## 5 Numerical Simulation and Validation Analysis  
The controlled object is given by Equation (3), taking the related parameters of the robot leg as \(p= \begin{bmatrix}2.35&0.9&1&1.75&1\end{bmatrix} ^{T}\) (this parameter is determined via the computation outlined in Equation (23) and the tabulated data presented in Table 1), and the initial state of the controlled objects \(\theta_{1}\) and \(\dot{\theta}_{1}\) are \([1.5,0]\). The position command is \(\theta=0.5+0.8\text{sin}t\) (reference input), and the ideal contact force is \(f=50 cost\) (referenceinput). In this algorithm simulation, the disturbance input is a common step or uniform random number disturbance. In the simulation, the selected parameters were as follows. In Table 2, the parameters of various controllers were selected based on the control target of this study, the controllers' principles, and the tuning experience. The parameter indexed as 1 was obtained through the synthesis of Equation (17) and the exposition provided in Section 4.1.1 in tandem with the empirical controller configurations. The parameter indexed as 2 was derived from the integration of Equation (19), supplemented by the discussion in Section 4.1.2 along with the empirical controller settings. The parameter indexed as 3 was acquired using an amalgamation of Equations (25)-(27) and elucidation in Section 4.2.2, in conjunction with empirical controller adjustments. The parameter indexed as 4 was obtained using Equations (25)-(27), the discussion in Section 4.2.3, and empirical controller configurations. Based on the theoretical derivation sequence of the controller described earlier, three sets of comparative simulations were conducted to validate the effectiveness of the LADRC-SMC-SP scheme effectiveness further. Table 2 lists the three sets of comparative simulations. The initial comparison considered schemes in indexes 1 and 2, the second focused on schemes in indexes 2 and 3, and the third addressed schemes in indexes 3 and 4. The performance of the control systems is evaluated through comparative analysis, error rates, and metrics such as ISE. In control engineering, the ISE is commonly employed to assess the performance of closed-loop control systems, reflecting the extent to which the outputs of the system deviate from the desired values over the entire period. A smaller ISE value indicates better control system performance [36]. Numerical Simulation and Comparative Optimization of MPC and SMC Schemes under the Step or Random Disturbances  
To evaluate the control effectiveness of the multi-loop controller, we conducted the first set of comparative simulations to evaluate the control effectiveness of the MLC. The MPC was reconstructed based on the literature mentioned in the introduction, specifically for the research object of this study. The SMC scheme was designed to address this research problem and is supported by theoretical proof. Because \(\theta_{2}\) can be represented by \(\theta_{1}\) (see Equation (4)), the tracking results for \(\theta_{1}\) are emphasized. Through an analysis of the ISE index, the joint-angle tracking performance under the unit step and random disturbance inputs are presented separately to assess the tracking performance of the joint angles (see Figures 6 and 7) and contact forces (see Figure 8) between the two controllers. Subsequently, a controller demonstrating a superior joint-angle tracking performance was selected to evaluate its contact-force tracking performance in the presence of diverse disturbances. According to the contact-force tracking situation, the controller with a strong position tracking performance was optimized and improved through the second group of comparison schemes.