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Due to the harsh space environment, burdensome tasks, and complex structure, space manipulators are vulnerable to joint failure, especially the free-swing joint failure during on-orbit operation. The unactuated units, including free-swing joint and free-floating base, lack driving source and need to be controlled by actuated joints. To determine the control input while executing the task, it is necessary to analyze the controllable degree of system input to the unactuated unit. In this paper, comprehensively considering the coupling effects of velocity and acceleration between free-floating base, fault joint, and actuated joint, the kinematic and dynamic coupling relationship of the space manipulator with free-swing joint failure is established first. Then, the controllability of the space manipulator is analyzed. Considering the factors affecting the coupling motion between the actuated joint and the unactuated unit, the controllable degree index of the actuated joint to the unactuated unit is constructed. Finally, taking a space manipulator with seven degrees of freedom as the object, the numerical simulation is carried out to analyze the influence of joint configuration, actuated joint combination, and joint velocity on the controllability of the space manipulator, so as to provide guidance for the subsequent fault-tolerant motion control of the space manipulator. [ABSTRACT FROM AUTHOR]