## 4 Deployment Test
### 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. \[y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{12}x_{1}x_{2}+ \beta_{13}x_{1}x_{3}+\beta_{23}x_{2}x_{3}+\beta_{11}x_{1}^{2}+\beta_{22}x_{2}^ {2}+\beta_{33}x_{3}^{2} \tag{15}\]  
where \(x_{i}\), \(\beta_{i}\) and \(\beta_{ij}\) 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\({}^{\circ}\)) at joint O and the outer panel (180\({}^{\circ}\)) 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, \(R^{2}\) (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\(\cdot\)m, 0.472 N\(\cdot\)m, and 0.065 N\(\cdot\)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 10(b). The coefficients of the polynomials were obtained in the Equation (16). \[ \begin{array}{lcl}\beta&=&[\beta_{0},\beta_{1},\beta_{2},\beta_{3},\beta_{12 },\beta_{13},\beta_{23},\beta_{11},\beta_{22},\beta_{33}]\\ &=&[16.329,\,3.713,\,-1.703,\,1.107,\,0.823,\,-1.290,\,0.658,\,2.040,\,1.529,\, 1.262]\end{array}  \tag{16}\]  
Figure 10: Definition of the equivalent friction torques. 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 11: Comparison of the rotation angle history between test and analysis before/after friction compensation: (**a**) case 1, (**b**) case 2, (**c**) case 3.