*3.2. Deployment Dynamics Analysis*

Deployment behavior was analyzed by RecurDyn. Table 7 shows the mass properties of the two panels, and Figure 6 shows the rotational stiffness of the 90◦ and 180◦ torsional spring hinges, which were measured and realized through the spline function of Recur-Dyn [23]. The initial rotation angles were set to 90◦ and 180◦, respectively, forcing the rotation joints to have driving torques in the stowed configuration.

**Table 7.** Dimensions and mass properties of inner and outer panels.

**Figure 6.** Configuration of rotational spring hinges and the relationship between torque and rotation angles: (**a**) 90◦ hinge (Joint O), (**b**) 180◦ hinge (Joint I).

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

#### **4. Deployment Test**

#### *4.1. Test Configuration and Test Cases*

Dummy panels accommodating rotational torsion spring hinges between panels were established to verify the design of the deployment performance. To make a weightless environment for structures, a zero-G device was used [27] by hanging the structure with spring-loaded wires to minimize deflection and friction against gravity on the air bearings (Figure 7). Before the test, alignment of the test fixture was adjusted within 0.01◦ using a two-axis digital protractor, to place the panel close to horizontal from the ground. As shown in Table 8, three test cases were chosen to determine whether the panel was fully deployed according to changes of the driving torque. The video camera was installed at the ceiling of the test facility. During the tests, we employed a set of red markers on top of the frames to record the deployment angle history of the inner and outer panels using the MATLAB color extraction algorithm (see Figure 8). Finally, the locations of color were used to evaluate the angles of panels using inverse kinematics from the position history. The deployment test showed the dummy panels were fully deployed with the joints of each truss-link latched successfully (see Figure 9).

**Figure 7.** Test configuration of panels with truss-link mechanism [27].

**Figure 8.** Configuration of red markers on dummy panels.


**Table 8.** Test cases.

**Figure 9.** Fully deployed configuration (case 2 and case 3).

## *4.2. Modeling of Friction and Trajectory Error*

For friction compensation, a rotational friction model considering the Coulomb friction torque was employed as given below [27,28]:

$$EFT \approx T\_C \cdot \tanh\left(\frac{\omega}{\omega\_{coul}}\right) \tag{13}$$

where *TC*, *ω* and *ωCoul* are the Coulomb friction torque, relative angular velocity, and Coulomb threshold velocity, respectively; *ωCoul* is a parameter used to alleviate numerical instability caused by Coulomb friction. In this work, 0.01 rad/s was used as the reference value [28]. As seen in Figure 10, EFT1, EFT2, and EFT3 are the equivalent friction torques at joints O, I, and F; *φ*<sup>1</sup> and *φ*<sup>2</sup> indicate the rotation angles of joint O and I; trajectory error was defined and computed using Equation (14) as follows:

$$\operatorname{Trajectory\,error}(\phi^a) = \frac{1}{n} \sum\_{t=0}^n \left( \frac{\phi^e(t\_i) - \phi^a(t\_i)}{\phi^e(t\_i)} \times 100 \right) \tag{14}$$

where *φ<sup>a</sup>* = *φa*(*EFT*1, *EFT*2, *EFT*3, *t*) in degrees are the rotation angles according to the time evolution; the initial time was *t*<sup>0</sup> = 0 s and the time step Δ*t* = 0.05 s; *n*, *a*, *e* denote the number of time data used, analysis, and experiment, respectively. The data were compared for four seconds until all panels were fully deployed in the test. We added the test frames in the MBD model to consider the effect of test fixture. Contact stiffness of 37,000 N/mm1.5 and contact damping of 3.7 N·s/mm1.25 were applied based on Hertz contact theory [29] between panel to panel and between truss-links to panels to allow for movement within a limited area in case of contact.

**Figure 10.** Definition of the equivalent friction torques.

### *4.3. Response Surface Methodology for Friction Identification*

From comparison results between the test and simulation, as shown in Figure 11, it was observed that all simulations deployed earlier than those of the test. This is believed to be caused by a slight misalignment of the test frame, friction at the joints, and air drag from the panels and truss-links during the deployment test. Therefore, we attempted to adjust the deployment simulation model by adding the equivalent friction torques (EFT) at the three revolute joints defined as EFT1, EFT2, and EFT3, including those from all sources of friction, as defined in Figure 10. The EFT values can be easily determined by solving an optimization problem with the response surface method (RSM), making the rotation angle history of the analysis results similar to that of the test. The response surface *y* was generated with three design parameters, as in Equation (15). It is a polynomial function having ten terms corresponding to unknown coefficients, and the optimal response is found through the response function.

$$\mathbf{y} = \beta\_0 + \beta\_1 \mathbf{x}\_1 + \beta\_2 \mathbf{x}\_2 + \beta\_3 \mathbf{x}\_3 + \beta\_{12} \mathbf{x}\_1 \mathbf{x}\_2 + \beta\_{13} \mathbf{x}\_1 \mathbf{x}\_3 + \beta\_{23} \mathbf{x}\_2 \mathbf{x}\_3 + \beta\_{11} \mathbf{x}\_1^2 + \beta\_{22} \mathbf{x}\_2^2 + \beta\_{33} \mathbf{x}\_3^2 \tag{15}$$

where *xi*, *β<sup>i</sup>* and *βij* are the values of the design variables and the coefficients of the polynomials of design variables, respectively. Here, the design variables indicate the equivalent friction torques: EFT1, EFT2, and EFT3. To determine the coefficients of the polynomials, 15 design cases were determined by the central composite design, popularly adopted in the design of experiments (DOE). The response variables were assigned to the trajectory error of the angles history of the inner panel (90◦) at joint O and the outer panel (180◦) at joint I. After deployment simulation for 15 design cases for test case 2, two output responses were computed and summarized, shown in Table 9. Finally, the coefficients of the polynomials of design variables were determined by regression analysis. To check the suitability of the derived response surface, *R*<sup>2</sup> (coefficient of determination) was obtained as 0.978, indicating a high correlation between the design variables and response variables. By determining the minimum point of the response surface, it was found that EFT1, EFT2, and EFT3 were 0.54 N·m, 0.472 N·m, and 0.065 N·m, respectively, showing an excellent correlation between the test and analysis for test case 2 with the corresponding trajectory error 1 and 2 as 7.69%, 6.62%, respectively, as presented in Figure 11b. The coefficients of the polynomials were obtained in the Equation (16).

$$\begin{array}{rcl} \mathfrak{B} & = & [\beta\_0, \beta\_1, \beta\_2, \beta\_3, \beta\_{12}, \beta\_{13}, \beta\_{23}, \beta\_{11}, \beta\_{22}, \beta\_{33}] \\ & = & [16.329, 3.713, -1.703, 1.107, 0.823, -1.290, 0.658, 2.040, 1.529, 1.262] \end{array} \tag{16}$$

**Figure 11.** Comparison of the rotation angle history between test and analysis before/after friction compensation: (**a**) case 1, (**b**) case 2, (**c**) case 3.


**Table 9.** Design cases of central composite design.

Although there remains a slight gap between the analysis and the test after friction compensation, this is explained by all the sources of friction occurring in the test which cannot be considered. In addition, these EFT values were applied in test case 1 and test case 3, which additionally produce a good correlation between the test and analysis, as shown in Figure 11a,c. A detailed comparison is summarized in Figures 12–14.

**Figure 12.** Comparison of the deployed motion history for case 1.

**Figure 13.** Comparison of the deployed motion history for case 2.

**Figure 14.** Comparison of the deployed motion history for case 3.

## *4.4. Torque Margin Evaluation*

From the deployment test result, for all test cases the deployment of the truss-link mechanism was successful. However, we re-validated the EFTs with torque margin analysis [30]. If the torque margin was negative, the deployment would fail due to inadequate driving torque. If the torque margin was positive, the selected driving torque would be appropriate and completely deployed. To analyze this phenomenon, the torque margin was calculated as

$$E\_D^{90^\circ} = \int\_0^{\frac{\pi}{2}} T\_D d\theta \tag{17}$$

$$E\_D^{180^\circ} = \int\_0^\pi T\_D d\theta \tag{18}$$

$$E\_{\mathbb{R}}^{90^\circ} = \int\_0^{\frac{\pi}{2}} T\_{\mathbb{R}} d\theta \tag{19}$$

$$E\_R^{180^\circ} = \int\_0^\pi T\_R d\theta \tag{20}$$

$$Torque\,Margin = \left(\frac{E\_D}{E\_R} - 1\right) \times 100\,\text{(\%)}\tag{21}$$

where *TD*, *TR* are the driving torque and the resistance torque of Joint O (90◦) and Joint I (180◦), respectively. Moreover, *ED*, *ER* are the driving torque energy and the resistance torque energy value of Joint O and Joint I. The torque margin calculation results for Joint O and I are summarized in Tables 10 and 11. If the torque margin value was negative, then the deployment of panels at the corresponding joints would not be achieved. For the test cases, all torque margin values were positive. As seen in Figures 12–14, all the test cases were fully deployed. Through this evaluation, it can be predicted that the worst case cannot be deployed in advance. Therefore, the EFT estimation method developed in this work was verified with excellent reliability.

**Table 10.** Driving and resistance torque energy according to the test cases.


**Table 11.** Torque margin calculation for all test cases.


#### **5. Conclusions**

In this paper, the modeling and validation of a passive truss-link mechanism applicable to a large-scale deployable structure for small satellites were designed to achieve a successful deployment. To design the truss-link mechanism, we referred to the truss-link structure of RADARSAT-2 satellites. A detailed design of the 3D truss-link structure followed based on the 2D configuration defined by the equations consisting of geometric construction methods and bar-groups methods.

Note that in contrast to the many truss-link mechanisms [11–17], we do not accommodate any active driving mechanisms as they lack simplicity, are not highly reliable, and are

expensive. As driving mechanisms, only conventional torsion spring hinges with proper latching mechanisms were employed in this work.

To demonstrate the performance of the deployable structure with truss-links, a deployment simulation with the MBD model was conducted with proper joint modeling and rotational spring stiffness. Furthermore, a deployment test was conducted to check the correlation between the test and analysis. The results revealed that the time history of the deployment test was slightly behind that of the deployment analysis. To compensate for this, an inverse identification technique for equivalent friction torque (EFT) was proposed based on the results of the response surface method combined with the central composite design technique. Finally, with the EFTs, the angular rotation history in the deployment analysis was much closer to that of the deployment test with a trajectory error of less than 8%. Therefore, the adequacy of the proposed design of the deployable structure with truss-links was verified. This kind of truss-links mechanism has strong potential to extend a multi-modular deployable structure for small satellites. It will be further discussed in our future works.

**Author Contributions:** Conceptualization, J.H.L.; methodology, H.-S.C., T.S.J.; software, J.-H.P., H.-S.C.; validation, H.-S.C., J.-H.P.; investigation, D.-Y.K., H.-S.C.; data curation, H.-S.C.; writing original draft preparation, J.H.L.; writing—review and editing, T.S.J.; supervision, J.H.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF-2018M1A3A02065478).

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

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

### **References**

