## 5 Experiment  
The real hydraulic system was used to demonstrate the trajectory tracking performance of TSMCSPO. The experiment was implemented on link 1 of the hydraulic system using TSMCSPO and SMCSPO. We expected the hydraulic system had many nonlinearities and uncertainties, resulting in the controller without the perturbation observer having a bad performance. Assuming the boundary of perturbation in the real system is less than \(\Gamma_{1}=1V\) shown as the unit of voltage. It can be shown in the unit of torque as \(\Gamma_{1}\cdot ratio_{servo-value}(\frac{N}{V})\approx 2500\)\(N\). Hence, the maximum of perturbation was assumed as 2500 N. With the purpose of how the gain \(\lambda_{d}\) affects the controller, the experiments were carried out twice with different \(\lambda_{d}\) values. The design parameters of TSMCSPO and SMCSPO were the same as the previous simulation, only with different \(\lambda_{d}\) values (i.e., 10 and 20). In the real experiment, chattering was reduced by using the saturation function \(sat(\hat{s}_{1})\) instead of the term \(sign(\hat{s}_{1})\) in Equation (34), which is defined as \(if\big{|}\hat{s}_{j}\big{|}\geq\varepsilon_{cj}sat(\hat{s}_{j})=\frac{\hat{s} _{j}}{|\hat{s}_{j}|}\) and \(if\big{|}\hat{s}_{j}\big{|}\leq\varepsilon_{cj}\frac{\hat{s}_{j}}{\varepsilon _{cj}}\), \(\varepsilon_{cj}\) is positive constant, and \(\varepsilon_{c1}\) was selected as 0.2 to balance the steady state error and the chattering. Detailed analysis is shown in the Appendix A.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Controllers** & **SMC a** & **SMCSPO b** & **TSMC c** & **TSMC** \\ \hline Maximum error (degree) & 0.134 & 0.0372 & 0.133 & 0.0042 \\ Steady state error (degree) & 0.0432 & 0 & 0.0021 & 0 \\ Undershoot max. error (degree) & - & 0.0466 & - & 0.0053 \\ Settling time (s) & 25.28 & 33.59 & 24.18 & 29.75 \\ \hline \end{tabular}  
* SMC: Sliding Mode Control; b SMCSPO: Sliding Mode Control with Sliding Perturbation Observer; c TSMC: Terminal Sliding Mode Control; d TSMCSPO: Terminal Sliding Mode Control with Sliding Perturbation Observer.  
\end{table}
Table 2: Performance parameters.  
Figure 8: Comparison between error trajectories of SMC.  
The experimental result of the comparison between error trajectories of TSMCSPO and SMCSPO at the value of \(\lambda_{d}\) was 10, as shown in Figure 9. The red line shows the tracking error result of TSMCSPO, and the blue dotted line shows the result of the tracking error using SMCSP. The maximum error of TSMCSPO was 0.292 degrees at 6 s. The maximum error of SMCSPO was 0.985 degrees at 6.5 s. In the step response area, the small chattering was caused by the revolution of the encoder.  
Figure 10 shows the experimental result of comparing the error trajectories of TSMCSPO and SMCSPO, where the value of \(\lambda_{d}\) was 20. The red line shows the experimental result of TSMCSPO error, and the blue dotted line shows the experimental result for the error of SMCSPO. The maximum error of TSMCSPO was 0.2285 degrees at 17.15 s, and the maximum error of SMCSPO was 0.6864 degrees at 17.4 s.  
Both SMCSPO and TSMCSPO had a 0 steady state error (the continuous oscillation is caused by the resolution of the encoder whose least measurable unit is 0.018 degree.). The trajectory tracking error of TSMCSPO was significantly less compared to that of SMCSPO in both experiments. The error of the trajectory also decreased when the value of \(\lambda_{d}\) increased, with a small increase in chattering. The reason for this was that the phenomenon was affected by the increased switching of control input gain (i.e., \(k_{1}\) in Equation (34)). These data indicate that TSMCSPO has a fast convergence time and less undersho  
Figure 10: Comparison between error trajectories of TSMCSPO and SMCSPO (\(\lambda_{d}=20\)).  
Figure 9: Comparing error trajectories of TSMCSPO and SMCSPO (\(\lambda_{d}=10\)).