## 1 Introduction  
The use of robotic systems to carry out on-orbit operations is highly desirable because it limits the need for astronaut intervention in extreme environments, meaning missions can be executed within shortened timescales and with higher levels of human safety. Therefore, the study of related theories and technologies has received considerable attention [1, 2, 3, 4, 5, 6], etc. The first space robot is the \"Canadian Arm\" developed in Canada. In the early 1980s, it began to work in space. It is used for loading and unloading loads from the US space shuttle, capturing floating loads, inspecting the space shuttle's insulating outer layer, and providing mobile platforms for astronauts. Until 2011, when the space robot retired and stopped working, five identical robotic arms completed 34 different tasks in space, including assisting astronauts in repairing the Hubble Telescope. In the 1990s, Canada developed two more space robots. One is the Canadian two-arm (also known as SSRMS or Canadam2) mainly used to assemble the International Space Station, and the other is the Canadian dexterous arm (also known as SPDM or DEXIRE) for repairing the space station. With the gradual development of space manipulators, the structure and performance of space robots are also becoming mature. The space robot is mainly composed of a floating base and robotic links. To expand its working range and improve the work efficiency,a parallel guide rail is usually installed on the base of the space robot. The robotic links move on the guide rail, which vibrates easily [7]. To reduce the launch cost and improve the dexterity of work, the robotic link is usually designed into a lightweight, slender structure that is driven by harmonic flexible wheels. The slender link easily causes vibrations at the end effector [8]. The flexible driver causes asynchrony between the rotation angle of the motor rotor and the actual rotation angle of the joint, resulting in joint vibration [9]. Therefore, it is of practical significance to consider the flexibility of the base, joints and links of the space robot. The application scope of space robots mainly includes repairing or recycling ineffective satellites, adding fuel to target satellites, building space stations, and cleaning orbital garbage. More and more space junk is being left as a result of space exploration. According to NASA statistics, there are about 500,000 spacecraft fragments larger than a marble, and about 20,000 larger than a baseball. At present, space junk is still being produced. For example, the collision between Iridium 33 and the Cosmos-2251 satellite in 2009 created another wave of hazardous waste. When performing the above tasks, which include capturing space junk, the end effector of the robotic link inevitably has to have a contact collision with the target. The impact effect causes the vibration of the base, joints, and links of the space robot, which causes instability in the base attitude and joint, and even damage to the structure of the space robot. The collision impact and post-capturing motion control of the system are complex processes. [PERSON] et al. [10] mainly studied the effect of collision distance before robot collision on the safe motion of robots, but did not explain the force transfer relationship during capture. [PERSON] et al. [11] mainly discussed the guidance and control methods of space robots when approaching non-cooperative spacecraft, but did not analyze stable control after capturing. In this study, the process of capturing the satellite is broadly divided into three stages: the pre-capturing stage of robots and the satellite, the capturing persistence stage, and the system stage after capturing. The motion analysis of the space robot and satellite in the first two stages provides theoretical support for the modeling and motion control of the system after capturing. Many control algorithms have been proposed for the robot base attitude and joint trajectory tracking control. [PERSON] et al.