## 2 Mechanism Analysis
### Calculation of Degree of Freedom
#### 2.3.2 2T1R Motion Mode  
When branched chain 1 is in the RPS mode, the kinematic screw of each kinematic pair on branched chain 1 is  
\[ \begin{array}{l}\mathbf{\xi}_{11}=(0\ \ 1\ \ 0\ \ 0\ \ 0\ \ 0)\\ \mathbf{\xi}_{12}=(0\ \ 0\ \ 0\ \ e_{21}\ \ \ 0\ \ f_{21})\\ \mathbf{\xi}_{13}=(0\ \ 1\ \ 0\ \ -\ f_{22}\ \ 0\ \ e_{22})\\ \mathbf{\xi}_{14}=(1\ \ 0\ \ 0\ \ 0\ \ f_{22}\ \ 0)\\ \mathbf{\xi}_{15}=(0\ \ 0\ \ 1\ \ 0\ \ -\ e_{22}\ \ 0)\\ \end{array}  \tag{7}\]  
The reciprocal screw of branched chain 1 in the RPS mode can be obtained as  
\[\mathbf{\xi}_{11}^{\prime}=(0\ \ 1\ \ 0\ \ -f_{22}\ \ 0\ \ e_{22}) \tag{8}\]  
Branched chain 1 provides a constraint force parallel to the \(Y\) axis, which passes through the rotation center of the spherical pair, as shown in Figure 3b. When the vA pair is in the revolute-pair mode, the kinematic screw of each kinematic pair on branched chain 4 is  
\[ \begin{array}{l}\mathbf{\xi}_{21}=(0\ \ 0\ \ 0\ \ 1\ \ 0\ \ 0)\\ \mathbf{\xi}_{22}=(1\ \ 0\ \ 0\ \ 0\ \ 0\ \ e_{23})\\ \mathbf{\xi}_{23}=(0\ \ 0\ \ 0\ \ e_{24}\ \ f_{24}\ \ g_{24})\\ \mathbf{\xi}_{24}=(1\ \ 0\ \ 0\ \ 0\ \ g_{25}\ \ -f_{25})\\ \end{array}  \tag{9}\]  
When the vA pair is in the revolute-pair mode, the reciprocal screw of branched chain 4 can be obtained as  
\[ \begin{array}{l}\mathbf{\xi}_{21}^{\prime}=(0\ \ 0\ \ 0\ \ 0\ \ 1\ \ 0)\\ \mathbf{\xi}_{22}=(0\ \ 0\ \ 0\ \ 0\ \ 0\ \ 1)\\ \end{array}  \tag{10}\]  
Branched chain 4 provides two constraint couples parallel to the \(Y\) axis and \(Z\) axis, as shown in Figure 3b. In the above equations, \(e_{21},f_{21},e_{22},f_{22},e_{23},e_{24},f_{24},f_{25}\) and \(g_{25}\) are different real numbers. By combining Equations (8) and (10), the kinematic screw system of the moving platform can be obtained as  
\[ \begin{array}{l}\mathbf{\xi}_{1}=(0\ \ 0\ \ 0\ \ 1\ \ 0\ \ 0)\\ \mathbf{\xi}_{2}=(0\ \ 0\ \ 0\ \ 0\ \ 0\ \ 1)\\ \mathbf{\xi}_{3}=(1\ \ 0\ \ 0\ \ 0\ \ f_{22}\ \ 0)\end{array}  \tag{11}\]  
From the kinematic screw system in Equation (11), it can be concluded that the moving platform has 2T1R degrees of freedom, which encompass two translational degrees of freedom along the \(X\) axis and \(Z\) axis, and a rotational degree of freedom that bypasses the origin \(O_{1}\) of the moving coordinate system and is parallel to the \(X\) axis. When branched chain 1 is in the RPS mode and the vA pair is in the revolute-pair mode, the modified G-K formula is used to calculate the degree of freedom of the mechanism as follows:  
\[M_{2}=d(n-g-1)+\sum_{i=1}^{S}f_{i}+v-\zeta=6(10-12-1)+21=3 \tag{12}\]  
It is concluded that when branched chain 1 is in the RPS mode and the vA pair is in the revolute-pair mode, the degree of freedom of the mechanism is 3, and the moving platform is constrained by two constraint couples and one constraint force. The two constraint couples are parallel to the \(Y\) axis and the \(Z\) axis in order to constrain the rotation of the moving platform around the \(Y\) axis and the \(Z\) axis, respectively. The constraint force that bypasses the origin \(O_{1}\) of the moving coordinate system and is parallel to the \(Y\) axis constrains the translation of the moving platform along the \(Y\) axis. When in motion, the positional relationship between the two constraint couples and the one constraint force remains unchanged, and the constraint property of the moving platform does not change. Therefore, when branched chain 1 is in the RPS mode and the vA pair is in the revolute-pair mode, the motion mode of the mechanism is 2T1R, and the degree of freedom is full-cycle. Based on the above analysis, when branched chain 1 is in the PS mode and the vA pair is in the universal joint mode, the mechanism is in the 1T2R motion mode, and the degree of freedom is full-cycle. When branched chain 1 is in the RPS mode and the vA pair is in the revolute-pair mode, the mechanism is in the 2T1R motion mode, and the degree of freedom is full-cycle. When the mechanism is in the 1T2R motion mode, the prismatic pairs of the four branched chains are used as the driving pairs.