## 4 Simulation
At this time, the system-elastic vibration caused by the flexible-joint is not suppressed, so the joint angle trajectories have seriously deviated from the expected trajectories in only 1.2 s. The output error and the angle error were large. So, the control failed. The effectiveness of the proposed fast subsystem controller \(\mathbf{\tau}_{\text{f}}\) is verified. At the same time, the simulation results also prove that the strategy proposed in this paper, which uses a singular perturbation method to decompose the system and control the slow and fast subsystems separately, will be beneficial to the effective control of the strong rigid-flexible coupling system. Figure 8: The trajectories of the joint angle \(\mathbf{q}_{\theta}\) when \(\mathbf{u}_{\text{p}}\) is off. Figure 10: The angle error \(\sigma\) when \(\mathbf{u}_{\text{p}}\) is off. The comparison results of the three cases in Simulation 1 are shown in Table 3. It can also be seen from Table 3 that RBFTHC is effective and feasible. The robust controller \(u_{\text{p}}\) and the fast subsystem's controller \(\tau_{\text{f}}\) in RBFTHC can guarantee the accuracy and stability of the system effectively. Figure 11: The joint angular velocity \(\tilde{q}_{\theta}\) when \(u_{\text{p}}\) is off. Figure 12: The trajectories of the joint angle \(q_{\theta}\) when \(\tau_{\text{f}}\) is off. Figure 13: The output error \(\tilde{x}\) when \(\tau_{\text{f}}\) is off. **Simulation 2**: The actuator is slightly faulty. The change of \(\rho\) is as follows:  
\[\rho_{1}=\left\{ \begin{array}{rr}&1&0<t\leq 5s\\ 0.7&5s<t\leq 10s\end{array} \right.\quad,\,\rho_{2}=\left\{ \begin{array}{rr}&1&0<t \leq 4s\\ 0.6&4s<t\leq 12s\\ 0.5&12s<t\leq 20s\end{array} \right.\]  
In the simulation, we compare the control effect with the proposed RBFTHC and the computed torque control method (CTC) proposed by [PERSON] and [PERSON] [PERSON] and [PERSON] (2018), which does not consider the fault-tolerant control. The CTC is:  
\[\mathbf{\tau}_{\rm s}=\mathbf{D}_{\rm 0s}(\mathbf{\eta})\left(\tilde{q}_{\rm 0d}-\mathbf{K}_{ \rm v}\tilde{\mathbf{x}}_{1}-\mathbf{K}_{\mathbf{p}}\tilde{x}_{1}\right)+\mathbf{C}_{\rm 0s}\]  
The comparison simulation results are shown in Figures 16-19. The numerical comparison results of \(\tilde{\mathbf{x}}\) and \(\sigma\) under the two control methods are shown in Table 4. According to the results, we find that when actuator failure occurs, RBFTHC can still ensure that the system has good control quality. The motion trajectory of the joint angle of the system can still track the desired motion trajectory, and the vibration is suppressed. However, CTC has  
Figure 14: The angle error \(\sigma\) when \(\mathbf{\tau}_{\rm t}\) is off. Figure 15: The joint angular velocity \(\tilde{q}_{\theta}\) when \(\mathbf{\tau}_{\rm t}\) is off. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **The Output Error \(\tilde{\mathbf{x}}(\times 10^{-3}\text{rad})\)** & **The Angle Error \(\sigma(\times 10^{-3}\text{rad})\)** \\ \hline RBFTHC & \(-4.6\)–\(5.1\) & \(-4.6\)–\(5.8\) \\ \(u_{p}\) is off & \(-100\)–\(80\) & \(-10\)–\(10\) \\ \(\tau_{f}\) is off & \(\infty\) & \(\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The simulation results. a poor trajectory-tracking effect. The output error \(\widetilde{\mathbf{x}}\) in CTC is 12 times larger than that in RBFTHC. The angle error \(\sigma\) in CTC is 2 times larger than that in RBFTHC. **Simulation 3**: The actuator has a serious fault. The change of \(\rho\) is as follows:  
\[\rho_{1}=\left\{ \begin{array}{cc}1&0<t\leq 5\text{s}\\ 0.5&5\text{s}<t\leq 10\text{s}\\ 0.3&10\text{s}<t\leq 20\text{s}\end{array} \right.\]  
The comparison simulation results are shown in Figures 20-23. The numerical comparison results of \(\widetilde{\text{x}}\) and \(\sigma\) under the two control methods are shown in Table 5. According to the results, we find that when an actuator has a serious fault, RBFTHC can also ensure that the system has good control quality.