## 5 Numerical Simulation and Validation Analysis
### Simulation and Verification of LADR-SMC with Smith Predictor  
To address the issue of poor tracking performance in the joint angle and contact force under interference delay, we developed a multi-loop control scheme called LADR-SMC-SP. This scheme is based on the LADR-SMC approach, demonstrating superior tracking performance discussed in Sections 4.1.2, 4.2.2, and 5.2. We proposed a fourth-order LADR-SMC-SP scheme in the theoretical part of Section 4.2.3 to tackle the challenges related to force-position tracking and system delay. To further evaluate the effectiveness of the LADR-SMC-SP scheme, we conducted simulations to compare and discuss the tracking performance (see Figures 12 and 13), joint torque (control law) (see Figure 14), force-tracking performance (see Figures 15 and 16), and force-tracking errors of joints \(\theta_{1}\) and \(\theta_{2}\) under a uniform random disturbance. Subsequently, we evaluated the performance of the finalized multi-loop controller scheme and summarized its theoretical contributions to the application scenarios. In addition, we determined the significance and practical implications of the proposed scheme. The position command was given by \(\theta=0.5+0.8\mathrm{sin}t\). Figure 12 shows the tracking results of the joint angle \(\theta_{1}\) for the LADR-SMC scheme with a uniform random disturbance and considering the system delay. To ensure the comparability of the three schemes, the parameters \(K_{P},K_{f},\mathrm{and}\ \Delta\) in the three schemes were consistent (refer to Table 2 in Section 5). \(\theta_{1\_\mathrm{delay}}\) represents the tracking result of the joint angle \(\theta_{1}\) for the LADR-SMC scheme without SP, whereas \(\theta_{1\_\mathrm{Smith}}\) represents the tracking result of the joint angle \(\theta_{1}\) for the LADR-SMC-SP scheme. Ideal \(\theta_{1d}\) represents the ideal tracking result of joint angle \(\theta_{1}\). A system comparison with and without the predictor was performed using MATLAB/Simulink(Version information of simulation software: MATLAB R2023b) (Figure 4). The control delay was set to zero, and the sensor delay was set to \(2\ \mathrm{s}\). The simulation results are shown in Figure 12. It can be observed that the control system without the Smith predictor (\(\theta_{1\_\mathrm{delay}}\)) exhibited a lag of \(2\ \mathrm{s}\) and began to diverge after \(10\ \mathrm{s}\), whereas the control system with the Smith predictor (\(\theta_{1\_\mathrm{Smith}}\)) exhibited a stable response, and the delay was effectively alleviated. The LADR-SMC scheme diverges because the sensor time delay leads to a phase change in the feedback signal and a frequency response distortion (Equations (29) and (30) in Section 4.2.3). Under random disturbance and delay, the angle \(\theta_{1}\) trajectory for LADR-SMC-SP is more accurate than that for LADR-SMC, as evidenced by the ISE values of \(1.6636\times 10^{-28}\ \mathrm{rad}^{2}\cdot\mathrm{s}\) for LADR-SMC-SP and \(2.4734\times 10^{-25}\ \mathrm{rad}^{2}\cdot\mathrm{s}\) for LADR-SMC. Therefore, the LADR-SMC-SP scheme is superior to the LADR-SMC scheme when considering the system delay under a uniform random disturbance. Figure 12: Tracking of joint angle \(\theta_{1}\) in the LADR-SMC scheme with or without the SP under random disturbance. Next, we checked the tracking results for the joint angle \(\theta_{2}\) (see Figure 13). Because joint angle \(\theta_{2}\) can be expressed in terms of joint angle \(\theta_{1}\) (see Equation (4)), the tracking behavior of joint angle \(\theta_{2}\) under the same conditions as those in Figure 11 is similar to that of joint angle \(\theta_{1}\). Figure 13 shows the tracking results of the joint angle \(\theta_{2}\) for the LADRC-SMC scheme with a uniform random disturbance and considering the system delay.