In order to solve the joint tracking control and vibration problem of flexible-joint space robot system in the pre-grasping stage when the position and pose are not controlled, the dynamic equations of a free-floating three-bar flexible-joint space robot system are established using the Lagrangian equation in conjunction with the principle of momentum conservation. In order to improve the equivalent stiffness of flexible-joint, a joint flexible compensation method was introduced. The flexible-joint space robot system was divided into slow and fast systems by singular perturbation theory. On this basis, a sliding mode control method with time delay estimation as the main framework was designed for the slow variable systems, while combining it with a low-pass filter to eliminate the system chattering problem caused by sliding mode control. A linear velocity difference feedback control system was designed for the fast-changing system to suppress the flexible vibration problems introduced by the flexible joints. Simulation verification demonstrates that the space robotic arm is capable of quickly and steadily tracking the desired trajectory within a limited time, confirming that the control scheme possesses good robustness and reliability.