## 1 Introduction
[15] studied the passive finite dimensional repetitive control and vibration suppression of flexible jointed space robots. [PERSON] et al. [16] investigated adaptive output feedback control algorithms for flexible base robots. [PERSON] et al. [17] investigated a hybrid trajectory-based terminal sliding mode control scheme for a flexible space robot arm under the influence of base vibration. However, research on motion control and vibration suppression of base, arm, and joint fully flexible space robots is not yet available and needs to be accumulated and improved further. The fully flexible space robot is a highly coupled and nonlinear time-varying system with rigid motion and triple-flexible vibration, which faces four outstanding difficulties: complicated dynamics modelling, difficulty of rigid-flexible decoupling, low accuracy of motion control and difficulty of triple-flexible simultaneous vibration suppression. Thus, this paper proposes a dynamics modeling method for elastic base, flexible arm and flexible joint space robots, as well as a rigid-flexible decoupling scheme based on the dual time-scale scaling method, and designs an integrated control scheme for the motion and vibration of fully flexible space robots based on the decomposed model. Tracking control is particularly important for space robotics applications, where the goal is to enable the base and joints to track up the desired trajectory with the desired dynamic performance. In recent decades, various space robot control methods have emerged [18; 19]. However, because the fully flexible space robot has time-varying, rigid-flexible strong coupling, highly nonlinear and other characteristics, traditional PID control, feedback control and other factors struggle to meet its high-precision requirements, and for the ordinary rigid motion controller, because it cannot suppress the system multiple flexible vibration, control failure of fully flexible system is easy to trigger. Although sliding mode control and adaptive control have high control accuracy, most of the stabilization is asymptotic stabilization, which takes an infinitely long time to converge to the equilibrium point of the system and the different initial states of the system can significantly affect the tracking performance. To address the above problems, this paper proposes a fixed-time sliding mode control for motion vibration integration, and uses Lyapunov theory to demonstrate that the system tracking error can converge in fixed time, while achieving the suppression of flexible vibration of the base, arm and joint. Simulation analysis was performed to verify the effectiveness of the proposed algorithm. This paper is organized as follows. In Section 2, the base, arm, and joint fully flexible space robot dynamics models are developed. In Section 3, the rigid-flexible decomposition of the model is performed for the fully flexible spatial robot system based on the dual time-scale scaling principle. In Section 4, the motion vibration integrated fixed-time sliding mode control is proposed to achieve convergence of the base and joints to the desired trajectory in a fixed time and to suppress the base, arm, and joint flexible vibration simultaneously. In Section 5, numerical simulations are performed to verify this control strategy. Finally, conclusions are given in Section 6.