## 2 Mechanism Description
The mobility analysis and the degrees of freedom of the mechanism have been conducted in our previous work [40], and the degrees of freedom of the mechanism are analyzed as follows:  
The method based on screw theory is used to analyze the degrees of freedom of the mechanism. 1. [label=(0)]
2. Whether the mechanism has local degrees of freedom: the mechanism has no local degrees of freedom. 3. Establish the reference coordinate system and construct the twist screw system of each limb. Since the four limbs have the same structure, take out limb 1 for analysis, as shown in Figure 3, the coordinate system of limb 1 \(O1x1y1z1\) is shown in the figure, and the center  
Figure 2: Kinematical diagrams of 4-SPPP parallel mechanism. (**a**) 3D diagram of 4-SPPP parallel mechanism; (**b**) 3D diagram of the limb; (**c**) Kinematical diagram of 4-SPPP parallel mechanism. point of the spherical pair at the end of the limb is chosen as the origin of the coordinate system. The \(x1\)-axis is parallel to the \(x\)-axis of the moving platform, and the \(y1\)-axis is parallel to the \(y\)-axis of the moving platform. The twist screw system of limb 1 is:  
\[ \begin{array}{l}\$_{11}=(1,0,0;0,0,0)\\ \$_{12}=(0,1,0;0,0,0)\\ \$_{13}=(0,0,1;0,0,0)\\ \$_{14}=(0,0,0;d_{1},e_{1},f_{1})\\ \$_{15}=(0,0,0;-d_{2},e_{2},f_{2})\\ \$_{16}=(0,0,0;d_{3},e_{3},f_{3})\end{array}  \tag{1}\]  
where \(\$_{1i}\) is the \(i\)-th twist screw in limb 1, \(i=1\)-6.
3. According to the reversibility of screws, \(\$_{bi}\Delta\$_{bi}^{r}=0\), the constraint wrench screw system of limb 1 is as follows: \[\$_{b1}^{r}=(0,0,0;0,0,0)\] (2) Similarly, the constraint wrench screw system of limbs 2, 3, and 4 is: \[\$_{b2}^{r}=\$_{b3}^{r}=\$_{b4}^{r}=(0,0,0;0,0,0)\] (3) (4) The constraint wrench screw system \(\$^{r}\) of the moving platform can be obtained according to \(\$^{r}=\cup\$_{bi}^{r}(i=1-4)\): \[\$^{r}=(0,0,0;0,0,0)\] (4) (5) Therefore, the twist screw system \(\$_{f}\) of the moving platform can be obtained: \[ \begin{array}{l}\$_{f1}=(1,0,0;0,0,0)\\ \$_{f2}=(0,1,0;0,0,0)\\ \$_{f3}=(0,0,1;0,0,0)\\ \$_{f4}=(0,0,0;1,0,0)\\ \$_{f5}=(0,0,0;0,1,0)\\ \$_{f6}=(0,0,0;0,0,1)\end{array} \] (6) According to the twist screw system \(\$_{f}\) of the moving platform, it is known that the degrees of freedom of the mechanism are three rotations and three translations. Figure 3: The twist screw system of limb 1. Since the geometrical relationship between the kinematic pairs of limbs does not change after the change in the mechanism position, it is known that the degrees of freedom of the mechanism are the real degrees of freedom. Finally, the correctness of the degree of freedom analysis is verified:  
According to the degree of freedom analysis, the number of members of the mechanism is \(n=14\), the number of kinematic pairs is \(g=16\), the total number of constraints is \(\sum\limits_{i=1}^{g}f_{i}=24\), and the public constraints and redundant constraints are all 0; therefore, the degree of freedom of the mechanism is:  
\[S=6(n-g-1)+\sum\limits_{i=1}^{g}f_{i}=6\times(14-16-1)+24=6 \tag{6}\]  
In this paper, the redundant drive is chosen, and the number of driving pairs is 8. As shown in Figure 2, the driving pairs are located on the first and the third Prismatic joint of each limb, i.e., \(P_{i1}\) and \(P_{i3}\), \(i=1\)-4.