## 4 Clinical Experimental and Validation
### Data Processing  
The active range of movement, movement deviation and movement smoothness of the subject were calculated by the real trajectory, which was mainly calculated by the forward kinematics formula of the robot. The data of each training period were normalized to 100 data points by the interpolation method to analyze the changes in each period. (1) The active range of movement  
The active range of movement mainly reflected the range of motion of the subject's shoulder and elbow joints. In the movement of circular trajectory, the general equation of elliptic curve was obtained by fitting the actual trajectory, shown in Figure 7. The ratio of the short semi-axis to the long semi-axis was calculated as the characteristic parameter of the active range of movement. The general equation of the ellipse was obtained by the following formula. \[Ax^{2}+Bxy+Cy^{2}+Dx+By+1=0 \tag{10}\]  
The central coordinates \((x_{c},y_{c})\) of the ellipse geometry were described as follows:  
\[\left\{ \begin{array}{l}x_{c}=\frac{Bx-2 CD}{4 AC-Bx}\\ y_{c}=\frac{B0-2 AE}{4 AC-B^{2}}\end{array} \right. \tag{11}\]  
\[r=Ax_{c}^{2}+Bx_{c}y_{c}+By_{c}^{2}-1 \tag{12}\]  
The length of long semi-axis a, the length of short semi-axis \(b\), and the ratio of two semi-axis were respectively described as follows:  
\[a=\sqrt{r/A} \tag{13}\]  
\[b=\sqrt{\frac{4 Ar}{4 AC-B^{2}}} \tag{14}\]  
\[R_{ratio}=\frac{b}{a} \tag{15}\]  
Figure 6: The experimental setup of subjects. The movement deviation between the actual trajectory and the expected trajectory can reflect the accuracy of the subject's movement. In circular trajectory tracking training, trajectory deviation was defined as the distance between the actual point and the nearest point on the target circle trajectory along the diameter direction. The average trajectory deviation in a movement period was calculated as follows:  
\[dev_{average}=\frac{\sum_{i=1}^{n}Deviation}{n} \tag{16}\]  
where \(n\) represents the number of data points recorded within a period, Deviation represents the deviation of any point in actual movement. \[Deviation=\sqrt{\left(x_{p}-x_{c}\right)^{2}-\left(y_{p}-y_{c}\right)^{2}}-r \tag{17}\]  
where \(\left(x_{c},y_{c}\right)\) represents the coordinates of the center of the target circular track, \(\left(x_{p},y_{p}\right)\) represents the coordinates of the actual movement track, \(r\) represents the radius of the target circular track. The smoothness of movement was mainly used to describe the coordination of movement of stroke patients. The change of movement velocity and movement acceleration can describe the smoothness of movement. The mean value of the speed divided by the peak speed in a movement period can also be calculated using the parameter of movement smoothness. These parameters are calculated as follows:  
\[v_{i}=\frac{\sqrt{\left(x_{i+1}-x_{i}\right)^{2}-\left(y_{i+1}-y_{i}\right)^{ 2}}}{t_{i+1}-t_{i}} \tag{18}\]  
where \(\left(x_{i+1},y_{i+1}\right)\) represents the coordinates of the end-point position at time \(t_{i+1}\), \(\left(x_{i},y_{i}\right)\) represents the end-point position at time \(t_{i}\). \[a_{i}=\frac{v_{i+1}-v_{i}}{t_{i+1}-t_{i}} \tag{19}\]  
Figure 7: The fitted ellipse trajectory during circular trajectory tracking training. where \(v_{i}\) represents the movement velocity at time \(t_{i}\) and \(v_{i+1}\) represents the movement velocity at time \(t_{i+1}\), \(a_{i}\) represents the movement acceleration at time \(t_{i}\). \[a=\sqrt{r/A} \tag{13}\]0  
where the average velocity during the movement was calculated as follows:  
\[a=\sqrt{r/A} \tag{13}\]1  
(4) The interactive force. Interactive force is mainly used to describe the subjects' abilities to exert force. In the circular tracking training, the average force was calculated as a characteristic parameter during a period [26]. As required by the circular trajectory tracking task, the patient needs to move the handle along the tangent of the circle. In fact, the interactive force, collected by the force sensor, is the resultant force between the patient and the robot.