## 5 Numerical Simulation and Validation Analysis
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Index** & **Controller** & **Parameters** \\ \hline
1. & MPC & \(k_{1}=15.42,k_{2}=5.55,k_{3}=k_{1}\) \\ \hline
2. & SMC & \(K_{p}=\begin{bmatrix}5&0\\ 0&2\end{bmatrix},K_{f}=140,\Delta=0.3\) \\ \hline
3. & LADR-SMC & \(\begin{bmatrix}N,\omega_{0},\omega_{c},\omega_{c},b_{0}\end{bmatrix}=[2,15,21,1.04],\)\(LESO=[1.5,-6.5,-15];\) \\  & \(Pa_{SMC}=\begin{bmatrix}K_{p},K_{f},\Delta\end{bmatrix}=\begin{bmatrix}[5&0 \\ 0&2\end{bmatrix},140,0.3\end{bmatrix}\) \\ \hline
4. & LADR-SMC-SP & \(\begin{bmatrix}N,\omega_{0},\omega_{c},b_{0}\end{bmatrix}=[4,5,10,100],\)\(LESO=[0,0,1,1,1];\) \\  & \(Pa_{SMC}=\begin{bmatrix}[5&0\\ 0&2\end{bmatrix},140,0.3\end{bmatrix};\)\(\sum\limits_{i=1}^{4}c_{i}=[5,1,3,5]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Set physical parameters based on actual conditions. values of \(9.0265\times 10^{-29}\) rad\({}^{2}\cdot\) s for SMC and \(0.0142464\) rad\({}^{2}\cdot\) s for MPC. Because joint angle \(\theta_{2}\) can be expressed in terms of joint angle \(\theta_{1}\), the SMC also surpasses the MPC in the studied scenarios with a uniform random disturbance. Consequently, this study opts for an SMC scheme for further analysis and optimization by comparing these disturbances. Subsequently, we investigated the performance of the SMC scheme in tracking the contact force under these two disturbances (Figure 8). The ideal contact force is represented by \(f=50\)cos\(f\). In Figure 8, at time \(t=22.3\) s, the maximum tracking error of the contact force under unit step disturbance is \(0.82167\) N. By calculation, the contact force at this time is \(46.26048\) N, with an error peak ratio of \(1.776\%\). At time \(t=24.5\) s, the maximum tracking error of the contact force under a uniform random disturbance was \(8.34877\) N. By calculation, the contact force at this time was \(45.49806\) N, and the error peak ratio was \(18.350\%\). In the first group of winning SMC schemes, the ISE value of the contact-force tracking error under random disturbance is smaller than that under step disturbance, as evidenced by the ISE values of \(2.2205\times 10^{-28}\) N\({}^{2}\cdot\) s for random disturbance and \(8.0582\times 10^{-30}\) N\({}^{2}\cdot\) s for step disturbance. Consequently, the contact-force error of the SMC exhibits significant differences between the two types of disturbances. Figures 6-8: In the first set of comparisons, the results indicate that while the SMC scheme outperforms the MPC scheme (compare References [13; 14] in the introduction), there is room for improvement in reducing response time for joint-angle tracking and error rate for contact-force tracking under uniform random disturbance. To address this issue, we have combined the robust control capability of LADRC in suppressing external disturbances and model uncertainties with the ability of SMC to suppress parameter variations and unmodeled errors. The integration of LADRC and SMC enables each method to leverage its strengths, enhancing overall disturbance rejection performance and improving control effectiveness. For specific details and effects analysis, please refer to Equation (28) and Section 5.2 for comparative simulation results. Simulation Verification and Comparative Analysis of LADRC-SMC and SMC Schemes under Uniform Random Disturbance  
To implement the LADRC-SMC-SP scheme designed for the research problem, a comparative simulation of the second scheme was conducted based on the conclusions drawn from the comparative verification of the first scheme in Sections 4.1, 4.2, and 5.1. A second-order LADR-SMC scheme is proposed in the theoretical section to improve the poor contact-force tracking performance of the SMC scheme. The joint angle tracking performance is shown in Figure 9.  
Figure 8: Comparison of \(e_{f}\) of SMC schemes under two disturbances.