## 1 Introduction
Although the impedance control strategy allows a patient's arm to deviate from the training trajectory, the patient's spatial freedom is limited. Assistance from the robot is always given as a restoring force to assist the patients once they are deviating from the planned trajectory. Therefore, scholars have developed an impedance-based AAN strategy to increase the patient's freedom, which can be mainly divided into two categories: the band-type controller and the window-type controller. The window-type controller defines a predetermined trajectory and a moving window that allows the patient's arm to move freely. When the patient's arm lies outside the moving window, the patient's arm would be corrected into the moving window by the robot's assistance force [26; 27]. Another band-type controller is based on the fault-tolerant region (FTR), which constructs a fault-tolerant band that allows the patient to move freely by establishing the inner and outer boundary of the FTR. The patient's spatial freedom is limited in the window-type AAN controller, while the band-type AAN controller allows the patient to have complete freedom and choose his/her own trajectory without being assisted by the robot. The patient's arm has complete freedom when moving in the FTR, while the robot will apply an assistance force to assist the patient's arm to finish the rehabilitation training task when deviating from FTR. Research studies have also been conducted to explore increasing the patient's freedom. [PERSON] et al. [28] first developed a rope-driven exoskeleton rehabilitation robot and then added a fault-tolerant area around the predetermined trajectory to increase the patient's spatial freedom. [PERSON] al. [29] validated the feasibility by adding a plane channel around the predetermined trajectory based on a two-link rehabilitation robot. Although the band-type AAN controller has shown positive results in clinical trials, its tendency or \"slacking\" for the patient to rely heavily on the assistance force has been recorded [30]. The robot will not apply any assistance or correction as long as the patient's arm is moving within the FTR, which will greatly reduce the active participation of patients. A velocity curve could be added in the FTR which will build a virtual movement in the FTR. When the patient's arm is stagnant or moving too slowly in the FTR, the controller will generate a moving wall to push the patient's arm to catch up with the virtual movement to complete the training task and avoid the slacking. To better adjust the assistance force of the AAN controller, it is necessary to evaluate the patient's real-time motor ability or functional ability. One can conclude that there are mainly two classifications for the evaluation of a patient's functional ability based on the current literatures: the sensor-based [31; 32] and model-based [33] evaluation techniques. Although there is more of a theoretical basis for the evaluation of upper limb motor ability for model-based methods, the modeling results are difficult to use to accurately evaluate the patient's true motion intention, while sensor-based evaluation methods can more intuitively express the patient's motion state. Some studies have already used the position deviation between the actual trajectory of a patient's hands and the expected trajectory which was calculated to represent the functional ability of patients during rehabilitation training [34]. Besides, the AAN control strategies are mainly applied to the exoskeleton upper limb rehabilitation robots currently, while there is relatively little research based on the end-effector rehabilitation robot. So far, only a small number of planar end-effector rehabilitation robots such as MIT-Manus [6] and the Braccio Di Ferro rehabilitation robot [35] have realized the AAN rehabilitation strategy on the specific plane trajectory. In this paper, an impedance-based AAN controller is proposed and implemented in an end-effector upper limb rehabilitation robot, which can carry out three-dimension trajectory rehabilitation training.