## 3 Analysis of the Posture Adjustment Range
### Effect of Mechanism Parameters on Posture Adjustment Range  
(1) Effect of limb layout on posture adjustment range  
The upper and lower endpoints of each limb intersect in the moving and fixed platforms to form two rectangles, as shown in Figure 2. The rectangular length and width of the fixed platform are \(M\) and \(N\), and the rectangular length and width of the moving platform are \(m\) and \(n\), respectively. The effect on the posture adjustment range is shown in Figure 8. From the figure, it can be seen that with the increase in the length and width of the moving and fixed platforms, the posture adjustment range increases and then decreases.  
(2) Effect of geometrical relations between the driving pairs axes on posture adjustment range  
The geometrical relationship between the driving pairs axes is shown in the positional relationship diagram in Figure 9. \(\psi\) is the angle between the plane formed by the second and third driving pairs and the first driving pair. \(\theta\) is the angle between the second and third driving pairs. \(\phi\) is the angle between the third driving pairs and the long side of moving platform where \(\phi=D,\theta=E,\psi=G\), and its effect on the posture adjustment range is shown in Figure 10. From Figure 10, it can be seen that with the increase in \(\theta\) and \(\psi\), the posture adjustment range increases and then decreases, and with the increase in \(\phi\), the posture adjustment range decreases and then increases.  
Figure 7: Conversion relationship between polar and right-angle coordinate systems.  
(3)  
Effect of the limb inclination angle \(\alpha\) on posture adjustment range  
As shown in Figure 9, the angle between the plane formed by the second and third driving pairs and the first driving pair is \(\psi\), which is related to the limb inclination angle \(\alpha\) as follows:  
\[\alpha=\arccos(d/l_{1})\text{ or }\alpha=\pi-\psi \tag{11}\]  
That is, \(\alpha\) is complementary to \(\psi\). Therefore, the effect of the limb inclination angle \(\alpha\) on the range of posture adjustment is similar to \(\psi\).  
(4)  
Effect of drive trajectory of the driving pair of limb on posture adjustment range  
Figure 8: Effect of limb layout on posture adjustment range. (**a**) Effect of the width of the moving and fixed platforms on the range of posture adjustment. (**b**) Effect of the length of the moving and fixed platforms on the range of posture adjustment.  
Figure 10: Effect of geometrical relations between the driving pairs axes on posture adjustment range.  
Figure 9: Positional relationship diagram.  
In this paper, only the effect of the active driving pairs (the first and second driving pair) on the posture adjustment range is considered, as shown in Figure 11. From Figure 11, it can be seen that the range of posture adjustment is approximately linearly negatively correlated with the minimum limit of the two active driving pairs, whereas the maximum limit of the two active driving pairs affects the range of posture adjustment differently. The maximum limit of the second active driving pair is approximately linearly positively correlated with the range of posture adjustment, while the maximum limit of the first active driving pair has a smaller impact on the range of posture adjustment.