## 4 Simulation and Experiment  
### Simulation  
In this section, two simulation cases were built to verify the validity of the obtained controller design method in the exoskeleton robot. By employing a one-DOF PAM exoskeleton system as the control plant in Equation (41), the relative parameters were obtained from the relative reference [47]. The reference trajectory was selected as a sinusoidal curve given by Equation (42):  
\[0.75\dot{q}+0.17q+0.84\sin(q)+0.05 sign(q)=\tau \tag{41}\]  
\[y_{d}=30\sin(0.2\pi t) \tag{42}\]  
In the first group of the simulation, the initial joint positions from the two controllers (the original MFAC-SMC-PP method and the proposed method) were set to 0 degrees. The proposed controller parameters were selected as \(\eta\)=10, \(\mu\)= 0.6, \(k_{in}=5\) s, \(\rho=0.5\) degrees, \(K=0.5\), \(\Phi_{1}(0)\)=\(\Phi_{2}(0)\)=0.75, \(\Gamma_{1}=0.05\) and \(\Gamma_{2}=0.01\). In addition, the original MFAC-SMC-PP controller was designed as the comparison method, and its controller parameters are selected to be identical to the proposed controller except for the prescribed performance function (i.e., \(\rho_{0}=5\), \(v=0.05\), and \(\rho_{\infty}=0.05\)). Moreover, to further demonstrate that the proposed method was relieved from the requirement of the initial tracking error, the initial joint position was set to \(-10\) degrees in the second group. The tracking trajectory and controller parameters remain.  
The results were obtained from the discussed simulations (in Figures 5-10). In group 1, the system output and tracking error generated from the proposed method are consistent with the results achieved using the original method (i.e., the root mean squared error: 0.15 degrees with the original method and 0.18 degrees with the proposed method; the maximal absolute value of the tracking error: 3.81 degrees with the original method and 3.25 degrees with the proposed method). Since the two methods cover prescribed performance functions, the tracking error is finally converged to the upper boundary and the lower boundary (i.e., (\(-0.5\), 0.5) degrees). Furthermore, the tracking error obtained by the proposed method enters the boundary in finite time (nearly 0.05 s), thereby satisfying the time requirement (i.e., 0.05 < \(k_{in}\)).  
Figure 5: Output signals of the simulation in group 1 (original method).  
Moreover, when the original method is applied, the initial errors should be confined in the predesigned boundary. Once a novel tracking task is assigned or the system needs to restart with an unknown position, certain design parameters of the controllers should be reselected--which is inconvenient for the user. For instance, in group 2, the initial position was set to \(-10\) degrees. After several adjustments, the tracking error converged into the predesigned boundary before \(k_{in}\) in the proposed method, as shown in Figure 10  
Figure 8: Tracking errors of the simulation in group 1 (proposed method).  
Figure 6: Tracking errors of the simulation in group 1 (original method).  
Figure 7: Output signals of the simulation in group 1 (proposed method).  
Figure 9: Output signals of the simulation in group 2 (proposed method).  
(i.e., the gray area). The root mean squared of the tracking error is equal to 0.41 degrees, and the maximal absolute value of the tracking error is 10 degrees (i.e., \(|e(0)|\)). However, since \(|e(0)|>|\rho_{0}|\), the original control method cannot work until the parameters of the prescribed performance function are reselected (particularly in \(\rho_{0}\) and \(v\)). Based on the discussed results, the joint position tracking trajectory task for the upper-limb exoskeleton can be fulfilled with the proposed method.