## 4 Simulation and Experimental Analysis
### Simulation Parameters  
The simulation time was 5 s, the simulation step length was 0.001 s, and the gait period was 2 s. The walking gait data of a normal person with a walking speed of 1.2 m/s was selected as the expected trajectory of each joint. The fitting curve is as follows.  
\[\left\{\begin{array}{l}\theta_{RH}=7.52-12.94\sin(2\pi t-0.29)+4.78\sin(4 \pi t-0.64),\\ \theta_{RK}=-22.7+23.69\sin(2\pi t+1.02)-18.54\sin(4\pi t-0.41),\\ \theta_{LH}=7.52+12.94\sin(2\pi t-0.29)+4.78\sin(4\pi t-0.64),\\ \theta_{LK}=-22.7-23.69\sin(2\pi t+1.02)-18.54\sin(4\pi t-0.41).\end{array}\right. \tag{22}\]  
where \(\theta_{RH}\) is the desired trajectory of the right hip joint; \(\theta_{RK}\) is the desired trajectory of the left hip joint; and \(\theta_{LK}\) is the desired trajectory of the left knee joint.  
From Equation (21), the self-adaptive-coefficient double-power sliding mode control method proposed in this paper was recorded as SACDP-SMC. We set \(\overline{Y}_{ij}^{\prime}=\left|Y_{ij}^{\prime}\right|+0.1\), \(\mathbf{\varepsilon}_{j}=\left|\overline{p}_{j}\right|+0.5\), \(\widetilde{p}_{j}=0.05p_{j}\), \(\overline{p}_{j}=0.95p_{j}f=\left|\mathbf{\tau}_{d}\right|+0.5\), and the control parameter as the optimal parameter obtained after 120 online optimizations. \(\mathbf{\Lambda}=\text{diag}[100,100,100,100]\)  
Figure 4: The control simulation in MATLAB/Simulink.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Segments**} & **Mass** & **Length** & \multicolumn{2}{c}{**Centroid Position**} & **Inertia** \\ \cline{2-6}  & \(m_{i}\)/**kg** & \(l_{i}\)/**mm** & \(l_{Ci}\)/**mm** & \(h_{i}\)/**mm** & \(\mathbf{I_{i}}\)/**(kg\(\cdot\)m\({}^{2}\))** \\ \hline Torso & 48.21 & 874.0 & 358.8 & 16.3 & 2.4093 \\ Thigh & 14.05 & 418.0 & 178.5 & 14.6 & 0.2440 \\ Shank \& Foot & 6.72 & 481.0 & 281.8 & 17.8 & 0.2142 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The parameters of each body segment.  
\(\mathbf{\Omega}=\text{diag}[15,15,15,15]\), \(\mathbf{k_{\alpha}}=[6,6.5,6,6.5]^{\text{T}}\), \(\mathbf{\alpha}=[5/3,5/3,5/3,5/3]^{\text{T}}\), \(\mathbf{k_{\beta}}=[1,1.5,1,1.5]^{\text{T}}\), and \(\mathbf{\beta}=[3/5,3/5,3/5,3/5]^{\text{T}}\).  
To demonstrate the superiority of SACDP-SMC, we named it Method 1 and compared it with the following two control methods:  
Method 2: To highlight the effect of the self-adaptive-coefficient \(\mathbf{\Lambda}\), the double-power sliding mode control method was adopted and recorded as DP-SMC. The actual control torque of each joint of the robot is shown in Equation (13); the control parameters were the same as those described for SACDP-SMC.  
Method 3: To highlight the compensation effect of the double-power sliding mode control, a traditional computed torque method based on PID was adopted, and recorded as PID-CTM. The actual control torque of each joint of the robot is expressed as follows.  
\[\mathbf{u}=\overset{-}{\mathbf{M}}(\mathbf{\theta})\bigg{(}\ddot{\mathbf{\theta}}_{d}+\mathbf{K}_ {P}e+\mathbf{K}_{D}\dot{e}+\mathbf{K}_{I}\int_{0}^{t}ed_{t}\bigg{)}+\overset{-}{\mathbf{C }}(\mathbf{\theta},\dot{\mathbf{\theta}})\dot{\mathbf{\theta}}+\mathbf{G}(\mathbf{\theta}) \tag{23}\]  
where \(\mathbf{K}_{P}\), \(\mathbf{K}_{D}\), and \(\mathbf{K}_{I}\) are control parameters of the computed torque method. \(\mathbf{K}_{P}=[5500,6500,5500,6500]^{\text{T}}\), \(\mathbf{K}_{I}=[200,200,200,200]^{\text{T}}\), \(\mathbf{K}_{D}=[80,80,80,80]^{\text{T}}\).