## 1 Introduction
An observer is used to estimate the unavailable states. [PERSON] [34] proposed a finite-time-command-filtered adaptive fault-tolerant control for a class of uncertain nonlinear systems. The control method is an effective method to compensate for the influence of an actuator fault in nonlinear systems; it can also guarantee the tracking performance and closed-loop stability of the system. However, most of the research objects are ground-based rigid robotic systems. They cannot be directly applied to the motion control of the space robot, and their joint flexibility is not considered. There are few studies on fault-tolerant control of space robots. [PERSON] [35] designed an adaptive fault-tolerant controller for space robot systems with uncertain parameters and local control failure of the joint actuator. The method can estimate and give feedback on the effective fault factors, so it does not require the definite value of the fault-effective factor. It is more suitable for practical engineering applications. [PERSON] [36] presented an adaptive sliding-mode fault-tolerant control method for a six-DOF space robot with flexible panels. This method can ensure that the space robot with the actuator fault reaches the predetermined position, and that it can suppress the vibration of the flexible plate. [PERSON] [37] proposed a hybrid controller consisting of a decentralized neural network fault-tolerant controller for a slow subsystem and a PD feedback controller for a fast subsystem. It can compensate unknown actuator faults and uncertain dynamics, obtain H\(\alpha\) convergence performance, and suppress residual vibration of the elastic base. [PERSON] [38] proposed an adaptive sliding mode fault-tolerant controller based on a fault observer. The fault observer can accurately estimate the gain faults. Yet the subject of this study is the rigid space robot system. The possibility of the existence of flexible joints in the robot structure is not considered. In conclusion, research into flexible-joint or fault-tolerant control has been carried out, but most studies are aimed at the ground-based robot system. Although the modeling and control method can be used as a reference for the free-floating flexible-joint space manipulator, there are great differences because of the space robot's free-floating base,which will make the nonlinear and coupling characteristics of the system more intense, and system control more difficult. More importantly, current research on flexible-joint space robots rarely considers the case of actuator failure, and some consider that the actuator failure is mostly for rigid space robots. In this paper, we consider both flexible-joints and actuator faults, and study the fault-tolerant control of the free-floating space manipulator with uncertain system parameters and external disturbances. Firstly, the system-dynamics equation is established by combining the flexible-joint model and the Lagrange equation. Then the system-dynamics equation is decoupled into two subsystems according to singular perturbation theory. For the slow subsystem, a speed filter is designed, based on the system-state equation, to reconstruct the speed signal. Therefore, it is not necessary to measure the real speed signal during the control process, which improves the robustness and reliability of the system. Instead, a robust sliding mode fault-tolerant controller based on the backstepping technique is proposed. The proposed control method can compensate for the influence of the uncertainty parameters and external interference, and ensure the asymptotic tracking of the expected trajectory when the actuator fails. A linear-velocity feedback control is proposed for the fast subsystem. It can suppress the vibration of the system and ensure the stability of the system. Finally, the feasibility of the proposed hybrid control method is verified by comparison simulation.