## 5 Discussion and Conclusions  
This paper proposed an advanced SMC control strategy for PAMs in antagonistic configuration. First, the discrete-time SOPDT is chosen to describe the dynamic behaviour of the antagonistic actuator. The chosen model demonstrated a good approximation of nonlinear characteristics of the actuator: the root mean square errors between estimated and measured values are less than 2.5\({}^{\circ}\). Based on the built-in model, an DFISMC controller, which employed a fractional order integral of tracking error together with a disturbance observer, was proposed for the tracking purpose. The implemented approximation of FOI and DO was able to reduce the _\"chattering\"_ phenomenon, which often occurs in SMC implementations. The reduction of the _\"chattering\"_ phenomenon is very important for applications of the PAMs in rehabilitation robot field. Finally, multi-scenario experiments were carried out to compare the tracking performances between the DFISMC and the conventional DSMC. In comparison with the three-elements model [5], hysteresis model [13; 14; 15], and mechanism-based model [22; 23], the identification procedure of the proposed method is simplified. Besides, this procedure does not need to measure the load's acceleration, which is very sensitive to noise. Experiments show that, in comparison with DSMC, the DFISMC was able to significantly enhance the tracking performance of 20\({}^{\circ}\) amplitude sinusoidal signals with frequency up to 1.0 Hz. In particular, when the actuator drove a load of m = 2.5 kg, the RMSTEs of DFISMC were about two times less than those of conventional DSMC in most of the desired trajectory frequencies. For example, with a frequency of 0.2 Hz, the RMSTEs are 0.93\({}^{\circ}\) and 1.67\({}^{\circ}\) for DFISMC and DSMC, respectively. The proposed controller achieves a performance comparable to the experimental results with similar configuration and desired trajectory in [23; 24]. In [23], when tracking a 0.4 Hz frequency and 5\({}^{\circ}\) sinusoidal signal, the residual error amplitude is 0.5 \({}^{\circ}\) equivalent to 10%. When tracking a 0.5 Hz frequency and 20\({}^{\circ}\) amplitude sinusoidal signal, the RMSTE of the DFISMC is 1.47 \({}^{\circ}\), equivalent to \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 & 3.94 & 2.16 & 1.67 & 0.93 \\
0.5 Hz & 5.11 & 5.39 & 2.31 & 1.47 \\
0.8 Hz & 8.13 & 7.13 & 2.64 & 1.56 \\
1.0 Hz & 10.56 & 11.13 & 3.28 & 2.61 \\
4 s of GC & 4.09 & 2.20 & 1.38 & 1.16 \\
2.5 s of GC & 5.23 & 3.41 & 1.68 & 1.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: MTE and RMSTE of the proposed control method and conventional DSMC control method with load m = 2.5 kg. Figure 9: MTE and RMSTE of the proposed controller and conventional DSMC controller with 0.2 Hz, 0.5 Hz, 0.8 Hz, and 1.0 Hz of the desired signal frequency and load m = 2.5 kg. 7.35% of amplitude. This result was better than the one in [24], in which a sinusoidal signal with 40\({}^{\circ}\) amplitude and frequency 0.25 Hz is used as a desired trajectory. The experiments also show that the proposed controller can track a human-gait pattern with the MTE of less than 6\({}^{\circ}\). This result is in accordance with the commercial gait training system LOKOMAT [30], in which the MTE is 15\({}^{\circ}\). It is shown that the built-in model and proposed controller can be applied in robot gait training system. Furthermore, the proposed DFISMC is designed in discrete-time domain, so that it is convenient for implementing in any digital industrial controller, e.g., the NI instrument in this research. In summary, this paper presents the control of an antagonistic actuator powered by PAMs. The dynamic behaviour of the antagonistic actuator is described by a discrete-time SOPDT model, which requires a simpler identification procedure.