## 5 Discussion and Conclusions  
In this paper, an impedance-based AAN controller for robot-aided rehabilitation training is proposed to provide training for all rehabilitation stages of upper limb and has been realized on an end-effector rehabilitation robot that was constructed by two commercial manipulators. The controller proposed in this work enables a subject's arm to have motion freedom within the FTR constructed around the predetermined training trajectory and provides as much assistance as needed when the patient is in need of help. Meanwhile, when patients are deviating from the FTR, the concept of the stiffness field is also proposed around the FTR to assist the patient's arm according to their functional ability. Furthermore, the controller contains five adjustable parameters: the radius \(R_{W}\) of the FTR, the upper and lower limits \(K_{0}\) and \(K_{1}\) of the stiffness field, the assistance factor \(K_{t}\) of the assistance wall to prevent slacking, and the stiffness inside the FTR \(K_{2}\). By adjusting these controller parameters, the AAN controller proposed in this paper could perform rehabilitation training of the passive mode, assistant mode, active mode, and resistant mode. Finally, the performance of the controller under different parameters was verified by experimental testing with a healthy elderly subject. Among the tests, the robot could accurately track the predetermined rehabilitation trajectory when the controller was activated in the passive training mode. When the controller was activated in the assistant training mode, the controller showed the AAN characteristic. The controller enabled the patient's arm to move freely in the allowed FTR, while the robot generated a series of columnar virtual stiffness walls according to the patient's functional ability to provide assistance when deviating from the FTR. Once slacking occurred in the patient, the moving assistance wall inside the FTR assisted the patient's arm to catch up with the virtual movement. Then, the robot would be in a completely backward-driven state when the controller is activated in the active training mode, and the patient has complete freedom, which can be used for stability training in the later rehabilitation stage. Finally, the controller could create a resistance force that hinders the patient's movement when activated in the resistant training mode. Therefore, the patient needed to overcome this obstacle to finish the training task, which can be used to train a participant's upper limb muscle strength. The preliminary experimental results under different training modes show the correctness of the controller proposed in this paper and the potential ability of providing passive, assistant, active, and resistant training modes by adjusting several controller parameters, which covers all rehabilitation stages of the upper limb. We expect that the impedance-based AAN controller proposed in this paper could be extended to develop similar controllers for the multi-DOF exoskeleton rehabilitation robot or bilateral rehabilitation robots. As a prospective study, only one healthy elderly subject was tested in this paper to validate the correctness of the proposed impedance-based AAN controller in passive, assistant, active, and resistant rehabilitation training modes. Nonetheless, the effectiveness of the proposed controller for patients with upper limb motor dysfunction needs to be validated through clinical studies in future work. In the next step, the controller with different rehabilitation training modes is planned to be applied on more subjects and also patients to demonstrate the performance of the controller with different parameters and to help to increase the training intensity. In this work, [PERSON] conceived, designed, and performed the experiments and wrote the paper; [PERSON] guided the writing of the article and gave some suggestions; [PERSON] and [PERSON] analyzed the data. All authors have read and agreed to the published version of the manuscript.