## 2 Configuration Design of the Spatial 8R Mechanism Configuration
### Manipulability Analysis of the Spatial 8R Mechanism  
The manipulability \(\omega\) of the mechanism \(\omega\) is used to describe the comprehensive motion and overall flexibility of the mechanism in all directions. The determinant of the product of the Jacobian matrix and its transpose matrix can be defined as the metric index of robot manipulability, which is  
\[\omega=\sqrt{\det(J)^{\mathrm{T}}} \tag{17}\]  
Further, the manipulability \(\omega\) of this mechanism can be obtained as  
\[\omega=\sqrt{\det(J)^{\mathrm{T}}}=|\lambda_{2}|=\left|\frac{\cos\beta\cos \gamma}{\sqrt{1-t^{2}}}\right| \tag{18}\]  
The distribution of \(\omega\) in the workspace of \(-90^{\circ}\)-\(90^{\circ}\) is show in Figure 3.  
As seen in Figure 3, the manipulability degree of (\(\gamma\), \(\beta\)) near the points (\(-90^{\circ}\), \(-90^{\circ}\)), (\(-90^{\circ}\), \(90^{\circ}\)), (\(90^{\circ}\), \(-90^{\circ}\)) and (\(90^{\circ}\), \(90^{\circ}\)) approaches 0, and the value of \(\omega\) near (0, \(\beta_{i}\)) approaches infinity; thus, these positions are the singular configuration of the mechanism, where \(\beta_{i}\) represents the coordinate position on the \(\beta\) axis.  
Table 3 shows the extreme motion angles of abduction and adduction, forward flexion and backward extension of the hip joint during the walking gait, corresponding to the motion ranges of \(\gamma\) and \(\beta\).  
Figure 4 shows the distribution of \(\omega\) in the motion space, whose value ranges from 0.58 to 1.02, and the closer it is to the initial equilibrium position, the closer it is to 1. We concluded that the distribution of the manipulability \(\omega\) value of the mechanism is between 0.58 and 1.02, and the variation is stable and continuous in the motion space.  
To sum up, manipulability is one of the kinematic performance indices of the mechanism, and the closer the value of the index is to 1, the better the movement performance of the mechanism is. Figures 3 and 4 show that the movement performance of this spatial 8R mechanism is improved.  
Figure 5: Wearable hip joint mechanism.  
Figure 6: Cont.  
Figure 6: _Cont._  
As seen in Figure 5, the variation laws of various movements are relatively stable, and there is no break point, indicating that the movement performance of this mechanism is good and can be applied for hip rehabilitation. The coordinate system setting of the variation laws follow the coordinate system direction detailed in the first section, and around a certain axis, the inverse becomes positive, and the positive becomes negative. In the legend, V\({}_{x}\) represents the velocity curve and \(\omega_{x}\) represents the angular velocity curve. In forward flexion of the hip joint, the leg is lifted forward; thus, the velocity curve along the \(y\) axis changes considerably, and the velocity curve along the \(x\) axis changes as a joint linkage to ensure smoother forward flexion movement. There is no velocity in the \(z\) axis direction, so the velocity is 0. Since the forward flexion movement rotates around the \(x\) axis, the change in the angular velocity curve in the \(x\) axis direction is more obvious. The change in the angular velocity curve around the \(y\) axis is smaller than that around the \(x\) axis, and there is no rotation around the \(z\) axis; thus, the corresponding angular velocity curve is 0. The analysis of the velocity and angular velocity curves of other movements was carried out in the same manner. As seen in Figure 6, the movement of this mechanism can conform to the regular motion of the human hip joint, and therefore can be used to help patients recover.