## 4 Simulation Results  
The validity of the proposed control strategy is confirmed through simulation studies in this part. The parameters of the exoskeleton mechanical system are shown in Table 1. The desired trajectory for rehabilitation is given as \(\theta_{d}=43.2\cos(\pi t)/\pi+64.8\sin(2\pi t)/\pi+90\). Due to the fact that the state variable of the exoskeleton EHA (5) is the rod displacement, the geometric transformation (10) and (11) should be solved to acquire the desired rod displacement before implementing the controller. The geometric transformation is shown in Figure 3. Consequently, the control objective of the following simulation is to track the desired rod displacement \(y_{pd}\) (the red curve). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Symbol** & **Value** \\ \hline \(l_{1}\) & Mechanical length 1 & 6 cm \\ \(l_{2}\) & Mechanical length 2 & 26 cm \\ \(l_{3}\) & Mechanical length 3 & 7 cm \\ \(l_{4}\) & Mechanical length 4 & 5 cm \\ \(l_{7}\) & Mechanical length 7 & 13 cm \\ \(m\) & Equivalent load acted on the piston rod & 15 kg \\ \(c\) & Combined friction coefficient & 950 \\ \(S_{1}\) & The piston area of the non-rod chamber & 7 cm\({}^{2}\) \\ \(S_{2}\) & The piston area of the rod chamber & 3 cm\({}^{2}\) \\ \(\beta_{e}\) & Effective bulk modulus & \(9\times 10^{4}\) N/cm\({}^{2}\) \\ \(\rho\) & The areas ratio & \(S_{1}/S_{2}\) \\ \(k_{s}\) & The valve flow/signal gain & 0.95 \\ \(V_{1}\) & Initial volume of the non-rod chamber & 85 cm\({}^{3}\) \\ \(V_{1}\) & Initial volume of the rod chamber & 6.5 cm\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the exoskeleton mechanical system. Figure 2: The proposed control structure. The external disturbance of the rehabilitation exoskeleton is given as \(f_{e}(t)=1.5\sin(2\pi t)\)\(+0.5\cos(\pi t)+2\), and the human-exoskeleton interaction is \(f_{h}(t)=2\sin(2\pi t)+3\cos(\pi t)+1.5\). The error constraint is set to \(k_{c}=0.05\). The unknown leakages of the EHA are  
\[\left\{ \begin{array}{rcl}Q_{li}(t)&=&0.2\pi\sin(2.5\pi t)+1.5\text{rand}(1), \\ \Delta_{li\epsilon 1}(t)&=&1.2\sin(1.5\pi t)+2\text{rand}(1),\\ \Delta_{l\epsilon 2}(t)&=&2\cos(\pi t)+\text{rand}(1).\end{array} \right. \tag{58}\]  
The actuator fault is supposed to happen at \(t_{h}=2\) and \(t_{\delta}=4\) with both enduring for \(1\) s. The healthy indicator and the bias faults are given as  
\[h=\left\{ \begin{array}{l}0.4,\,2\leq t\leq 3,\\ 1,\,0<t<2,3<t.\end{array} \right.\quad\delta=\left\{ \begin{array}{l}5,\,4\leq t \leq 5,\\ 0,\,0<t<4,5<t.\end{array} \right. \tag{59}\]  
To perform the proposed controller, a RBF-NN with \(11\) nodes is applied. The input of the NN is \(X=[x_{1}^{T},x_{2}^{T},e_{1}^{T},e_{2}^{T},\mu_{2}]\). The network gain matrix is set to \(\Xi=\text{diag}[30]_{11\times 11}\) with the initial weight \(\Psi(0)=[0]_{11\times 1}^{T}\). The control parameters are selected to \(k_{1}=60\), \(k_{2}=20\), \(k_{3}=1.5\), and \(k_{o}=20\). The initial value of the state is \(x=[8,0,0]\). Responses of the proposed controller are shown in Figures 4-6. Figure 4 displays the curves of the desired rod displacement \(y_{pd}\) and the actual rod displacement \(y_{p}\). It is clearly indicated that the trajectory tracking of the rod displacement is achieved by using the proposed control scheme. Figure 5 shows the tracking error and the bounds of the error constraint. The error constraint is not violated with the aid of the barrier Lyapunov function. Figure 6 shows the commanded controller and the actual controller. We can clearly see that the multiplicative fault happens at \(t=2\) and disappears at \(t=3\), and the bias fault happens at \(t=4\) and disappears at \(t=5\). These figures demonstrate that the control objective is reached by the proposed control scheme although the system is facing the difficulty of modeling uncertainties, unknown human-robot interaction, external disturbance, and actuator fault.