## 4 Numerical Simulation and Discussion
### Case Study 1: Non-Smooth Reference Trajectory  
Firstly, the three controllers are tested with the reference trajectory (in radian unit) which has the following form  
\[x_{1d}(t)=\left\{ \begin{array}{ll}0,&\text{if}\quad t<5\,\text{s}\\ \pi/8,&\text{if}\quad t\geq 5\,\text{s}\end{array} \right. \tag{64}\]  
The tracking performances of all controllers are illustrated in Figure 5. To clearly demonstrate the tracking capabilities of all controllers in the steady-state regime, two zoom-in figures in the top-left region (from the 1 st second to 3 rd second) and bottom-left region (from the 7 th second to 9 th second) of Figure 5 are given. As shown, the proposed control strategy possesses the smallest tracking error during these periods. Meanwhile, due to the lack of the mismatched disturbance compensation mechanism, the SESO-BC controller performs the worst with the largest tracking error. In addition, regarding the transient performances of all controllers, it can be seen from the sub-figure (from the 4.8 th second to the 5.5 th second) in the bottom right corner of this figure that although the proposed control method exhibits the longest transient time compared to the DESO-BC and SESO-BC controllers, the smallest overshoot is achieved by the recommended controller. Moreover, the original reference trajectory is smoothened by the designed nonlinear filter-based trajectory planner (32) and (33), hence, the recommended approach guarantees the prescribed tracking performance as depicted in Figure 6 even when the desired trajectory changes abruptly and under the influences of the lumped mismatched and matched disturbances on the mechanical part and hydraulic dynamic. The predefined time-varying tracking performance is given by \(\rho(t)=(\pi/4-0.002)e^{-10t}+0.002\), \(\underline{v}=1\), and \(\overline{v}=1\). It should be emphasized that only the tracking-error curve of the proposed controller produced by the deviations between the system output and customized reference trajectory is given for fair comparison since the remaining controllers completely cannot fulfill the prescribed performance in the case of step desired trajectory. The performance indexes of all controllers in the steady-state are given in Table 2. It should be noted that the recommended control approach outperforms the remaining controllers in terms of all performance indexes. Specifically, the DESO-BC and SESO-BC controllers violate the output tracking error constraint with the maximal absolute tracking errors being \(23.0193\times 10^{-4}\) (rad) and \(57.6372\times 10^{-4}\) (rad), respectively. Meanwhile, this index attained by the suggested controller is only \(7.5795\times 10^{-4}\) (rad). The better results are obtained by the suggested controller in terms of the remainder indexes. It indicates the effectiveness of the recommended control framework in the disturbance attenuation ability and tracking capability. The control actions generated by all controllers are demonstrated in Figure 7. It can be seen from this figure that due to the influences of disturbances on system dynamics, the non-zero control actions are always generated to compensate for their effects to maintain the HRA at the desired position. Furthermore, it should be mentioned that when the reference trajectory suddenly varies at the 5 th second, because of the lack of the trajectory planner, \begin{table}
\begin{tabular}{c c c c} \hline
**Controller** & \(\mathbf{M_{e}}\) **(rad)** & \(\mathbf{\mu_{e}}\) **(rad)** & \(\mathbf{\sigma_{e}}\) **(rad)** \\ \hline Proposed & \(7.5795\times 10^{-4}\) & \(4.7642\times 10^{-4}\) & \(2.0903\times 10^{-4}\) \\ DESO-BC & \(23.0193\times 10^{-4}\) & \(15.3353\times 10^{-4}\) & \(7.0569\times 10^{-4}\) \\ SESO-BC & \(57.6372\times 10^{-4}\) & \(38.8567\times 10^{-4}\) & \(17.4823\times 10^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Performance indexes of all controllers in the case of step reference trajectory in the steady-state regime. Figure 5: Tracking performances of all controllers in case of step reference trajectory.