## 1 Introduction  
With the development of space technology, space robots have begun to undertake more and more important tasks, such as the construction of space stations, the installation and maintenance of space equipment, and the refueling of spacecraft. So research into the dynamics and control of space robots is of great significance. The research in this field has become a hotspot and has made remarkable progress [1; 2; 3; 4; 5; 6; 7; 8; 9]. For most manipulators, the motor is mounted at the joint to drive the link to complete the motion. Therefore, research on the motion control of manipulator joints is extremely important. Considering engineering practice, the connection between the motor rotor and the link is connected through a harmonic-gear drive, so the joint shows flexibility. The flexible joint can alleviate the impact force when the manipulator performs a task, reducing the collision damage. However, flexible joints also introduce additional system errors and unwanted vibrations, which will affect the control quality of the system [10; 11]. The free-floating flexible-joint space manipulator system is a nonlinear and strongly-coupled system [12]. The interaction between rigid and flexible motion makes system-dynamics analysis and control more difficult. The control method of ground-based robot systems cannot be directly applied to the control of free-floating robot systems. At the same time, the dynamic model of free-floating space manipulator systems is often uncertain due to the influence of parameter perturbation, load variation, fuel consumption, external disturbance, and other uncertain factors. The uncertainty will affect the system's control quality. At present, the modelling and the control of the flexible-joint have received extensive attention, and many control methods have been proposed [13; 14; 15; 16; 17; 18; 19; 20]. However, less research exists on free-floating flexible-joint space manipulators. [PERSON] [21] proposed an extended state-observer-based adaptive controller for flexible-joint space manipulators with dynamic uncertainties and joint-stiffness uncertainties. [PERSON] [22] studied the compliant control problem of a free-flying flexible-joint space robot. A non-singular composite impedance controller with finite-time convergence based on backstepping method is designed. [PERSON] [23] studied the problems of dynamic modeling, motion control, and the vibration suppression of free-floating flexible-links and flexible-joint space manipulators, and proposed a robust fuzzy-sliding mode control method to achieve an accurate and stable control of the system. [PERSON] [24] proposed a robust control law, based on the backstepping technique for the case of unknown time-varying disturbances and input saturation. A performance function and a transformation function are introduced to improve the tracking performance. Based on the singular perturbation method, [PERSON] [25] designed a double adaptive control method, which combines the multi-parameter adjustable non-deterministic-equivalent adaptive control method of the slow subsystem and the adaptive control method of the fast subsystem, to realize the motion control and flexible vibration suppression of the flexible-joint space robot on the flexible base with unknown parameters. At the same time, we also consider that the actuator of the space robot may break down after long-term work. The specific performance aspects drive the failure. Especially in the harsh space environment, the probability of actuator failure is greatly increased. However, due to the limitation of physical conditions, and the high cost of raw materials and maintenance, maintenance of the actuator is quite difficult. Therefore, the space robot system requires strong fault tolerance. At present, some achievements have been made in the research of fault-tolerant algorithms [26; 27; 28; 29; 30; 31; 32]. [PERSON] [33] developed an output-based adaptive neural-tracking control strategy for the considered system against an actuator fault.