With the rapidly advancing space exploration in recent years, the number of non-cooperative space objects left in space is growing. Some non-cooperative objects, such as space debris and defunct satellites, potentially threaten the safety of the space station and many satellites in service. Thus, the capture of non-cooperative objects in space has attracted significant attention from scholars. Even though astronauts can walk out of spacecraft to capture dangerous non-cooperative objects, capturing non-cooperative objects by space robots costs less, has higher efficiency, and has better safety than astronauts. Compared with single-arm space robots, dual-arm space robots have a higher probability of success in carrying out complex capturing missions. Due to the coupled dynamics between the base and the two arms of a dual-arm space robot, the coordinated control of the base attitude and the motion of the end-effectors is addressed in this article. The smooth desired trajectories of the end-effectors to approach and synchronize with a defunct tumbling space object in 3D space are planned based on the Bezier curves. Due to the nonlinear properties and possible model uncertainties of the dual-arm space robot and the non-cooperative target, a sliding mode controller (SMC) with better robustness against uncertainties than conventional PID controllers is developed. The SMC can control the space robot to follow the desired trajectories to approach and synchronize with the tumbling target. Simulation results show the feasibility and effectiveness of the proposed motion control strategy to approach a tumbling target in space. Moreover, the tracking errors in following the desired trajectories have high accuracy when the model uncertainties are considered.