## 5 Numerical Simulation and Validation Analysis
### Simulation and Verification of LADR-SMC with Smith Predictor
By considering the joint-angle tracking results in Figures 12 and 13, as well as the joint-torque variations in Figure 14, it is evident that under uniform random disturbances and considering the system delay and motor threshold, the LADRC-SMC-SP scheme outperforms the scheme without the SP and demonstrates better feasibility. By compensating for the delay in advance, the LADR-SMC-SP mitigates the impact of the system delay on the control performance. The controller can accurately estimate and counteract system disturbances and improve the system's robustness and control precision. In the next step, we will analyze the contact-force tracking results to validate the effectiveness of the LADRC-SMC-SP scheme. Figure 14: Joint-torque tracking of LADRC-SMC scheme with or without SP under random disturbance. Figure 15: Contact-force tracking of LADRC-SMC scheme with or without SP under random disturbance. The ideal contact force is represented by \(f=50\)cos\(t\). This study assumes that the contact force at the foot end of a lightweight-legged robot ranges from \(-50\) N to \(50\) N. The tracking of the contact force and ideal contact force is illustrated in Figure 15 for both schemes with and without SP. Although both algorithms exhibit similar overall trends in tracking the ideal contact force, local fluctuations are observed. By zooming in on the highlighted fluctuation points in the figure, we can clearly observe the characteristics and trends of these fluctuations. Specifically, the \(f_{Smith}\) of the LADRC-SMC-SP scheme exhibited the maximum fluctuation amplitude at time \(t=15.8\) s, whereas the \(f_{delay}\) of the LADRC-SMC scheme exhibited the maximum fluctuation amplitude at time \(t=26.34\) s. The specific differences are shown in Figure 16.  
Under uniform random disturbances and considering the system delay, as shown in Figure 16, it can be observed that at t = 15.8 s, the maximum tracking error of the contact force for the LADRC-SMC-SP scheme is 0.343534 N. By calculation, the maximum contact force at this time is 48.11088996 N, resulting in an error rate of 0.687%. At t = 26.34 s, the maximum tracking error for the LADRC-SMC scheme is 3.30296 N. According to calculations, the top contact force at this time is 44.808845 N, resulting in an error rate of 7.371%. Figure 16 clearly illustrates that the LADRC-SMC-SP scheme significantly reduces the peak contact-force error rate by more than tenfold compared to the LADRC-SMC scheme. In the third comparison scheme, the ISE values of the contact-force tracking error are respectively 1.8062 \(\times\) 10\({}^{-30}\) N\({}^{2}\cdot\) s and 4.6799 \(\times\) 10\({}^{-29}\) N\({}^{2}\cdot\) s, which are close to 0. In Figures 12 and 13, the ISE metric of the LADRC-SMC-SP scheme is also smaller than that of the LADRC-SMC, indicating a higher control accuracy and more minor cumulative error for the LADRC-SMC-SP scheme. In conjunction with the analysis presented in Figures 12-15, this finding demonstrates that the LADRC-SMC-SP scheme outperforms the LADRC-SMC scheme regarding joint angle, contact-force tracking, and control-law output amplitude. The simulation results in Section 5.3 validate the superior performance of the LADRC-SMC-SP scheme compared to the LADRC-SMC scheme. This multi-loop controller aims to address the force-position tracking and system delay in legged robots under disturbances. The LADRC-SMC-SP system exhibited a stable response and effective delay suppression, whereas the LADRC-SMC system demonstrated instability and lag. The results of the analysis regarding the tracking behavior of joint angles \(\theta_{1}\) and \(\theta_{2}\) consistently support the superiority of the LADRC-SMC-SP scheme over the LADRC-SMC scheme. Analysis of the joint-torque variations revealed that the LADRC-SMC-SP scheme maintained stable periodic changes within the limits of the motor, whereas the LADRC-SMC scheme exhibited oscillatory instability and exceeded the motor saturation threshold.