## 4 Validation Experiments and Results
### Validation Experiment of Assistant Rehabilitation Training Mode of the Controller  
The AAN assistant training is applicable to patients with certain control ability of their upper arm. When the patient's motor ability is better (moving in the FTR), the robot does not provide any assistance to the patient's movement. Conversely, the robot would provide corresponding assistance force according to the patient's functional ability to assist the patient's arm to finish the training task. The controller will show AAN characteristics as long as the parameters of the controller are properly adjusted. In order to illustrate the AAN characteristics of the controller, the parameters of the controller were set as \(R_{W}\) = 25 mm, \(K_{0}\) = [300, 300, 300] N/m, \(K_{1}\) = [500, 500, 500] N/m, \(K_{t}\) = [500, 500, 500] N/m, and \(K_{2}\) = [10, 10, 10] N/m. The pose of the robot was fixed during the test, and a total of seven turns of counterclockwise movement were tested. In the first three laps, the subject was required to be relaxed, and the moving wall within the controller would push the patient's arm to catch up with the virtual movement to complete the training task. In the following three laps, the subject takes the initiative to complete the exercise. In the last lap, the subject was required to deliberately deviate from the FTR to test and verify the variation trend of the assistance force provided by the robot. The Unity scene provided the subject with visual guidance and recorded the number of laps of the movement during the testing. Figure 14 shows the trajectory of the subject's arm and the training trajectory, in which the blue curve is the actual trajectory of the subject's arm, the black curve is the desired trajectory, and the red channel is for the virtual FTR. Figure 15 shows the filtered interaction force and wrench between the subject's hand and the handle. Figure 16 shows the filtered distance between the subject's hand and the training trajectory, as well as the total interactive force between the subject's hand and the handle. In the first three laps, the subject was required to be relaxed, and her movement lagged behind the virtual movement in the FTR (considered as slacking occurred). Therefore, the robot would generate a moving wall to assist the subject to catch up with the virtual movement. It can be seen from Figure 14c,d that the trajectory of the subject's hand is below the predetermined trajectory due to the influence of the weight of the subject's arm. It can be seen from Figure 15a that the RMS of the total interaction force between the subject's hand and the handle is 10.2 N, which is mainly caused by the weight of the arm. In the following three laps, the subject was required to be active and the moving trajectory mainly concentrated within the FTR. One can conclude from Figure 16 that the deviation between the actual trajectory and the predetermined trajectory of the subject's hand was smaller than that of the first three laps obviously. The RMS of the total interactive force between the subject's hand and the handle is 2.01 N, as shown in Figure 15a, which is mainly caused by the motion speed and the error of the torque sensor. During the last lap, it can be seen from Figure 14c,d that the trajectory of the subject's hand intentionally deviated from FTR significantly. Figure 15a shows that the robot will provide assistance when the subject's arm deviates from the FTR. The greater the degree of deviation, the stiffer the stiffness wall will be and the greater the assistance force will be. It can be seen from Figure 16 that when the subject's arm moves in the FTR, the robot does not provide assistance force, while the robot will provide corresponding assistance force according to the deviation degree when deviating from the FTR. Since the robot's pose remained fixed during the testing, the RMS of the interactive wrench is 2.05 Nm, as shown in Figure 15b. Figure 16: The total interaction force and the deviation between the handle and the subject’s hand.