## 1 Introduction  
Space exploration is of great significance to the development of resources exploration, meteorological observation, navigation, and positioning, so a large number of satellites are launched into space every year. It is inevitable that a small part of satellites will fail to enter the intended orbit or damage in orbit. If the satellites can be recovered, the cost of space exploration will be greatly saved. At present, it is feasible to use a space robot to complete the capture task. This has therefore become one of the research hot spots of space exploration [1; 2; 3; 4; 5; 6; 7; 8]. Generally, the process of capture operation can be divided into four stages: (1) the space robot observes the target satellite; (2) a pre-operation stage before capture operation, such as deceleration and detumbling control of the target satellite; (3) contact and collision between the space robot and target satellite; and (4) the stabilization control of the closed chain hybrid system. The impact force makes it easy to damage the joints of space robot in the third stage, and the impact effect will make the hybrid system unstable in the fourth stage. Therefore, these two stages represent the focus and challenges of this study. For the third stage, [PERSON] et al. [9] analyzed the dynamic evolution for dual-arm space robot with capturing a spin satellite and calm control for unstable closed chain composite system are discussed. [PERSON] et al. [10] presented the impedance-based contact control of a free-flying space robot utilizing a compliant wrist for non-cooperative satellite capture operation. An open loop impedance control law based on contact dynamics model is introduced to realize the desired coefficient of restitution defined between the manipulator hand of a space robot and a contact point on a free-flying target. [PERSON] et al. [11] established the contact dynamic model of a space robot capturing satellite and analyzed the momentum exchange problem in the capture process. [PERSON] et al. [12] studied the collision dynamics and kinematics of a space robot capturing satellite based on momentum conservation. [PERSON] et al. [13] approximately described the elastic deformation of flexible manipulator by using the assumed mode method, and then analyzed the collision dynamics of space robot capturing satellite by combining the momentum impulse. It is worth noting that scholars mainly focus on the analysis of collision dynamics, and they do not pay attention to the protection of the joints. In fact, if the joints are not protected during the capture operation, they will be damaged by impact force and could even cause damage to the space robot. A series elastic actuator (SEA) is usually added between the joint motor and manipulator in the ground robot, which can avoid joint being damaged by impact force when the robot collides with the external environment [14; 15; 16; 17; 18; 19; 20; 21][PERSON] However, the space robot is a rootless system, and it works in both microgravity and a vacuum. If the SEA is added in the joints of space robot (refer to our published previously work [3; 22]), the flexible vibration caused by SEA will be difficult to suppress. Therefore, the controller was divided into a fast subsystem controller and slow subsystem controller, wherein the slow subsystem controller realizes trajectory tracking and the fast subsystem controller suppresses the flexible vibration. This means that the controller becomes more complex, and the joint motor needs to provide additional torque to suppress flexible vibration. Sometimes this torque will exceed the load capacity of the motor. In order to cope with the problem, this paper adds a spring-damper device (SDD) between the joint motor and manipulator. Compared with SEA, The SDD can not only absorb and digest the impact energy, but also quickly attenuate the flexible vibration to achieve the suppression of flexible vibration. For the fourth stage, [PERSON] et al. [23] used impedance control to stabilize the hybrid system after space robot captures satellite.