The dynamics modeling, motion control design and joint-link double flexible vibration active suppression of free-floating flexible-joint flexible-link space robot with uncertain parameters and external disturbance are discussed. The system’s dynamic equations are established according to linear momentum conservation, angular momentum conservation and Lagrange-Assumed mode method. Based on singular perturbation method, the system is decomposed into a slow subsystem, a fast subsystem 1 and a fast subsystem 2. For the slow subsystem, a saturation robust fuzzy sliding controller is proposed to compensate the influences of the uncertain parameters, rotation angles’ errors and the external disturbance, and thus realize the asymptotic tracking of the system desired trajectory. The saturation function is used to recede the chattering of the sliding control; for the fast subsystem 1, a velocity difference feedback controller is used to suppress the flexible vibration caused by flexible-joint; for the fast subsystem 2, linear quadric regulator is used to suppress the flexible vibration caused by flexible-link. The simulation results prove the controller’s efficiency.