For the elderly, disabled and hemiplegic patients who have mild weakness of lower extremity muscle strength but can still walk independently, a mobile rehabilitation robot accompanied by the user side is designed to meet their needs of daily assistance and rehabilitation training. In order to accurately estimate human walking intention, a human walking intention estimation algorithm based on Kinect vision sensor and ellipsoid set member filter is proposed. In order to solve the problem of limited detection range of the vision sensor, a robot following control algorithm with prescribed performance is proposed, which not only guarantees that the robot will smoothly follow the user, but also keeps the robot in a relative position where the user can comfortably grasp the armrest without collision risk. Simulation and experimental results verify the effectiveness of the proposed intention estimation algorithm and the following control algorithm.