# Coordinated Control after Grasping the Space Targets Using Controllable Damping Mechanism
## 4 Numerical Simulation
**Remark 2. ** From Table 4, we can see that the developed control input of BNFITSMC has many parameters such as \(c_{1}\), \(c_{2}\), \(p\), \(q\), \(\alpha_{1}\), \(\alpha_{2}\) for the sliding surface and \(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\) for the backstepping gains; the selection of these parameters greatly affects the performance of the system. The parameters of sliding surface have been widely studied in many works, so these parameters are chosen based on experience from similar research and our experiments. However, studies on the combination of the backstepping method and SMC are limited. Therefore, the backstepping gains need to be effectively selected. In this paper, the backstepping gain parameters are chosen based on experiments. However, the values chosen based on the experiments are usually not optimal, so the design of a parameter optimization algorithm is necessary. This is also our research orientation in future works. From Figures 6 and 7, we can see that the PID controller performance was worse because of the largest steady-state error and the longest convergence time. The SMC and NTSMC both provide better performance compared with the PID with smaller steady-state error. However, it should be noted that the convergence of ABNFITSMC is obviously faster than SMC and NTSMC. In other words, the proposed method provides the best performance in terms of low steady-state error and fast convergence time. It is assumed that the center of mass of a rigid body is at its geometric center; the base and joint are adjusted to initial posture \(q=\left[ \begin{array}{cc}0&10&15\end{array} \right]^{\text{T}}\) before the capture. Then, the space robot approaches the space target slowly, and the initial angular velocity of the base and each joint is 0, that is, only translational motion during the approach process. In order to reduce the chattering of the sliding surface in SMC, we replace symbolic function with saturation function. The impact forces and torques obtained in ADMAS are used as input for collision experiments in MATLAB, and the time of contact between the end effector (gripper) and the target is set at 2.2 s. The desired attitude of the stabilized system is \(q=\left[ \begin{array}{cc}0&10&15\end{array} \right]^{\text{T}}\); the angle and angular velocity changes of the base and each joint with the NFITSMC after collision are shown in Figure 8. The tracking errors of base and 2-DOF joints are shown as Figure 9.  
Figure 7: Attitude tracking error for different controllers of joint 2. Figure 8: (**a**) The attitude of base and 2-DOF joints. (**b**) The angle velocity of base and 2-DOF joints. Figure 9: The trajectory tracking errors of base and 2-DOF joints. From Figure 8, we can see the attitude of the base and 2-DOF joints converging to the desired pose in a short finite time. Furthermore, the angle velocity converging to zero ensures the system reaches steady state. The tracking errors of the base and 2-DOF joints are shown in Figure 9, and they demonstrate the superior performance of the proposed method in terms of low steady-state error and fast convergence time. The damping torque outputs by the rotating damper of FDCDM are shown in Figure 10 and the joint active torque of SMS are shown in Figure 11. From Figures 10 and 11, we can see that the proposed ABNFITSMC eliminates the chattering significantly, so the frequency and amplitude of the chattering is almost invisible, which results  
The results of Figures 8-11 demonstrate that the full-dimensional damping mechanism and coordinated control system can greatly reduce the vibration caused by the contact force, and the attitude of SMS is quickly stabilized after capture using the proposed NFITSMC. From the above experimental results, we can conclude that the proposed BNFITSMC provides superior performance in terms of tracking accuracy, fast finite-time convergence, chattering elimination, and low steady-state error.