## 1 Introduction  
In recent years, the number of subjects with upper extremity motor dysfunctions caused by stroke, spinal cord injuries, and accidents has been dramatically increasing year by year [1], severely limiting their motor and activities of daily living (ADL) abilities. This not only brings physical inconvenience to the subjects but also brings financial and mental burden to their families. Research results have showed that compared with the traditional manual rehabilitation therapy, robot-aided rehabilitation training could carry out high-intensity repetitive and task-oriented training tasks [2], and clinical research studies have also shown that robot-aided rehabilitation therapy could effectively improve the rehabilitation effect on patients' upper extremity motor abilities [3, 4]. To satisfy the patient's requirements at different rehabilitation stages, different types of upper limb rehabilitation robots have been developed, which can be divided into two categories [5]; the end-effector type [6; 7] and the exoskeleton type [8; 9]. Generally, exoskeleton type upper limb rehabilitation robots could implement rehabilitation training of a single joint; however, additional interactive forces and torques between the patient's arm and the robot would occur [10]. Compared with the traditional rehabilitation training methods, the end-effector type upper limb rehabilitation robot has better superiority during upper limb rehabilitation training [11]. Clinical comparative trial results have also showed that compared with the exoskeleton upper limb rehabilitation robot, intervention with the end-effector rehabilitation robot was more effective in terms of the active participation of patients with mild and severe stroke [12]. According to the Brunstorm theory, patients with an upper limb motor disorder in the early recovery stage experience difficulties in moving their arm, and therapists or robots are required to carry out repeated passive training. Therefore, a stiff position controller is required to be applied in such passive training. The rehabilitation effect of the passive training method would be not great with the recovery of the patient's motor ability because the active participation of the patients will be ignored in those position controllers. For patients with partial motor control ability, robot intervention to a minimum extent could stimulate and promote the brain neuroplasticity of patients [13; 14]. Studies have shown that task-oriented rehabilitation training is only effective when it is associated with task-oriented movements involving effort by the subject actively [15]. Therefore, a controller needs to be developed to provide necessary assistance or correction to complete rehabilitation training tasks according to the patient's functional ability. Controllers with such characteristics are called assist-as-needed (AAN) controllers, also known as assistance or corrective controllers. At present, AAN controllers have been used in lower limb rehabilitation [16], finger rehabilitation [17], and upper limb rehabilitation [18] to stimulate the patient's active participation during rehabilitation training. The results of a clinical trial proved that an upper limb rehabilitation robot with an AAN controller could provide intensive and repeated rehabilitation training and could also promote the active participation and motion performance of patients, promoting the motor recovery [19]. Regarding robot-aided training with the AAN characteristic, impedance control is often applied to assist patients during robot-aided rehabilitation training tasks [20]--the schematic diagram is shown in Figure 1. Clinical studies on stroke patients [21; 22] showed that rehabilitation robots with the impedance control strategy have a better effect on reducing the upper limb muscle strength and improving the motor function of patients than robot-assisted therapy that simply mimics traditional therapy [23; 24; 25].