## 4 Results and Discussion
### Friction Torque Evaluation by Inverse Analysis  
In the deployment test, the deployment time history of angular displacement was compared with the deployment analysis results for all joints. As seen in case T01 and T02 (see Figure 9), the deployment time of testing was later than that of analysis, no critical overshoot of panels was found. In the test frames, there were various friction sources such as the slight misalignment of the deployment test device, the resistance forces caused by the air drag and air-bearing of the dummy panels were needed in the MBD model to compensate such deviation. Thus, in our work, the equivalent friction torque defined as EFT1, EFT2, and EFT3 including all friction sources is proposed. The EFT values can be easily determined by solving an optimization problem with the response surface method [2] to find EFT values that makes the rotation angle history of analysis results similar to those of testing for 2 s. In our work, the rotation angle history of three rotation joints \(\phi_{1}\), \(\phi_{2}\), and \(\phi_{3}\) were used as the target variables for the model correlation between test and analysis. For the friction compensation [35], a rotation friction model was employed as,  
\[EFT=\sqrt{2e}(T_{brk}-T_{C})\cdot\exp\left(-\left(\frac{\omega}{\omega_{st}} \right)^{2}\right)\cdot\frac{\omega}{\omega_{st}}+T_{C}\cdot\tanh\left(\frac{ \omega}{\omega_{Coul}}\right)+f\omega, \tag{4}\]  
where, \(T_{brk}\), \(T_{c}\), and \(f\) indicates the breakway friction, Coulomb friction, and viscous friction coefficient, respectively; \(\omega\), \(\omega_{Coul}\), and \(\omega_{st}\) are the relative velocity, Coulomb threshold velocity, and Stribeck velocity threshold, respectively; And \(\omega_{Coul}\) is a parameter used to alleviate numerical instability caused by Coulomb friction. In this work, 0.01 rad/s was used in reference value [35]. It reduced to a simple Equation (5) by setting \(T_{brk}\) to \(T_{c}\) and neglecting the effect of viscous friction as,  
\[\approx T_{C}\cdot\tanh\left(\frac{\omega}{\omega_{Coul}}\right), \tag{5}\]  
The error function in Equation (6) is defined in terms of the deviation of deployment angles \(\phi_{1}\), \(\phi_{2}\), and \(\phi_{3}\) of the three panels that were derived referring to the test data for the first 2 s while no interaction between the panels occurred. The deployment angles varied with friction torques: EFT1,  
Figure 9: Comparison of the time history of angular displacements for all test cases: (**a**) T01, (**b**) T02. EFT2, and EFT3. Therefore, the optimal design solution using various combinations of friction torques was obtained by minimizing the error function as,  
\[Error(\phi_{1}^{a},\phi_{2}^{a},\phi_{3}^{a})=\sum_{i=0}^{200}\bigl{(}\phi_{1}^{a }(t_{i})-\phi_{1}^{e}(t_{i})\bigr{)}^{2}+\sum_{i=0}^{200}\bigl{(}\phi_{2}^{a}(t_ {i})-\phi_{2}^{e}(t_{i})\bigr{)}^{2}+\sum_{i=0}^{200}\bigl{(}\phi_{3}^{a}(t_{i} )-\phi_{3}^{e}(t_{i})\bigr{)}^{2} \tag{6}\]  
where \(\phi^{a}{}_{i}=\phi^{a}{}_{i}(\text{EFT2, EFT3})\), \(t_{i+1}=t_{i}+\Delta t\) (\(t_{0}=0s,\ \Delta t=0.01s\)). Superscript a and e denote analysis and experiment, respectively. As a result, the values of \(T_{C}\) for the revolute joint 1, revolute joint 2, and revolute joint 3 were 2.63 N-m, 1.7 N-m, and 0.001 N-m to show excellent correlation between test and analysis for the T01 and T02 case for 2 s, respectively. The friction torque of Hinge-1 is higher than those of Hinge-2 and Hinge-3. It seems that inner hinges have to experience more friction caused by accumulated misalignments of friction elements from panels, air bearings, and panels. After considering of EFTs, the simulation result for \(\phi_{1}\) value was quite consistent with that of testing, as seen in Figure 10. For \(\phi_{2}\), and \(\phi_{3}\) values, during the first 2 s, the deployed configuration of the dummy solar panels with EFTs was very similar to that of testing.