## 5 Numerical Simulation and Validation Analysis
### Simulation and Verification of LADR-SMC with Smith Predictor
The tracking effectiveness of the contact force and the error analyses further validated the efficacy of the  
Figure 16: Comparison of contact-force tracking error of LADRC-SMC scheme with or without SP under random disturbance. LADR-SMC-SP scheme. In conclusion, introducing SP in the LADR-SMC scheme enhances the control accuracy, improves system robustness, and is valuable for applying single-leg force-position tracking under interference delay and limited motor amplitude. The LADRC-SMC-SP scheme effectively addresses the challenges associated with joint-angle and contact-force tracking control in a single-legged, multijoint robot. This LADRC-SMC-SP scheme overcomes the problems of disturbances, system delays, and motor limitations (Compare References [7; 8; 9] in the introduction). Through a comprehensive analysis of three different sets of controller designs and corresponding experimental simulations, the following conclusions can be drawn regarding our research topic:  
* The LADRC-SMC-SP scheme improves the controller capabilities of each loop in a multi-loop system. LADRC addresses the challenges of external perturbations and model uncertainties in single-legged systems, SMC handles parameter variations and unmodelled errors, and SP resolves sensor delays. The multi-loop controller enhances the system's robustness, stability, and convergence;
* The first set of comparison experiments demonstrated that the SMC scheme performed better in joint-angle tracking than the MPC scheme under step or uniform random interference. This implies that the robustness (Equations (18) and (19)), the response speed (Equations (17) and (19)), and the system structure (Equations (16) and (20)) of the designed SMC scheme are superior to those of the MPC scheme. However, the shortcomings of contact-force tracking also reveal the limitations and challenges of the SMC scheme, including the accuracy of contact-force modeling (Equations (8)-(10)), ability to handle nonlinear contact forces (refer to \(f_{\uparrow}\) in Section 4.1.2), and the empirical nature of parameter adjustment (Table 2);
* The conclusion drawn from the second experiment is that the devised LADRC-SMC scheme not only guarantees accurate joint-angle tracking (see Equation (28) and Figure 9) but also significantly enhances the contact-force tracking performance (see Equations (25)-(27) and Figure 11). The LADRC-SMC scheme enhances the performance of single-legged systems by mitigating the two types of interference and achieving precise force-position tracking. Nevertheless, single-legged systems commonly encounter two types of delays (see Section 4.2.3), and the LADRC-SMC scheme is ineffective in addressing these delays (see Figures 12-16);
* The findings of the third comparative experiment suggested that MPC, LADR-SMC, and LADR-SMC-SP exhibit progressive levels of optimality. The LADR-SMC-SP scheme effectively resolves multiple challenges in single-legged systems such as dual interference (refer to Figures 12-16 and Equations (27) and (32)), dual delay (refer to Figures 12 and 13 and Equations (29) and (30)), and motor limitations (Figure 14). In a single-legged system model, the LADR-SMC-SP scheme is well suited for contact-force control and position control of multijoint objects experiencing interference delay and limited motor amplitude;
* The design of this scheme has a strong portability. Regarding model design, the derivation of the reduced-order model is based on actual and reasonable physical scenarios. The derivation process is rigorous and can reflect the actual physical situation. In terms of parameter selection, the parameters in Table 1 were selected according to the physical knowledge and application scenarios with certain randomness. The parameters in Table 2 were manually adjusted according to the controller design theory and corresponding parameter design laws. The algorithm design and ISE index of the tracking error performance verify that the model design is reasonable and can adapt to various controllers.