## 4 Experimental Results and Discussion
### Experimental Results  
In this subsection, the proposed DTSMC is tested for the high-precision trajectory tracking control of the DEAP actuator. First, we select the optimal control parameters of the proposed controller. If \(\alpha\) is fixed, a lager \(\kappa_{2}\) can guarantee a better performance. In addition, if choosing \(\beta=1/2\) which implies \(1/\beta=1/(1-\beta)=2\) and \(\sigma=O(T^{2})\), we can obtain the smallest boundary of the final sliding mode motion. Disturbances can be well suppressed with the increasing of \(\xi_{1}\) and \(\xi_{2}\); however, the chattering phenomenon will be also increased in the control signal. Through trial and error, the selected optimal control parameters are presented in Table 5.  
This work employs the real experimental data to verify the effectiveness of the proposed controller. For comparison, the conventional DSMC with the hysteresis compensator is employed. The control parameters of the DSMC are the same as those of the proposed controller, which are presented in Table 5.  
During experimental studies, different frequencies' input signals are generated by S15-3P and applied to drive the DEAP actuator. The corresponding tracking results of the DEAP actuator are shown in Figures 10-15. The desired displacement has a frequency of 0.1 Hz, the tracking results of the DSMC and the proposed DTSMC are depicted in Figure 10. The RMSE of the DSMC is 0.0565 mm, and the MAE is 0.0207 mm. In contrast, for the proposed DTSMC, the RMSE is 0.0536 mm, and the MAE is 0.0122 mm. Compared with the DSMC, the proposed DTSMC reduces the RMSE and MAE by 5% and 41%. In addition, from Figure 10c,d, one can see that the hysteresis phenomenon has been substantially compensated by the DTSMC, and the DSMC can also basically compensate the hysteresis phenomenon at a low frequency of 0.1 Hz. Figure 9: Experimental platform of the DEAP actuator. In Figure 11, the desired displacement has a frequency of 0.2 Hz. The RMSE of the DSMC is 0.0942 mm, and the MAE is 0.0504 mm. In contrast, for the proposed DTSMC, the RMSE is 0.0612 mm, and the MAE is 0.0215 mm. Compared with the DSMC, the proposed DTSMC reduces the RMSE and MAE by 35% and 57%. In addition, compared with the desired displacement of 0.1 Hz, the proposed DTSMC increases the RMSE and MAE by 12% and 43%. However, the DSMC increases the RMSE and MAE by 40% and 59%, which are all obviously larger than those of the proposed DTSMC. Furthermore, from the Figures 12-14, it can be seen that the hysteresis compensations of the proposed DTSMC gradually exceeds those of the DSMC with increasing frequency of the input signal. For more detail trajectory tracking-error results, see the computational results in Table 6. In addition, the overshoots of the DSMC at the turning points are significantly larger than those of the proposed DTSMC. Therefore, it can be concluded that the proposed DTSMC can improve the tracking control accuracy of the DEAP actuator with increasing frequency to 0.5 Hz. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Controller** & \(\mathbf{\kappa_{1}}\) & \(\mathbf{\kappa_{2}}\) & \(\mathbf{\alpha}\) & \(\mathbf{\xi_{1}}\) & \(\mathbf{\xi_{2}}\) & \(\mathbf{\beta}\) & \(\mathbf{\sigma}\) \\ \hline DSMC & 3.50 & - & 7/11 & 0.25 & 0.50 & 1/2 & 0.01 \\ DTSMC & 3.50 & 0.55 & 7/11 & 0.25 & 0.50 & 1/2 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The control parameters of different controllers. Figure 10: The tracking results under 0.1 Hz input signal. (**a**) Displacement output. (**b**) Tracking error. (**c**) Desired displacement versus actual displacement of the DSMC. (**d**) Desired displacement versus actual displacement of the DTSMC. Figure 11: The tracking results under 0.2 Hz input signal. (**a**) Displacement output. (**b**) Tracking error. (**c**) Desired displacement versus actual displacement of the DSMC. (**d**) Desired displacement versus actual displacement of the DTSMC. Figure 12: The tracking results under 0.3 Hz input signal. (**a**) Displacement output. (**b**) Tracking error.