## 6 Simulation Results  
Taking the process of a satellite captured by a flexible-base, flexible-link and flexible-joint space robot as an example, as shown in Figure 1, and \(t_{0}\) as the initial time, the simulation studies are carried out. The physical parameters of the flexible-base, flexible-link and flexible-joint space robot system and satellite are: \(l_{0}=l_{1}=l_{2}=1.5\) m and \(l_{3}=1\) m. The mass, moment of inertia, and linear density of the flexible link are selected as \(m_{0}=40\) kg, \(m_{3}=3\) kg, \(J_{0}=30\) kg\(\cdot\)m\({}^{2}\), \(J_{\text{m}1}=J_{\text{m}2}=J_{3}=0.1\) kg\(\cdot\)m\({}^{2}\), \(\rho_{1}=3.5\) kg/m, and \(\rho_{2}=1.1\) kg/m. The bending stiffness of the flexible link and the elastic coefficients of the base and joints are selected as \(EI_{1}=EI_{2}=100\) N/m\({}^{2}\), \(k_{\text{b}}=500\) N/m, and \(k_{\text{m}1}=k_{\text{m}2}=50\) Nm/rad. Before collision, the space robot is in a static waiting state, and its initial state is \(q_{z}(t_{0})=[0\text{m},0\text{m},0\text{m},2\text{rad},1\text{rad},-2\text{ rad},0\text{m},0\text{m},0\text{m},1\text{rad},-2\text{rad}]^{\mathsf{T}}\). The satellite flies to the robot end effector with a movement speed of \(\dot{x}_{3}=0.5\) m/s, \(\dot{y}_{3}=0.5\) m/s and roll speed \(\dot{\theta}_{3}=0.5\) rad/s. The collision occurs at time \(t_{0}\), lasts for a very short time \(\Delta t\), and is tightly locked after time \(t_{0}+\Delta t\) to form a system. Then, the controller is turned on to stabilize the base attitude and joints of the system in the following expected states:  
\[\boldsymbol{q}_{\text{d}}=[1.7\,\text{rad},0.7\,\text{rad},-1.4\,\text{rad}]^ {\mathsf{T}}\]  
In order to compare and illustrate the stabilization control effect of the designed controller for the unstable system, this study carries out simulation analysis in two cases. (1) After contact and collision, the motion control of the system is carried out without vibration suppression, that is, turning off the fast sub-controller and virtual control force. The repetitive learning sliding mode control without vibration suppression (RSC-NV) is composed of Equations (17) and (28)-(30). RSC-NV simulation is used to reveal the impact of collision on the vibration and motion of the base, joint and links of the space robot. (2) After contact and collision, the stabilization control of the motion and vibration of the system is carried out, that is, turning on the fast sub-controller and virtual control force. The repetitive learning sliding mode stabilization control (RSSC) is composed of Equations (17), (37)-(39), (41) and (43). RSSC is used for simulation, to verify the effectiveness of the system stabilization control. The relevant parameters of the controller are selected as \(a=1\), \(b=1\), \(q/p=2\), \(\mathbf{K}_{1}=\left[1000,0.1,1\right]^{\mathsf{T}}\), \(\mathbf{K}_{2}=\left[1,1,1\right]^{\mathsf{T}}\), \(\mathbf{K}_{3}=\left[1,1,1\right]^{\mathsf{T}}\), \(\mathbf{K}_{d}=\left[1,1,1\right]^{\mathsf{T}}\), \(\mathbf{K}_{m}=\left[1,1,100\right]^{\mathsf{T}}\), \(\mathbf{K}_{c}=\left[104,72\right]^{\mathsf{T}}\), \(\mathbf{Q}_{0}=\left[1,1,1\right]^{\mathsf{T}}\), \(\mathbf{Q}_{k}=\left[1,1,1\right]^{\mathsf{T}}\), and \(N=3\). The trajectory tracking curves of the system base attitude under two control algorithms shown in Figures 2-4 are the trajectory tracking curves of the system joint \(O_{1}\) and joint \(O_{2}\) under two control algorithms. From Figures 2-4, we can observe that the RSC-NV algorithm makes the base attitude and joint of the system stable in the desired trajectories. However, owing to the mutual coupling of the system motion and vibration, the uninhibited vibration of the base, links, and joints will lead to different amplitude vibrations of the base attitude and joint of the system, which will affect the control accuracy.