## 5 Numerical Simulation and Validation Analysis
The joint-angle tracking performance, contact-force tracking ability (see Figure 10), and force tracking error (see Figure 11) in the second scheme were comprehensively analyzed by comparing the performances of the joint \(\theta_{1}\) tracking and contact-force tracking under the influence of a uniform random disturbance. The effectiveness of the upgraded second-order LADR-SMC scheme was evaluated, and new challenges were discussed. The position command is \(\theta=0.5+0.8\)sin\(t\). To ensure the effective tracking of joint angle \(\theta_{1}\), the two schemes were compared, and their corresponding joint-angle \(\theta_{1}\) tracking results are presented in Figure 9. To ensure comparability of the second group of schemes, the values of the parameters \(K_{p},K_{f},\) and \(\Delta\) in the LADRC-SMC scheme were consistent with those in the SMC scheme (refer to Table 2 in Section 5). Under the same uniform random disturbances, both controllers exhibited similar overall trends in tracking joint angle \(\theta_{1}\). Regarding the response speed and convergence rate, the SMC scheme demonstrated significantly poorer tracking performance than LADRC-SMC. Although the tracking trend of the LADRC-SMC initially deviated from the reference trajectory for the first 0.5 s, it achieved comprehensive tracking of the reference trajectory around 1 s, while the SMC scheme only caught up with the reference trajectory after 7.5 s. Under the same uniform random disturbances, the angle \(\theta_{1}\) trajectory for LADRC-SMC is more accurate than that for SMC, as evidenced by the ISE values of \(7.4355\times 10^{-29}\) rad\({}^{2}\cdot\)s for LADRC-SMC and \(9.0265\times 10^{-29}\) rad\({}^{2}\cdot\)s for SMC. By comparing the joint angle \(\theta_{1}\) under these two disturbance scenarios, we can infer a similar situation for joint angle \(\theta_{2}\), thus indicating the superiority of the LADRC-SMC method in joint-angle tracking. Next, we analyze the performance differences between the contact-force tracking schemes. 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 and ideal contact forces is illustrated in Figure 10 for both schemes. 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 LADRC-SMC algorithm exhibited the maximum fluctuation amplitude at time \(t=24.5\) s, whereas the SMC algorithm showed the maximum fluctuation amplitude at time \(t=29.1\) s. The specific differences are shown in Figure 11.  
Under uniform random perturbations, Figure 11 shows that at \(t=24.5\) s, the maximum tracking error of the contact force for the SMC scheme was 8.34877 N. Thus, the contact force at this time was 45.49806 N, resulting in an error peak ratio of 18.350%. At \(t=29.1\) s, the LADRC-SMC scheme exhibited a maximum tracking error of 0.69991 N. The calculations indicate that the contact force was 43.68861 N, resulting in an error peak ratio of 1.602%. In the second comparison scheme, the contact-force tracking error ISE values  
Figure 9: Joint angle \(\theta_{1}\) tracking of LADRC-SMC and SMC schemes under uniform random disturbance. were \(2.2205\times 10^{-28}\) N\({}^{2}\cdot\) s and \(1.7293\times 10^{-29}\) N\({}^{2}\cdot\) s, close to 0. In Figure 9, the ISE metric of the LADRC-SMC scheme is also smaller than that of the SMC, indicating a higher control accuracy and a minor cumulative error for the LADRC-SMC scheme. Consequently, it can be concluded that the LADRC-SMC scheme addresses the high peak contact-force error rate observed in the winning SMC scheme of the first group, while ensuring the superiority of the ISE metric.