## 4 Simulation  
We use the proposed robust backstepping fault-tolerant hybrid control (RBFTHC) to simulate the system shown in Figure 1. The parameters of the system are shown in Table 1. In the simulation, it is assumed that the load's mass \(m_{\mathrm{p}}\) and the moment of inertia \(J_{\mathrm{p}}\) are uncertain parameters. The estimate values are: \(m_{\mathrm{p}}^{\prime}=1.5\,\mathrm{kg}/J_{\mathrm{p}}=1\,\mathrm{kg}\cdot \mathrm{m}^{2}\). The parameters of the control and simulation are shown in Table 2.  
Figure 3: The control system. **Simulation 1**: The actuators work without fault, that is, \(\rho_{1}=1\), \(\rho_{2}=1\). **Case 1**: Figures 4-7 are the simulation results by using the proposed RBFTHC. The simulation results show that the system can achieve the asymptotic tracking of the joint desired motion trajectory accurately and stably. The output error \(\widetilde{\mathbf{x}}\) is very small (only from \(-6\) to \(3\times 10^{-3}\) rad), which reflects the excellent control precision of the system. The error angles caused by the flexible-joint are only from \(-1\) to \(7\times 10^{-3}\) rad, which indicates that the influence of the flexible-joints on the system is compensated, and that motion accuracy and stability of the system are achieved. The observation of the joint velocity based on the speed filter is also accurate, which reflects the validity and correctness of the proposed speed filter. In conclusion, the effectiveness of RBFTHC is verified. **Case 2**: To prove the control effect of the robust controller \(u_{\mathrm{P}}\) in RBFTHC, we turn off \(u_{\mathrm{P}}\). The simulation results are shown in Figures 8-11. From the results we can find that when \(u_{\mathrm{P}}\) is off, the tracking accuracy of the joint angle's desired trajectory is not good. The output error \(\widetilde{\mathbf{x}}\) becomes larger (from \(-116\) to \(70\times 10^{-3}\) rad) than the case of \(u_{\mathrm{P}}\) is on (from \(-6\) to \(3\times 10^{-3}\) rad). The system angle error \(\sigma\) becomes larger (from \(-5.8\) to \(10\times 10^{-3}\) rad) than the case of \(u_{\mathrm{P}}\) is on (from \(-1\) to \(7\times 10^{-3}\) rad). The observation of the joint velocity is not accurate. The reason for these results is that the uncertain parameters and the external disturbances of the system are not effectively compensated for. So, the simulation results \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Body** & **Mass (kg)** & **Length (m)** & \
\begin{tabular}{c} **Moment of** \\ **Inertia (kg\(\cdot\)m\({}^{2}\))** \\ \end{tabular} &
\begin{tabular}{c} **Stiffness Coefficient** \\ **(N\(\cdot\)m/rad)** \\ \end{tabular} \\ \hline base & 40 & 1.5 & 34.17 & \\ Link \(B_{1}\) & 2 & 3 & 1.5 & \\ Link \(B_{2}\) & 1 & 3 & 0.75 & \\ Motor rotor & 0 & 0 & 0.5 & 100 \\ load & uncertain & uncertain & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the system. Figure 4: The trajectories of the joint angle \(q_{\theta}\). \begin{table}
\begin{tabular}{l l} \hline \hline
**The control parameters** & \(\mathbf{\Lambda_{1}=diag(0.1,0.1)}\), \(K_{\mathrm{P}}=diag(50,50)\), \(K_{\mathrm{v}}=diag(10,10)\), \(K_{\mathrm{s}}=diag(10,10)\), \(A=diag(30,30)\), \\  & \(B=diag(100,100)\), \(a=0.05\), \(p=50.1\)\(\left\|\widetilde{\mathbf{x}}_{2}\right\|\) \\ The desired trajectories & \(q_{1d}=\sin(\frac{T}{10}t)\)\(q_{2d}=cos\big{(}\frac{T}{10}t\big{)}\) \\ The system initial values & \(q(0)=[1.9\)\(0.1\)\(1.5]^{T}rad\), \(\theta(0)=[0.1\)\(1.5]^{T}rad\) \\ The external disturbance & \(\tau_{d}=\left[\sin(t)\right.\)\(\cos(t)\)\(\big{]}^{T}N\cdot m\) \\ The simulation time & \(t=40s\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of the control. **Case 3:** In order to verify the control effect of the fast subsystem's controller \(\mathbf{\tau}_{\text{f}}\) in RBFTHC, we turn off \(\mathbf{\tau}_{\text{f}}\). The simulation results are shown in Figures 12-15.