## 5 Kinematic Performance Analysis
The output transmission performance index \(\eta_{1g}\) of the parallel mechanism can be obtained:  
\[\eta_{1i}=\frac{\left|\mathbf{S}_{Ti}\circ\mathbf{S}_{Oi}\right|}{\left|\mathbf{S}_{Ti} \circ\mathbf{S}_{Oi}\right|_{\max}},(i=1,\,2,\,\cdots,\,n). \tag{31}\]  
Combinations of the mechanism's input twist screw, transmission wrench screw, and output twist screw under different motion branches are shown in Table 2.  
The output transmission performance of the 3(Rc)PU PM under different motion modes can be obtained as  
\[\eta_{k}=\frac{1}{N}\sum_{g=1}^{N}\eta_{kg}\,(k=1,\ 2,\ \cdots,\ n). \tag{32}\]  
The local minimization transmission index (LMTI for short) can be used to evaluate the motion/force transmission performance of the redundantly actuated PM. According to Equations (28) and (32), the LMTI can be expressed as follows:  
\[\gamma=\min\{\psi_{1},\ \psi_{2},\ \cdots,\ \psi_{n},\ \eta_{k}\},(k=1,\ 2,\ \cdots,\ n). \tag{33}\]  
where \(\psi_{1},\ \psi_{2},\ \cdots,\ \psi_{n}\) are the input transmission performance indices of the 3(Rc)PU PM. With the structural parameters set as \(R=360\) mm and \(r=220\) mm, the motion/force transmission performance of the 3(Rc)PU parallel mechanism under different positions was analyzed. When the parallel mechanism is in the 3T motion branch, let \(\alpha=0,\ \beta=0,\) and \(\gamma=0,\) and take the center of the moving platform in the fixed coordinate system \(Y\) direction \(Y=400\) mm and \(Y=600\) mm as an example to calculate the motion/force transmission performance of the (Rc)PU PM in the 3T motion mode, as shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Motion Mode** & **Input Twist Screw** & **Transmission Wrench Screw** & **Output Twist Screw** \\ \hline
3T & S\({}_{41}\), S\({}_{42}\), S\({}_{43}\) & S\({}_{71}\), S\({}_{72}\), S\({}_{73}\) & S\({}_{61}\), S\({}_{62}\), S\({}_{63}\) \\
1R3T & S\({}_{41}\), S\({  
As can be seen from Figure 6, when the mechanism is in the 3T motion branch, the LMTI value gradually increases with the increase in the \(Y\) value. In the \(XO_{n}Z\) plane, the LMTI value is the largest when the 3(Rc)PU PM is in the area near (\(X\), \(Z\))= (0, \(-\)65), and the motion/force transmission efficiency of the mechanism is the highest. Similarly, when the parallel mechanism is in the 1R3T motion branch, let \(\alpha\) = 0, \(\beta\) = 0, and \(\gamma\) = 30\({}^{\circ}\), and take the center of the moving platform in the fixed coordinate system \(Y\) direction \(Y\) = 400 mm and \(Y\) = 600 mm as an example to calculate the motion/force transmission performance of the parallel mechanism in the 1R3T motion branch. The performance distribution curve is shown in Figure 7. The motion/force transmission performance distribution curves are shown in Figures 8 and 9 for when the parallel mechanism is in the 2R3T moving branch and the 3R3T motion branch, respectively. Figure 9: _Cont._  
It can be seen from Figures 6-9 that the LMTI value of the 3(Rc)PU PM gradually increases with the \(Y\) value in the accessible working space of the mechanism. At the same time, there is no aggregation of the LMTI distribution isolines in each motion branch, and the LMTI values are all greater than 0.5, indicating that the 3(Rc)PU PM has good motion/force transmission performance.