Actuation failure and fault-tolerant control under the actuation failure scenario have drawn more attention in accordance with the recent increasing demand for reliable robot control applications such for long-term and remote operation. The emergence of control torque loss, i.e., the free-swinging failure, is particularly challenging when the robot performs dynamic operational space tasks due to complexities stemming from redundancies in the kinematic structure as well as a dynamical disturbance in the under-actuated multi-body system. To reinforce robustness and accuracy of task-space control under the failure condition, this letter proposes a performance index, named generalized failure-susceptibility (GFS), which is formulated to render thorough dynamic and kinematic effects caused by the un-actuated joints. The GFS index is then exploited with the hierarchical task controller, where self-motion is controlled to minimize the index in real-time. Several experiments with a seven-degrees-of-freedom torque-controlled robot verify that the proposed control strategy with the GFS index effectively improves fault tolerance against anticipating actuation failure.