## 2 Mechanism Analysis
### Calculation of Degree of Freedom
#### 2.3.1 1T2R Motion Mode  
When branched chain 1 is in the PS mode, the kinematic screw of each kinematic pair on branched chain 1 is  
\[\begin{array}{l}\mathbf{\xi}_{11}=(0\ \ 0\ \ 0\ \ 0\ \ 0\ \ 1)\\ \mathbf{\xi}_{12}=(0\ \ 1\ \ 0\ \ -e_{11}\ \ 0\ \ 0)\\ \mathbf{\xi}_{13}=(1\ \ 0\ \ 0\ \ 0\ \ e_{11}\ \ 0)\\ \mathbf{\xi}_{14}=(0\ \ 0\ \ 1\ \ 0\ \ 0\ \ 0)\end{array} \tag{1}\]  
The reciprocal screw of branched chain 1 in the PS mode can be obtained as  
\[\begin{array}{l}\mathbf{\xi}_{11}^{\prime}=(1\ \ 0\ \ 0\ \ 0\ \ e_{11}\ \ 0)\\ \mathbf{\xi}_{12}^{\prime}=(0\ \ 1\ \ 0\ \ -e_{11}\ \ 0\ \ 0)\end{array} \tag{2}\]  
Branched chain 1 provides two constraint forces parallel to the \(X\) axis and \(Y\) axis, and both pass through the rotation center of the spherical pair, as shown in Figure 3a.  
When the vA pair is in the universal joint mode, the kinematic screw of each kinematic pair on branched chain 4 is  
\[\begin{array}{l}\mathbf{\xi}_{21}=(0\ \ 0\ \ 0\ \ 1\ \ 0\ \ 0)\\ \mathbf{\xi}_{22}=(1\ \ 0\ \ 0\ \ 0\ \ 0\ \ e_{12})\\ \mathbf{\xi}_{23}=(0\ \ 0\ \ 0\ \ e_{13}\ \ \ f_{13}\ \ g_{13})\\ \mathbf{\xi}_{24}=(1\ \ 0\ \ 0\ \ 0\ \ g_{14}\ \ \ -f_{14})\\ \mathbf{\xi}_{25}=(0\ \ 1\ \ 0\ \ -g_{14}\ \ 0\ \ e_{14})\end{array} \tag{3}\]  
When the vA pair is in the universal joint mode, the reciprocal screw of branched chain 4 can be obtained as  
\[\mathbf{\xi}_{21}^{\prime}=(0\ \ 0\ \ 0\ \ 0\ \ 0\ \ 1) \tag{4}\]  
In the above equations, \(e_{11}\), \(e_{12}\), \(g_{12}\), \(e_{13}\), \(e_{13}\), \(e_{14}\), \(f_{14}\) and \(g_{14}\) are different real numbers.  
Branched chain 4 provides a constraint couple that is parallel to the \(Z\) axis, as shown in Figure 3a.  
Branched chain 2 and branched chain 3 pose no constraints on the moving platform; thus, they are not analyzed.  
By combining Equations (2) and (4), the kinematic screw system of the moving platform can be obtained as  
\[\begin{array}{l}\mathbf{\xi}_{1}=(1\ \ 0\ \ 0\ \ 0\ \ e_{11}\ \ 0)\\ \mathbf{\xi}_{2}=(0\ \ 1\ \ 0\ \ e_{11}\ \ 0\ \ 0)\\ \mathbf{\xi}_{3}=(0\ \ 0\ \ 0\ \ 0\ \ 0\ \ 1)\end{array} \tag{5}\]  
From the kinematic screw system in Equation (5), it can be concluded that the moving platform has 1T2R degrees of freedom, which encompass two rotational degrees of freedom that bypass the origin \(O_{1}\) of the moving coordinate system and are parallel to the \(X\) axis and \(Y\) axis, together with one translational degree of freedom along the \(Z\) axis.  
\begin{table}
\begin{tabular}{c c When branched chain 1 is in the PS mode and the vA pair is in the universal joint mode, the modified G-K formula is used to calculate the degree of freedom of the mechanism as follows:  
\[M_{1}=d(n-g-1)+\sum_{i=1}^{g}f_{i}+v-\zeta=6(9-11-1)+21=3 \tag{6}\]  
where \(d\) is the order of the mechanism, \(n\) is the number of mechanism links, \(g\) is the number of kinematic pairs of the mechanism, \(f_{i}\) is the degree of freedom of the kinematic pairs of the mechanism, \(v\) is the number of parallel redundant constraints, and \(\zeta\) is the number of passive degrees of freedom.  
It is concluded that when branched chain 1 is in the PS mode and the vA pair is in the universal joint mode, the degree of freedom of the mechanism is 3, and the moving platform is constrained by two constraint forces and one constraint couple. The two constraint forces are parallel to the \(X\) axis and \(Y\) axis, and always intersect vertically at the center point of the moving platform, which restricts the translation of the moving platform along the \(X\) axis and \(Y\) axis. The constraint couple is parallel to the \(Z\) axis and constrains the rotation of the moving platform around the \(Z\) axis. When in motion, the positional relationship between the two constraint forces and the one constraint couple remains unchanged, and the constraint property of the moving platform does not change. Therefore, when branched chain 1 is in the PS mode and the vA pair is in the universal joint mode, the motion mode of the mechanism is 1T2R, and the degree of freedom is full-cycle.