Bandaging is a regular and indispensable nursing task in daily medical care. The task is challenging for robots because it requires simultaneous control of the bandaging trajectory and tension. The patient’s involuntary swaying due to pain and other reasons may cause further complications. To solve these problems, a compliant robot arm based on the serial elastic actuator (SEA) is selected as the hardware solution for this task. In this article, a method for a force–position-decoupling control strategy is developed for achieving the full process of bandage tension control as well as bandaging trajectory adjustment and following. Experiments are conducted to evaluate the performance of the proposed design and methods, using five subjects and a collaborative robot. The results show that the proposed robot arm successfully performs the bandaging task and that the proposed methods can effectively enable the robot to estimate the movement direction and position of the injured arm. The bandage tension can be effectively maintained even when the arm is swaying.