## 5 Numerical Simulation and Validation Analysis
### Simulation and Verification of LADR-SMC with Smith Predictor
\(\theta_{2\_\text{delay}}\) represents the tracking result of the joint angle \(\theta_{2}\) for the LADRC-SMC scheme without the SP, whereas \(\theta_{2\_\text{Smith}}\) represents the tracking result of the joint angle \(\theta_{2}\) for the LADRC-SMC-SP scheme. The ideal \(d\theta_{1d}\) represents the ideal tracking result of the joint angle \(\theta_{1}\). Compared with the tracking behavior of joint angle \(\theta_{1}\), the LADRC-SMC scheme without SP exhibits overall instability and irregular oscillations in the tracking of joint angle \(\theta_{2}\). This phenomenon was more pronounced in the control law, as shown in Figure 14.  
The LADRC-SMC-SP scheme shows the tracking behavior of joint angle \(\theta_{2}\), which closely follows the trend of the ideal angle. Compared to LADRC-SMC, the ISE for LADRC-SMC-SP to the \(\theta_{2\_\text{Smith}}\) is \(1.7983\times 10^{-28}\text{ rad}^{2}\cdot\text{s}\) and is \(9.686\times 10^{-26}\text{ rad}^{2}\cdot\text{s}\) for LADRC-SMC to the \(\theta_{2\_\text{delay}}\), which indicates that the \(\theta_{2\_\text{Smith}}\) trajectory for LADRC-SMC-SP is more accurate than LADRC-SMC. In conclusion, based on the tracking behavior of the joint angles in Figures 12 and 13, it is evident that the LADRC-SMC-SP scheme outperforms the LADRC-SMC scheme under a uniform random disturbance and considers the system delay. Next, we examine the variation trajectory of the joint torque, that is, the change in the control parameter \(\tau\) (see Figure 14), to further analyze the feasibility of the LADRC-SMC scheme with the SP. In this experiment, the position command was represented by the equation \(\theta=0.5+0.8\text{sin}t\), whereas the ideal contact force was expressed as \(f=50\text{cos}t\). Under the same experimental conditions as those shown in Figures 12 and 13, the joint torque was simulated using the SMC law (Equation (32) in Section 4.2.3). In Figure 14, \(\tau_{1\_\text{delay}}\) and \(\tau_{2\_\text{delay}}\) illustrate the torque variation curves of joints 1 and 2, respectively, in the LADRC-SMC scheme. \(\tau_{1\_\text{Smith}}\) and \(\tau_{2\_\text{Smith}}\) curves represent the torque variations of joints 1 and 2 in the LADRC-SMC-SP scheme. Notably, the lengths of the joints in the single-legged robot designed in this study were one meter each, which does not fall within the category of small-legged robots. Therefore, we take the example of an electric servo drive of a mechanical arm called \"ABB IRB 6700-150/2.85: maximum torque of \(285\text{ N}\cdot\text{m}\).\" This series of mechanical arms is equipped with different motor models, resulting in variations in their maximum torque. Considering factors related to the robot's materials and structure, such as safety, stability, and reliability, we cautiously set the technical parameters of the joint drive motor in this study as follows: a saturation threshold torque of \(\pm 200\text{ N}\cdot\text{m}\). The maximum  
Figure 13: Tracking of joint angle \(\theta_{2}\) in the LADRC-SMC scheme with or without the SP under random disturbance. output limit for the joint torque shown in Figure 14 is denoted as \(L_{+}=200\)\(\mathrm{N}\cdot\mathrm{m}\), whereas the minimum output limit is denoted as \(L_{-}=-200\)\(\mathrm{N}\cdot\mathrm{m}\). After the analysis, we can draw the following conclusions: considering the system sensor delay, the fourth-order LADRC-SMC-SP scheme is more stable and reliable than the scheme without [PERSON]'s compensation. The experimental results demonstrated that after 20 s of system operation, \(\tau_{1\_\mathrm{delay}}\) and \(\tau_{2\_\mathrm{delay}}\) in the control system without [PERSON] compensation exhibited oscillatory instabilities. In particular, the \(\tau_{2\_\mathrm{delay}}\) exhibited oscillations at approximately 29 s and exceeded the servomotor saturation threshold of 200\(\mathrm{N}\cdot\mathrm{m}\). However, the joint torques in the control system with [PERSON] compensation exhibited stable periodic variations without oscillations or exceeding the motor threshold.