A space robot will violently collide with satellites in the capture operation. If its fragile joints are not protected, the space robot may be damaged. Therefore, a spring-damper device is added between a joint motor and a manipulator, which can not only digest and absorb the impact energy in the collision process, but also design a compliance strategy matched with the buffer device to realize the calm control of the hybrid system. Firstly, the dynamic modes of the dual-arm space robot open-loop system and target satellite system before capturing are established using the Lagrange function and newton-Euler. Combined with the momentum theorem, velocity constraints, closed-chain geometric constraints and Newton's third law, the closed-chain dynamic model of the hybrid system after capture is obtained, and impact effect and impact force are calculated. Then, in order to control the hybrid system, an integral terminal sliding mode control matched with the spring-damper device is proposed. The parameters of the sliding mode controler are optimized by the grey wolf optimizer to realize fast and stable calm control of the hybrid system. Finally, the stability of the hybrid system is proved using the Lyapunov theorem, and the anti-impact performance of the spring-damper device and the effectiveness of compliance strategy are verified by numerical simulation.