**4. Results and Discussion**

In this section, the results of the simulation of the estimation model of the parameter estimation algorithm are presented. The results of the estimation model are compared to those of the real model and the simulation model. The initial covariance **P**<sup>0</sup> used in the augmented state estimator includes *σ*<sup>2</sup> *<sup>s</sup>* = <sup>1</sup> × <sup>10</sup>−<sup>4</sup> m2 for the actuator position, *<sup>σ</sup>*<sup>2</sup> *<sup>s</sup>*˙ = <sup>1</sup> × <sup>10</sup>−<sup>4</sup> <sup>m</sup>2/s2 for the actuator velocity, and three pressure terms of *<sup>σ</sup>*<sup>2</sup> *<sup>p</sup>* <sup>=</sup> 22.50 <sup>×</sup> <sup>10</sup><sup>7</sup> Pa2 in the diagonal. For the hydraulic parameters, the initial covariance values *σ*<sup>2</sup> *kp* <sup>=</sup> <sup>1</sup> <sup>×</sup> 1014 Pa2, *σ*2 *<sup>k</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>×</sup> 102 and *<sup>σ</sup>*<sup>2</sup> **Cv** <sup>=</sup> <sup>9</sup> <sup>×</sup> <sup>102</sup> *<sup>m</sup>*<sup>6</sup> *<sup>s</sup>*2*Pa* are used in the diagonal. The numerical values of the plant noise *σ*<sup>2</sup> **x¨** = 0.8 m2/s4 and *<sup>σ</sup>*<sup>2</sup> *<sup>p</sup>* <sup>=</sup> 259.81 <sup>×</sup> <sup>10</sup><sup>7</sup> Pa2 for Equation (31) are obtained through trial and error. All models are run with a time step of 1 ms and provide sensor data to the parameter estimation algorithm at 1000 Hz.

#### *4.1. Estimating the Characteristic Curve of the Valve*

In the real model, as only the minimum point cmin and the maximum point cmax on the characteristic curves are known at the *a*, *b*, *c*, and *d* ports of the directional control valve, the characteristic curves are generally unclear in the working cycles of the real model. The characteristic curves may vary from one valve to another and can be highly non-linear in the working cycle. In Figure 3, *Spline* 1 and *Spline* 2, which are in the cyan colour, are used to demonstrate the non-linear behaviour of the directional control valve in the real model.

**Figure 3.** The estimation of the characteristic curves of the directional control valve by using the ADEKF with third-order B-spline interpolation. (**a**) Three-point B-spline estimation. (**b**) Four-point B-spline estimation. (**c**) Five-point B-spline estimation. (**d**) Six-point B-spline estimation.

The proposed parameter estimation algorithm can be used to estimate the characteristic curves of the real model with this limited information. To this end, the semi-empiric flow rate coefficients **Cv***<sup>a</sup>* , **Cv***<sup>b</sup>* , **Cv***<sup>c</sup>* , and **Cv***<sup>d</sup>* are defined with the data control points between cmin and cmax in Equation (37). Equation (37) is further used in terms of the control point vector **N** in Equations (20)–(28) to estimate the characteristic curves.

For instance, in the case of Figure 4, the control point vector **N***<sup>a</sup>* at port *a* of the directional control valve can be defined in terms of c1, c2, c3, and c4 as

$$\mathbf{N}\_{\mathfrak{d}} = \begin{bmatrix} \mathbf{c}\_{\min} & \mathbf{c}\_1 & \mathbf{c}\_2 & \mathbf{c}\_3 & \mathbf{c}\_4 & \mathbf{c}\_{\max} \end{bmatrix}^T. \tag{39}$$

To estimate the characteristic curve, three, four, five, and six control points are used in the control point vector **N***a*. As an example, these data control points for *Spline* 1 in each case are presented in Table 3.

The abscissa of vector **N***<sup>a</sup>* represents the spool position *U*, whereas the ordinate of vector **N***<sup>a</sup>* indicates the semi-empiric flow rate coefficients **Cv***<sup>a</sup>* at port *a* of the directional control valve. The results of the second-order B-spline are described in Appendix A. The results of the third-order B-spline demonstrate the characteristic curve of the directional control valve relatively better in a working cycle, as shown in Figure 3. As can be seen, *Spline* 1 and *Spline* 2 of the third-order B-spline are drawn in each data control point estimation case. The dashed red-coloured line indicates the characteristic curve of the simulation model. The dashed black-coloured line demonstrates the estimation model. In Figure 3a, three points, cmin, c1, and cmax, are used to estimate *Spline* 1 and *Spline* 2 of the real model. The characteristic curve of the estimation model precisely follows the real model in the case of three points. Further, the percentages of the root mean square error (RMSE) are described in the Table 3 for *Spline* 1 and *Spline* 2 to verify the observations.

**Figure 4.** Data control points between cmin and cmax on the characteristic curve of the directional control valve.



Figure 3b shows the estimation of the characteristic curve when using four points cmin, c1, c2, and cmax. As can be seen in Figure 3b, the semi-empiric flow rate coefficient **Cv***<sup>a</sup>* for *Spline* 2 changes with small increments until 52% opening of the spool as compared to *Spline* 1 in the real model. After this point, the parameter **Cv***<sup>a</sup>* increases sharply towards the maximum point cmax. The difference of the estimated curve from the real model's curve is indistinguishable. The RMSEs of these curves are given in Table 3. The relatively complicated non-linear behaviours of the directional control valve can be estimated by using five control points and six control points. This can be seen in Figure 3c,d. By using the estimated characteristic curves, the working conditions of the directional control valve can be predicted.

#### *4.2. Convergence of the Vector Data Control Points*

The convergence rate of the data control points in the parameter vector **Cv***<sup>a</sup>* is further explained in Figure 5 to describe the estimation process. These plots demonstrate the convergence rate of data control points in the case of *Spline* 2, as presented in Figure 3. For instance (see Figure 5b), c1 and c2 converge towards the corresponding point on the curve of the real model at 0.22 s. However, during the estimation process, c1 briefly becomes negative, and shortly thereafter converges smoothly to the real model.

**Figure 5.** Convergence of the control points in the vector **Cv***<sup>a</sup>* in the case of *Spline* 2. (**a**) Convergence of c1 in the three-point estimation process. (**b**) Convergence of c1 and c2 in the four-point estimation process. (**c**) Convergence of c1, c2, and c3 in the five-point estimation process. (**d**) Convergence of c1, c2, c3, and c4 in the six-point estimation process.

The curves of *Spline* 2 change into an S-shape during the working cycle, as shown in Figure 3c,d. In these cases, c2, c3, and c4 converge at different simulation times according to the corresponding order in the vector **Cv***<sup>a</sup>* . Through the ADEKF algorithm, unknown curves start converging within a range of 0 < *t* ≤ 0.3 s when using the three-, four-, five-, and six-point estimation techniques.

#### *4.3. Accuracy Requirements of State Estimations*

The successful application of the parameter estimation algorithm requires the accurate estimation of the system states **x**. To demonstrate this requirement, the errors in the estimated actuator position *s*, estimated actuator velocity *s*˙, estimated pump pressure *pp*, estimated piston side pressure *p*<sup>1</sup> , estimated piston-rod side pressure *p*2, and estimated parameter **Cv***<sup>a</sup>* in the case of *Spline* 2 (described in Figure 3d) are shown in Figure 6. The

errors in the estimated parameters *kp* and *k*<sup>0</sup> are presented in Appendix B. The average of the parameter vector **Cv***<sup>a</sup>* at each time step is considered in Figure 6.

The errors are computed from ±1.96*σ*. Here, *σ* is the standard deviation calculated from the covariance matrix **P**<sup>+</sup> *<sup>k</sup>* at each time step. These plots demonstrate the requirement of an accurate estimation of the system's states to estimate the system's parameters. As can be seen in Figure 6, the 95% confidence interval (CI) is used by the system states in the 5 s simulation period. The errors in *s*, *s*˙, *pp*, *p*1, and *s*˙ fluctuate in the confidence interval. As indicated earlier, *s*, *s*˙, and *pp* are measured in this example. The errors in the parameters **Cv***<sup>a</sup>* , *kp*, and *k*<sup>0</sup> are also in the CI, as can be seen in the corresponding plots. The key to the parameter estimation is that the estimated system states should be in the 95% CI during the working cycle.

**Figure 6.** Requirements for the accuracy in the system states for parameter estimation. (**a**) Error in *s* with 95% CI. (**b**) Error in *s*˙ with 95% CI. (**c**) Error in *pp* with 95% CI. (**d**) Error in *p*<sup>1</sup> with 95% CI. (**e**) Error in *p*<sup>2</sup> with 95% CI. (**f**) Error in parameter **Cv** with 95% CI.

#### **5. Conclusions**

This work proposes the estimation of the parameters of a system by combining parameter estimation theories and curve-fitting methods. The ADEKF algorithm is introduced in the framework of a B-spline curve-fitting method. Using the proposed algorithm, the parameters can be defined as a vector containing a set of data control points. This algorithm is applied on a hydraulically driven four-bar mechanism to estimate the characteristic curves of a directional control valve. The double-step semi-recursive formulation and lumped fluid theories are used to model the four-bar mechanism and the hydraulic system, respectively. The measurements taken from the real system include the actuator position, pump pressure, and piston side pressure. The semi-empiric flow rate coefficient vector **Cv***<sup>a</sup>* is defined with three to six data control points in order to define the characteristic curve of the directional control valve.

The unknown non-linear nature of the characteristic curves of the directional control valve are precisely estimated. The maximum RMSE observed in the estimation of the characteristic curves is 0.08%. This implies that the characteristic curves are accurately estimated. The data control points in the parameter vector **Cv***<sup>a</sup>* converge in the range of 0 < *t* ≤ 0.3 s in these estimation cases. To account for the system's response, the estimation of the system's state vector variables should be located in the 95% confidence interval. By using the estimated characteristic curves, important information about the discharge coefficient, pressure losses, and flow characteristics of the directional control valve can be interpreted. With this valuable information, manufacturers and users can monitor the condition of a system and make decisions about the repair and maintenance of hydraulically driven systems.

Applying the parameter estimation algorithm in the real world by using a multibodybased estimation model can enable the estimation of important parameters. This can be challenging, as the estimation model might not be as accurate as the real world necessitates. However, despite implementation challenges, the application of this parameter estimation algorithm will provide an interesting area for manufacturers and researchers. Manufacturers can use these parameters in condition monitoring, repair and maintenance, and the anticipation of product life cycles. With a product's application history, important design changes can be introduced in future designs of the product. This will ultimately lead to more efficient MBS-based digital-twin applications through the use of real-time simulations and more sustainable future products.

**Author Contributions:** Conceptualization, M.K.-O. and A.M.; Funding acquisition, A.M.; Investigation, Q.K. and M.K.-O.; Methodology, Q.K. and M.K.-O.; Resources, A.M.; Software, Q.K. and M.K.-O.; Supervision, M.K.M. and A.M.; Visualization, Q.K. and M.K.-O.; Writing—original draft, Q.K.; Writing—review & editing, Q.K., M.K.-O., S.J., M.K.M. and A.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported in the Research Project of DigiBuzz at Lappeenranta University of Technology, Lappeenranta, Finland.

**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.
