of the FEPS employed by the lightweight fixed-wing UAV named TERSA has been designed and verified in terms of dynamic performances by simulating major electrical faults during relevant flight manoeuvres. The reference FEPS includes a dual-stator AFPMSM operating in active/active mode during climb and in standby/active mode in other flight phases, and it is designed as a finite-state machine to switch the closed-loop system from speed-tracking control during the flight, to position-tracking control during flight termination/landing, when the propeller blades are aligned with the wings to avoid interference with the landing parachute opening. Thanks to a high-fidelity

modelling approach, the impacts of failure transients from component levels up to UAV levels during climb, cruise and flight termination/landing manoeuvres are evaluated, by simulating a phase-to-ground fault on a phase of one of the two AFPMSM stators. The results show that if the fault is not detected, it strongly impacts on both FEPS and UAV performances, up to causing the UAV to fall, the generation of large-amplitude highfrequency torque and current oscillations, and dangerous interactions with the drivetrain structural resonance. On the other hand, if the health monitoring is applied and the fault is detected and compensated, the FEPS permits maintaining the climb capability (even if it is with reduced performance), holding the UAV altitude and speed during cruise, as well as safely managing the flight termination/landing manoeuvre.

**Author Contributions:** Conceptualization, methodology and investigation, A.S. and G.D.R.; software, data curation and writing—original draft preparation, A.S.; validation, formal analysis and writing review and editing, G.D.R.; resources, supervision and visualization, G.D.R. and R.G.; project administration and funding acquisition, R.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was co-funded by the Italian Government (*Ministero Italiano dello Sviluppo Economico*, MISE) and by the Tuscany Regional Government, in the context of the R&D project "*Tecnologie Elettriche e Radar per SAPR Autonomi* (TERSA)", Grant number: F/130088/01-05/X38.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors wish to thank Luca Sani, from the University of Pisa (*Dipartimento di Ingegneria dell'Energia, dei Sistemi, del Territorio e delle Costruzioni*), for the support in the definition of the AFPMSM model parameters, and Ing. Francesco Schettini, from Sky Eye Systems (Italy), for the information about the aero-mechanical data and the general performances of the UAV.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Appendix A**

This section contains tables reporting parameters and data related to the propulsion system model (Table A1) and the UAV model (Table A2).


**Table A1.** FEPS model parameters.


**Table A2.** UAV model parameters.

**Table A3.** FEPS and UAV regulators parameters.


## **References**


## *Article* **Design and Simulation Analysis of an Electromagnetic Damper for Reducing Shimmy in Electrically Actuated Nose Wheel Steering Systems**

**Chenfei She <sup>1</sup> , Ming Zhang 1,\*, Yibo Ge <sup>2</sup> , Liming Tang <sup>1</sup> , Haifeng Yin <sup>2</sup> and Gang Peng <sup>2</sup>**


**Abstract:** Based on the technical platform of electrically actuated nose wheel steering systems, a new type of damping shimmy reduction technology is developed to break through the limitations of traditional hydraulic damping shimmy reduction methods, and an electrically actuated nose wheel steering structure scheme is proposed. The mathematical model of the electromagnetic damper is established, the derivation of skin depth, damping torque and damping coefficient is completed, and the design of the shape and size of the electromagnetic damper is combined with the derivation results and the technical index of shimmy reduction. The electromagnetic field finite element simulation results show that the mathematical modeling method of the electromagnetic damper has good accuracy, and its application to the shimmy reduction module of the electrically actuated nose wheel steering system is also feasible and superior. Finally, the key factors influencing the performance of electromagnetic damper shimmy reduction are studied and analyzed, thus forming a complete electromagnetic damper shimmy reduction technology for the electrically actuated system, and laying the foundation for the design of novel all-electric aircraft and landing gear.

**Keywords:** electrically actuated nose wheel steering; all-electric aircraft; electromagnetic damper; electromagnetic simulation; landing gear shimmy reduction

## **1. Introduction**

The concept of more-electric aircraft or even all-electric aircraft has been proposed and rapidly developed in order to improve energy efficiency, cut operating costs and reduce the take-off weight. In the process of progressive replacement of mechanical, hydraulic or pneumatic power sources by electromechanical actuators (EMA) [1], the validation of electromechanical nose wheel steering mechanisms has already started worldwide. In 2009, a project called Distributed and Redundant Electro-mechanical nose wheel Steering System (DRESS), jointly completed by several European aviation industry units, predicted that electromechanical integration may significantly improve the reliability and availability of nose wheel steering mechanisms [2]. In 2010, Bennett applied a fault-tolerant electromechanical actuator to the aircraft nose wheel steering system and theoretically investigated and experimentally validated it [3]. In 2010, Liao et al. studied the all-electric nose wheel steering system and introduced a design method for an electric turning control system for small aircraft landing gear based on DSP, which was experimentally verified to have the advantages of miniaturization, reliability and control flexibility [4].

Landing gear shimmy is a kind of self-excited vibration dominated by the oscillation of the wheels [5]. Understanding the shimmy phenomenon and proposing reasonable shimmy reduction measures have been a pressing problem for experts in aircraft structural dynamics,

**Citation:** She, C.; Zhang, M.; Ge, Y.; Tang, L.; Yin, H.; Peng, G. Design and Simulation Analysis of an Electromagnetic Damper for Reducing Shimmy in Electrically Actuated Nose Wheel Steering Systems. *Aerospace* **2022**, *9*, 113. https://doi.org/10.3390/ aerospace9020113

Academic Editor: Gianpietro Di Rito

Received: 24 January 2022 Accepted: 16 February 2022 Published: 19 February 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

all-electric aircraft are no exception. Currently, the most effective measure to cope with the phenomenon of aircraft landing gear shimmy is the installation of a shimmy damper [6]. Dampers enable most aircraft to eliminate shimmy, or to improve shimmy problems. Most modern aircrafts use oleo dampers, except for some small aircraft and helicopters with dry friction dampers on the tail landing gear. In recent years, the rapid development of magnetorheological dampers has led to the research on landing cushioning as well as shimmy control using magnetorheological fluids. In 2021, Luong and Jo et al. mounted magnetorheological dampers on the aircraft landing gear to reduce landing impact. They also designed an intelligent controller based on supervised neural networks and verified the performance and stability of the magnetorheological dampers by crash tests [7,8]. In 2010, Chen et al. used magnetorheological dampers to suppress aircraft landing gear shimmy and designed a semi-active control strategy [9]. In 2019, Zhu et al. optimized the structure of an external magnetorheological damper on the aircraft landing gear and verified its shimmy reduction performance by means of damping characteristics tests [10]. In addition, with the development of electromechanical technology, electromagnetic dampers are proposed to be applied in the electrically actuated nose wheel steering system to realize the shimmy reduction function. Compared with the traditional friction dampers and oleo dampers, electromagnetic dampers have advantages in maintenance, environmental protection, service life, structural principle and so on.

In 2019, Jia et al. proposed an electromagnetic damper design for a soft contact robotic arm joint to cushion and unload the impact of the robotic arm in contact with the grasping target [11]. In 2013, Kou et al. used electromagnetic dampers to simulate the load force to which the motor is subjected during normal operation under laboratory conditions, and this load force could be arbitrarily adjusted within a certain range to detect some technical performance indicators of the motor [12]. In 2009, Ebrahimi designed and developed a new type of electromagnetic damper for active vehicle suspension control systems. Unlike traditional passive and semi-active control system dampers, this damper could not only adjust the damping coefficient but also convert the mechanical energy of vibration into electrical energy for reuse [13]. In 2011, Liu et al. conducted theoretical and experimental studies on electromagnetic dampers applied to auxiliary braking of large vehicles [14].

The abovementioned work studied and optimized only the nose wheel landing gear dampers of modern aircraft, which do not meet the innovative development goals of aircraft electrification, or studied and optimized electromagnetic dampers in engineering fields such as vehicle suspension systems, space docking mechanisms, and braking of high-speed trains. However, few articles have reported on the design and research of electromagnetic dampers for shimmy reduction in electrically actuated nose wheel steering systems. In view of this, this paper takes the nose landing gear of a certain jet fighter as the research object. We establish the mathematical model and 3D model of the electromagnetic damper based on the electrically actuated nose wheel steering system, and verify the feasibility and superiority of the electromagnetic damper applied to the electrically actuated nose wheel steering system by using the finite element simulation method in combination with various factors affecting the shimmy reduction performance.

## **2. Demand Analysis of Electromagnetic Dampers for Shimmy Reduction Performance** *2.1. Operating Principle*

From the theorem of electromagnetic induction, it is known that the movement of a wire cutting magnetic induction lines in a magnetic field will produce an induced current, and according to the Lenz's law, the magnetic field of the induced current will impede the movement of the wire in the magnetic field. In addition, due to the resistance of the wire itself, the induced current flowing through the wire will cause the kinetic energy of the wire to be dissipated as heat energy, thus achieving the effect of impeding the movement [15], as shown in Figure 1.

*Aerospace* **2022**, *9*, x 3 of 22

[15], as shown in Figure 1.

[15], as shown in Figure 1.

Electromagnetic dampers are electromagnetic devices based on the abovementioned principles. According to the different excitation methods, electromagnetic dampers can be divided into three types: electrically excited electromagnetic dampers, permanent magnet electromagnetic dampers and hybrid excited electromagnetic dampers [16]. Based to the differences in structure form, electromagnetic dampers can also be divided into rotary electromagnetic dampers, single rotor disk electromagnetic dampers and double rotor disks electromagnetic dampers. Considering the working environment of the nose landing gear and the actual demand of shimmy reduction, the double rotor disks permanent magnet electromagnetic damper is finally selected to eliminate shimmy, and Electromagnetic dampers are electromagnetic devices based on the abovementioned principles. According to the different excitation methods, electromagnetic dampers can be divided into three types: electrically excited electromagnetic dampers, permanent magnet electromagnetic dampers and hybrid excited electromagnetic dampers [16]. Based to the differences in structure form, electromagnetic dampers can also be divided into rotary electromagnetic dampers, single rotor disk electromagnetic dampers and double rotor disks electromagnetic dampers. Considering the working environment of the nose landing gear and the actual demand of shimmy reduction, the double rotor disks permanent magnet electromagnetic damper is finally selected to eliminate shimmy, and its structure is shown in Figure 2. principles. According to the different excitation methods, electromagnetic dampers can be divided into three types: electrically excited electromagnetic dampers, permanent magnet electromagnetic dampers and hybrid excited electromagnetic dampers [16]. Based to the differences in structure form, electromagnetic dampers can also be divided into rotary electromagnetic dampers, single rotor disk electromagnetic dampers and double rotor disks electromagnetic dampers. Considering the working environment of the nose landing gear and the actual demand of shimmy reduction, the double rotor disks permanent magnet electromagnetic damper is finally selected to eliminate shimmy, and its structure is shown in Figure 2.

wire to be dissipated as heat energy, thus achieving the effect of impeding the movement

wire to be dissipated as heat energy, thus achieving the effect of impeding the movement

 **Figure 2.** Structural diagram of the electromagnetic damper: (**a**) Rotor disk; (**b**) Stator disk. **Figure 2.** Structural diagram of the electromagnetic damper: (**a**) Rotor disk; (**b**) Stator disk.

(**a**) (**b**) **Figure 2.** Structural diagram of the electromagnetic damper: (**a**) Rotor disk; (**b**) Stator disk. The double rotor disks permanent magnet electromagnetic damper mainly includes the stator disk, rotor disks and permanent magnet poles and other components. Six permanent magnets are fixed on each side of the stator disk and they are distributed along the circumference. The two adjacent poles are arranged at intervals of N and S to form three independent groups of poles. The rotor disks are commonly composed of two circular conductor disks on the top and bottom, which will leave a very small air gap when The double rotor disks permanent magnet electromagnetic damper mainly includes the stator disk, rotor disks and permanent magnet poles and other components. Six permanent magnets are fixed on each side of the stator disk and they are distributed along the circumference. The two adjacent poles are arranged at intervals of N and S to form three independent groups of poles. The rotor disks are commonly composed of two circular conductor disks on the top and bottom, which will leave a very small air gap when installed with the magnetic poles. This not only ensures that the relative rotation between the upper and lower rotor disks and the stator does not generate friction, but also allows the performance of the electromagnetic damper to be changed by adjusting the size of the The double rotor disks permanent magnet electromagnetic damper mainly includes the stator disk, rotor disks and permanent magnet poles and other components. Six permanent magnets are fixed on each side of the stator disk and they are distributed along the circumference. The two adjacent poles are arranged at intervals of N and S to form three independent groups of poles. The rotor disks are commonly composed of two circular conductor disks on the top and bottom, which will leave a very small air gap when installed with the magnetic poles. This not only ensures that the relative rotation between the upper and lower rotor disks and the stator does not generate friction, but also allows the performance of the electromagnetic damper to be changed by adjusting the size of the air gap [17].

installed with the magnetic poles. This not only ensures that the relative rotation between the upper and lower rotor disks and the stator does not generate friction, but also allows the performance of the electromagnetic damper to be changed by adjusting the size of the air gap [17]. When the electromagnetic damper starts to operate, the magnetic field generated by the magnetic poles forms a circuit between the stator disk, the air gap and the upper and air gap [17]. When the electromagnetic damper starts to operate, the magnetic field generated by the magnetic poles forms a circuit between the stator disk, the air gap and the upper and lower rotor disks. The polarity of the two adjacent magnets on each side is opposite, and the magnitude of the magnetic flux is related to the magnetic inductance strength of the When the electromagnetic damper starts to operate, the magnetic field generated by the magnetic poles forms a circuit between the stator disk, the air gap and the upper and lower rotor disks. The polarity of the two adjacent magnets on each side is opposite, and the magnitude of the magnetic flux is related to the magnetic inductance strength of the magnetic poles. When the rotor disks rotate, the magnetic flux through the rotor disks changes, thus inducing a swirl-shaped induced current on the rotor disks, also known as

lower rotor disks. The polarity of the two adjacent magnets on each side is opposite, and the magnitude of the magnetic flux is related to the magnetic inductance strength of the

induced eddy current. As shown in Figure 3, adjacent eddy currents in opposite directions are generated on the rotor disks during operation, and the corresponding induced magnetic field generates a damping torque in the opposite direction of the rotor disks' rotation, thus impeding the rotation of the rotor disks. From the perspective of energy conversion, the induced currents on the rotor disks transform the kinetic energy of the rotor disks into thermal energy, so as to achieve the effect of shimmy reduction. are generated on the rotor disks during operation, and the corresponding induced magnetic field generates a damping torque in the opposite direction of the rotor disks' rotation, thus impeding the rotation of the rotor disks. From the perspective of energy conversion, the induced currents on the rotor disks transform the kinetic energy of the rotor disks into thermal energy, so as to achieve the effect of shimmy reduction.

magnetic poles. When the rotor disks rotate, the magnetic flux through the rotor disks changes, thus inducing a swirl-shaped induced current on the rotor disks, also known as induced eddy current. As shown in Figure 3, adjacent eddy currents in opposite directions

*Aerospace* **2022**, *9*, x 4 of 22

**Figure 3.** The damping torque and induced current of electromagnetic damper. **Figure 3.** The damping torque and induced current of electromagnetic damper.

Compared with the traditional oleo damper, the electromagnetic damper does not need to consider the need for gas tightness during the assembly process, nor does it need regular inspection of oil quality and quantity during the working process, which greatly Compared with the traditional oleo damper, the electromagnetic damper does not need to consider the need for gas tightness during the assembly process, nor does it need regular inspection of oil quality and quantity during the working process, which greatly reduces the production costs and maintenance costs, reflecting its superiority.

### reduces the production costs and maintenance costs, reflecting its superiority. *2.2. Analysis of Shimmy Motion Parameters*

*2.2. Analysis of Shimmy Motion Parameters*  Combining the engineering practice with the empirical data in the literature [5], the angular amplitude of shimmy is generally around 2 to 20° and the frequency is around 5 to 30 Hz. It shows the characteristics of high frequency at low amplitude and low Combining the engineering practice with the empirical data in the literature [5], the angular amplitude of shimmy is generally around 2 to 20◦ and the frequency is around 5 to 30 Hz. It shows the characteristics of high frequency at low amplitude and low frequency at high amplitude. Assuming that the nose wheel shimmy amplitude is *A* (between 2 and 20◦ ) and the frequency is *f* (between 5 and 30 Hz). The angle of the nose wheel shimmy varies with time in accordance with the sine function relationship:

$$a = A \sin 2\pi ft \tag{1}$$

wheel shimmy varies with time in accordance with the sine function relationship: Then the angular velocity of the nose wheel shimmy is:

$$
\omega\_{\rm NW} = \frac{d\mathfrak{a}}{dt} = 2\pi f A \cos 2\pi ft \tag{2}
$$

Then the angular velocity of the nose wheel shimmy is: *NW* = = 2 cos2 *<sup>d</sup><sup>α</sup> ω πfA <sup>π</sup>ft dt* (2) From this, it can be seen that there is a phase difference of 1/4 cycle between the From this, it can be seen that there is a phase difference of 1/4 cycle between the shimmy angle and angular velocity of the nose wheel in line with the above hypothetical law, that is, the angular velocity of the nose wheel is maximum when it passes through the neutral position where the shimmy angle is 0, and the angular velocity is 0 at the position of the maximum shimmy angle (amplitude). The maximum angular velocity is:

$$
\omega\_{\text{max}} = 2\pi f A \tag{3}
$$

the neutral position where the shimmy angle is 0, and the angular velocity is 0 at the position of the maximum shimmy angle (amplitude). The maximum angular velocity is: *ω π* max <sup>=</sup> <sup>2</sup> *fA* (3) According to the assumption that 2◦ shimmy amplitude corresponds to 30 Hz frequency, meawhile, 20◦ shimmy amplitude corresponds to 5 Hz frequency and the frequency decreases linearly with the increase of shimmy amplitude, the shimmy motion parameters in the above-mentioned range of amplitude and frequency can be deduced as shown in Table 1.

According to the assumption that 2° shimmy amplitude corresponds to 30 Hz frequency, meawhile, 20° shimmy amplitude corresponds to 5 Hz frequency and the

parameters in the above-mentioned range of amplitude and frequency can be deduced as

shown in Table 1.


2.0 30.00 6.58 4.0 27.22 11.94 6.0 24.44 16.08

**A (°) F (Hz) Angular Velocity of Strut** 

**(rad/s)** 

**Table 1.** Motion parameters of shimmy. 8.0 21.67 19.01

**Table 1.** Motion parameters of shimmy.

*Aerospace* **2022**, *9*, x 5 of 22

### *2.3. Damping Requirements for Shimmy Reduction* coefficient of shimmy reduction for the research object in this paper is not less than 40 Nms/rad. The output damping coefficient of the electromagnetic damper may be

Combined with the actual needs of the project, the design target of the damping coefficient of shimmy reduction for the research object in this paper is not less than 40 Nms/rad. The output damping coefficient of the electromagnetic damper may be amplified by the transmission mechanism and transmitted to the strut. The amplified damping coefficient should meet the design target under different shimmy motion states described in Table 1. Based on this, the design of the steering structure under the electrically actuated nose wheel steering system and the scheme of the electromagnetic damper are developed in the next sections. amplified by the transmission mechanism and transmitted to the strut. The amplified damping coefficient should meet the design target under different shimmy motion states described in Table 1. Based on this, the design of the steering structure under the electrically actuated nose wheel steering system and the scheme of the electromagnetic damper are developed in the next sections. **3. Research on Electromagnetic Damping and Shimmy Reduction Scheme of Electri-**

### **3. Research on Electromagnetic Damping and Shimmy Reduction Scheme of Electrically Actuated Nose Wheel Steering Systems cally Actuated Nose Wheel Steering Systems**  *3.1. Structural Scheme Design of the Electrically Actuated Nose Wheel Steering System*

### *3.1. Structural Scheme Design of the Electrically Actuated Nose Wheel Steering System* The overall design of the structure of the electrically actuated nose wheel steering

The overall design of the structure of the electrically actuated nose wheel steering system uses a gear-driven steering mechanism, and the electromagnetic damper is reasonably installed to better achieve the shimmy reduction function. As shown in Figure 4, the electrically actuated nose wheel steering system includes the transmission mechanism, power output source and other components. The power of the steering mechanism is provided by a servo motor, while the gear transmission mode is selected for the transmission mechanism. Other components include an electromagnetic damper, angle sensors, load sensors, etc. system uses a gear-driven steering mechanism, and the electromagnetic damper is reasonably installed to better achieve the shimmy reduction function. As shown in Figure 4, the electrically actuated nose wheel steering system includes the transmission mechanism, power output source and other components. The power of the steering mechanism is provided by a servo motor, while the gear transmission mode is selected for the transmission mechanism. Other components include an electromagnetic damper, angle sensors, load sensors, etc.

**Figure 4.** The design of transmission mechanism. **Figure 4.** The design of transmission mechanism.

The schematic diagram of the gear reducer is shown in Figure 5 and the specific structural scheme design is shown in Figure 6. The nose landing gear shimmy reduction channel and the steering channel are connected in parallel in the same gear transmission The schematic diagram of the gear reducer is shown in Figure 5 and the specific structural scheme design is shown in Figure 6. The nose landing gear shimmy reduction channel and the steering channel are connected in parallel in the same gear transmission mechanism. The reducer is a fixed axis gear train during the steering operation of the nose landing gear. Motor shaft gear 1 engages with gear 2, gear 2 is coaxial with 3, gear 3 engages with 4, gear 4 and 5 are duplex gears, gear 6 engages with gear 5, gear 7 is coaxial

with 6 and engages with gear 8, gear 8 is coaxial with gear 9, and finally passes to the front landing gear strut gear 12. When the shimmy reduction maneuver is performed, the reducer is a compound gear train, meanwhile, gears 6–7 are planetary gears, and gears 4, 5, 8, 9 are center gears. The motor shaft is braked, the electromagnetic damper gear shaft is the input shaft, gear 13 engages with 12, gear 12 is coaxial with 11, and gear 11 engages with gear 10, driving gears 6–7 to rotate around the center gear and to be able to rotate at the same time, eventually passing to the nose landing gear strut gear 12. with 6 and engages with gear 8, gear 8 is coaxial with gear 9, and finally passes to the front landing gear strut gear 12. When the shimmy reduction maneuver is performed, the reducer is a compound gear train, meanwhile, gears 6–7 are planetary gears, and gears 4, 5, 8, 9 are center gears. The motor shaft is braked, the electromagnetic damper gear shaft is the input shaft, gear 13 engages with 12, gear 12 is coaxial with 11, and gear 11 engages with gear 10, driving gears 6–7 to rotate around the center gear and to be able to rotate at the same time, eventually passing to the nose landing gear strut gear 12. ducer is a compound gear train, meanwhile, gears 6–7 are planetary gears, and gears 4, 5, 8, 9 are center gears. The motor shaft is braked, the electromagnetic damper gear shaft is the input shaft, gear 13 engages with 12, gear 12 is coaxial with 11, and gear 11 engages with gear 10, driving gears 6–7 to rotate around the center gear and to be able to rotate at the same time, eventually passing to the nose landing gear strut gear 12. 8, 9 are center gears. The motor shaft is braked, the electromagnetic damper gear shaft is the input shaft, gear 13 engages with 12, gear 12 is coaxial with 11, and gear 11 engages with gear 10, driving gears 6–7 to rotate around the center gear and to be able to rotate at the same time, eventually passing to the nose landing gear strut gear 12.

mechanism. The reducer is a fixed axis gear train during the steering operation of the nose landing gear. Motor shaft gear 1 engages with gear 2, gear 2 is coaxial with 3, gear 3 engages with 4, gear 4 and 5 are duplex gears, gear 6 engages with gear 5, gear 7 is coaxial

mechanism. The reducer is a fixed axis gear train during the steering operation of the nose landing gear. Motor shaft gear 1 engages with gear 2, gear 2 is coaxial with 3, gear 3 engages with 4, gear 4 and 5 are duplex gears, gear 6 engages with gear 5, gear 7 is coaxial with 6 and engages with gear 8, gear 8 is coaxial with gear 9, and finally passes to the front landing gear strut gear 12. When the shimmy reduction maneuver is performed, the re-

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*Aerospace* **2022**, *9*, x 6 of 22

*Aerospace* **2022**, *9*, x 6 of 22

mechanism. The reducer is a fixed axis gear train during the steering operation of the nose landing gear. Motor shaft gear 1 engages with gear 2, gear 2 is coaxial with 3, gear 3 engages with 4, gear 4 and 5 are duplex gears, gear 6 engages with gear 5, gear 7 is coaxial

ducer is a compound gear train, meanwhile, gears 6–7 are planetary gears, and gears 4, 5,

**Figure 5.** Schematic diagram of the gear reducer. **Figure 5.** Schematic diagram of the gear reducer. **Figure 5.** Schematic diagram of the gear reducer.

**Figure 6.** Specific structure of electric nose wheel steering system. . **Figure 6.** Specific structure of electric nose wheel steering system. **Figure 6.** Specific structure of electric nose wheel steering system.

The total transmission ratio of the shimmy reduction channel of the gear reducer is 9.07, and the transmission ratio from the gear reducer to the nose wheel strut is 6, so the total transmission ratio of the shimmy reduction channel is 54.4. A 3D model of the electrically actuated steering shimmy reduction mechanism mounted on the nose landing gear of a jet fighter is shown in Figure 7. **Figure 6.** Specific structure of electric nose wheel steering system. The total transmission ratio of the shimmy reduction channel of the gear reducer is 9.07, and the transmission ratio from the gear reducer to the nose wheel strut is 6, so the total transmission ratio of the shimmy reduction channel is 54.4. A 3D model of the elec-The total transmission ratio of the shimmy reduction channel of the gear reducer is 9.07, and the transmission ratio from the gear reducer to the nose wheel strut is 6, sothe total transmission ratio of the shimmy reduction channel is 54.4. A 3D model of the electrically actuated steering shimmy reduction mechanism mounted on the nose landing gear of a jet fighter is shown in Figure 7. The total transmission ratio of the shimmy reduction channel of the gear reducer is 9.07, and the transmission ratio from the gear reducer to the nose wheel strut is 6, so the total transmission ratio of the shimmy reduction channel is 54.4. A 3D model of the electrically actuated steering shimmy reduction mechanism mounted on the nose landing gear of a jet fighter is shown in Figure 7.

trically actuated steering shimmy reduction mechanism mounted on the nose landing

**Figure 7.** 3D model of the electro-mechanical nose wheel steering system. **Figure 7.** 3D model of the electro-mechanical nose wheel steering system.

**Figure 7.** 3D model of the electro-mechanical nose wheel steering system.

## *3.2. Mathematical Modeling of the Electromagnetic Damper*

## 3.2.1. Calculation of Eddy Current Skin Depth

When an alternating current, especially a high frequency current, flows through a conductor, it will tend to the conductor surface and flow in a thin layer near the surface, which is the skin effect in electromagnetic field theory [18]. As can be seen from Figure 3, the rotor disks generates eddy currents with alternating directions during rotating around the stator disk.

The analysis of the practical application of electromagnetism shows that the axial distribution of the eddy current density, the electric field strength and the induced magnetic field strength of the alternating eddy current of the electromagnetic damper on the rotor disk decays according to the exponential law *e* − √ *ωµγ*/2·*h* [19], where *ω* is the angular frequency of the alternating eddy current, *µ* is the magnetic permeability of the rotor disk material, *γ* is the electrical conductivity (inverse of the resistivity) of the rotor disk material, and *h* is the vertical depth from the surface of the rotor disk. In engineering, the depth at which the amplitude of the abovementioned alternating field quantity drops to the surface value of 1/e is generally defined as the penetration depth of skin effect:

$$\begin{aligned} \varepsilon^{-} \sqrt{\omega \mu \gamma / 2} \,\Delta &= \varepsilon^{-1} \\ \Delta &= \sqrt{2 / \omega \mu \gamma} \end{aligned} \tag{4}$$

As can be seen from the above equation, the higher the angular frequency of the alternating eddy current or the better the conductivity and permeability of the rotor disk material, the shallower the penetration depth of the skin effect.

In addition, let the rotor disk be arranged with *n* magnets (for an even number to ensure that adjacent poles are reversed), then the fixed position on the rotor disk rotates for one circle and the eddy current direction changes *n* times, that is, it experiences *n*/2 complete alternating periods. Let the rotating angular velocity of the rotor disk be *ωn*, then the rotor disk rotation period and frequency are:

$$T\_{\text{Rotor}} = \frac{2\pi}{\omega\_n} \tag{5}$$

$$f\_{\text{Rotor}} = \frac{1}{T\_{\text{Rotor}}} = \frac{\omega\_n}{2\pi} \tag{6}$$

The eddy current alternating period, frequency and angular frequency at a fixed position on the rotor disk are:

$$T\_{\rm Ec} = \frac{T\_{\rm rotor}}{n/2} = \frac{4\pi}{n\omega\_n} \tag{7}$$

$$f\_{\rm Ec} = \frac{1}{T\_{\rm Ec}} = \frac{n\omega\_n}{4\pi} \tag{8}$$

$$
\omega = 2\pi f\_{\text{Ec}} = \mathfrak{n}\omega\_{\text{n}}/2 \tag{9}
$$

Therefore, the penetration depth of skin effect can be further expressed as:

$$
\Delta = \sqrt{2/\omega\mu\gamma} = 2/\sqrt{n\omega\_n\mu\gamma} \tag{10}
$$

## 3.2.2. Calculation of Damping Torque

Assuming that the magnetic flux density of a single magnet passing through a rotor disk is *B*, the rotor disk is regarded as consisting of countless small iron rods centered on the circle and having length *R*2*-R*1. When the rotor disk rotates, the small iron rod cuts the magnetic lines of force to excite the electromotive force, thus forming an eddy current on the surface of the rotor disk [20], as shown in Figure 8.

width *dl* is:

this micro-ring are:

pole can be calculated as follows:

**Figure 8.** Analysis of the rotor disk potential and damping torque. **Figure 8.** Analysis of the rotor disk potential and damping torque.

Let the rotor disk rotate with linear velocity *V* and angular velocity *ωn*. Then the induced electromotive force generated on the inner and outer sides of the micro-ring with Let the rotor disk rotate with linear velocity *V* and angular velocity *ωn*. Then the induced electromotive force generated on the inner and outer sides of the micro-ring with width *dl* is:

$$d\varepsilon = (V \times B)dl = \omega\_n Bldl\tag{11}$$

*<sup>d</sup><sup>ε</sup>* =× = ( ) *V B dl <sup>ω</sup>nBldl* (11) The resistance value of the inner and outer sides of the micro-ring with width *dl* and The resistance value of the inner and outer sides of the micro-ring with width *dl* and the current value flowing along the radial direction through the inner and outer sides of this micro-ring are:

$$d\mathbf{R} = \frac{\rho dl}{2\pi l \Delta} \tag{12}$$

$$I\_{dR} = \frac{d\varepsilon}{dR} = \frac{2\pi\omega\_n B l^2 \Delta}{\rho} \tag{13}$$

<sup>2</sup> <sup>2</sup> *<sup>n</sup>* <sup>Δ</sup> *dR <sup>d</sup>ε πω Bl <sup>I</sup> dR <sup>ρ</sup>* = = (13) On the premise of uniform magnetic field, combined with ampere force formula, the On the premise of uniform magnetic field, combined with ampere force formula, the damping force and damping torque of the micro-ring under the action of the magnetic pole can be calculated as follows:

$$dF = BI\_{dR}dl = \frac{2\pi\omega\_n\mathcal{B}^2l^2\Delta}{\rho}dl\tag{14}$$

$$dT = ldF = \frac{2\pi\omega\_n B^2 \Delta}{\rho} l^3 dl\tag{15}$$

<sup>2</sup> <sup>2</sup> <sup>Δ</sup> <sup>3</sup> *πωnB dT ldF l dl <sup>ρ</sup>* = = (15) For the integration of the above-mentioned micro-ring torque in the entire rotor disk (from *R*<sup>1</sup> to *R*2), the damping torque of a single rotor disk under the action of magnet can be preliminarily calculated, that is:

For the integration of the above-mentioned micro-ring torque in the entire rotor disk

$$T\_1 = \int\_{R\_1}^{R\_2} dT = \int\_{R\_1}^{R\_2} \frac{2\pi\omega\omega\_n B^2 \Delta}{\rho} l^3 dl = \frac{\pi \left(R\_2^{-4} - R\_1^{-4}\right) \omega\_n B^2 \Delta}{2\rho} \tag{16}$$

2 2 ( ) 1 1 44 2 <sup>2</sup> 2 1 <sup>3</sup> <sup>1</sup> <sup>2</sup> <sup>Δ</sup> <sup>Δ</sup><sup>2</sup> *R R <sup>n</sup> <sup>n</sup> R R πω B π R R ω B T dT l dl ρ ρ* <sup>−</sup> == = (16) It should be noted that in the derivation of the above equation, it is assumed that the entire rotor disk is located within the uniform magnetic field. However, the actual area of the magnetic poles acting on the rotor disk is approximately equal to the permanent magnet cross-sectional area (assuming that the air gap spacing is small enough). Therefore, the damping torque generated by a single rotor disk should be the above equation multiplied by the ratio of the real magnetic flux area to the rotor disk area. Setting It should be noted that in the derivation of the above equation, it is assumed that the entire rotor disk is located within the uniform magnetic field. However, the actual area of the magnetic poles acting on the rotor disk is approximately equal to the permanent magnet cross-sectional area (assuming that the air gap spacing is small enough). Therefore, the damping torque generated by a single rotor disk should be the above equation multiplied by the ratio of the real magnetic flux area to the rotor disk area. Setting the radius of the magnet cross section as *rMag*, the expression of the damping torque of the double rotor disks permanent magnet electromagnetic damper is:

$$T\_n = 2T\_1 \cdot \frac{n \pi r\_{\rm Mg} \,^2}{\pi \left(R\_2^2 - R\_1^2\right)} = \frac{n \pi r\_{\rm Mg} \,^2 \left(R\_2^2 + R\_1^2\right) \omega\_n \mathcal{B}^2 \Delta}{\rho} \tag{17}$$

( )

2 1

*π R R ρ*

Based on the derivation of the penetration depth ∆ of the skin effect in Equation (10), it can be further obtained that:

$$T\_n = \frac{n\pi r\_{\rm Mg} \,^2 \Big( \mathcal{R}\_2^{-2} + \mathcal{R}\_1^{-2} \Big) \omega\_n \mathcal{B}^2}{\rho} \cdot 2 \sqrt{\frac{\rho}{n\omega\_n \mu}} = 2\pi r\_{\rm Mg} \,^2 \Big( \mathcal{R}\_2^{-2} + \mathcal{R}\_1^{-2} \Big) \mathcal{B}^2 \sqrt{\frac{n\omega\_n}{\mu \rho}} \tag{18}$$

From the above derivation results, it can be seen that if we want to increase the damping torque, we need to increase the magnetic flux density through the rotor disk or choose rotor disk materials with lower permeability and lower resistivity and larger size permanent magnets, provided that the rotor disk shape size is determined. At the same time, the greater the angular speed of the rotor disk, the larger the output damping torque.In the structural scheme of the electrically actuated nose wheel steering system proposed in Section 3.1, the nose wheel strut is connected to the electromagnetic damper through a transmission device with transmission ratio *i* = 54.4. The transmission efficiency of commonly used 8-grade cylindrical spur gears is 0.97, i.e., *η*Gear = 0.97. The transmission efficiency of the bearings is 0.99, i.e., *η*Bearing = 0.99, so the total transmission efficiency is 0.81, i.e., *η* = *η*Gear 5 ·*η*Bearing <sup>5</sup> = 0.81. The angular velocity and damping torque of the nose wheel shimmy are related to the angular velocity and damping torque of the electromagnetic damper itself as:

$$
\omega\_n = \dot{\mathbf{u}} \omega\_{\text{Strut}} \tag{19}
$$

$$T\_{Strust} = i\eta T\_n \tag{20}$$

According to Equation (18), the damping torque of strut shimmy is:

$$T\_{\rm Strut} = \dot{\eta} \, T\_{\rm n} = 2 \dot{i}^2 \eta \, \pi r\_{\rm Mag} \, ^2 \left( R\_2 \, ^2 + R\_1 \, ^2 \right) B^2 \sqrt{\frac{m \omega\_{\rm Strut}}{\mu \rho}} \tag{21}$$

The relationship between the nose landing gear damping torque and the design parameters of the electromagnetic damper and the nose wheel shimmy speed is thus established.

## 3.2.3. Calculation of Damping Coefficient

Whether the nose wheel dampers can meet the damping requirements of aircraft taxiing on the ground is judged mainly by how many damping coefficients they can provide. The damping coefficient of dampers or struts is usually defined in engineering applications by the ratio of damping torque and angular velocity, and then the damping coefficients of the dampers and struts are:

$$\hbar\_{\rm ll} = \frac{T\_{\rm n}}{\omega\_{\rm n}} = 2\pi r\_{\rm Mag}^{-2} \left( R\_2^{-2} + R\_1^{-2} \right) B^2 \sqrt{\frac{n}{\mu \rho \omega\_{\rm n}}} \tag{22}$$

$$\hbar\_{\rm Strut} = \frac{T\_{\rm Strut}}{\omega\_{\rm Strut}} = \dot{\imath}^2 \eta \hbar\_n = 2\pi \dot{\imath}^2 \eta r\_{\rm Mag} \left(\mathcal{R}\_2^2 + \mathcal{R}\_1^2\right) \mathcal{B}^2 \sqrt{\frac{n}{\mu \rho \omega\_n}} \tag{23}$$

From Equations (22) and (23), it can be seen that the damping coefficient decreases if the rotor disk angular velocity increases. The next step is to design the specific scheme of the electromagnetic damper based on the demand of the damping coefficient of the shimmy reduction, the structural scheme of the electrically actuated nose wheel steering system and the derivation of the theoretical calculation.

## *3.3. Design of the Electromagnetic Damper*

Based on the operating principle of the electromagnetic damper, the materials selected for each component are shown in Table 2.


**Table 2.** Material selection of electromagnetic damper components.

The resistivity of steel 1010 is 5 <sup>×</sup> <sup>10</sup>−<sup>7</sup> <sup>Ω</sup>m and the permeability is 1.8 <sup>×</sup> <sup>10</sup>−<sup>3</sup> H/m. This scheme uses a permanent magnet with the magnetic flux density of 0.86 T. Assuming that the air gap between the magnetic pole of the permanent magnet and the rotor disk is small enough. In order to meet the target of the damping coefficient of not less than 40 Nms/rad, according to Equations (18) and (21), dimensions of electromagnetic damper components can be obtained which are shown in Table 3.

**Table 3.** Dimensions of electromagnetic damper components.


The damping torque and damping coefficient of the electromagnetic damper and the strut under the shimmy motion distribution for this set of design parameters are shown in Table 4.


**Table 4.** Distribution of damping torque and damping coefficient with different shimmy state.

From the data in Table 4, it can be seen that the theoretical calculated values of the damping coefficient of the electromagnetic damper amplified to the strut by the transmission mechanism are all satisfactory according to the design index. In the actual installation, there is an air gap between the rotor disk and the magnetic poles, which will cause the magnetic flux density of the poles to be attenuated on the rotor disks. In order to verify the correctness of the mathematical model and theoretical calculations, and to study in depth the shimmy reduction characteristics of the double rotor disks permanent magnet electromagnetic damper applied to the electrically actuated nose wheel steering system, finite element simulation in the electromagnetic field is required.

## **4. Electromagnetic Field Simulation of the Electromagnetic Damper**

## *4.1. Static Simulation Results and Analysis of the Electromagnetic Damper*

The finite element simulation software generally used for electromagnetic devices such as electromagnetic dampers is Maxwell. Maxwell is an interactive software package that uses finite element analysis (FEA) to solve 3D electrostatic, magnetostatic, eddy current, and transient problems. Using Maxwell we can compute:


The magnetostatic solver in Maxwell software is selected for the static magnetic field simulation of the electromagnetic damper. In a magnetostatic solution, the magnetic field is produced by DC currents flowing in conductors/coils and by permanent magnets. The electric field is restricted to the objects modeled as real (non-ideal) conductors. The electric field existing inside the conductors as a consequence of the DC current flow is totally decoupled from the magnetic field. Thus, as far as magnetic material properties are concerned, the distribution of the magnetic field is influenced by the spatial distribution of the permeability. There are no time variation effects included in a magnetostatic solution, and objects are considered to be stationary. The energy transformation occurring in connection with a magnetostatic solution is only due to the ohmic losses associated with the currents flowing in real conductors.

The magnetostatic field solution verifies the following two Maxwell's equations:

$$
\nabla \times \overrightarrow{H} = \overrightarrow{J} \tag{24}
$$

$$\begin{array}{c} \stackrel{\rightharpoonup}{B} = 0 \end{array} \tag{25}$$

with the following constitutive (material) relationship being also applicable:

$$
\overrightarrow{B} = \mu\_0 \left(\overrightarrow{H} + \overrightarrow{M}\right) = \mu\_0 \cdot \overrightarrow{H} + \mu\_0 \cdot \mu\_r \cdot \overrightarrow{M}\_p \tag{26}
$$

where \* *H*(*x*, *y*, *z*) is the magnetic field strength; \* *B*(*x*, *y*, *z*) is the magnetic flux density; \* *J* (*x*, *y*, *z*) is the conduction current density; \* *Mp*(*x*, *y*, *z*) is the permanent magnetization; *<sup>µ</sup>*<sup>0</sup> <sup>=</sup> <sup>4</sup> · *<sup>π</sup>* · <sup>10</sup>−<sup>7</sup> H/m is the permeability of vacuum; *<sup>µ</sup><sup>r</sup>* is the relative permeability.

The 3D numerical model of the electromagnetic damper is established according to the form dimensions in Table 3. To simplify the calculation, the central cross section of the stator disk is selected as the even symmetry surface, as shown in Figure 9, and the distribution of magnetic flux density on the rotor disk when it is stationary is analyzed. *Aerospace* **2022**, *9*, x 12 of 22

**Figure 9.** Finite element simulation model of the electromagnetic damper: (**a**) Simplified model; (**b**) Central cross section of the stator disk. **Figure 9.** Finite element simulation model of the electromagnetic damper: (**a**) Simplified model; (**b**) Central cross section of the stator disk.

**Figure 10.** Magnetic flux density distribution of permanent magnet poles(static).

netic poles close to the real conditions is shown in Figure 11.

The distribution of magnetic flux density on the rotor disks under the action of mag-

Firstly, the parameters of relative permeability, conductivity and coercivity of the

of the adjacent magnetic poles are opposite, and the magnetic flux density provided by each pole is calculated between 0.852–0.865 T by the fields calculator, which is basically consistent with the properties of the permanent magnet actually selected for this scheme.

Firstly, the parameters of relative permeability, conductivity and coercivity of the permanent magnet poles are corrected, and the simulation results of the magnetic flux density of magnetic poles are shown in Figure 10. The magnetic field strength directions of the adjacent magnetic poles are opposite, and the magnetic flux density provided by each pole is calculated between 0.852–0.865 T by the fields calculator, which is basically consistent with the properties of the permanent magnet actually selected for this scheme. Firstly, the parameters of relative permeability, conductivity and coercivity of the permanent magnet poles are corrected, and the simulation results of the magnetic flux density of magnetic poles are shown in Figure 10. The magnetic field strength directions of the adjacent magnetic poles are opposite, and the magnetic flux density provided by each pole is calculated between 0.852–0.865 T by the fields calculator, which is basically consistent with the properties of the permanent magnet actually selected for this scheme.

**Figure 9.** Finite element simulation model of the electromagnetic damper: (**a**) Simplified model; (**b**)

*Aerospace* **2022**, *9*, x 12 of 22

 (**a**) (**b**)

Central cross section of the stator disk.

**Figure 10.** Magnetic flux density distribution of permanent magnet poles(static). **Figure 10.** Magnetic flux density distribution of permanent magnet poles(static). *Aerospace* **2022**, *9*, x 13 of 22

The distribution of magnetic flux density on the rotor disks under the action of magnetic poles close to the real conditions is shown in Figure 11. The distribution of magnetic flux density on the rotor disks under the action of magnetic poles close to the real conditions is shown in Figure 11.

It can be seen from the static magnetic flux density distribution on the rotor disks: It can be seen from the static magnetic flux density distribution on the rotor disks:


The dynamic simulation of the electromagnetic damper is also completed in Maxwell

The formulation used by the Maxwell transient module supports Master-Slave boundary conditions and motion induced eddy currents everywhere in the model, in the stationary as well as in the moving parts of the model. Mechanical equations attached to the rigid-body moving parts allows a complex formulation with the electric circuits being strongly coupled with the finite element part and also coupled with the mechanical elements whenever transient mechanical effects are included by users in the solution. In this case the electromagnetic force/torque is calculated using the virtual work approach. For problems involving rotational type of motion a "sliding band" type of approach is fol-

The following two Maxwell's equations are relevant for transient (low frequency)

*<sup>B</sup> <sup>E</sup>*

<sup>∂</sup> ∇× =−

*t*

∇× = *H σ*( ) *E* (27)

<sup>∂</sup> (28)

mulation. Motion (translational or cylindrical/non-cylindrical rotation) is allowed, excitations-currents and/or voltages-can assume arbitrary shapes as functions of time, nonlinear BH material dependencies are also allowed. For a simpler formulation of problems where motion is involved, Maxwell uses a particular convention and uses the fixed coordinate system for the Maxwell's equations in the moving and the stationary part of the model. Thus the motion term is completely eliminated for the translational type of motion while for the rotational type of motion a simpler formulation is obtained by using a cylindrical

*4.2. Dynamic Simulation Results and Analysis of the Electromagnetic Damper* 

coordinate system with the z axis aligned with the actual rotation axis.

lowed and thus no re-meshing is done during the simulation.

applications:

reach 1.26T, and gradually decays in all directions.

## *4.2. Dynamic Simulation Results and Analysis of the Electromagnetic Damper*

The dynamic simulation of the electromagnetic damper is also completed in Maxwell finite element software, and the difference with the static simulation is that the solver is changed to transient. In the 3D transient (time domain), the solver uses the \* *T* − Ω formulation. Motion (translational or cylindrical/non-cylindrical rotation) is allowed, excitations-currents and/or voltages-can assume arbitrary shapes as functions of time, nonlinear BH material dependencies are also allowed. For a simpler formulation of problems where motion is involved, Maxwell uses a particular convention and uses the fixed coordinate system for the Maxwell's equations in the moving and the stationary part of the model. Thus the motion term is completely eliminated for the translational type of motion while for the rotational type of motion a simpler formulation is obtained by using a cylindrical coordinate system with the z axis aligned with the actual rotation axis.

The formulation used by the Maxwell transient module supports Master-Slave boundary conditions and motion induced eddy currents everywhere in the model, in the stationary as well as in the moving parts of the model. Mechanical equations attached to the rigidbody moving parts allows a complex formulation with the electric circuits being strongly coupled with the finite element part and also coupled with the mechanical elements whenever transient mechanical effects are included by users in the solution. In this case the electromagnetic force/torque is calculated using the virtual work approach. For problems involving rotational type of motion a "sliding band" type of approach is followed and thus no re-meshing is done during the simulation.

The following two Maxwell's equations are relevant for transient (low frequency) applications:

$$
\nabla \times H = \sigma(E) \tag{27}
$$

$$
\nabla \times E = -\frac{\partial B}{\partial t} \tag{28}
$$

The following equation directly results from the above two equations: The following equation directly results from the above two equations:

$$
\nabla \times \frac{1}{\sigma} \nabla \times H + \frac{\partial B}{\partial t} = 0 \tag{29}
$$

The final result is a formulation where vector fields are represented by first order edge elements and scalar fields are represented by second order nodal unknowns. The final result is a formulation where vector fields are represented by first order edge elements and scalar fields are represented by second order nodal unknowns.

The rotor disks' output damping torque is taken as the target parameter for the study. Because there are many operating conditions involved in the distribution of the shimmy motion, the set of motion parameters *ω<sup>n</sup>* = 1034.14 rad/s in Table 4 is used as an example to analyze the dynamic magnetic field simulation results of the electromagnetic damper. The output damping torque of the double rotor disks under this operating condition is shown in Figure 12. The rotor disks' output damping torque is taken as the target parameter for the study. Because there are many operating conditions involved in the distribution of the shimmy motion, the set of motion parameters *ωn* = 1034.14 rad/s in Table 4 is used as an example to analyze the dynamic magnetic field simulation results of the electromagnetic damper. The output damping torque of the double rotor disks under this operating condition is shown in Figure 12.

*Aerospace* **2022**, *9*, x 14 of 22

**Figure 12.** Output damping torque of the electromagnetic damper. **Figure 12.** Output damping torque of the electromagnetic damper.

As can be seen from Figure 12, when the rotor disks start to move suddenly in the magnetic field generated by the poles of the permanent magnet, vibration is generated As can be seen from Figure 12, when the rotor disks start to move suddenly in the magnetic field generated by the poles of the permanent magnet, vibration is generated and

and the value of the output damping torque temporarily fluctuates. After 6 ms, the motion of the rotor disk stabilizes and the output damping torque remains constant around 18.8

this condition with small deviation. It also shows that in engineering practice, electromagnetic dampers can quickly respond to and suppress the instantaneous shimmy of nose wheels, which demonstrates its superiority compared with traditional oleo dampers.

The following is an in-depth analysis of the simulation results of the rotor disks magnetic field strength distribution, magnetic flux density and current density during stable

As shown in Figure 13, the magnetic field strength on the surface of the rotor disks during the stable operation of the electromagnetic damper is in the range of 8517–127690 A/m. The magnetic field strength distribution is more concentrated in the area close to the permanent magnets, and the magnetic poles of the two adjacent concentrated areas are

operation.

opposite.

the value of the output damping torque temporarily fluctuates. After 6 ms, the motion of the rotor disk stabilizes and the output damping torque remains constant around 18.8 Nm, which is close to the theoretical calculated value of the mathematical model under this condition with small deviation. It also shows that in engineering practice, electromagnetic dampers can quickly respond to and suppress the instantaneous shimmy of nose wheels, which demonstrates its superiority compared with traditional oleo dampers.

The following is an in-depth analysis of the simulation results of the rotor disks magnetic field strength distribution, magnetic flux density and current density during stable operation.

As shown in Figure 13, the magnetic field strength on the surface of the rotor disks during the stable operation of the electromagnetic damper is in the range of 8517–127690 A/m. The magnetic field strength distribution is more concentrated in the area close to the permanent magnets, and the magnetic poles of the two adjacent concentrated areas are opposite. *Aerospace* **2022**, *9*, x 15 of 22

**Figure 13.** Transient magnetic field strength distribution of the rotor disks: (**a**) Vector distribution; (**b**) Field distribution. **Figure 13.** Transient magnetic field strength distribution of the rotor disks: (**a**) Vector distribution; (**b**) Field distribution.

As shown in Figure 14, the distribution of magnetic flux density on the rotor disks is also concentrated in the area close to the permanent magnets. When the electromagnetic damper is operating, the induced magnetic field on the rotor disks will be generated, and after superimposing the magnetic field of the permanent magnet poles through the rotor disks, the magnetic flux density is about 0.86–2.15 T, which is significantly larger than the As shown in Figure 14, the distribution of magnetic flux density on the rotor disks is also concentrated in the area close to the permanent magnets. When the electromagnetic damper is operating, the induced magnetic field on the rotor disks will be generated, and after superimposing the magnetic field of the permanent magnet poles through the rotor disks, the magnetic flux density is about 0.86–2.15 T, which is significantly larger than the static magnetic flux density of the permanent magnet pole in Figure 10.

static magnetic flux density of the permanent magnet pole in Figure 10. As shown in Figure 15, the induced eddy current density of the rotor disks is in the range of 1.25 <sup>×</sup> <sup>10</sup>7–4.69 <sup>×</sup> <sup>10</sup><sup>7</sup> A/m<sup>2</sup> . Observing the vector distribution, it can be found that the adjacent eddy currents on the surface of the rotor disks are in opposite directions, which proves the correctness of the electromagnetic damper using the induced eddy currents to convert kinetic energy into thermal energy. Meanwhile, observing the field distribution, it can be found that the induced eddy currents on the rotor disks are mainly concentrated on the surface near the pole side, which also proves the existence of skin effect and is consistent with the mathematical model.

The above simulation results are based on the set of motion parameters *ω<sup>n</sup>* = 1034.14 rad/s. In order to study the shimmy reduction performance of the electromagnetic damper in the electrically actuated nose wheel steering system, all shimmy motion states listed in Table 4 need to be considered, and the output damping torque of the electromagnetic damper corresponding to each operating condition is shown in Figure 16.

**Figure 14.** Transient magnetic flux density distribution of the rotor disks: (**a**) Vector distribution; (**b**)

As shown in Figure 15, the induced eddy current density of the rotor disks is in the range of 1.25 × 107–4.69 × 107 A/m2. Observing the vector distribution, it can be found that the adjacent eddy currents on the surface of the rotor disks are in opposite directions, which proves the correctness of the electromagnetic damper using the induced eddy currents to convert kinetic energy into thermal energy. Meanwhile, observing the field distribution, it can be found that the induced eddy currents on the rotor disks are mainly concentrated on the surface near the pole side, which also proves the existence of skin effect

(**a**) (**b**)

and is consistent with the mathematical model.

Field distribution.

static magnetic flux density of the permanent magnet pole in Figure 10.

 (**a**) (**b**)

(**b**) Field distribution.

**Figure 14.** Transient magnetic flux density distribution of the rotor disks: (**a**) Vector distribution; (**b**) Field distribution. **Figure 14.** Transient magnetic flux density distribution of the rotor disks: (**a**) Vector distribution; (**b**) Field distribution.

**Figure 13.** Transient magnetic field strength distribution of the rotor disks: (**a**) Vector distribution;

As shown in Figure 14, the distribution of magnetic flux density on the rotor disks is also concentrated in the area close to the permanent magnets. When the electromagnetic damper is operating, the induced magnetic field on the rotor disks will be generated, and after superimposing the magnetic field of the permanent magnet poles through the rotor disks, the magnetic flux density is about 0.86–2.15 T, which is significantly larger than the

**Figure 16.** Distribution of damping torque with angular velocity of the rotor disks. **Figure 16.** Distribution of damping torque with angular velocity of the rotor disks.

By comparing the simulation curve and the theoretical calculation curve of the output damping torques of the electromagnetic damper under various operating conditions, it can be found that the simulation value and the theoretical value are closer, which shows

2) The application of electromagnetic dampers to the shimmy reduction function of electrically actuated nose wheel steering systems is also feasible, but there are some differences in the values at some data points and the slopes of the two curves are not

**Figure 16.** Distribution of damping torque with angular velocity of the rotor disks.

1) The mathematical model of the electromagnetic damper is reasonable;

By comparing the simulation curve and the theoretical calculation curve of the output damping torques of the electromagnetic damper under various operating conditions, it can be found that the simulation value and the theoretical value are closer, which shows

• When calculating the output damping torques of the electromagnetic damper using Equation (18), the magnetic flux density of the permanent magnet source is assumed to be the magnetic flux density on the surface of the rotor disks, and the magnetic losses caused by the air gap and resistance and the induced magnetic field generated by the rotor disks themselves during rotation are ignored; • The magnetic permeability of the permanent magnet material is simply taken as a constant value in the numerical calculation. However, from the hysteresis curve of the permanent magnet material in the finite element simulation, the

2) The application of electromagnetic dampers to the shimmy reduction function of electrically actuated nose wheel steering systems is also feasible, but there are some differences in the values at some data points and the slopes of the two curves are not

• When calculating the output damping torques of the electromagnetic damper

using Equation (18), the magnetic flux density of the permanent magnet source is assumed to be the magnetic flux density on the surface of the rotor disks, and the magnetic losses caused by the air gap and resistance and the induced magnetic field generated by the rotor disks themselves during rotation are ignored; • The magnetic permeability of the permanent magnet material is simply taken

as a constant value in the numerical calculation. However, from the hysteresis curve of the permanent magnet material in the finite element simulation, the

that:

that:

identical, because:

identical, because:

By comparing the simulation curve and the theoretical calculation curve of the output damping torques of the electromagnetic damper under various operating conditions, it can be found that the simulation value and the theoretical value are closer, which shows that:

	- The magnetic permeability of the permanent magnet material is simply taken as a constant value in the numerical calculation. However, from the hysteresis curve of the permanent magnet material in the finite element simulation, the magnetic field strength is not linearly related to the magnetic flux density, and the relative permeability is also not linear; magnetic field strength is not linearly related to the magnetic flux density, and the relative permeability is also not linear; • The demagnetization effect of the permanent magnet is not taken into account
		- The demagnetization effect of the permanent magnet is not taken into account in the whole mathematical modeling process of the electromagnetic damper, while the experimental results in the literature [21] show that the demagnetization effect is actually real. The magnetic flux density distribution of the permanent magnet magnetic pole at the relative angular velocity of 1034.14 rad/s is shown in Figure 17, and comparing the static magnetic flux density distribution of the pole in Figure 10, it can be found that the finite element simulation takes this demagnetization effect into account; in the whole mathematical modeling process of the electromagnetic damper, while the experimental results in the literature [21] show that the demagnetization effect is actually real. The magnetic flux density distribution of the permanent magnet magnetic pole at the relative angular velocity of 1034.14rad/s is shown in Figure 17, and comparing the static magnetic flux density distribution of the pole in Figure 10, it can be found that the finite element simulation takes this demagnetization effect into account;

**Figure 17.** Magnetic flux density distribution of permanent magnet poles (at 1034.14 rad/s). **Figure 17.** Magnetic flux density distribution of permanent magnet poles (at 1034.14 rad/s).

By observing Figure 16, it can be found that when the rotor disk rotates at a low speed, the theoretical value of its output damping torque is greater than the simulation value, because in the process of theoretical derivation, the mechanical power of the rotor disk is considered to be fully converted into electromagnetic power, while the electromagnetic loss is ignored. In addition, the offsetting effect of opposite poles when multiple permanent magnets work simultaneously is not taken into account in the theoretical derivation. When the rotor disk rotates at high speed, the simulation value of the output damping torque is closer to the theoretical value because the induced magnetic field superim-By observing Figure 16, it can be found that when the rotor disk rotates at a low speed, the theoretical value of its output damping torque is greater than the simulation value, because in the process of theoretical derivation, the mechanical power of the rotor disk is considered to be fully converted into electromagnetic power, while the electromagnetic loss is ignored. In addition, the offsetting effect of opposite poles when multiple permanent magnets work simultaneously is not taken into account in the theoretical derivation. When the rotor disk rotates at high speed, the simulation value of the output damping torque is closer to the theoretical value because the induced magnetic field superimposed on the surface of the rotor disk is large enough, which can not only ignore the demagnetization

posed on the surface of the rotor disk is large enough, which can not only ignore the demagnetization effect of the magnetic poles at high speed, but also make up for the power

damper amplified by the transmission mechanism and its own angular velocity, as shown

**Figure 18.** Distribution of damping coefficient with angular velocity of rotor disk.

in Figure 18.

effect of the magnetic poles at high speed, but also make up for the power loss in the theoretical calculation. loss in the theoretical calculation. We can draw the curves between the damping coefficient of the electromagnetic

**Figure 17.** Magnetic flux density distribution of permanent magnet poles (at 1034.14 rad/s).

By observing Figure 16, it can be found that when the rotor disk rotates at a low speed, the theoretical value of its output damping torque is greater than the simulation value, because in the process of theoretical derivation, the mechanical power of the rotor disk is considered to be fully converted into electromagnetic power, while the electromagnetic loss is ignored. In addition, the offsetting effect of opposite poles when multiple permanent magnets work simultaneously is not taken into account in the theoretical derivation. When the rotor disk rotates at high speed, the simulation value of the output damping torque is closer to the theoretical value because the induced magnetic field superimposed on the surface of the rotor disk is large enough, which can not only ignore the demagnetization effect of the magnetic poles at high speed, but also make up for the power

We can draw the curves between the damping coefficient of the electromagnetic damper amplified by the transmission mechanism and its own angular velocity, as shown in Figure 18. damper amplified by the transmission mechanism and its own angular velocity, as shown in Figure 18.

*Aerospace* **2022**, *9*, x 17 of 22

the relative permeability is also not linear;

this demagnetization effect into account;

magnetic field strength is not linearly related to the magnetic flux density, and

• The demagnetization effect of the permanent magnet is not taken into account in the whole mathematical modeling process of the electromagnetic damper, while the experimental results in the literature [21] show that the demagnetization effect is actually real. The magnetic flux density distribution of the permanent magnet magnetic pole at the relative angular velocity of 1034.14rad/s is shown in Figure 17, and comparing the static magnetic flux density distribution of the pole in Figure 10, it can be found that the finite element simulation takes

**Figure 18. Figure 18.** Distribution of damping coefficient with angular velocity of rotor disk. Distribution of damping coefficient with angular velocity of rotor disk.

It can be seen from Figure 18 that the damping coefficient of shimmy reduction provided by the electromagnetic damper varies with the shimmy frequency when the transmission mode is determined. Figures 16 and 18 show that the damping torque increases while the damping coefficient decreases as the angular velocity of the rotor disk increases, which is consistent with the derivation of Equations (18) and (23). Therefore, when evaluating the damping performance of electromagnetic dampers, we should pay more attention to the damping coefficient under high frequency shimmy.

## *4.3. Effect of Various Factors on the Performance of Electromagnetic Dampers*

According to the operating principle and magnetic field simulation of electromagnetic dampers, there are many factors that affect the performance of electromagnetic dampers to reduce shimmy, here the two key factors, the dimensions *R*<sup>2</sup> and *R*<sup>1</sup> of the rotor disks and air gap width *δ*, are studied in detail, the former has direct reference value for the structural design of electromagnetic dampers under the premise of known shimmy reduction index requirements, the latter is the most important way to adjust the output damping torque after the structural scheme of the electromagnetic damper is determined.

As discussed in Section 4.2, because there are more operating conditions involved in the shimmy motion distribution, and the shimmy reduction design index usually requires the minimum shimmy reduction damping coefficient of the damper, the set of motion parameters *ω<sup>n</sup>* = 1153.82 rad/s in Table 4 is used as an example to analyze the influence of the rotor disks' outer diameter *R*<sup>2</sup> and air gap width *δ* on the shimmy reduction performance of the electromagnetic damper, respectively.

## 4.3.1. Study of the Effect of Rotor Disks' Dimensions *R*<sup>2</sup> and *R*<sup>1</sup> on Electromagnetic Damping

The rotor disks with *R*<sup>2</sup> = 80 mm and *R*<sup>1</sup> = 20 mm are used as the starting point and scaled by 1.25, 1.5, 1.75, 2, 2.25 and 2.5, respectively. The results of the damping coefficients after being enlarged by the transmission mechanism are shown in Figure 19. The outer dimensions of the rotor disks enlarged by 1.75 are the rotor disks' outer diameter *R<sup>2</sup>* = 140 mm and the inner diameter *R*<sup>1</sup> = 35 mm selected for this scheme.

From Figure 18, it can be seen that by changing the dimensions of the electromagnetic damper, the damping coefficient amplified by the transmission mechanism ranges from 2.83 to 195.29 Nms/rad, and there is an approximate quadratic power relationship with the dimensions of the rotor disks of the electromagnetic damper, which once again verifies the correctness of the derivation of the damping coefficient in the mathematical model.

mance of the electromagnetic damper, respectively.

Damping

**Figure 19.** Distribution of damping coefficient with the outer diameter of rotor disk. **Figure 19.** Distribution of damping coefficient with the outer diameter of rotor disk.

= 140 mm and the inner diameter *R1* = 35 mm selected for this scheme.

4.3.2. Study of the Effect of Air Gap Width δ on Electromagnetic Damping

From Figure 18, it can be seen that by changing the dimensions of the electromagnetic damper, the damping coefficient amplified by the transmission mechanism ranges from 2.83 to 195.29 Nms/rad, and there is an approximate quadratic power relationship with the dimensions of the rotor disks of the electromagnetic damper, which once again verifies the correctness of the derivation of the damping coefficient in the mathematical model. 4.3.2. Study of the Effect of Air Gap Width δ on Electromagnetic Damping The width of the air gap between the rotor disks and the permanent magnet poles is generally obtained within the range of 0.5–3 mm. Too small will lead to the thermal ex-The width of the air gap between the rotor disks and the permanent magnet poles is generally obtained within the range of 0.5–3 mm. Too small will lead to the thermal expansion and collision of the rotor disks and the permanent magnet poles during operating, while too large will lead to a sharp decline in the performance of the electromagnetic damper shimmy reduction. Since the air gap is composed of air, whose relative permeability is 1, and other parts such as the rotor disks and the stator disk are composed of high permeability materials, so the magnetic resistance of other structures can be neglected relative to the air gap. Taking the air gap width *δ* as 0.2, 0.3, 0.5, 1, 1.5, 2, 2.5, 3 and 3.5 mm, respectively, the corresponding output torque of the electromagnetic damper is obtained as shown in Figure 20. pansion and collision of the rotor disks and the permanent magnet poles during operating, while too large will lead to a sharp decline in the performance of the electromagnetic damper shimmy reduction. Since the air gap is composed of air, whose relative permeability is 1, and other parts such as the rotor disks and the stator disk are composed of high permeability materials, so the magnetic resistance of other structures can be neglected relative to the air gap. Taking the air gap width *δ* as 0.2, 0.3, 0.5, 1, 1.5, 2, 2.5, 3 and 3.5 mm, respectively, the corresponding output torque of the electromagnetic damper is obtained as shown in Figure 20.

It can be seen from Figure 18 that the damping coefficient of shimmy reduction provided by the electromagnetic damper varies with the shimmy frequency when the transmission mode is determined. Figures 16 and 18 show that the damping torque increases while the damping coefficient decreases as the angular velocity of the rotor disk increases, which is consistent with the derivation of Equations (18) and (23). Therefore, when evaluating the damping performance of electromagnetic dampers, we should pay more atten-

According to the operating principle and magnetic field simulation of electromagnetic dampers, there are many factors that affect the performance of electromagnetic dampers to reduce shimmy, here the two key factors, the dimensions *R2* and *R1* of the rotor disks and air gap width *δ*, are studied in detail, the former has direct reference value for the structural design of electromagnetic dampers under the premise of known shimmy reduction index requirements, the latter is the most important way to adjust the output damping torque after the structural scheme of the electromagnetic damper is determined. As discussed in Section 4.2, because there are more operating conditions involved in the shimmy motion distribution, and the shimmy reduction design index usually requires the minimum shimmy reduction damping coefficient of the damper, the set of motion parameters *ωn* = 1153.82 rad/s in Table 4 is used as an example to analyze the influence of the rotor disks' outer diameter *R2* and air gap width *δ* on the shimmy reduction perfor-

4.3.1. Study of the Effect of Rotor Disks' Dimensions R2 and R1 on Electromagnetic

The rotor disks with *R2* = 80 mm and *R1* = 20 mm are used as the starting point and

scaled by 1.25, 1.5, 1.75, 2, 2.25 and 2.5, respectively. The results of the damping coefficients after being enlarged by the transmission mechanism are shown in Figure 19. The outer dimensions of the rotor disks enlarged by 1.75 are the rotor disks' outer diameter *R2* 

tion to the damping coefficient under high frequency shimmy.

*4.3. Effect of Various Factors on the Performance of Electromagnetic Dampers* 

**Figure 20.** Distribution of damping torque with the air gap width of the rotor disks. **Figure 20.** Distribution of damping torque with the air gap width of the rotor disks.

For permanent magnet electromagnetic dampers, the magnetoresistance increases as the air gap width increases, while the flux is the ratio of the magneto motive force to the magnetoresistance. The flux decreases with increasing magnetoresistance and constant magneto motive force, producing a flux density in the magnetic saturation region, which decreases more slowly, and then in the non-saturation region, which decreases rapidly. From Figure 20, it can be seen that during the reduction of the air gap width from 3.5mm to 0.2mm, the damping coefficient of shimmy reduction gradually increases from 17.55 Nms/rad to 49.79 Nms/rad, while the change of the damping torque of shimmy reduction slows down during the continued reduction from 1.5 mm, indicating that the induced magnetic field on the rotor disks has started to enter the saturation state. Therefore, within a certain range, the magnetic flux density can be increased by reducing the air gap to increase the output damping torque of the electromagnetic damper, but the air gap should For permanent magnet electromagnetic dampers, the magnetoresistance increases as the air gap width increases, while the flux is the ratio of the magneto motive force to the magnetoresistance. The flux decreases with increasing magnetoresistance and constant magneto motive force, producing a flux density in the magnetic saturation region, which decreases more slowly, and then in the non-saturation region, which decreases rapidly. From Figure 20, it can be seen that during the reduction of the air gap width from 3.5 mm to 0.2 mm, the damping coefficient of shimmy reduction gradually increases from 17.55 Nms/rad to 49.79 Nms/rad, while the change of the damping torque of shimmy reduction slows down during the continued reduction from 1.5 mm, indicating that the induced magnetic field on the rotor disks has started to enter the saturation state. Therefore, within a certain range, the magnetic flux density can be increased by reducing the air gap to increase the output damping torque of the electromagnetic damper, but the air gap should not be too small considering the influence of mechanical processing precision and material thermal expansion and contraction.

not be too small considering the influence of mechanical processing precision and material

wheel steering system is studied, a structural scheme of the electrically actuated nose wheel steering system is proposed, and a new electromagnetic damping shimmy reduction device integrated in this system is designed. In order to verify the shimmy reduction performance of the proposed electromagnetic damper, the corresponding mathematical model of the electromagnetic damper is established, and then the specific scheme design of the electromagnetic damper is carried out and parametric modeling and electromagnetic field simulation are conducted according to the design index of shimmy reduction

1) After correcting the assumption that the rotor disk is located in the uniform magnetic field, which was commonly used in the previous derivation process, and combining the calculation of the skin depth, the derived equation for the output damping torque

2) The electromagnetic damper designed in this paper can provide a damping coeffi-

cient of not less than 40 Nms/rad under the conditions of shimmy amplitude between 2–20° and frequency between 5–30 Hz, which not only meets the requirements of the index, but also overcomes the disadvantages of relying on the hydraulic power

damping coefficient, and the following conclusions are obtained:

of the electromagnetic damper is closer to the simulated value;

thermal expansion and contraction.

**5. Conclusions** 

## **5. Conclusions**

In this paper, an electromagnetic damping method for an electrically actuated nose wheel steering system is studied, a structural scheme of the electrically actuated nose wheel steering system is proposed, and a new electromagnetic damping shimmy reduction device integrated in this system is designed. In order to verify the shimmy reduction performance of the proposed electromagnetic damper, the corresponding mathematical model of the electromagnetic damper is established, and then the specific scheme design of the electromagnetic damper is carried out and parametric modeling and electromagnetic field simulation are conducted according to the design index of shimmy reduction damping coefficient, and the following conclusions are obtained:


**Author Contributions:** Conceptualization: C.S. and M.Z.; Investigation: M.Z., Y.G., H.Y. and G.P.; Methodology: C.S., M.Z. and Y.G.; Project administration: M.Z.; Software: C.S. and L.T.; Supervision: M.Z. and Y.G.; Visualization: C.S., H.Y. and G.P.; Writing—original draft: C.S. and L.T.; Writing—review & editing, M.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Aeronautical Science Foundation of China under Grant No. 20182852021.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The author would like to thank the reviewers and the editors for their valuable comments and constructive suggestions that helped to improve the paper significantly.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Nomenclature**


## **References**


## *Article* **Optimization of the Wire Diameter Based on the Analytical Model of the Mean Magnetic Field for a Magnetically Driven Actuator**

**Zhangbin Wu , Hongbai Bai, Guangming Xue \* and Zhiying Ren**

School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China

**\*** Correspondence: yy0youxia@163.com; Tel.: +86-150-0514-1625

**Abstract:** A magnetic field induced by an electromagnetic coil is the key variable that determines the performance of a magnetically driven actuator. The applicability of the empirical models of the coil turns, static resistance, and inductance were discussed. Then, the model of the mean magnetic field induced by the coil was established analytically. Based on the proposed model, the sinusoidal response and square-wave response were calculated with the wire diameter as the decision variable. The amplitude and phase lag of the sinusoidal response, the time-domain response, steady-state value, and the response time of the square-wave response were discussed under different wire diameters. From the experimental and computational results, the model was verified as the relative errors were acceptably low in computing various responses and characteristic variables. Additionally, the optimization on the wire diameter was carried out for the optimal amplitude and response time. The proposed model will be helpful for the analytical analysis of the mean magnetic field, and the optimization result of the wire diameter under limited space can be employed to improve the performance of a magnetically driven actuator.

**Keywords:** mean magnetic field; wire diameter; coil; sinusoidal response; square-wave response

## **1. Introduction**

The magnetically driven actuator has been widely used in plenty of engineering fields, including vibration reduction or control, ultra-precision machining, acting fluidic valves, etc. [1–4]. Magnetically driven actuators have also been introduced to quite commonly actuate an aerospace device [5–15], including the electro-hydraulic servo valve.

Optimization of the actuator is quite important to improve the actuator's performance. A magnetic field was generally chosen as the optimization objective function as it influences the output performance of the actuator directly, and is the simplest variable to optimize the actuator, compared to the magnetization/magnetic induction intensity or the displacement. Taking the giant magnetostrictive actuator which employs the giant magnetostrictive material (GMM) as its actuation core as an example, Figure 1 summarizes the generally used optimization methods. From the point of view of magnetic fields, the optimization of actuator performance was generally converted to the promotion of the mean magnetic field in the GMM area, which is equivalent to the maximization of the magneto motive force (MMF) distributed on GMM. Additionally, two methods were used to promote the MMF on GMM, respectively, improving the MMF ratio occupied by GMM and increasing the total MMF.

The first optimization method was accomplished based on some magnetic field models from a "field" or "circuit" method [16–19]. Liang Yan et al. [20] and HyoYoung Kim et al. [21] proposed a mathematic model based on the Biot–Savart law and the finite element model to formulate the three-dimensional magnetic field distribution in a spherical actuator. Abdul Ghani Olabi et al. [22] also established the finite element model of a magnetostrictive actuator for analyzing the magnetic field in the actuator. The proposed

**Citation:** Wu, Z.; Bai, H.; Xue, G.; Ren, Z. Optimization of the Wire Diameter Based on the Analytical Model of the Mean Magnetic Field for a Magnetically Driven Actuator. *Aerospace* **2023**, *10*, 270. https:// doi.org/10.3390/aerospace10030270

Academic Editor: Gianpietro Di Rito

Received: 26 December 2022 Revised: 7 March 2023 Accepted: 8 March 2023 Published: 10 March 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

models supplied the mean values and distribution characters of magnetic devices, which were quite helpful for the magnetic circuit optimization. Due to the complex magnetic circuit of the hybrid excitation generator used in an energy conversion system, Huihui Geng et al. [23] proposed an analytical method of the main magnetic field, where the Carter coefficient and rotor magnetomotive force were taken as the objective variables. Compared with traditional methods, the proposed method can improve the accuracy of the outputted magnetic field. Jaewook Lee et al. [24] adopted a simplified finite element model to execute structural topology optimization for the high magnetic force of a linear actuator, and they found that the use of a periodic ladder structure was best for magnetic field manipulation. Kim Tien, Xulei Yang et al. [14,25] utilized the finite element model to analyze the distribution of the magnetic field in a giant magnetostrictive actuator separately. By adjusting the permeability of the parts appropriately, the uniformity and mean intensity of the magnetic field within the material could be improved. Some other modeling and optimizing methods for the magnetic field within specified structures can also supply effective references [17,26–29]. On the whole, the circuit model was always used to form a magnetic field model for an analytical analysis. The finite element model [19,30,31] was commonly used to promote magnetic field uniformity. For the mean magnetic field applied to the giant magnetostrictive material, the positively proportional model vs. the coil current [1–3,16,19,28,32–35] was quite commonly used. Then, the closed circuit was verified to be helpful for higher magnetic field intensity [15] as it improved the proportional factor. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 3 of 16 sively for an optimal selection of the wire diameter. During analysis, the relative errors in computing various variables were also given to verify the precision of the proposed model and effectiveness of the optimization. For the magnetically driven actuator, optimized results can be employed to promote the amplitude and response speed of the mean magnetic field, and then to improve the actuator performance.

**Figure 1.** Generally used optimization methods for the giant magnetostrictive actuator: Refs. [16– 31] based on the first method and Refs. [36–44] based on the second one. **Figure 1.** Generally used optimization methods for the giant magnetostrictive actuator: Refs. [16–31] based on the first method and Refs. [36–44] based on the second one.

field model in an infinitely long solenoid [3,5,14,16,19,33–35]

The dynamic magnetic field intensity or magnetic flux density was always measured "indirectly" based on Ampere's circuital theorem or Faraday's law of induction.

positive proportion relationship, which has been a commonly used analytical model of the mean magnetic field in some magnetically driven actuators, especially the giant magnetostrictive actuator. Based on this measuring principle, as long as the coil current is measured, the accurate mean of the magnetic field in a dynamic type can be obtained. The model was easily given by adding a proportional coefficient to the magnetic

**2. Experimental Methods** 

*2.1. Test Principle* 

For the second optimization method, an appropriate voltage waveform or the winging method was considered to directly promote the total MMF and the magnetic field induced by a coil [2]. A high threshold with a low-holding voltage has been widely used in an electromagnetic injector. C.B. Britht et al. [36] and G. Xue et al. [37,38] introduced this type of voltage to stimulate a giant magnetostrictive device. Additionally, it was comprehensively verified that the introduced voltage promoted the response time of the coil current, and the magnetic field, quite efficiently. Manh Cuong Hoang et al. [39] proposed an optimization method of the magnetic field for an electromagnetic actuation system. The maximum magnetic and gradient fields were significantly enhanced by the proposed algorithm compared to the conventional independent control. Haoying Pang et al. [40] proposed a novel spherical coil for the atomic sensor, where the magnetic field uniformity was improved along the axis. Yiwei Lu et al. [18] introduced the magnetically shielded room to enhance the coil magnetic field and reduce power loss for a multi-coil system. Cooperated with the non-dominated sorting genetic algorithm, the design reached prominent reductions in total current and power loss. Yundong Tang et al. [41] introduced two correcting coils to improve the uniformity of the magnetic field for a solenoid coil, while it was not so convenient when the coil space was limited as the correcting coils should have occupied some axial spaces. Some other optimization methods for the coil or contactor [42–44] can also provide useful references for optimizing the magnetic field induced by an electromagnetic coil.

Based on the second optimization method, this paper focuses on coil optimization when the volume of the magnetically driven actuator suitable for an electro-hydraulic servo valve is limited. In this paper, the dynamic magnetic field was tested based on the linear relationship between the magnetic field and coil current. Then, the dimension parameter, static resistance, and static induction were modeled based on empirical equations or mathematical fitting. The mean magnetic field within the coil was modeled, especially its functional relationship with respect to the wire diameter. Then, the sinusoidal and square-wave responses were calculated, and the important characteristic parameters of these responses were extracted. From the calculated and tested results, the influence of the wire diameter on the mean magnetic field was discussed comprehensively for an optimal selection of the wire diameter. During analysis, the relative errors in computing various variables were also given to verify the precision of the proposed model and effectiveness of the optimization. For the magnetically driven actuator, optimized results can be employed to promote the amplitude and response speed of the mean magnetic field, and then to improve the actuator performance.

## **2. Experimental Methods**

## *2.1. Test Principle*

The dynamic magnetic field intensity or magnetic flux density was always measured "indirectly" based on Ampere's circuital theorem or Faraday's law of induction. Based on the former principle, the induced magnetic field and coil current have an ideal positive proportion relationship, which has been a commonly used analytical model of the mean magnetic field in some magnetically driven actuators, especially the giant magnetostrictive actuator. Based on this measuring principle, as long as the coil current is measured, the accurate mean of the magnetic field in a dynamic type can be obtained.

The model was easily given by adding a proportional coefficient to the magnetic field model in an infinitely long solenoid [3,5,14,16,19,33–35]

$$H = \mathbb{C}\_{HI} \frac{NI}{L} \tag{1}$$

where *H* is the magnetic field intensity and *I* is the current intensity within the coil, *CHI* is the proportional coefficient of the mean magnetic field intensity; its value belongs to (0,1), *N* is the number of the coil turn, and *L* is the coil length.

The following optimization was based on Equation (1)—the optimization is effective as long as the mean magnetic field in the magnetically driven actuator is in direct proportion to the product of the number of coil turns and current intensity. For a hollow coil, Equation (1) was not only capable of computing the mean magnetic field within homogeneous medium, but was also suitable to the local mean magnetic field as long as the whole magnetic circuit was filled locally uniformly and did not have too many reluctance numbers. Considering Equation (1) is suitable for most giant magnetostrictive actuators and some micro-displacement electromagnetic actuators; the optimization proposed in this paper is suitable to these types of actuators.

## *2.2. Experiment Setup and Parameters*

The experimental system was shown in Figure 2. As illustrated in Figure 2a,c, the computer controlled PS3403D digital oscilloscope (with an embedded signal generator) to generate the required waveform signals. The generated signals were then amplified by an ATA304 power amplifier and inputted into the two ends of the coil. The input voltage at both ends of the coil was differentially collected and the coil current was measured by a TA189A current clamp. The measured voltage and current data were delivered into the digital oscilloscope and then into the computer for processing. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 5 of 16

**Figure 2.** Experimental system and wound coils: (**a**) block diagram of the experimental system; (**b**) dimensioned sectional drawing of the coil, (**c**) photograph of the experimental system; (**d**) photograph of the coils. **Figure 2.** Experimental system and wound coils: (**a**) block diagram of the experimental system; (**b**) dimensioned sectional drawing of the coil, (**c**) photograph of the experimental system; (**d**) photograph of the coils.

core. Then, a certain functional relationship can be supplied between the enameled wire diameter *Dwire* and the copper core diameter *Dcore*. Figure 3 shows the actual values of

**3. Data Processing and Analysis** 

*3.1. Inherent Characteristic Parameters of Coils* 

Figure 2b,d supplied the sectional drawing and photograph of the coils, where *La*, *L<sup>b</sup>* , *Lf* represented the coil length, coil thickness, and diameter of the skeleton shaft, respectively, and *Dwire* and *Dcore* were the enameled wire diameter and copper core diameter, respectively. The coils were tightly wound by the use of standard enameled wires. Since this article focuses on the optimization of the coil itself, it is not necessary to consider the influence of the iron core or other parts in an actuator. The parameters of the coils are given in Table 1, and some necessary parameters of the skeleton and material are supplied in Table 2. Considering the value of *CHI* does not affect the increasing or decreasing relationship between the variables; *CHI* will have no effect on the optimization results. *CHI* is specified as 0.8 here.



**Table 2.** The main parameters of the skeleton and material.


## **3. Data Processing and Analysis**

## *3.1. Inherent Characteristic Parameters of Coils*

## 3.1.1. Dimension Parameters

Standard enameled wire has a nominal diameter of the external wire or the copper core. Then, a certain functional relationship can be supplied between the enameled wire diameter *Dwire* and the copper core diameter *Dcore*. Figure 3 shows the actual values of *Dwire* and *Dcore* and the fitted results using linear functions. It can be seen from Figure 3 that the diameter of copper core is approximately linear vs. the external diameter of enameled wire. With and without an intercept, the fitted linear equations were determined as *Dcore* = 0.9687 *Dwire* − 0.03214 and *Dcore* = 0.9394 *Dwire*, respectively. The linear function with an intercept was quite accurate as the relative error was lower than 1.52% when *Dwire* was higher than 0.3 mm and lower than 2.55 mm. In contrast, the linear function without an intercept was not so accurate since the relative error was higher than 5% under some conditions, especially when *Dwire* was quite low.

Though the positively proportional relationship was not suitable to a wide range of dimensions, it may be feasible when the *Dwire* changed within a relatively narrow interval. The coils used in this paper were wound by the wires with diameters of 0.3~0.8 mm. Executing a simple linear fitting, Table 3 supplies the results and relative errors of the two line equations. From computation, the linear equation with intercept was *Dcore* = 0.962 *Dwire* − 0.0277 and had a relative error lower than 0.84%. In comparison, the linear equation without an intercept *Dcore* = 0.898 *Dwire* also had high precision as the relative error was lower than 3.2%. Thus, it is acceptable to use a positively proportional function to describe the relationship between *Dcore* and *Dwire* when *Dwire* changes within a narrow interval, which is quite convenient for the following optimization.

val, which is quite convenient for the following optimization.

**Figure 3.** Actual and fitted values of *Dcore* simultaneously supplied the relative errors of the fitting lines with and without an intercept. **Figure 3.** Actual and fitted values of *Dcore* simultaneously supplied the relative errors of the fitting lines with and without an intercept.

*Dwire* and *Dcore* and the fitted results using linear functions. It can be seen from Figure 3 that the diameter of copper core is approximately linear vs. the external diameter of enameled wire. With and without an intercept, the fitted linear equations were determined as *Dcore* = 0.9687 *Dwire* − 0.03214 and *Dcore* = 0.9394 *Dwire*, respectively. The linear function with an intercept was quite accurate as the relative error was lower than 1.52% when *Dwire* was higher than 0.3 mm and lower than 2.55 mm. In contrast, the linear function without an intercept was not so accurate since the relative error was

Though the positively proportional relationship was not suitable to a wide range of dimensions, it may be feasible when the *Dwire* changed within a relatively narrow interval. The coils used in this paper were wound by the wires with diameters of 0.3~0.8 mm. Executing a simple linear fitting, Table 3 supplies the results and relative errors of the two line equations. From computation, the linear equation with intercept was *Dcore* = 0.962 *Dwire* − 0.0277 and had a relative error lower than 0.84%. In comparison, the linear equation without an intercept *Dcore* = 0.898 *Dwire* also had high precision as the relative error was lower than 3.2%. Thus, it is acceptable to use a positively proportional function to describe the relationship between *Dcore* and *Dwire* when *Dwire* changes within a narrow inter-

higher than 5% under some conditions, especially when *Dwire* was quite low.

**Table 3.** Linear fitting between *Dcore* and *Dwire* when *Dwire*∈ [0.3, 0.8]. **Table 3.** Linear fitting between *Dcore* and *Dwire* when *Dwire* ∈ [0.3, 0.8].


0.69 0.64 0.6361 −0.6125 0.6196 −3.1844 0.8 0.74 0.7419 0.2568 0.7184 −2.9189 From the sectional drawing shown in Figure 2b, it can be observed that winding a coil was equivalent to arranging the cross-sectional area of the wire in the rectangular area supplied by the coil skeleton. The coil turns must be an integer; while *L<sup>a</sup>* or *L<sup>b</sup>* was From the sectional drawing shown in Figure 2b, it can be observed that winding a coil was equivalent to arranging the cross-sectional area of the wire in the rectangular area supplied by the coil skeleton. The coil turns must be an integer; while *L<sup>a</sup>* or *L<sup>b</sup>* was not exactly the integral multiple of *Dwire*, the effective length *La*' and thickness *L<sup>b</sup>* ' were a little lower than *L<sup>a</sup>* and *L<sup>b</sup>* , respectively. Coil length or thickness was not fully utilized, and the available area was *La*'×*L<sup>b</sup>* ', which was slightly less than the actual area.

From Figure 2b, the turn number per layer was b*La*/*Dwire*c and the number of layers was <sup>j</sup> √ *Lb*−*Dwire* <sup>3</sup>*Dwire*/2 <sup>k</sup> + 1, so that the accurate value of coil turns was

$$\begin{split} N &= \mathcal{C}'\_f \left\lfloor \frac{L\_a}{D\_{wire}} \right\rfloor \left( \left\lfloor \frac{L\_b - D\_{wire}}{\sqrt{3}D\_{wire}/2} \right\rfloor + 1 \right) \\ &\leq \mathcal{C}'\_f \left( \frac{L\_a L\_b}{\sqrt{3}D\_{wire}^2/2} - 0.155 \frac{L\_a}{D\_{wire}} \right) \\ &\approx \mathcal{C}\_f \frac{L\_a L\_b}{\pi \mathcal{D}\_{wire}^2/4} \end{split} \tag{2}$$

where *C<sup>f</sup>* <sup>0</sup> was introduced to describing the winding effect, *C<sup>f</sup>* was the filling factor of the enameled wire. *LaL<sup>b</sup>* was the axis-sectional area of the coil and π*Dwire* <sup>2</sup>/4 was the cross-sectional area of single enameled wire.

From Equation (2), the assumption that *N* was positively proportional to the ratio of the cross-sectional area of the coil skeleton to *Dwire* was conditional. That was, with the effectiveness of Equation (2), determined by the weight of *C<sup>f</sup>* '0.155 *La*/*Dwire* in the total coil turns. Additionally, the relative error of the positively proportional function was 0.155/(1.155 *Lb*/*Dwire* − 0.155) × 100%, which was determined by *Lb*/*Dwire*. should be wound with three layers at least. When *L<sup>b</sup>* < 2.8 *Dwire*, one should use *Lb*' =1 3 2 *b wire wire L D D* − + instead of *L<sup>b</sup>* for computations. For the coils in this paper, the values of

0.155/(1.155 *Lb*/*Dwire* − 0.155) × 100%, which was determined by *Lb*/*Dwire*.

in Figure 4. From the calculation results, the relative error of *CfLaLb*/(π*Dwire*<sup>2</sup>

*Aerospace* **2023**, *10*, x FOR PEER REVIEW 7 of 16

From Figure 2b, the turn number per layer was

*N*

1

cross-sectional area of single enameled wire.

+

 2  

 

 

 *wire*

*wire*

 *D*

was

3

*D*

−

*b*

*L*

not exactly the integral multiple of *Dwire*, the effective length *La*' and thickness *Lb*' were a little lower than *L<sup>a</sup>* and *Lb*, respectively. Coil length or thickness was not fully utilized,

, so that the accurate value of coil turns was

 

 

 

2

*wire*

*a b*

*D*

*L L*

4

where *Cf*' was introduced to describing the winding effect, *C<sup>f</sup>* was the filling factor of the

the cross-sectional area of the coil skeleton to *Dwire* was conditional. That was, with the effectiveness of Equation (2), determined by the weight of *Cf*'0.155 *La*/*Dwire* in the total coil turns. Additionally, the relative error of the positively proportional function was

*N* decreased with *Lb*/*Dwire* increasing. To guarantee that the relative error of Equation (2) is lower than 5.0% in computing *N*, it should be met that *L<sup>b</sup>* > 2.8 *Dwire*. That is, the coil

From Equation (2), the assumption that *N* was positively proportional to the ratio of

*Lb*/*Dwire* determined the number of layers and the relative error, which are displayed

 2

2

*wire*

*a b*

*L L*

*D*

π

enameled wire. *LaL<sup>b</sup>* was the axis-sectional area of the coil and π*Dwire*<sup>2</sup>

3

*a*

*L*

*D*

*wire*

*f*

 *C*

=

*f*

*C*

*f*

*C*

*a*

 2

 *wire*

0.155

*wire*

 *D* *L D*

3

 − *D*

−

 *b*

 *L*

 *wire*  

and the number of layers

(2)

/4 was the

/4) computing

 

1

 

 

 

 

 

 

 *a*

 +

 *L*

 

 

 

*D*

*wire*

and the available area was *La*'×*Lb*', which was slightly less than the actual area.

*Lb*/*Dwire* determined the number of layers and the relative error, which are displayed in Figure 4. From the calculation results, the relative error of *CfLaLb*/(π*Dwire* <sup>2</sup>/4) computing *N* decreased with *Lb*/*Dwire* increasing. To guarantee that the relative error of Equation (2) is lower than 5.0% in computing *N*, it should be met that *L<sup>b</sup>* > 2.8 *Dwire*. That is, the coil should be wound with three layers at least. When *L<sup>b</sup>* < 2.8 *Dwire*, one should use *Lb* ' = <sup>j</sup> √ *Lb*−*Dwire* <sup>3</sup>*Dwire*/2 <sup>k</sup> + 1 instead of *L<sup>b</sup>* for computations. For the coils in this paper, the values of *Lb*/*Dwire* under different *Dwire* were higher than 6.8/0.8 = 8.5 so that the approximate expression in Equation (2) has enough precision. *Lb*/*Dwire* under different *Dwire* were higher than 6.8/0.8 = 8.5 so that the approximate expression in Equation (2) has enough precision. For convenience, the value of *C<sup>f</sup>* was determined by the mean values of *N* and *Dwire* so that *C<sup>f</sup>* = 0.57. The effect of Equation (2) computing *N* is shown in Table 4. From the calculation results, the model of coil turn can predict the practical coil turn effectively as the relative error was lower than 2.8%.

**Figure 4.** Relative errors of the positively proportional function of the available area of coil skeleton vs. the cross-sectional area of enameled wire. **Figure 4.** Relative errors of the positively proportional function of the available area of coil skeleton vs. the cross-sectional area of enameled wire.

For convenience, the value of *C<sup>f</sup>* was determined by the mean values of *N* and *Dwire* so that *C<sup>f</sup>* = 0.57. The effect of Equation (2) computing *N* is shown in Table 4. From the calculation results, the model of coil turn can predict the practical coil turn effectively as the relative error was lower than 2.8%.


**Table 4.** Coil turns from the test and model.

<sup>1</sup> Cannot be an integer.

From above analysis, the empirical equations in describing the relationships between *Dcore*, *Dwire* and *N* were written as

$$\begin{cases} \begin{array}{l} \text{ $\dot{D}\_{core} = 0.962D\_{wire} - 0.0277$  or  $0.898D\_{wire}$ }\\ \text{ $N = C\_f \frac{L\_b L\_b}{\pi D\_{wire}^2/4}$ } \end{array} \tag{3} $$

#### 3.1.2. Static Resistance and Static Inductance 0 4 4π *L L CN* = =

Static resistance and inductance are the key parameters to determine the current response of a coil with an unobvious skin effect. Based on the empirical expression of inductance *L* and the basic equation of resistance *R*, the model can be easily established as 2 22 () () <sup>16</sup> 4 *wire f b ab f b f D NL L LL L L R C D DD* ρ ρ + + = = 

2

2

*D*

= −

*f*

=

*L L N C*

π 4

*wire*

*a b*

*Aerospace* **2023**, *10*, x FOR PEER REVIEW 8 of 16

**Coil Label Coil Turns from Test Coil Turns from Model 1 Relative Error (%)**  1 837 847.33 1.23 2 537 535.36 −0.30 3 342 339.15 −0.83 4 229 226.19 −1.23 5 175 171.03 −2.27 6 124 127.23 2.61

From above analysis, the empirical equations in describing the relationships be-

*core wire wire*

Static resistance and inductance are the key parameters to determine the current

2

*L f*

*C C*

response of a coil with an unobvious skin effect. Based on the empirical expression of inductance *L* and the basic equation of resistance *R*, the model can be easily established

(3)

(4)

ˆ 0.962 0.0277 or 0.898

*DD D*

**Table 4.** Coil turns from the test and model.

tween *Dcore*, *Dwire* and *N* were written as

 

3.1.2. Static Resistance and Static Inductance

1 Cannot be an integer.

as

$$\begin{cases} \text{ } L = 4\pi \mathbf{C}\_{Lb} \mathbf{N}^2 = \frac{\mathbf{C}\_L \mathbf{C}\_f^2}{D\_{\text{wire}}^4} \\\ R = \rho \frac{\mathbf{N}(L\_f + L\_b)}{D\_{\text{wire}}^2 / 4} = 16\rho \mathbf{C}\_f \frac{L\_b \mathbf{L}\_b (L\_f + L\_b)}{\pi D\_{\text{wire}}^2 D\_{\text{wire}}^2} \end{cases} \tag{4}$$

π

where *CL*<sup>0</sup> and *C<sup>L</sup>* were two parameters dependent on *La*, *L<sup>b</sup>* , *L<sup>f</sup>* while independent of other variables and met *C<sup>L</sup>* = 64 *CL*0(*LaL<sup>b</sup>* ) <sup>2</sup>/π; *ρ* was the resistivity of copper. computation. From the results, it was easily reached that both *L* and *R* were monotonically decreasing functions vs. *Dwire*. More specifically, as concluded from the expression

Figure 5 displays the relationships between *L*, *R*, and *Dwire* from the experiment and computation. From the results, it was easily reached that both *L* and *R* were monotonically decreasing functions vs. *Dwire*. More specifically, as concluded from the expression of *N* in Equation (3) and *Dcore* = 0.898 *Dwire*, both *R* and *L* were inversely proportional functions vs. *Dwire* 4 (also *N*<sup>2</sup> ). The model was in good agreement with the experiment as the relative errors of the model in computing *R* and *L* were lower than 3.1% and 2.8%, respectively. of *N* in Equation (3) and *Dcore* = 0.898 *Dwire*, both *R* and *L* were inversely proportional functions vs. *Dwire*4 (also *N*2). The model was in good agreement with the experiment as the relative errors of the model in computing *R* and *L* were lower than 3.1% and 2.8%, respectively.

**Figure 5.** Curves of the static resistance and static inductance vs. the enameled wire diameter. **Figure 5.** Curves of the static resistance and static inductance vs. the enameled wire diameter.

## *3.2. Sinusoidal Response*

Equivalent to the series connection of an inductor and a resistor, the electromagnetic coil was generally modeled as a first-order linear time-invariant system model. Additionally, the amplitude-frequency and phase-frequency characteristics are the most important characteristics of the sinusoidal response of the coil.

Stimulated by a sinusoidal voltage *U*(*t*) = *Uamp*sin(*ωt*), the current response within the coil can be calculated by *I*(*t*) = *Iamp*sin(*ωt* – *ϕ<sup>I</sup>* ), where *ω* is the angular frequency of the input and *ϕ<sup>I</sup>* is the phase lag of the coil current compared to the voltage. From the theory of the linear time-invariant system, the amplitude ratio function is *A<sup>I</sup>* = *Iamp*/*Uamp* = 1/(*R* <sup>2</sup> + *ω*2*L* 2 ) 1/2 , and tan*ϕ<sup>I</sup>* = *ωL*/*R*. Substituting Equation (4) into these expressions, one obtains

$$\begin{cases} \begin{aligned} \boldsymbol{A}\_{I} &= \frac{\pi D\_{\text{core}}^{2} D\_{\text{wire}}^{2}}{\mathsf{C}\_{f} L\_{\text{t}} \sqrt{\left[\begin{array}{c} 16 \rho L\_{b} \cdot \\ (D\_{f} + L\_{b}) \end{array}\right]^{2} + \left(\frac{\omega \mathsf{C}\_{L} \mathsf{C}\_{f} D\_{\text{wire}}^{2}}{D\_{\text{wire}}^{2}}\right)^{2}} \approx \frac{0.806 \pi D\_{\text{wire}}^{4}}{\mathsf{C}\_{f} L\_{\text{t}} \sqrt{\left[\begin{array}{c} 16 \rho L\_{b} \cdot \\ (D\_{f} + L\_{b}) \end{array}\right]^{2} + (0.806 \omega \mathsf{C}\_{L} \mathsf{C}\_{f})^{2}}} \end{aligned} \end{cases}$$
 
$$\begin{aligned} \begin{aligned} \begin{aligned} \begin{aligned} \varphi\_{I} &= \frac{\omega \mathsf{C}\_{L} \mathsf{C}\_{f}}{\rho L\_{\text{wire}} \cdot \mathsf{C}\_{f}} \end{aligned} \end{aligned} \end{cases} \end{cases}$$
 
$$\begin{aligned} \begin{aligned} \varphi\_{I} &= \arctan \frac{0.0504 \omega \mathsf{C}\_{L} \mathsf{C}\_{f}}{D\_{\text{wire}}^{2}} \end{aligned} \end{cases}$$

From the empirical equation of the mean magnetic field given in Equation (1), the amplitude radio to inputted voltage of the magnetic field *A<sup>H</sup>* and the lagging phase of the magnetic field *ϕ<sup>H</sup>* can be easily reached as *CHINAI*/*L<sup>a</sup>* and *ϕ<sup>H</sup>* = *ϕ<sup>I</sup>* . By substituting Equation (5) into these two equations, one obtains

$$\begin{cases} \begin{aligned} A\_{H} &= \frac{4\mathcal{C}\_{HI}L\_{b}D\_{\text{core}}^{2}}{\omega\_{d}\sqrt{\left[\begin{array}{c} 16\rho L\_{b} \cdot \\ \left(D\_{f} + L\_{b}\right) \end{array}\right]^{2} + \left(\frac{\omega\_{L}\mathcal{C}\_{f}D\_{\text{core}}^{2}}{\mathcal{D}\_{\text{wire}}^{2}}\right)^{2}} \approx \frac{3.226\mathcal{C}\_{HI}L\_{b}D\_{\text{wire}}^{2}}{L\_{d}\sqrt{\left[\begin{array}{c} 16\rho L\_{b} \cdot \\ \left(D\_{f} + L\_{b}\right) \end{array}\right]^{2} + \left(0.806\omega\mathcal{C}\_{L}\mathcal{C}\_{f}\right)^{2}}} \end{aligned} \end{cases} $$
 
$$\begin{aligned} \rho\_{H} = \arctan\frac{\omega\mathcal{C}\_{L}\mathcal{C}\_{f}}{16\rho L\_{b}(D\_{f} + L\_{b})} \cdot \frac{D\_{\text{core}}^{2}}{D\_{\text{wire}}^{2}} \approx \arctan\frac{0.0504\omega\mathcal{C}\_{L}\mathcal{C}\_{f}}{\rho L\_{b}(D\_{f} + L\_{b})} \end{aligned} \tag{6}$$

Changing the frequency from 10 Hz to 1000 Hz, Figure 6 shows the tested and calculated amplitude ratios and phase lags of the magnetic field with respect to the inputted voltage. To demonstrate the influence of the wire diameter more clearly, the wire diameter was plotted on the horizontal axis. From the tested and calculated results, a wider wire diameter is quite helpful for a higher magnetic field amplitude as the amplitude ratio increased faster with an increase in wire diameter. On the contrary, the wire diameter has little influence on the phase lag of the magnetic field, which represents the response time of the magnetic field from 0 to some required proportion of a steady-state value. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 10 of 16 diameter was verified as the tested points under a certain frequency were roughly plotted in a line passing through the origin.

**Figure 6.** Amplitude ratio and phase lag of the magnetic field under different wire diameters: (**a**) curves of the amplitude ratio vs. enameled wire diameter; (**b**) curves of the phase lag vs. enameled wire diameter. **Figure 6.** Amplitude ratio and phase lag of the magnetic field under different wire diameters: (**a**) curves of the amplitude ratio vs. enameled wire diameter; (**b**) curves of the phase lag vs. enameled wire diameter.

10<sup>1</sup> 10<sup>2</sup> 10<sup>3</sup> − 8 − 6 − 4 − 2 0 2 4 0.31mm 0.39mm 0.49mm 0.60mm 0.69mm 0.80mm Figure 7 shows the relative errors of the model under various frequencies. For predicting the amplitude ratio, the calculation error was lower than 2.0% when the wire diameter was between 0.39 mm and 0.69 mm. The model accuracy was a little lower when the wire diameter was wider than 0.8 mm or narrower than 0.31 mm, as the relative errors at these points were higher than 5%; this was acceptable as the errors were still lower than 6.4%. For computing the lagging phase, the relative errors under different parameters, including various frequencies and wire diameters, were lower than 3.2%, which showed high precision of the model in predicting the lagging phase of the magnetic field. A low calculation accuracy regarding the computing amplitude ratio was mainly caused by poor winding when the coil wire was quite thin or thick. On the whole, the proposed models for the magnetic field amplitude and lagging phase were verified by the low relative errors under most conditions.

 (**a**) (**b**) **Figure 7.** The relative errors of the model in computing the coil current under a harmonic voltage: (**a**) the relative errors of computing the amplitude ratio; (**b**) the relative errors of computing the

**Figure 8.** Linear curves of *AH*-*Dwire*2 from the model and test under different frequencies.

Frequency [Hz]

phase lag.

Amplitude ratio of magnetic

field [(A/m)/V]

(**a**) (**b**) **Figure 6.** Amplitude ratio and phase lag of the magnetic field under different wire diameters: (**a**) curves of the amplitude ratio vs. enameled wire diameter; (**b**) curves of the phase lag vs. enameled

0

0.5

1

Test 10Hz Model 10Hz Test 100Hz Model 100Hz

1.5

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Enameled wire diameter[mm]

Test 500Hz Model 500Hz Test 1000Hz Model 1000Hz

(**a**) (**b**)

diameter was verified as the tested points under a certain frequency were roughly plot-

diameter was verified as the tested points under a certain frequency were roughly plot-

ted in a line passing through the origin.

ted in a line passing through the origin.

Test 10Hz Model 10Hz Test 100Hz Model 100Hz Test 500Hz Model 500Hz Test 1000Hz Model 1000Hz

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Enameled wire diameter[mm]

*Aerospace* **2023**, *10*, x FOR PEER REVIEW 10 of 16

wire diameter.

0

6

12

18

**Figure 7.** The relative errors of the model in computing the coil current under a harmonic voltage: (**a**) the relative errors of computing the amplitude ratio; (**b**) the relative errors of computing the phase lag. **Figure 7.** The relative errors of the model in computing the coil current under a harmonic voltage: (**a**) the relative errors of computing the amplitude ratio; (**b**) the relative errors of computing the phase lag. 10<sup>1</sup> 10<sup>2</sup> 10<sup>3</sup> Frequency [Hz] − 8 0.60mm 0.69mm 0.80mm

Figure 8 shows the relationships between *A<sup>H</sup>* and *Dwire* 2 . From Figure 8, the linear relationship between the amplitude ratio of the magnetic field and the square of the wire diameter was verified as the tested points under a certain frequency were roughly plotted in a line passing through the origin. (**a**) (**b**) **Figure 7.** The relative errors of the model in computing the coil current under a harmonic voltage: (**a**) the relative errors of computing the amplitude ratio; (**b**) the relative errors of computing the phase lag.

**Figure 8.** Linear curves of *AH*-*Dwire*2 from the model and test under different frequencies. **Figure 8.** Linear curves of *AH*-*Dwire* 2 from the model and test under different frequencies.

## *3.3. Square-Wave Response*

## 3.3.1. Time-Domain Response

In addition to the sinusoidal voltage, the direct current (DC) square-wave voltage is frequently used, especially to drive an on–off-type actuator.

For the square-wave response, more attention should be paid to the transient-state process. Additionally, based on the first-order linear time-invariant system, the transientstate current within the coil is

$$I(t) = \frac{\mathcal{U}\_{\rm st}}{\mathcal{R}} + (I\_0 - \frac{\mathcal{U}\_{\rm st}}{\mathcal{R}}) \mathbf{e}^{-\frac{\mathcal{R}}{\mathcal{L}}t} \tag{7}$$

where *I*<sup>0</sup> is the initial value of the coil current, *Ust* is the steady-state amplitude of the voltage. Equation (7) was suitable to both the charging and discharging process of the coil. For charging, *I*<sup>0</sup> = 0. For discharging, *Ust* = 0.

From Equation (4), the reciprocal of the time-constant used in Equation (7) was

$$\frac{\mathcal{R}}{L} = \frac{\rho (L\_f + L\_b) D\_{\text{wire}}^2}{4 \mathcal{C}\_{L0} \mathcal{C}\_f L\_a L\_b D\_{\text{core}}^2} \tag{8}$$

By substituting Equations (7) and (8) into *H*(*t*) = *CHINI*(*t*)/*La*, one obtains the transientstate response of the magnetic field ent-state response of the magnetic field *Rt HI f b st st <sup>L</sup>* − = +− 

By substituting Equations (7) and (8) into *H*(*t*) = *CHINI*(*t*)/*La*, one obtains the transi-

( )

*f b wire L f a b core*

In addition to the sinusoidal voltage, the direct current (DC) square-wave voltage is

For the square-wave response, more attention should be paid to the transient-state process. Additionally, based on the first-order linear time-invariant system, the transi-

−

2 2

= +− (7)

<sup>+</sup> <sup>=</sup> (8)

<sup>0</sup> ( ) ( )e *Rt U U st st <sup>L</sup> It I R R*

where *I*0 is the initial value of the coil current, *Ust* is the steady-state amplitude of the voltage. Equation (7) was suitable to both the charging and discharging process of the

From Equation (4), the reciprocal of the time-constant used in Equation (7) was

*R L LD L C C LLD*

0

4

ρ

*H*(*t*)= *CH IC<sup>f</sup> L<sup>b</sup> πD*<sup>2</sup> *wire*/4 *Ust R* + (*I*<sup>0</sup> − *Ust R* )*e* − *R L t* = 4*CH IC<sup>f</sup> L<sup>b</sup> πD*<sup>2</sup> *wire πD*<sup>2</sup> *coreD*<sup>2</sup> *wireUst* 16*ρC<sup>f</sup> LaL<sup>b</sup>* (*L<sup>f</sup>* + *L<sup>b</sup>* ) + *I*<sup>0</sup> − *πD*<sup>2</sup> *coreD*<sup>2</sup> *wireUst* 16*ρC<sup>f</sup> LaL<sup>b</sup>* (*L<sup>f</sup>* + *L<sup>b</sup>* ) ! *e* − *ρ*(*L f* +*Lb* )*D*2 *wire* 4*CL*0 *Cf LaLbD*<sup>2</sup> *core t* = *CH ID*<sup>2</sup> *core* 4*ρLa*(*L<sup>f</sup>* + *L<sup>b</sup>* ) *Ust* + 4*CH IC<sup>f</sup> L<sup>b</sup> πD*<sup>2</sup> *wire I*0e − *ρ*(*L f* +*Lb* )*D*2 *wire* 4*CL*0 *Cf LaLbD*<sup>2</sup> *core t* − *CH ID*<sup>2</sup> *core* 4*ρLa*(*L<sup>f</sup>* + *L<sup>b</sup>* ) *Ust*e − *ρ*(*L f* +*Lb* )*D*2 *wire* 4*CL*0 *Cf LaLbD*<sup>2</sup> *core t* (9) 2 2 0 ( ) 22 22 4 2 0 2 () ( ) <sup>4</sup> 4 16 ( ) 16 ( ) 4( ) *f b wire L f a b core wire L LD <sup>t</sup> HI f b core wire st core wire st C C LLD wire f a b f b f a b f b HI core af bC CL U U Ht I e DR R C CL DDU DDU I e D C LL L L C LL L L C D <sup>U</sup> LL L* ρ π π π πρ ρ ρ <sup>+</sup> <sup>−</sup> = +− + + <sup>=</sup> <sup>+</sup> 2 2 2 2 0 0 () () <sup>2</sup> 4 4 2 0 4 e e 4( ) *f b wire f b wire L f a b core L f a b core L LD L LD t t HI f b C C LLD C C LLD HI core st st wire a f b C CL C D I U D LL L* ρ ρ π ρ + + − − + − + (9)

*Aerospace* **2023**, *10*, x FOR PEER REVIEW 11 of 16

frequently used, especially to drive an on–off-type actuator.

*3.3. Square-Wave Response*  3.3.1. Time-Domain Response

2 0

ent-state current within the coil is

coil. For charging, *I*0 = 0. For discharging, *Ust* = 0.

The inputted voltage was generated with an amplitude of 2 V and a high-voltage duration of 20 ms to guarantee the coil current reaching the steady state. The time-domain magnetic fields are shown in Figure 9. From the test and model, the proposed model precisely calculated the amplitudes and effectively described the curve shapes under different wire diameters as the transient-state results were also quite close. The inputted voltage was generated with an amplitude of 2 V and a high-voltage duration of 20 ms to guarantee the coil current reaching the steady state. The time-domain magnetic fields are shown in Figure 9. From the test and model, the proposed model precisely calculated the amplitudes and effectively described the curve shapes under different wire diameters as the transient-state results were also quite close.

**Figure 9.** The dynamic magnetic field under square-wave input: (**a**) *Dwire* = 0.31 mm; (**b**) *Dwire* = 0.39 mm; (**c**) *Dwire* = 0.49 mm; (**d**) *Dwire* = 0.60 mm; (**e**) *Dwire* = 0.69 mm; (**f**) *Dwire* = 0.80 mm. **Figure 9.** The dynamic magnetic field under square-wave input: (**a**) *Dwire* = 0.31 mm; (**b**) *Dwire* = 0.39 mm; (**c**) *Dwire* = 0.49 mm; (**d**) *Dwire* = 0.60 mm; (**e**) *Dwire* = 0.69 mm; (**f**) *Dwire* = 0.80 mm.

3.3.2. Steady-State Value and Response Time 3.3.2. Steady-State Value and Response Time

The steady-state value and response time are the most important characteristic parameters of the on–off-type actuator. When the square-wave voltage maintains a high level for a long enough duration, the steady-state response of the magnetic field *Hst* can be easily acquired from Equation (9), as The steady-state value and response time are the most important characteristic parameters of the on–off-type actuator. When the square-wave voltage maintains a high level for a long enough duration, the steady-state response of the magnetic field *Hst* can be easily acquired from Equation (9), as

By exacting the mean value of the magnetic field in the steady-state stage, Figure 10 shows the curves of *Hst* vs. *Dwire* from the test and model. From the tested and calculated results, a higher *Dwire* is helpful for a higher *Hst*. More specifically, *Hst* was positively

the square wave was the same as one of the magnetic field amplitudes under the sinusoidal voltage. It was easily illustrated that both the functions of 1/(*R*2 + *ω*2*L*2)1/2 and 1/*R* can be expressed by the quartic function vs. the wire diameter approximately. In addition, the relatively errors under different wire diameters were less than 1.2% thus, the model

**Figure 10.** Steady-state magnetic field of the square-wave response from the test and model.

(or the duty cycle was adjusted but not the amplitude in the voltage wave).

Equation (7), the response time *tIv* can be reached, as

There are two commonly used response times—the time from 0 to the required intensity and the time from 0 to the required proportion of the steady-state value. The former was especially concerned when the high-opening voltage was employed and the latter was generally concerned when the standard square-wave voltage was employed

By imposing *Ireq* the required intensity of the coil current and substituting *Ireq* into

$$H\_{\rm st} = \frac{\mathcal{C}\_{HI}\mathcal{D}\_{\rm core}^2}{4\rho L\_a (L\_f + L\_b)} \mathcal{U}\_{\rm st} \approx \frac{0.2016 \mathcal{C}\_{HI}\mathcal{D}\_{\rm wire}^2}{\rho L\_a (L\_f + L\_b)} \mathcal{U}\_{\rm st} \tag{10}$$

can predict the steady-state magnetic field quite effectively.

0.3 0.4 0.5 0.6 0.7 0.8 Enameled wire diameter [mm]

0.3 0.4 0.5 0.6 0.7 0.8 Enameled wire diameter [mm]

−1.2 −0.8 −0.4 0 0.4 Test Model

Steady-state

Relative error [%]

value [kA/m]

By exacting the mean value of the magnetic field in the steady-state stage, Figure 10 shows the curves of *Hst* vs. *Dwire* from the test and model. From the tested and calculated results, a higher *Dwire* is helpful for a higher *Hst*. More specifically, *Hst* was positively proportional to *Dwire* 2 , as explained in Equation (10). Thus, the change law of the *Hst* under the square wave was the same as one of the magnetic field amplitudes under the sinusoidal voltage. It was easily illustrated that both the functions of 1/(*R* <sup>2</sup> + *ω*2*L* 2 ) 1/2 and 1/*R* can be expressed by the quartic function vs. the wire diameter approximately. In addition, the relatively errors under different wire diameters were less than 1.2% thus, the model can predict the steady-state magnetic field quite effectively. shows the curves of *Hst* vs. *Dwire* from the test and model. From the tested and calculated results, a higher *Dwire* is helpful for a higher *Hst*. More specifically, *Hst* was positively proportional to *Dwire*<sup>2</sup> , as explained in Equation (10). Thus, the change law of the *Hst* under the square wave was the same as one of the magnetic field amplitudes under the sinusoidal voltage. It was easily illustrated that both the functions of 1/(*R*<sup>2</sup> + *ω*2*L*<sup>2</sup> ) 1/2 and 1/*<sup>R</sup>* canbe expressed by the quartic function vs. the wire diameter approximately. In addition, the relatively errors under different wire diameters were less than 1.2% thus, the model can predict the steady-state magnetic field quite effectively.

 )  *U*

 *b*  *st*

By exacting the mean value of the magnetic field in the steady-state stage, Figure 10

 

(**e**) (**f**) **Figure 9.** The dynamic magnetic field under square-wave input: (**a**) *Dwire* = 0.31 mm; (**b**) *Dwire* = 0.39

The steady-state value and response time are the most important characteristic parameters of the on–off-type actuator. When the square-wave voltage maintains a high level for a long enough duration, the steady-state response of the magnetic field *Hst* can

> 2

> > )

 *b*  *st* (10)

 *U*

0.2016

 *L L L*

   (

 *a f*  *HI wire*

 +

 *C D*

mm; (**c**) *Dwire* = 0.49 mm; (**d**) *Dwire* = 0.60 mm; (**e**) *Dwire* = 0.69 mm; (**f**) *Dwire* = 0.80 mm.

2

+

 (

*a f*

*C D*

*L L L*

*HI core*

4

*st*

=

*H*

3.3.2. Steady-State Value and Response Time

be easily acquired from Equation (9), as

*Aerospace* **2023**, *10*, x FOR PEER REVIEW 12 of 16

**Figure 10.** Steady-state magnetic field of the square-wave response from the test and model. **Figure 10.** Steady-state magnetic field of the square-wave response from the test and model.

There are two commonly used response times—the time from 0 to the required intensity and the time from 0 to the required proportion of the steady-state value. The former was especially concerned when the high-opening voltage was employed and the latter was generally concerned when the standard square-wave voltage was employed There are two commonly used response times—the time from 0 to the required intensity and the time from 0 to the required proportion of the steady-state value. The former was especially concerned when the high-opening voltage was employed and the latter was generally concerned when the standard square-wave voltage was employed (or the duty cycle was adjusted but not the amplitude in the voltage wave).

(or the duty cycle was adjusted but not the amplitude in the voltage wave). By imposing *Ireq* the required intensity of the coil current and substituting *Ireq* into Equation (7), the response time *tIv* can be reached, as By imposing *Ireq* the required intensity of the coil current and substituting *Ireq* into Equation (7), the response time *tIv* can be reached, as

$$\begin{split} t\_{Iv} &= \frac{L}{R} \ln \left( 1 + \frac{I\_{\text{req}}}{\mathcal{U}\_{\text{sl}}/R - I\_{\text{req}}} \right) \\ &= \frac{\mathbb{C}\_{L}\mathbb{C}\_{f}}{16\rho L\_{b} (D\_{f} + L\_{b})} \frac{D\_{\text{core}}^{2}}{D\_{\text{wire}}^{2}} \ln \left( 1 + \frac{I\_{\text{req}}}{\pi D\_{\text{core}}^{2} D\_{\text{wire}}^{2} \mathcal{U}\_{\text{sl}} / \left[ 16\rho \mathbb{C}\_{f} L\_{b} L\_{b} (D\_{f} + L\_{b}) \right] - I\_{\text{req}}} \right) \end{split} \tag{11}$$

Similarly, by substituting the required magnetic field *Hreq* into Equation (9), one obtains the response time to the specified magnetic field intensity *tHv,* as

$$\begin{split} t\_{Hv} &= \frac{L}{R} \ln \left( 1 + \frac{L\_d H\_{req}}{\mathcal{C}\_{Hl} \text{NLI}\_{tl} / R - L\_d H\_{req}} \right) \\ &= \frac{\mathcal{C}\_L \mathcal{C}\_f}{16 \rho L\_b (D\_f + L\_b)} \frac{D\_{core}^2}{D\_{wire}^2} \ln \left( 1 + \frac{L\_d H\_{req}}{\mathcal{C}\_{Hl} \mathcal{D}\_{core}^2 lL\_{tl} / \left[ 4 \rho (D\_f + L\_b) \right] - L\_a H\_{req}} \right) \\ &\approx \frac{0.0504 \omega \mathcal{C}\_L \mathcal{C}\_f}{\rho L\_b (D\_f + L\_b)} \ln \left( 1 + \frac{L\_d H\_{req}}{0.2016 \mathcal{C}\_{Hl} \mathcal{D}\_{wire}^2 lL\_{tl} / \left[ \rho \left( D\_f + L\_b \right) \right] - L\_a H\_{req}} \right) \end{split} \tag{12}$$

From the calculated result, it can be observed that a thicker wire is better for reducing both the response time of the coil current and one of the magnetic fields, while the change degree is different. The enameled wire diameter has more influence on the response time of the coil current than that of the magnetic field as *tIv* is in the function form of *<sup>a</sup>*ln [1 + *<sup>b</sup>*/(*cx*<sup>2</sup> <sup>−</sup> *<sup>b</sup>*)] vs. *<sup>D</sup>wire* while *<sup>t</sup>Hv* is expressed by *<sup>a</sup>*ln [1 + *<sup>b</sup>*/(*cx*<sup>4</sup> <sup>−</sup> *<sup>b</sup>*)] vs. *<sup>D</sup>wire*.

The response time to a specified proportion of the steady-state value can be easily deduced from Equations (10) and (12). For a given proportion *p*, the corresponding intensities of the coil current and magnetic field are, respectively, *Ireq* = *p*(*Ust*/*R*) or *Hreq* = *p*[*CHIN*(*Ust*/*R*)/*La*]. By substituting the two expressions into Equations (10) and (11), one obtains the response time to a specified proportion, as

$$\begin{split} t\_{Hp} = t\_{Ip} &= \frac{L}{R} \ln\left(\frac{1}{1-p}\right) \\ &= \frac{\mathbb{C}\_L \mathbb{C}\_f}{16\rho L\_b (D\_f + L\_b)} \frac{D\_{core}^2}{D\_{wire}^2} \ln\left(\frac{1}{1-p}\right) \\ &\approx \frac{0.0504\omega \mathbb{C}\_L \mathbb{C}\_f}{\rho L\_b (D\_f + L\_b)} \ln\left(\frac{1}{1-p}\right) \end{split} \tag{13}$$

where *p* is a constant belonging to (0, 1).

Compared to *tHv*, the factors influencing *tHp* were almost independent of *Dwire*. More specifically, *tHp* was just determined by the ratio of *L*/*R*. The value of *L*/*R* was only slightly influenced by *Dwire*; optimizing the wire diameter would be helpless to promote this type of response speed. *Aerospace* **2023**, *10*, x FOR PEER REVIEW 14 of 16

> Figure 11 shows the two types of response times from the tested and calculated results; the specified intensities *Hreq* were 3 kA/m, 3.5 kA/m, and 4 kA/m, and the specified proportions *p* were 0.7, 0.8, and 0.9. Just as predicted by the model, *Hreq* was effectively reduced by increasing *Dwire*. Furthermore, *Hreq* declined fast first and then slowly with *Dwire* increasing. For the value of *tHp*, it changed slightly with *Dwire* increasing. The model was verified as the calculated results were consistent with the experimental data. ue. Therefore, for an electromagnetic actuator stimulated by a high-open-low-hold-type voltage, a coil with a wider wire diameter will be stimulated more quickly to save the response time of the whole actuator, while when a traditional square-wave voltage is introduced, an adjustment in the wire diameter is helpless.

**Figure 11.** The response time of the square-wave response from the test and model: (**a**) the time from 0 to specified intensities, respectively, of 3 kA/m, 3.5 kA/m, and 4 kA/m; (**b**) the time from 0 to the specified proportions of the steady-state response, respectively, of 0.7, 0.8, and 0.9. **Figure 11.** The response time of the square-wave response from the test and model: (**a**) the time from 0 to specified intensities, respectively, of 3 kA/m, 3.5 kA/m, and 4 kA/m; (**b**) the time from 0 to the specified proportions of the steady-state response, respectively, of 0.7, 0.8, and 0.9.

**4. Conclusions**  An analytical model of the mean magnetic field for the hollow cylindrical coil used in a magnetically driven actuator was proposed in this paper. Additionally, the selection of the enameled wire diameter was optimized for a high-amplitude and fast-response magnetic field based on the model. (1) The resistance and inductance are inversely proportional functions vs. the quartic of the enameled wire diameter. Under the sinusoidal voltage, a wider wire diameter is On the whole, increasing the wire diameter is quite helpful for reducing the response time from 0 to a specified value of the coil current or magnetic field, while failing to improve the response speed from 0 to the steady-state or any other proportional value. Therefore, for an electromagnetic actuator stimulated by a high-open-low-hold-type voltage, a coil with a wider wire diameter will be stimulated more quickly to save the response time of the whole actuator, while when a traditional square-wave voltage is introduced, an adjustment in the wire diameter is helpless.

### quite helpful for a higher magnetic field amplitude while it has little influence on **4. Conclusions**

the phase lag of the magnetic field. Under the square-wave voltage, the steady-state magnetic field was positively proportional to the square of the wire diameter, as a wider wire diameter is helpful for a higher steady-state magnetic field. Regarding An analytical model of the mean magnetic field for the hollow cylindrical coil used in a magnetically driven actuator was proposed in this paper. Additionally, the selection of the

the response speed, increasing the wire's diameter is helpful for reducing the response time from 0 to the specified intensity, while it is helpless to improve the re-

This paper was devoted to the promotion of the output performance of the whole magnetically driven actuator without considering the coil quality factor or power loss.

(2) The proposed model was verified as the calculated results from the model were in good agreement with the experimental results. Specifically, the relative errors of the model in computing the resistance and the inductance were lower than 3.1% and 2.8%, respectively. For predicting the sinusoidal response, the errors were lower than 6.4% (lower than 2.0% under most conditions) in computing the amplitude and lower than 3.2% in computing the lagging phase. For predicting the square-wave response, the model calculated the amplitudes with errors lower than

1.2% and described the curve shape effectively.

Further work can focus on reducing the power loss of the coil.

enameled wire diameter was optimized for a high-amplitude and fast-response magnetic field based on the model.


This paper was devoted to the promotion of the output performance of the whole magnetically driven actuator without considering the coil quality factor or power loss. Further work can focus on reducing the power loss of the coil.

**Author Contributions:** Conceptualization. G.X.; Data curation, Z.W.; Formal analysis, Z.W.; Investigation: Z.W.; Software and visualization, Z.W. and G.X.; Methodology, G.X.; Validation, G.X. and H.B.; Funding acquisition, H.B. and G.X.; Writing—review and editing, Z.W. and G.X.; Writing—review and editing, H.B., G.X. and Z.R.; Supervision: Z.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Young and Middle-aged Teachers Education and Research Project (Science and Technology) of Fujian Province (No. JAT220016) and First Batch of Yin Ling Fund (No. ZL3H39).

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** The authors thank Zhaoshu Yang (in China Astronaut Research and Training Center) and Tuo Li (in Officers College of PAP) for their improvement in the English of this article.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


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## *Article* **A Preliminary Top-Down Parametric Design of Electromechanical Actuator Position Control**

**Jean-Charles Maré**

INSA-Institut Clément Ader (CNRS UMR 5312), 31400 Toulouse, France; jean-charles.mare@insa-toulouse.fr

**Abstract:** A top-down process is proposed and virtually validated for the position control of electromechanical actuators (EMA) that use conventional cascade controllers. It aims at facilitating the early design phases of a project by providing a straightforward mean that requires simple algebraic calculations only, from the specified performance and the top-level EMA design parameters. This makes it possible to include realistic control considerations in the preliminary sizing and optimisation phase. The position, speed and current controllers are addressed in sequence. This top-down process is based on the generation and use of charts that define the optimal position gain, speed loop second-order damping factor and natural frequency with respect to the specified performance of the position loop. For each loop, the control design formally specifies the required dynamics and the digital implementation of the following inner loop. A noncausal flow chart summarises the equations used and the interdependencies between data. This potentially allows changing which ones are used as inputs. The process is virtually validated using the example of a flight control actuator. This is achieved with resort to the simulation of a realistic lumped-parameter model, which includes any significant functional and parasitic effects. The virtual tests are run following a bottom–up approach to highlight the pursuit and rejection performance. Using low-, medium- and high-excitation magnitudes, they show the robustness of the controllers against nonlinearities. Finally, the simulation results confirm the soundness of the proposed process.

**Keywords:** actuator; aerospace; electromechanical; flight control; friction; modelling; position control; preliminary design; simulation; validation

## **1. Introduction**

The last decade has seen significant progress in electromechanical technology for actuation. In the range of some kilowatts or some tens of kilonewtons, they provide attractive solutions compared with the servohydraulic (or so-called conventional) technology [1]. This evolution is particularly observed in aerospace, which is looking for greener actuation for flight controls, landing gears and engines.

For many applications, electromechanical actuators (EMAs) have already reached the highest technology readiness level, TRL9, which enables them to be put into service. However, it appears that EMAs for aerospace cannot be standardised easily, as opposed to those devoted to industrial applications. This mainly comes from the specificity of requirements and constraints that concern the geometrical integration, the reliability, the mission profiles (including four-quadrant operation with numerous and rapid changes between quadrants) and the certifiability and development assurance level (DAL). The EMA control design itself is driven by these considerations.

Although commercially off-the-shelf drives for industrial applications include efficient self-tuning features [2], each aerospace actuation project requires a specific activity for control design, which must suit the application constraints and development timing in a systems-engineering (SE) frame [3]. There are potentially many candidate types of controllers that today offer extended possibilities: for example, R-S-T digital polynomial controllers (combining parallel R, series S and feedforward T corrections), state

**Citation:** Maré, J.-C. A Preliminary Top-Down Parametric Design of Electromechanical Actuator Position Control. *Aerospace* **2022**, *9*, 314. https://doi.org/10.3390/ aerospace9060314

Academic Editor: Gianpietro Di Rito

Received: 28 March 2022 Accepted: 6 June 2022 Published: 9 June 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

feedback controllers with an estimator, nonlinear controllers or adaptive controllers, for example [4–7]. On their side, EMAs have numerous technology imperfections (e.g., friction and backlash) that are highly sensitive to the operating point. This generally greatly penalises the applicability, robustness and certifiability of these advanced controllers for safety-critical applications such as flight controls. This is why production EMAs involve quite conventional control strategies, which are based on fixed-gain, cascade controllers.

In the preliminary design phases of a project, the concepts and options must be benchmarked rapidly. During these early phases, emphasis is generally put on power sizing under mass, envelope, reliability and thermal constraints [8–10]. Although the natural dynamics of the EMA power part is sometimes addressed, control is never considered in a realistic way. This puts a high penalty on the preliminary design process for two main reasons:


When the control is addressed in more detail, the well-established approach consists in using a bottom–up process [11,12]: the current loop is first addressed, and then the speed loop is considered. The position loop is rarely addressed in the literature because it is not present in many electric drives that aim to control speed (e.g., electric vehicles, fan or pump drives). For each internal loop, the bottom-up process allocates a flat-top target bandwidth that is related to the position loop specified dynamics. Unfortunately, this blind allocation deprives the control designer of a realistic and quantified view of the effective contribution of an inner loop to the stability and rapidity of its upper loops.

The research work that is reported hereafter has been driven by these considerations. It puts emphasis on the design and implementation of a top–down process that serves as a straightforward preliminary control design of a cascade position controller from top-level specifications. This work was driven by two major constraints:


Section 1 introduces the context. Section 2 details the proposed process and its implementation. The soundness of the proposal is shown in Section 3, which reports the control design validation through virtual testing. Section 4 provides important elements of discussion. The Appendices A and B merge all major resources that are used to generate the proposed preliminary control design process.

## **2. Top-Down Controller Design**

Given the specified dynamic performance of the position loop, the proposed process outputs the proportional and integral control gains that are defined sequentially for the position, speed and current loops. Additionally, it provides the sampling frequencies for the digital implementation of the controllers, the target dynamics of the measurement chains and some values of interest for analysis purposes.

This common architecture of a cascade position controller, Figure 1, takes the benefit of the current and speed measurements that are needed to implement the brushless motor control so as to feed the controller back with measured state variables.

control so as to feed the controller back with measured state variables.

This common architecture of a cascade position controller, Figure 1, takes the benefit of the current and speed measurements that are needed to implement the brushless motor

**Figure 1.** Cascade control of EMAs (notations are detailed in the Appendices A and B, full arrows and half arrows highlight the signal and power flows, respectively). **Figure 1.** Cascade control of EMAs (notations are detailed in the Appendices A and B, full arrows and half arrows highlight the signal and power flows, respectively).

The current controller computes the duty cycle setpoints for the motor power drive. Current sensors and their conditioning provide the required feedback signals. The speed controller determines the current setpoints according to the power-operating domain of the motor. The speed feedback signal is commonly acquired from a resolver sensor and its resolver-to-digital converter (RDC), which measures the relative motion between the motor rotor and the stator. This chain also provides the position and speed signals for the field-oriented control (FOC) and back electromotive force (BEMF) compensation. The position controller determines the motor speed setpoint. The position feedback signal is commonly provided by a linear variometer differential transformer (LVDT), which measures the relative position between the EMA rod and the housing. In addition to the three loops, a force loop is sometimes required to meet the specific requirements related to the force limitation or rejection of dynamic loads [13]. The current controller computes the duty cycle setpoints for the motor power drive. Current sensors and their conditioning provide the required feedback signals. The speed controller determines the current setpoints according to the power-operating domain of the motor. The speed feedback signal is commonly acquired from a resolver sensor and its resolver-to-digital converter (RDC), which measures the relative motion between the motor rotor and the stator. This chain also provides the position and speed signals for the field-oriented control (FOC) and back electromotive force (BEMF) compensation. The position controller determines the motor speed setpoint. The position feedback signal is commonly provided by a linear variometer differential transformer (LVDT), which measures the relative position between the EMA rod and the housing. In addition to the three loops, a force loop is sometimes required to meet the specific requirements related to the force limitation or rejection of dynamic loads [13].

### *2.1. Step 1: Design of the Position Controller and Specification of the Speed Loop Dynamics 2.1. Step 1: Design of the Position Controller and Specification of the Speed Loop Dynamics*

The power architecture of an aerospace EMA typically involves a three-phase inverter, which is supplied by the DC-link and drives a brushless motor of the permanent magnet synchronous machine (PMSM) type. The motor shaft power is transmitted to the driven load through a mechanical reducer (a nut-screw system in the most common direct-drive, linear EMA design). In the following, the PMSM is considered as its DC motor equivalent and the inverter is assumed to be perfect. Figure 2 displays the linear control model of the EMA that is used in the proposed process. The power architecture of an aerospace EMA typically involves a three-phase inverter, which is supplied by the DC-link and drives a brushless motor of the permanent magnet synchronous machine (PMSM) type. The motor shaft power is transmitted to the driven load through a mechanical reducer (a nut-screw system in the most common direct-drive, linear EMA design). In the following, the PMSM is considered as its DC motor equivalent and the inverter is assumed to be perfect. Figure 2 displays the linear control model of the EMA that is used in the proposed process. *Aerospace* **2022**, *9*, x FOR PEER REVIEW 4 of 24

**Figure 2.** Block-diagram representation of the equivalent linear control model of the EMA to control (see Appendices A and B for details). controller in the pursuit transfer function, as shown in Figure 3. **Figure 2.** Block-diagram representation of the equivalent linear control model of the EMA to control (see Appendices A and B for details).

produce a low-frequency limit cycle in the presence of the I action. The magnitude of the limit cycle is linked to the minimal position step that can be produced at the rod output. Therefore, it is not affected by any change in the I gain, which only acts on the frequency of this limit cycle. This explains why it is preferred to keep the position controller purely

<sup>∗</sup> = (

In the absence of friction or backlash (or compliance), the EMA internal mechanical transmission between the motor shaft and the EMA rod links the rotor and rod mechanical

> = ௧Ω = ௧

As given in Table A2, the speed loop behaves as a second-order system versus the speed demand and the rate of external load. When a first-order, low-pass filter of time constant ஐ = ஐ/ஐ is applied to the speed demand, the controller becomes of the I-P

 + <sup>1</sup> ஐ <sup>ଶ</sup> ଶ

<sup>∗</sup> − ) (1)

௧ = /2 (3)

(2)

(4)

According to the author's experience, using an integral action in the position controller is not welcome for several reasons. First, the rejection of disturbances is quite low because the I gain is hardly limited by stability considerations. Second, many nonlinear ef-

power variables by:

and:

proportional, as seen in Equation (1), however with output limitation.

2.1.1. Performance of the Position Loop with I-P Speed Controller

Ω =

type, and the speed loop transfer is given by:

Ω

൜

with as the nut-screw lead and as the reduction ratio of the intermediate gear.

Ω <sup>∗</sup> <sup>−</sup> <sup>1</sup> ஐ

1 + 2ஐ ஐ

The pole of the feedforward filter compensates the zero introduced by the speed P-I

The first step of the process deals with the position loop. It uses as inputs the dynamic requirements of the position control, either in the frequency domain (*f*<sup>3</sup> frequency for −3 dB magnitude or *f*<sup>45</sup> for 45◦ phase lag) or in the time domain (settling time *tsX*), the design margin parameter *DM* and the transmission ratio *n<sup>t</sup>* of the EMA. As a result, it provides the proportional gain *KpX* of the position controller and specifies the speed loop dynamics for the second step and the minimal sampling frequency of the position controller. It also outputs additional performance indicators, in particular the angular frequency for phase margin *ωPMX*, which is used to specify the sampling frequency of the position controller.

According to the author's experience, using an integral action in the position controller is not welcome for several reasons. First, the rejection of disturbances is quite low because the I gain is hardly limited by stability considerations. Second, many nonlinear effects (e.g., friction, compliance, backlash, measurement noise, quantisation) combine to produce a low-frequency limit cycle in the presence of the I action. The magnitude of the limit cycle is linked to the minimal position step that can be produced at the rod output. Therefore, it is not affected by any change in the I gain, which only acts on the frequency of this limit cycle. This explains why it is preferred to keep the position controller purely proportional, as seen in Equation (1), however with output limitation.

$$
\Omega\_m^\* = K\_{pX} (X\_L^\* - X\_L) \tag{1}
$$

In the absence of friction or backlash (or compliance), the EMA internal mechanical transmission between the motor shaft and the EMA rod links the rotor and rod mechanical power variables by:

$$\begin{cases} sX\_L = n\_l \Omega\_m \\ T\_L = n\_l F\_L \end{cases} \tag{2}$$

and:

$$m\_t = l/2\pi N\tag{3}$$

with *l* as the nut-screw lead and *N* as the reduction ratio of the intermediate gear.

## 2.1.1. Performance of the Position Loop with I-P Speed Controller

As given in Table A2, the speed loop behaves as a second-order system versus the speed demand and the rate of external load. When a first-order, low-pass filter of time constant *τ*<sup>Ω</sup> = *Kp*Ω/*Ki*<sup>Ω</sup> is applied to the speed demand, the controller becomes of the I-P type, and the speed loop transfer is given by:

$$
\Omega\_m = \frac{\Omega\_m^\* - \frac{1}{\mathcal{K}\_m \mathcal{K}\_{\text{if}\Omega}} s T\_L}{1 + \frac{2\frac{\pi}{\omega\_{\text{f}\Omega}} s + \frac{1}{\omega\_{m\Omega}^2} s^2}} \tag{4}
$$

<sup>ᇱ</sup> = )

(7)

<sup>ଶ</sup> ଶ൨ (5)

ଶ (8)

(1 + 2ஐഥ + ഥଶଶ) (9)

= ௧ (6)

The pole of the feedforward filter compensates the zero introduced by the speed P-I controller in the pursuit transfer function, as shown in Figure 3. *Aerospace* **2022**, *9*, x FOR PEER REVIEW 5 of 24

becomes, in the nonsaturated domain:

EMA transmission factor by:

which becomes:

loops.

**Figure 3.** I-P speed controller obtained by filtering the speed setpoint (continuous-time domain). **Figure 3.** I-P speed controller obtained by filtering the speed setpoint (continuous-time domain).

One can implement the speed controller in the I-P form because this makes the openloop position transfer simpler. This therefore enables the closed-loop position transfer to be expressed formally in a canonical form. In this case, the open-loop position transfer for One can implement the speed controller in the I-P form because this makes the openloop position transfer simpler. This therefore enables the closed-loop position transfer to be expressed formally in a canonical form. In this case, the open-loop position transfer for

<sup>∗</sup> − )−ி

 + <sup>1</sup> ஐ

the pure proportional control of gain and perfect position measurement (

 1 + 2ஐ ஐ

The position loop gain is linked to the position proportional gain and the

ி <sup>=</sup> ௧

ி <sup>=</sup> ௧

ஐ

ாஐ

It is only linked to the motor torque constant, the EMA transmission factor and the integral control gain of the speed loop. Therefore, the proportional position control gain depends only on the speed loop target dynamics ஐ given the EMA-specified dy-

The open-loop transfer function for position pursuit, Equation (5), combines a pure gain () with integral and second-order dynamics (ஐ, ஐ). It is therefore welcome to link the closed-loop performance to these parameters in a dimensionless manner by intro-

<sup>∗</sup> − )=/ε <sup>=</sup> ഥ

The key enabler of the proposed process is the chart that is generated once numerically. It calculates, e.g., using a control toolbox, the position closed-loop performance indicators as a function of the two parameters ഥ and ஐ, which maximises a given constrained objective. Figure 4 displays the chart obtained to secure the fastest closed-loop response to a step position demand (minimal settling time) without overshoot. The data are generated with 1% accuracy. Particular attention is paid to the −45° phase lag requirement because it is a major one regarding the stability of the upper aircraft flight control

<sup>=</sup> (

where is the external load applied at the EMA rod.

namics and design parameters (, ா, ௧).

/(

while the dynamic compliance of the position control is given by:

ducing the dimensionless angular frequency ഥ = /ஐ, which gives:

where ഥ = /ஐ is the dimensionless loop gain of the position loop.

the pure proportional control of gain *KpX* and perfect position measurement (*X* 0 *<sup>L</sup>* = *XL*) becomes, in the nonsaturated domain:

$$X\_L = \frac{K\_{lX}(X\_L^\* - X\_L) - K\_{XF}sF\_L}{s\left[1 + \frac{2\xi\_\Omega}{\omega\_{n\Omega}}s + \frac{1}{\omega\_{n\Omega}^2}s^2\right]}\tag{5}$$

where *F<sup>L</sup>* is the external load applied at the EMA rod.

The position loop gain *KlX* is linked to the position proportional gain *KpX* and the EMA transmission factor by:

$$\mathbf{K}\_{lX} = \mathbf{K}\_{pX}\mathbf{n}\_{l} \tag{6}$$

while the dynamic compliance of the position control is given by:

$$K\_{XF} = \frac{n\_t}{K\_m K\_{i\Omega} K\_{pX}} \tag{7}$$

which becomes:

$$K\_{XF} = \frac{n\_t}{K\_{pX} l\_E \omega\_{n\Omega}^2} \tag{8}$$

It is only linked to the motor torque constant, the EMA transmission factor and the integral control gain of the speed loop. Therefore, the *KpX* proportional position control gain depends only on the speed loop target dynamics *ωn*<sup>Ω</sup> given the EMA-specified dynamics and design parameters (*Km*, *JE*, *nt*).

The open-loop transfer function for position pursuit, Equation (5), combines a pure gain (*KlX*) with integral and second-order dynamics (*ξ*Ω, *ωn*Ω). It is therefore welcome to link the closed-loop performance to these parameters in a dimensionless manner by introducing the dimensionless angular frequency *ω* = *ω*/*ωn*Ω, which gives:

$$X\_L / (X\_L^\* - X\_L) = X\_L / \varepsilon\_X = \frac{K\_{lX}}{s \left(1 + 2\xi\_1 \overline{\omega} s + \overline{\omega}^2 s^2\right)}\tag{9}$$

where *KlX* = *KlX*/*ωn*<sup>Ω</sup> is the dimensionless loop gain of the position loop.

The key enabler of the proposed process is the chart that is generated once numerically. It calculates, e.g., using a control toolbox, the position closed-loop performance indicators as a function of the two parameters *KlX* and *ξ*Ω, which maximises a given constrained objective. Figure 4 displays the chart obtained to secure the fastest closed-loop response to a step position demand (minimal settling time) without overshoot. The data are generated with 1% accuracy. Particular attention is paid to the −45◦ phase lag requirement because it is a major one regarding the stability of the upper aircraft flight control loops.

Figure 4a displays the links among the dimensionless loop gain *KlX*, the phase margin and the dimensionless settling time *t<sup>s</sup>* = *tsX*.*ωn*<sup>Ω</sup> for a given value of the damping factor *ξ*Ω. The best compromise between stability and rapidity is found for *ξ*<sup>Ω</sup> = 0.54. Figure 4b summarises the closed-loop performance indicators expressed in the frequency domain. All the values are dimensionless, with reference to *ωn*Ω. Again, the best bandwidth is obtained when *ξ*<sup>Ω</sup> is close to 0.5. Figure 4c shows that the frequency for the phase margin varies in the range of 1 to 1.25 times the closed-loop bandwidth, while the phase margin is always greater than 65◦ (Figure 4a). On its side, Figure 4d confirms that for low values of the loop gain, the closed-loop system is equivalent to a first-order system of time constant 1/*KlX* . However, when the loop gain increases, the stability is affected by the closed-loop imaginary poles. The greatest dimensionless bandwidth at −45◦ phase is 0.287. It is obtained for *KlX* = 0.58, while the shortest dimensionless settling time of 5.89 is achieved for *Klx* = 0.54. Although *ξ*<sup>Ω</sup> = 0.54 minimises the settling time, such a damping generates 13% overshoot for the speed loop. Setting *ξ*<sup>Ω</sup> = 1 removes this overshoot. It is therefore welcome in the presence of backlash, and it still provides a good compromise for position

*Aerospace* **2022**, *9*, x FOR PEER REVIEW 6 of 24

loop stability and rapidity. However, it requires much faster speed (and current) loop dynamics than the first choices for a given dynamics of the position loop.

**Figure 4.** Performance indicators for position control for the fastest closed-loop step response without overshoot using dimensionless variables: (**a**) loop gain, settling time and phase margin; (**b**) performance indicators in the frequency domain; (**c**) frequency for the phase margin; and (**d**) closedloop rapidity versus loop gain. **Figure 4.** Performance indicators for position control for the fastest closed-loop step response without overshoot using dimensionless variables: (**a**) loop gain, settling time and phase margin; (**b**) performance indicators in the frequency domain; (**c**) frequency for the phase margin; and (**d**) closed-loop rapidity versus loop gain.

Figure 4a displays the links among the dimensionless loop gain ഥ, the phase margin and the dimensionless settling time ̅ <sup>௦</sup> = ௦. ஐ for a given value of the damping factor ஐ. The best compromise between stability and rapidity is found for ஐ = 0.54. Figure 4b summarises the closed-loop performance indicators expressed in the frequency domain. All the values are dimensionless, with reference to ஐ. Again, the best bandwidth In the implemented approach, the control design parameter is *ξ*Ω. The data plotted in Figure 4a or Figure 4b are first used to determine *ωn*<sup>Ω</sup> and then *KlX* given the specified dynamics of the position control. This enables the *KpX* P gain of the position controller to be calculated using Equation (6) from the EMA transmission factor *n<sup>t</sup>* .

### is obtained when ஐ is close to 0.5. Figure 4c shows that the frequency for the phase mar-2.1.2. Performance of the Position Loop with P-I Speed Controller

gin varies in the range of 1 to 1.25 times the closed-loop bandwidth, while the phase margin is always greater than 65° (Figure 4a). On its side, Figure 4d confirms that for low values of the loop gain, the closed-loop system is equivalent to a first-order system of time constant 1/ഥ . However, when the loop gain increases, the stability is affected by the closed-loop imaginary poles. The greatest dimensionless bandwidth at −45° phase is 0.287. It is obtained for ഥ = 0.58, while the shortest dimensionless settling time of 5.89 is achieved for ഥ௫ = 0.54. Although ஐ = 0.54 minimises the settling time, such a damping When the low-pass filter of Figure 3 is not implemented, a zero remains in the pursuit transfer function Ω*m*/Ω∗ *<sup>m</sup>*. This P-I implementation of the speed controller also has its merits. As it does not introduce any lowpass filtering of the speed setpoint that is generated by the position controller, it decreases the tracking error. The presence of the zero that remains in the pursuit transfer function of the closed-loop position, however, tends to introduce overshoot in the position step response. Nonetheless, it does not affect the *KXF* parameter, which quantifies the load position sensitivity to the rate of external load.

generates 13% overshoot for the speed loop. Setting ஐ = 1 removes this overshoot. It is therefore welcome in the presence of backlash, and it still provides a good compromise for position loop stability and rapidity. However, it requires much faster speed (and current) loop dynamics than the first choices for a given dynamics of the position loop. In the implemented approach, the control design parameter is ஐ. The data plotted in Figure 4a or Figure 4b are first used to determine ஐ and then given the speci-In this case, the performance chart is generated to obtain the smallest response time of the position loop, ensuring that all closed-loop poles are stable and purely real. This helps to avoid back and forth motion in the presence of backlash and limits the overshoot generated by the zero of the speed loop. The main data of this chart are presented graphically in Figure 5.

fied dynamics of the position control. This enables the P gain of the position control-

ler to be calculated using Equation (6) from the EMA transmission factor ௧.

2.1.2. Performance of the Position Loop with P-I Speed Controller

**Figure 5.** Performance indicators of position control with the P-I implementation of the speed controller giving the fastest response time under a purely real negative closed-loop poles constraint. **Figure 5.** Performance indicators of position control with the P-I implementation of the speed controller giving the fastest response time under a purely real negative closed-loop poles constraint.

When the low-pass filter of Figure 3 is not implemented, a zero remains in the pursuit

In this case, the performance chart is generated to obtain the smallest response time of the position loop, ensuring that all closed-loop poles are stable and purely real. This helps to avoid back and forth motion in the presence of backlash and limits the overshoot generated by the zero of the speed loop. The main data of this chart are presented graph-

its. As it does not introduce any lowpass filtering of the speed setpoint that is generated by the position controller, it decreases the tracking error. The presence of the zero that remains in the pursuit transfer function of the closed-loop position, however, tends to introduce overshoot in the position step response. Nonetheless, it does not affect the ி parameter, which quantifies the load position sensitivity to the rate of external load.

<sup>∗</sup> . This P-I implementation of the speed controller also has its mer-

transfer function Ω/Ω

ically in Figure 5.

It can be remarked that the constraint imposed on the closed-loop poles cannot be met when the damping ratio ஐ is lower than the unity. For values greater than 1.3, this constraint significantly impacts the speed loop natural frequency compared with the I-P It can be remarked that the constraint imposed on the closed-loop poles cannot be met when the damping ratio *ξ*<sup>Ω</sup> is lower than the unity. For values greater than 1.3, this constraint significantly impacts the speed loop natural frequency compared with the I-P implementation under the null overshoot constraint. For instance, when *ξ*<sup>Ω</sup> = 1.3:


cerning the rejection of disturbances, as shown by Equation (8).

## 2.1.3. Digital Implementation of the Position Controller

2.1.3. Digital Implementation of the Position Controller In this paper, the controllers are set in the continuous-time domain, using transfer functions as control models. Although this choice puts aside any advanced controller that does not exist in the continuous-time domain, it keeps a direct link with the physics through canonical parameters (time constants, damping factors and natural frequencies). Once designed, the controller is discretised for digital implementation. The phase lag introduced by filtering, sampling and computation is actively managed to specify the sam-In this paper, the controllers are set in the continuous-time domain, using transfer functions as control models. Although this choice puts aside any advanced controller that does not exist in the continuous-time domain, it keeps a direct link with the physics through canonical parameters (time constants, damping factors and natural frequencies). Once designed, the controller is discretised for digital implementation. The phase lag introduced by filtering, sampling and computation is actively managed to specify the sampling frequencies.

pling frequencies. When seen from the continuous-time domain, the zero-order sampling and hold function performed at the sampling frequency *f<sup>s</sup>* in the digital implementation of the controller is equivalent to a pure delay of ∆*<sup>s</sup>* = 1/2 *f<sup>s</sup>* . At frequency *f* , it introduces a phase of *ϕ*( ◦ ) = −180 *f* / *f<sup>s</sup>* . In a closed-loop system, this delay is, with rare exceptions, detrimental to the closed-loop stability. This is why it is important to select the sampling frequency consistently with the target dynamics of the considered closed-loop system. There are a few practical known recommendations to make this decision [14]:


However, these general rules are not directly driven by the stability of the loop under consideration. This is why the author prefers the following more direct approach that can be expressed as follows. In total, the digital control introduces at the frequency *f* a parasitic phase lag (phase lag stands for the opposite value of phase) *ϕd*( ◦ ) . It comes from the sampling delay plus the phase lag of the antialiasing filter and the ∆*<sup>c</sup>* time spent for conversions and processing. If it is assumed that the antialiasing filtering is achieved with a Butterworth second-order low-pass filter of cut-off frequency of *fs*/2, then the phase lag is given by:

$$\varphi\_{d(\,^\circ)} = \frac{180}{\pi} \left[ \pi / n\_s + \operatorname{atan} \left\{ \frac{2.8 / n\_s}{1 - \left( 2 / n\_s \right)^2} \right\} + 2 \pi f\_s \Delta\_c / n\_s \right] \tag{10}$$

where:

$$n\_{\rm s} = f\_{\rm s}/f \tag{11}$$

For outer loops having a low bandwidth, the delay ∆*<sup>c</sup>* is generally negligible in comparison with other contributors. Depending on the implementation of the digital control, it may, however, be significant for the most inner (e.g., current) loops. If ∆*<sup>c</sup>* is neglected, *ϕd*( ◦ ) already reaches 80.5◦ when *n<sup>s</sup>* drops to 4.35. Above this value, the phase lag of the antialiasing filter varies almost linearly versus frequency, and *ϕd*( ◦ ) can be approximated by:

$$
\varphi\_{d(^{\circ})} \cong 180 \, (1 + 2.8/\pi)/n\_{\mathrm{s}} \cong 340.4/\, n\_{\mathrm{s}} \tag{12}
$$

with less than 2.9% error of underestimation (a 360 factor, instead of 340.4, corresponds to the phase lag produced by a full sampling period delay). Antialiasing contributes to almost half the total phase lag, while the magnitude effect of the low-pass filter remains below ±0.003 dB. Of course, the *ϕd*( ◦ ) , as shown in Equation (12), can be adapted to the current context, e.g., for the antialiasing filter or if the processing time becomes the major source of phase lag. If necessary, Equation (12) can be modified to include the phase lag introduced by the sensor and measurement chain.

These results provide a straightforward means to quantify (or specify) the reduction of the open-loop phase margin given the digital implementation of a controller that has been designed in the continuous time domain. For example, if this contribution (including the antialiasing filter) must not exceed 10◦ parasitic phase lag, Equation (12) indicates that the sampling frequency must be at least 34 times the frequency at which the phase margin is determined. This is a really huge value.

If the dynamics of the position measurement can be neglected, the minimal sampling frequency of the position controller can be specified using Equation (12). This option has been anticipated when building the performance charts, Figures 4 and 5, which explicitly provide the angular frequency *ωPMX* = 2*π fPMX*, at which the phase margin of the position control is determined when parasitic phase lags are not considered.

Using, e.g., Equation (12), the sampling frequency *fsX* of the position controller must satisfy the constraint:

$$f\_{\rm sX} \ge \mathbf{340.4} \, f\_{\rm PMX} / \, \varphi\_{\rm dX(\circ)} \tag{13}$$

where *ϕdX*( ◦ ) is the allocated parasitic phase lag introduced by the digital implementation of the position controller.

Notes


## *2.2. Step 2: Design of the Speed Controller and Specification of the Current Loop Dynamics* 2.2.1. Viscous Friction vs. Real Friction

A pure viscous friction coefficient of coefficient *b<sup>E</sup>* is most of the time considered in the accounts dealing with setting the speed controllers of electric drives [15,16]. However, real friction is far different from pure viscous friction (where the friction force is proportional to the velocity). This is particularly true for motion control when the actuator drives a variable load at variable speed with frequent speed reversals. In this case, the friction force mainly depends on load, temperature, and, in a much lesser amount, relative speed [17]. This is clearly illustrated by the examples given in Figure 6. a variable load at variable speed with frequent speed reversals. In this case, the friction force mainly depends on load, temperature, and, in a much lesser amount, relative speed [17]. This is clearly illustrated by the examples given in Figure 6.

A pure viscous friction coefficient of coefficient ா is most of the time considered in the accounts dealing with setting the speed controllers of electric drives [15,16]. However, real friction is far different from pure viscous friction (where the friction force is proportional to the velocity). This is particularly true for motion control when the actuator drives

where ௗ(°) is the allocated parasitic phase lag introduced by the digital implementation

• The phase lag introduced by the position measurement is not considered. Although it is generally negligible, this assumption must be verified (when the measurement chain is known), ensured by relevant specification (when the measurement chain is to be defined) or removed by adding the position measurement dynamics in Equa-

• For LVDTs position sensors, the demodulation filter is welcome to avoid any fre-

*2.2. Step 2: Design of the Speed Controller and Specification of the Current Loop Dynamics* 

௦ ≥ 340.4 ெ/ௗ(°) (13)

*Aerospace* **2022**, *9*, x FOR PEER REVIEW 9 of 24

of the position controller.

Notes

tion (13).

quency aliasing.

2.2.1. Viscous Friction vs. Real Friction

**Figure 6.** Friction force in EMAs: (**a**) the influence of temperature on friction torque for a geared EMA at a rated output torque, calculated from harmonic drive efficiency data (size 14, ratio 100) [17]; (**b**) the influence of speed and load at room temperature (demonstration EMA sized for the Airbus A320 aileron–gear drive, ball screw, no preload) [18]. **Figure 6.** Friction force in EMAs: (**a**) the influence of temperature on friction torque for a geared EMA at a rated output torque, calculated from harmonic drive efficiency data (size 14, ratio 100) [17]; (**b**) the influence of speed and load at room temperature (demonstration EMA sized for the Airbus A320 aileron–gear drive, ball screw, no preload) [18].

This figure clearly shows that a pure viscous friction model totally fails to reproduce real friction. For linear control design, it is therefore preferred to consider friction as an unmodelled effect. This requires the controller to be robust enough against it. In this work, this robustness is assessed a posteriori, either by simulation (when validated models are available), from partial real tests or through former capitalised experience. This approach is not only applied to friction but also to backlash and compliance, whether they concern the EMA itself or the kinematics linking the EMA to the driven load. It works particularly well in the field of aerospace, e.g., for flight controls or landing gears actuation. Indeed, for such applications, the natural dynamics generated by the combination of moving bodies' inertance and the backlash/compliance of the mechanical transmissions is far greater than the specified bandwidth of the actuator position control. This figure clearly shows that a pure viscous friction model totally fails to reproduce real friction. For linear control design, it is therefore preferred to consider friction as an unmodelled effect. This requires the controller to be robust enough against it. In this work, this robustness is assessed a posteriori, either by simulation (when validated models are available), from partial real tests or through former capitalised experience. This approach is not only applied to friction but also to backlash and compliance, whether they concern the EMA itself or the kinematics linking the EMA to the driven load. It works particularly well in the field of aerospace, e.g., for flight controls or landing gears actuation. Indeed, for such applications, the natural dynamics generated by the combination of moving bodies' inertance and the backlash/compliance of the mechanical transmissions is far greater than the specified bandwidth of the actuator position control.

## 2.2.2. Setting the Speed Loop Controller

The first step of the proposed process has specified the second-order dynamics of the speed loop: the undamped natural frequency *ωn*<sup>Ω</sup> and the damping factor *ξ*Ω. These target values are used as inputs in Table A2 to obtain the proportional (*Kp*Ω) and integral (*Ki*Ω) gains of the speed controller from the total equivalent reflected inertia at the motor rotor *J<sup>E</sup>* and the EMA motor electromagnetic constant *Km*:

$$\mathcal{K}\_{\rm i} \boldsymbol{\Omega} = \mathcal{J}\_{\rm E} \omega\_{n\Omega}^2 / \mathcal{K}\_m \tag{14}$$

$$K\_{p\Omega} = 2J\_E \mathfrak{f}\_{\Omega} \omega\_{n\Omega} / K\_m \tag{15}$$

These settings are directly linked to the specified position loop dynamics. It is interesting to remark that the time constant of the P-I speed controller,

$$
\pi\_{\Omega} = \mathcal{K}\_{p\Omega} / \,\, \mathcal{K}\_{\dot{\Omega}\Omega} = 2\xi\_{\Omega} / \omega\_{\mathfrak{m}\Omega} \tag{16}
$$

is only linked to the dynamics specified for the speed loop, determined in Step 1, once the damping factor *ξ*<sup>Ω</sup> is chosen. It is therefore independent of the EMA parameters.

## 2.2.3. Digital Implementation of the Speed Controller

The sampling frequency *fs*<sup>Ω</sup> for the digital implementation of the speed controller is specified in the same manner as for the position loop:

$$f\_{s\Omega} \le 340.4 \frac{f\_{PM\Omega}}{\varphi\_{d\Omega(^{\circ})}} \tag{17}$$

This is constrained by the frequency *fPM*<sup>Ω</sup> = *ωPM*Ω/2*π*, at which the phase margin of the speed loop is determined, and by the parasitic phase lag *ϕd*Ω( ◦ ) introduced by the digital implementation of the speed controller.

Note

The motor speed measurement can generate significant phase lag. Allocating the accepted phase lag for motor angle measurement can add another constraint to specify the dynamics of the rotor speed/angle measurement chain.

## 2.2.4. Specification of the Current Loop Dynamics

Step 2 is also used to specify the current loop dynamics. Again, the objective is to limit the parasitic phase lag that the current loop introduces into the speed loop or, in other words, to ensure the validity of the results summarised in Table A2. This is achieved as follows.

It can be shown that the frequency at which the phase margin of the speed loop is given by:

$$
\omega\_{PM\Omega} = \omega\_{\text{n\Omega}} \sqrt{2\xi\_{\Omega}^2 + \sqrt{1 + 4\xi\_{\Omega}^4}} \tag{18}
$$

If the current controller is set as usual, its P-I time constant is made equal to that of the motor windings, leading to:

$$
\pi\_{\rm CI} = \,\,\mathbf{K}\_{pI} / \mathbf{K}\_{iI} = \mathbf{L} / \mathbf{R} \tag{19}
$$

In this case, the current loop behaves as a first-order lag of time constant:

$$
\pi\_{lI} = \mathcal{L} / \mathbb{K}\_{pI} \mathcal{U}\_{\text{DCE}} \tag{20}
$$

Thus, the dynamics of the current loop is specified by limiting the parasitic phase lag *ϕI* that it introduces in the speed loop at the *ωPM*<sup>Ω</sup> angular frequency at which the speed loop phase margin is determined:

$$
\pi\_{II} \le \tan(\varphi\_I) / \omega\_{PM\Omega} \tag{21}
$$

It is worth remarking that this constraint does not involve any EMA design parameter.

## *2.3. Step 3. Design of the Current Controller*

2.3.1. Setting the P-I Controller of the Current Loop

The proportional and integral gains of the current loop controller are set according to Appendix A, given the following two constraints:

$$\mathbf{K}\_{pI} \ge \mathbf{L} / \mathbf{U}\_{\text{DCE}} \mathbf{r}\_{II} \tag{22}$$

$$\mathbf{K}\_{iI} = \mathbf{R} \,\, \mathbf{K}\_{pI} / \,\, \mathbf{L} \tag{23}$$

Note

These equations involve quantities related to the EMA design (*L*, *R*, *UDCE*), which can vary significantly during the EMA operation and consequently alter the performance of the current loop. To make the EMA sufficiently robust, the setting of the current controller gains must consider the worst conditions and their effect on rapidity and stability.

## 2.3.2. Digital Implementation of the Current Controller

According to Appendix A, when the P-I time constant of the current controller compensates the electric time constant of the motor, the open-loop transfer function becomes a pure integrator of gain *Ki*<sup>I</sup> *UDCE*/*R*. In the presence of pure parasitic delays, the angular frequency *ωPMI* , at which the phase margin of the current loop is defined, is given by:

$$
\omega\_{\rm PMI} = 2\pi f\_{\rm PMI} = \mathcal{K}\_{\rm iI} \,\mathcal{U}\_{\rm DCE}/\mathcal{R} \tag{24}
$$

This frequency can be used to specify the sampling frequency *fsI* for the digital implementation of the current controller. Given the high dynamics required for the current loop, it may be important to consider not only sampling and antialiasing but also additional effects that can limit the allowable controller gains by alteration of the closed-loop stability: time spent for computation and conversions, and dynamics of the currents measuring chain.

All these effects increase the open-loop phase lag. Thus, they can be merged to consider their negative contribution to the phase margin globally. In the very common case, the dynamics of the current measurement chain is negligible compared with that introduced by the various delays. However, a simple conservative option consists of considering that the overall delay is equal to a full sampling period, giving the constraint:

$$f\_{\rm sI} \ge 360 \, \frac{f\_{\rm PMI}}{\varphi\_{dI(^{\circ})}} \,\tag{25}$$

### *2.4. Synthesis of the Top-Down Process Aerospace* **2022**, *9*, x FOR PEER REVIEW 12 of 24

All these results can be represented graphically to summarise the interdependencies among the parameters involved in the design of the EMA position control. This is achieved using the diagram shown in Figure 7. the ா, and parameters can be obtained from estimation models, for example, using scaling laws or metamodels [8], from the main design parameters ௧, and ா.

**Figure 7.** Noncausal representation of the interdependence among parameters for the design of the EMA position control. **Figure 7.** Noncausal representation of the interdependence among parameters for the design of the EMA position control.

The example of a wingtip, direct-drive, linear flight control actuator used for regional

The modelled and unmodelled phenomena are summarised in Table 1. The highfidelity model is implemented and simulated in the Simcenter-AMEsim (2020.1, Imagine, Roanne, France) environment. It involves 75 state variables, no implicit variable and +200 parameters. Iron losses and magnetic saturation at the motor are not modelled as they are not significant in this application. Any energy loss is made sensitive to temperature, enabling isothermal simulations to be run for various operating temperatures. Given the dynamics in presence and the sampling/switching frequencies, a 1 s simulation with integration accuracy of 10−7 typically takes a 290 s CPU on a 64-bit personal computer (Intel Core

and electromechanical units) and the driven load, which was developed in former studies

**3. Illustrative Example** 

*3.1. Virtual Prototype* 

I7-8550U CPU at 1.8 GHz).

[20,21].

The blue, red and green blocks use the equations related to the position, speed and current controllers, respectively. A noncausal representation is preferred (nonoriented signal lines) because this enables the calculation causality to be adapted to the current context. This possibility is particularly attractive, e.g., for EMA preliminary sizing when the control hardware is imposed. When read from the top down, the data flow implements the proposed top-down process, where each sequential step from 1 to 3 is dedicated to the setting of a given controller and the specification of the next inner controller. The eight controller parameters (right) are computed given the performance specification and control design choices (top), given the main EMA design parameters (left). When the process is combined with preliminary sizing and optimisation during the early phases of a project, the *JE*, *L* and *R* parameters can be obtained from estimation models, for example, using scaling laws or metamodels [8], from the main design parameters *n<sup>t</sup>* , *K<sup>m</sup>* and *UDCE*

## **3. Illustrative Example**

The example of a wingtip, direct-drive, linear flight control actuator used for regional aircraft [19] is used to illustrate the proposed control design. The process is validated through the simulation of an accurate lumped-parameter model of the actuator (control and electromechanical units) and the driven load, which was developed in former studies [20,21].

## *3.1. Virtual Prototype*

The modelled and unmodelled phenomena are summarised in Table 1. The highfidelity model is implemented and simulated in the Simcenter-AMEsim (2020.1, Imagine, Roanne, France) environment. It involves 75 state variables, no implicit variable and +200 parameters. Iron losses and magnetic saturation at the motor are not modelled as they are not significant in this application. Any energy loss is made sensitive to temperature, enabling isothermal simulations to be run for various operating temperatures. Given the dynamics in presence and the sampling/switching frequencies, a 1 s simulation with integration accuracy of 10−<sup>7</sup> typically takes a 290 s CPU on a 64-bit personal computer (Intel Core I7-8550U CPU at 1.8 GHz).


**Table 1.** Model used for the virtual validation of the controller design.

## *3.2. Virtual Validation*

The EMA position controller is virtually validated using a bottom–up incremental approach that follows the real validation process, i.e., the integration branch of the V-model of product lifecycle [3]. The operation of every (simulated) element of the EMA (motor, inverter, measurement chains and mechanical transmission) has been virtually validated, as should be done with partial tests for the real elements. The current loop is validated first, followed by the speed loop and finally the position loop.

The controllers have been designed with the following allocation of the parasitic phase lag because of digital implementation: 5◦ for the position loop, 10◦ + 10◦ for the speed loop and 20◦ for the current loop.

The loops are excited to assess both pursuit and rejection performances on the same response plot. As numerical simulation naturally provides time responses, a demand step is applied first, followed by a disturbance step. To make the virtual validation realistic, a random noise is introduced on each measured quantity, typically very few percent of the maximal or rated values. The time responses given in this section have been plotted using the realistic magnitudes that were identified during real tests of the power and signal electronics: 4% of the maximal RMS phase current, 3% of the rated rotor speed, 5◦ for the rotor angle and 6% of LVDT secondary voltage magnitude. Particular attention is also paid to the effect of nonlinearities and unmodelled dynamics on the performance expected from the linear continuous control model. In this attempt, the responses are analysed for various step magnitudes. High magnitude leads to saturations as a result of power and signal limitations. Medium magnitude generally enables the EMA to operate far from hard nonlinear effects and static imperfections. Very low magnitude points to the influence of static imperfection such as quantisation, breakaway friction and backlash.

The dimensionless responses are presented to provide on a single figure the demand, the response of the linear continuous control model and the response of the high-fidelity, nonlinear, high-order model. In the responses provided for the high-fidelity model, the EMA is assumed to operate at room temperature. The excitation magnitudes are referred to the rated values and to the noise magnitude (before the antialiasing filter). The time values are hidden for confidentiality. However, it can be mentioned that the time ranges of Figures 8–10 are in the ratio 1:8:100, respectively, to indicate the relative dynamics of the current, speed and position loops.

## 3.2.1. Current Loop

To obtain the current loop responses, the speed and position loops are opened. The motor is tested without connection to the nut-screw while externally imposing the rotor angular velocity. A current step demand (*I* ∗ *<sup>m</sup>* or *I* ∗ *q* ) is applied first with the rotor blocked, followed by a motor speed step disturbance Ω*m*. The response of the linear continuous control model is obtained from the last transfer function of Table A1. The responses are displayed in Figure 8.

This figure elicits the following comments:


transforms, although this is very fast, which generates an alias *I<sup>d</sup>* current proportionally to the *I<sup>q</sup>* current demand. control model is obtained from the last transfer function of Table A1. The responses are displayed in Figure 8.

<sup>∗</sup> ) is applied first with the rotor blocked,

*Aerospace* **2022**, *9*, x FOR PEER REVIEW 14 of 24

the current, speed and position loops.

angular velocity. A current step demand (

3.2.1. Current Loop

far from hard nonlinear effects and static imperfections. Very low magnitude points to the influence of static imperfection such as quantisation, breakaway friction and backlash.

The dimensionless responses are presented to provide on a single figure the demand, the response of the linear continuous control model and the response of the high-fidelity, nonlinear, high-order model. In the responses provided for the high-fidelity model, the EMA is assumed to operate at room temperature. The excitation magnitudes are referred to the rated values and to the noise magnitude (before the antialiasing filter). The time values are hidden for confidentiality. However, it can be mentioned that the time ranges of Figures 8–10 are in the ratio 1:8:100, respectively, to indicate the relative dynamics of

To obtain the current loop responses, the speed and position loops are opened. The motor is tested without connection to the nut-screw while externally imposing the rotor

<sup>∗</sup> or

followed by a motor speed step disturbance Ω. The response of the linear continuous

**Figure 8.** Time response of the current loop for step excitations: (**a**) small-magnitude excitations, noise effects magnified; (**b**) medium-magnitude excitations, operation close to linear; and (**c**) largemagnitude excitations, close to internal saturation.

## 3.2.2. Speed Loop

For the speed loop test, the position loop is opened and the translating part of the nut-screw is removed. A rotor speed step demand Ω∗ *<sup>m</sup>* is applied first for a free rotor shaft, followed by a step disturbance torque *T<sup>L</sup>* applied to the rotating part of the nut-screw. The response of the continuous linear control model is obtained from the last transfer function of Table A2. The most relevant time responses are displayed in Figure 9, which elicits the following comments:


**Figure 9.** Time response of the speed loop for step excitations: (**a**) small-magnitude excitations, noise effects magnified; (**b**) medium-magnitude excitations, at the limit of saturation; (**c**) large-magnitude excitations, introducing long saturation of the current and modulation ratio; and (**d**) the speed and current controller outputs for large-magnitude excitations. **Figure 9.** Time response of the speed loop for step excitations: (**a**) small-magnitude excitations, noise effects magnified; (**b**) medium-magnitude excitations, at the limit of saturation; (**c**) large-magnitude excitations, introducing long saturation of the current and modulation ratio; and (**d**) the speed and current controller outputs for large-magnitude excitations.

The simulations are run with all control loops active. According to the customer's

3.2.3. Position Loop

## 3.2.3. Position Loop

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The simulations are run with all control loops active. According to the customer's specification for the validation of the EMA performance, a pure mass equivalent to the reflected mass of the driven load is attached to the EMA rod. The rod position step setpoint *X* ∗ *L* is applied first from rest at the null position, without any external force. reflected mass of the driven load is attached to the EMA rod. The rod position step setpoint <sup>∗</sup> is applied first from rest at the null position, without any external force. To better highlight the combined effect of EMA friction and backlash, the external

To better highlight the combined effect of EMA friction and backlash, the external disturbance force *F<sup>L</sup>* is then applied at the EMA rod, without any change in the position demand, as two opposite and consecutive steps. The response of the continuous linear control model is obtained by a simulation of the speed closed-loop model combined with Equations (5) and (6). The most relevant time responses are displayed in Figure 10: disturbance force is then applied at the EMA rod, without any change in the position demand, as two opposite and consecutive steps. The response of the continuous linear control model is obtained by a simulation of the speed closed-loop model combined with Equations (5) and (6). The most relevant time responses are displayed in Figure 10: • At a very low magnitude of rod position demand (twice the EMA internal backlash),

	- Even in the presence of backlash, the rod force step is rejected with the same dynamics as that of the linear model (Figure 10b). For 100% force, the transient position error does not exceed 155% of the backlash or 10% of the nut-screw lead. This excellent capability of external force disturbance confirms the soundness of the proposed approach, which is intended to maximise the position loop gain for a given *ξ*Ω. ics as that of the linear model (Figure 10b). For 100% force, the transient position error does not exceed 155% of the backlash or 10% of the nut-screw lead. This excellent capability of external force disturbance confirms the soundness of the proposed approach, which is intended to maximise the position loop gain for a given ஐ.

**Figure 10.** *Cont*.

**Figure 10.** Time response of the position loop for step excitations: (**a**) small-magnitude excitations, noise effects magnified; (**b**) rejection of rod force disturbance, null position demand; (**c**) large-magnitude excitations, introducing long saturation of the current and modulation ratio; and (**d**) controller outputs for large-magnitude excitations. **Figure 10.** Time response of the position loop for step excitations: (**a**) small-magnitude excitations, noise effects magnified; (**b**) rejection of rod force disturbance, null position demand; (**c**) largemagnitude excitations, introducing long saturation of the current and modulation ratio; and (**d**) controller outputs for large-magnitude excitations.

#### **4. Discussion 4. Discussion**

Within the nonsaturating domain of operation, two main nonlinear effects act as disturbances in the linear model that is used for control synthesis. It is important to relate the excellent robustness of the position control to these unmodelled effects and to their magnitude: in the reported validation, the EMA internal backlash is 6.3% of the nut-screw lead, while the load-independent friction force represents 3% of the EMA rated force. At the rated output force, this percentage rises to 12.4% under the contribution of the loaddependent friction. When they are expressed as their equivalent at the EMA rod level, the motor rotor inertia is 44 times greater than that of the driven load. Within the nonsaturating domain of operation, two main nonlinear effects act as disturbances in the linear model that is used for control synthesis. It is important to relate the excellent robustness of the position control to these unmodelled effects and to their magnitude: in the reported validation, the EMA internal backlash is 6.3% of the nut-screw lead, while the load-independent friction force represents 3% of the EMA rated force. At the rated output force, this percentage rises to 12.4% under the contribution of the loaddependent friction. When they are expressed as their equivalent at the EMA rod level, the motor rotor inertia is 44 times greater than that of the driven load.

Besides this particular example, it is worth addressing more general comments: Besides this particular example, it is worth addressing more general comments:

As shown by Equation (8), the ி rod position sensitivity to the rate of external load is inversely proportional to ஐ <sup>ଶ</sup> . Given the proposed control design process, the selection of ஐ can offer a means to act on ி to meet a given pursuit dynamics. As shown by Equation (8), the *KXF* rod position sensitivity to the rate of external load is inversely proportional to *KpXω*<sup>2</sup> *n*Ω . Given the proposed control design process, the selection of *ξ*<sup>Ω</sup> can offer a means to act on *KXF* to meet a given pursuit dynamics.


can be met in practice. If not, the concerned subspecification has to be replaced by a constraint, and the data flow of the design process has to be revised accordingly. This is enabled by the noncausal representation used in Figure 7.

## **5. Conclusions**

The design of position control of EMAs has been addressed for aerospace safetycritical applications. The focus has been placed on the proposition and implementation of a top-down process requiring very few input data for the design of cascade controllers, when certifiability constraints and design assurance levels welcome common control techniques. This work was primarily intended to enable control considerations to be added to the preliminary sizing phases in order to accelerate the development process. This objective was achieved by several contributions. The first one comes from the generation of charts that link the position control gain to the speed loop dynamics and damping targets, when a position control performance criterion is maximised. For a given loop, the second one consists in determining the control parameters, the numerical implementation and the specification of the following loop, in a formal and simple way which calls upon a minimal number of EMA parameters. The third contribution lies in the graphical representation that synthetises in a noncausal way the interdependencies between the control specifications, the EMA key design parameters, the control choices and the controller parameters. The proposed process has been virtually validated using a very detailed high-fidelity model of the EMA. The EMA responses derived from the virtual testbench have shown the efficiency of the proposed process, even in the presence of significant noise, saturation, friction and backlash, which are unmodelled in the linear models used for control synthesis.

**Funding:** The reported work was funded by the Clean Sky 2 European project ASTIB (JTI-CS2-2014- CPW01-REG-01-01). This project aims at supporting the improvement of the Technological Readiness Level for a number of significant equipment items that are considered of critical importance for the future Green Regional Aircraft (GRA).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The author sincerely thanks the project partners of Work Package 2.4 for their constructive discussions during his development and implementation of EMA models and control laws.

**Conflicts of Interest:** The author decla es no conflict of interest.

## **Nomenclature**



Dimensionless

◦ Angle expressed in degree

## **Appendix A. Current Loop**

With reference to Figures 1 and 2, the elements involved in the motor current loop make a double-input, single-output dynamic system. The motor current *I<sup>m</sup>* is the controlled variable that must follow the demand *I* ∗ *<sup>m</sup>* (pursuit function) and reject the disturbance *E* (rejection function). The modelling and analysis of the current loop are summarised in Table A1.

The PMSM motor is assumed to be of three phases with star connection. Being controlled under the max torque per current (null direct current, *I* ∗ *<sup>d</sup>* = 0) strategy, it is considered as its equivalent brushed DC machine [26]. The motor constant *K<sup>m</sup>* stands for the torque constant *K<sup>T</sup>* (Nm/A), where the current is the *I<sup>q</sup>* quadrature current of the power conservative dq0 transform. The torque constant equals the motor BEMF constant *K<sup>E</sup>* (Vs/rad) when it is defined using the root mean square (RMS) line-to-line voltage. In the linear operating range of the PWM, the maximal line-to-line RMS voltage *UDCE* at the motor windings is defined from the DC-link supply voltage *UDC* as:

$$
\mathcal{U}\_{\rm DCE} = \frac{\sqrt{3}}{2\sqrt{2}} \mathcal{U}\_{\rm DC} = 0.612 \,\mathcal{U}\_{\rm DC} \tag{A1}
$$

Allowing the PWM to operate in the pseudo-linear range extends the 0.612 factor to 1/<sup>√</sup> 2 = 0.707 [27].

The last part of the table displays the main performance indicators for the very common setting that fixes the P-I time constant *τCI* of the controller to the motor electric time constant *τ<sup>e</sup>* . In this case, the zero introduced by the P-I current controller compensates exactly the pole corresponding to the motor electric time constant. Therefore, the pursuit dynamics is fixed by the *KiI* integral control gain, while the BEMF disturbance *E* is rejected at order 1 (s factor at the numerator) instead of order 0. The BEMF rate is rejected with the first-order dynamics of the electric time constant *τ<sup>e</sup>* , whose gain is fixed by the proportional control gain *KpI*. The BEMF disturbance can be theoretically removed thanks to feed-forward or compensation schemes. They involve the motor electromagnetic constant *K<sup>E</sup>* and the measurement (or the estimate) of the motor shaft angular velocity.

At this level, the gains of the P-I current controller only depend on two parameters (the resistance and inductance of the motor windings). They seem to be independent of the target dynamics of the current loop. However, although Table A1 does not explicitly show any limitation in these gains, several additional effects bound in practice the dynamics and accuracy of the current loop:


**Table A1.** Current loop model and analysis in continuous time domain.


## **Appendix B. Speed Loop**

With reference to Figures 1 and 2, the elements involved in the actuator speed loop make a double-input, single-output dynamic system. The motor shaft speed Ω*<sup>m</sup>* is the controlled variable that must follow the demand Ω∗ *<sup>m</sup>* (pursuit function) and reject the disturbance torque *T<sup>L</sup>* (rejection function). Table A2 summarises the simplified modelling and linear analysis of the speed loop in the continuous time domain. It is obtained under the following assumptions:


The P-I speed controller makes the speed closed loop behave as generalised secondorder dynamics. The *ωn*<sup>Ω</sup> natural undamped frequency is fixed by the *Ki*<sup>Ω</sup> integral control gain. Given this gain, the *ξ*<sup>Ω</sup> dimensionless damping factor is set linearly by the proportional gain *Kp*Ω. As for the current loop, the integral action of the controller removes the speed dependence on constant external loads, while the dependence on the load rate is directly proportional to the integral control gain.

Given the linear modelling assumptions, Table A2 indicates no limitation in setting the speed controller gains. They are only linked to three EMA parameters (equivalent inertia *JE*, equivalent viscous friction *b<sup>E</sup>* and motor torque constant *Km*) and to the target second order (damping factor *ξ*<sup>Ω</sup> and natural frequency *ωn*Ω).

**Table A2.** Speed loop model and analysis in continuous-time domain.


## **References**


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