## 1 Introduction
[12] presented the terminal sliding mode control of a robot sliding perturbation observer; however, the convergence rate was not considered. [PERSON] et al. [13] presented a fast terminal sliding mode control without higher tracking quality for repetitive tasks. [PERSON] [14] proposed an exponentially stable repetitive learning algorithm, and [PERSON] et al. [15] demonstrated the repetitive control method for minimum phase nonlinear systems, which is complex and inapplicable in the aerospace field. In this study, the desired signal is a Fourier series approximated by combining the idea of repetitive control. Considering that most signals account for a large proportion of power in the middle and low frequency bands, the analytical periodic signals were finite-dimensional Fourier series, which, when combined with the exponential approach law sliding mode algorithm and finite-dimensional repetition learning algorithm, enable the design of a repetitive learning sliding mode controller applicable in the space industry. The model uncertainty and its disturbance in the structural estimation system connected parallel to the \(N\) linear oscillators and an integrator differed from the traditional internal model-based repetitive controller [16; 17]. The repetitive learning sliding mode controller can effectively avoid strict stability conditions and slow convergence problems with simple control law, because it is not completely dependent on model information, has high control accuracy, and is easy to realize. The base attitude and joints of the system can reach the desired position after capturing using a repetitive learning sliding mode controller; however, in the capture operation, the collision easily leads to the vibration of the base, joints, and links of the space robot with different amplitudes. Particularly, in an undamped space environment, the attenuation is slow, and the vibration of each component may cause instability in the system after capturing. Thus, it is necessary to restrain the vibration of all flexible members of the system. [PERSON] et al. [18] discussed the motion control problem of ground flexible base robotsand verified the effectiveness of the augmented method in the design of adaptive output feedback control. [PERSON] et al. [19] discussed the problem of the robust control of flexible joint space robots, and [PERSON] et al. [20] discussed the problem of the adaptive control of flexible link robots. The above studies provide a theoretical basis for the study of flexible robots; however, they only consider the influence of the single structural flexibility of the base, joint, or links. [PERSON] et al. [21] considered the flexibility of the joint and introduced flexible links into the system for analysis, and [PERSON] [22] considered the influence of a flexible vibration of the base and joints when analyzing the motion control of a space robot. However, the above studies did not involve the capture operation. [PERSON] et al. [23] discussed the dynamics and control of the robot capture of tumbling satellites, [PERSON] et al. [24] discussed the minimum base disturbance control of the visual servo pre-capture process of free-floating space robots, and [PERSON] et al. [25] studied the trajectory planning and coordinated control of space robots in the post-capture stage. Although various problems of space robot capture have been addressed, the effect of flexible vibration has not been considered. In this study, the process of satellite capture by a fully flexible space robot is analyzed by considering the influence of the flexibility of the base, joints, and links of the space robot, and a dynamic model of the system after capturing is established.