## 5 Simulations and Analyses
### Algorithm Simulation
#### 5.2.2 Force Tracking Control Simulation  
In reality, when manipulators engage in tasks involving interactions with the environment, the contact with the environment is almost never flat, and it is likely to be accompanied by time-varying desired forces. Similarly, for the four control algorithms, simulations were conducted in four different environments (plane, inclined plane, sinusoidal surface, and complex surface) under three conditions for each (constant force, variable force tracking, and position tracking). Additionally, when it comes to setting simulated parameter values for different environmental conditions, we considered various combinations and factors. These factors include the theoretical considerations in Section 4, prior literature settings, and the constraints imposed by the manipulator used in the simulation experiments. Furthermore, for different environments and conditions, we incrementally introduced time-varying and increased complexity into the simulated parameter values. This approach enhances the persuasiveness of the simulation results and provides deeper insights for future research in this area. Assuming that the robot end effector is in contact with a plane, the environment stiffness is set to \(k_{e}\) = 5000 N/m. The environment position is set to \(x_{e}\) = 0.3 m, and the robot impedance parameters are configured as \(m_{r}\) = 0.15, \(b_{r}\) = 300, and \(k_{r}\) = 80. The estimated initial values of the environment parameters are \(x_{e}(0)\) = 0.295 m, \(k_{e}(0)\) = 4500 N/m, \(\eta_{1}\) = 75, and \(\eta_{2}\) = 13. The parameters for the FOAIC control are \(k_{p}\) = 0.85, \(k_{i}\) = 0.1, \(k_{d}\) = 0.3, \(\lambda\) = 0.5, and \(\mu\) = 0.8. The simulation time is set to 5 s with a step size of 0.0002. Additionally, the basic parameters for the subsequent force tracking simulations are kept consistent with the above-mentioned parameters. For constant force tracking, the desired force is set to \(f_{r}\) = 12 N, while for variable force tracking, the desired force is set to \(f_{r}\) = 12 + 3\(sin(\pi/3t)\). The simulation results are illustrated in Figure 9.  
From Figure 9a, it can be observed that during the flat force tracking with \(\dot{x}_{e}=\dot{x}_{e}=0\), when the parameter \(k_{r}\) of CIC is set to 0, CIC achieves excellent force tracking, confirming the derivation of steady-state tracking error in Section 3.1. According to the simulation results, on the plane environment, all four algorithms exhibit good force tracking performance with minimal overshoot, yet FOAIC demonstrates the fastest response speed. Figure 9: Simulation of force tracking control on the plane. (**a**) Comparison of constant force tracking control; (**b**) comparison of variable force tracking control; (**c**) comparison of constant force position tracking control. Assuming that the robot end effector is in contact with an inclined plane, the environment position is set as \(x_{e}=0.3+0.05t\), \(\dot{x}_{e}\
eq 0\), \(\ddot{x}_{e}=0\). The FOAIC control parameters are set as \(k_{p}=0.95\), \(k_{i}=0.15\), \(k_{d}=0.3\), \(\lambda=0.5\), \(\mu=0.8\). The desired force fr for constant force tracking and variable force tracking is set as 12 N and \(12+3 sin(\pi/3t)\)N, respectively. The simulation results are presented in Figure 10.  
As shown in Figure 10, during the force tracking control on the inclined plane, CIC exhibits a force tracking error of about 2N. On the other hand, AIC, VIC, and FOAIC maintain good force tracking performance. However, AIC and VIC have overshoots of 34.5% and 25.6%, respectively, while FOAIC demonstrates minimal overshoot. Larger overshoots can result in impacts when the robot interacts with the environment, increasing the difficulty of actual robot force control and potentially causing damage to the robot, leading to unnecessary losses. FOAIC significantly reduces overshoot while achieving a shorter time to reach a steady state, showcasing excellent force tracking performance.