## 4 Experimental Evaluation
### Experiment Result  
To quantitatively evaluate the tracking performance, the maximum tracking error (MTE) and root mean square tracking error (RMSTE) are computed. The RMSTE is calculated as follows:  
\[RMSTE=\sqrt{\frac{1}{N}\sum_{k=0}^{N}e_{k}^{2}}, \tag{22}\]  
where \(N\) is the total number of data samples, and \(e_{k}\) is the tracking error under \(k^{th}\) sample. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{2}{c}{**FDISMC**} & \multicolumn{2}{c}{**DSMC**} \\ \cline{2-5}  & \(\alpha\) & \(K_{I}\) & \(\lambda\) & \(K_{sw}\) \\ \hline Values & 0.8 & 0.01 & 0.1 & \(1.5\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the discrete-time fractional order integral sliding mode control (DFISMC) and conventional DSMC controller. Figure 4 depicts the experimental results when the actuator tracks the sine wave signals without load. The sinusoidal signal had an amplitude of 20\({}^{\circ}\) and frequency from 0.2 Hz to 1 Hz. In the second scenario, a knee gait pattern was given as a desired trajectory. The proposed controller was evaluated with two different gait cycle (GC) times: 2.5 s and 4 s. The experimental results of this scenario are shown in Figure 5. In both Figures 4 and 5, the upper sub-figure of each image includes the desired trajectory (back line), measured angle controlled by DFISMC (dash-blue line), and measured angle controller by conventional DSMC (dash-dot red line). The lower part of each figure shows the tracking errors of both proposed controller and DSMC controller. In comparison with the traditional DSMC, the DFISMC was able to provide a better performance in both transient and steady states. As demonstrated in Figure 6, in all scenarios of the experiment, both MTE and RMSTE of the proposed control approach are smaller than the ones of the conventional DSMC control method. For example, when tracking the 1.0 Hz frequency sinusoidal signal, the RMSTEs of the DFISMC and DSMC controllers are 1.63\({}^{\circ}\) and 1.43\({}^{\circ}\), respectively. It means that DFISMC is able to provide a better performance than the conventional DSMC controller. In particular, as seen in the error graphs in Figures 4 and 5, the finite amplitude oscillation of the tracking error in DFISMC is much smaller than in DSMC. It can be concluded that the inherent \"chattering\" phenomenon of SMC control is reduced with DFISMC. The numerical values of the experimental results in all scenarios are given in Table 4.  
Figure 4: Experiment results without a load for tracking a sinusoidal trajectory: (**a**) 0.2 Hz, (**b**) 0.5 Hz, (**c**) 0.8 Hz, and (**d**) 1.0 Hz of signal frequency. Figures 7 and 8 show the control performances of the system when tracking the sinusoidal signals and human-gait pattern with a load of 2.5 kg, respectively. When the antagonistic actuator carries a load m = 2.5 kg, the difference between the DFISMC and DSMC is not significant in terms of MTE. However, the RMSTE of the DFISMC controller are smaller than the ones of the DSMC controller, as shown in Figure 9. For example, when tracking the 2.5 s human-gait trajectory, the RMSTEs of the DFISMC and DSMC controllers are 1.22\({}^{\circ}\) and 1.68\({}^{\circ}\), respectively. Furthermore, the same conclusion \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Signal Frequency**} & \multicolumn{2}{c}{**MTE (\({}^{\circ}\))**} & \multicolumn{2}{c}{**RMSTE (\({}^{\circ}\))**} \\ \cline{2-5}  & **DSMC** & **DFISMC** & **DSMC** & **DFISMC** \\ \hline
0.2 Hz & 3.14 & 2.65 & 1.03 & 0.98 \\
0.5 Hz & 6.01 & 5.71 & 1.12 & 1.00 \\
0.8 Hz & 7.73 & 7.39 & 1.43 & 1.11 \\
1.0 Hz & 8.68 & 8.67 & 1.63 & 1.43 \\
4 s of GC & 2.40 & 2.31 & 1.30 & 1.04 \\
2.5 s of GC & 4.69 & 2.26 & 1.45 & 1.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum tracking error (MTE) and root mean square tracking error (RMSTE) of the proposed control method and conventional DSMC control method in case of no load.