## 5 Simulations and Analyses
### Algorithm Simulation
#### 5.2.2 Force Tracking Control Simulation
Assuming that the robot end effector is in contact with a sinusoidal surface, where \(x_{e}=0.3+0.01 sin(2\pi t/7)\), \(\dot{x}_{e}\
eq 0\), and \(\dot{x}_{e}\
eq 0\). The FOAIC control parameters are set as \(k_{p}=0.9\), \(k_{i}=0.23\), \(k_{d}=0.25\), \(\lambda=0.8\), and \(\mu=0.8\). The desired force \(f_{r}\) for constant force tracking and variable force tracking are set as 12 N and \(12+3 sin(\pi t/3)\)N, respectively. The simulation results for force tracking on the sinusoidal surface are presented in Figure 11.  
As shown in Figure 11, during the force tracking control on the sinusoidal surface, CIC exhibits poor force tracking performance, being unable to track both the desired force and the environmental position. On the other hand, the oth  
Figure 11: Simulation of force tracking control on the sinusoidal surface. (**a**) Comparison of constant force tracking control; (**b**) comparison of variable force tracking control; (**c**) comparison of constant force position tracking control. Figure 10: Simulation of force tracking control on the inclined plane. (**a**) Comparison of constant force tracking control; (**b**) comparison of variable force tracking control; (**c**) comparison of constant force position tracking control. manage to maintain good force tracking control, albeit with minor steady-state errors. AIC and VIC, however, exhibit force overshoots of 38.3% and 33.4%, respectively, while FOAIC demonstrates almost no overshoot. Furthermore, for a comprehensive evaluation of the force tracking control performance of AIC, VIC, and FOAIC, quantitative and visual comparisons were conducted using statistical indicators, including mean absolute error (MAE), root mean square error (RMSE), and mean square error (MSE), as indicated in Equation (45). These indicators were employed for analyzing the reliability of dynamic force tracking (1 s-5 s). The comparison results are presented in Table 2.  
\[\text{MAE}=\frac{1}{N}\sum_{i=1}^{N}\Bigl{|}e_{f}(t)\Bigr{|},\ \text{RMSE}= \sqrt{\frac{1}{N}\sum_{i=1}^{N}\Bigl{(}e_{f}(t)\Bigr{)}^{2}},\ \text{MSE}=\frac{1}{N}\sum_{i=1}^{N}\Bigl{(}e_{f}(t)\Bigr{)}^{2} \tag{45}\]  
Based on the specific quantified data in Table 2, it is evident that in terms of MAE, MSE, and RMSE metrics, the force tracking control performance of FOAIC surpasses that of AIC and VIC. Taking MSE as an example, during the sinusoidal surface force tracking control, FOAIC achieves a 26% improvement in force tracking accuracy compared to AIC and a 45% improvement compared to VIC. This demonstrates its exceptional force tracking control performance. Lastly, assuming that the robot end effector is in contact with a complex surface. The environmental position is set as \(x_{e}=0.3+(0.015t)sin(\pi t/2)e^{(-1/2t)}\), \(\hat{x}_{e}\
eq 0\), and \(\hat{x}_{e}\
eq 0\). The FOAIC control parameters are configured as \(k_{p}=0.96\), \(k_{i}=0.3\), \(k_{d}=0.3\), \(\lambda=0.6\), \(\mu=0.8\). The desired values for constant force tracking and variable force tracking are set at 12 N and \(12+3 sin(\pi t/3)\)N, respectively. The simulation results are depicted in Figure 12.  
As shown in Figure 12a, CIC still fails to achieve effective force and position tracking control, exhibiting no force steady-state error. Additionally, the overshoot of AIC and VIC rises to 43.6% and 39.2%, respectively. However, FOAIC remains unaffected, maintaining negligible overshoot and a shorter settling time. Yet, due to the rapid changes in the environment, compared to the sinusoidal surface, AIC, VIC, and FOAIC display more noticeable force steady-state tracking errors. Similarly, based on Equation (45), a \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Control Mode** & **AIC** & **VIC** & **FOAIC** \\ \hline MAE & 0.0891 & 0.0994 & 0.0742 \\ RMSE & 0.0963 & 0.1094 & 0.0812 \\ MSE & 0.0093 & 0.0120 & 0.0066 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Force tracking errors: MAE, RMSE, and MSE. Figure 12: Simulation of force tracking control on the complex surface.