## 4 Performance Analysis
### Workspace and Load Capacity Performance Index in the 2T1R Motion Mode  
When the mechanism is a redundantly actuated parallel mechanism, the prismatic pairs of branched chains 1, 2 and 3 and the cylindrical pair of branched chain 4 are the driving pairs of the mechanism; when the mechanism is a non-redundantly actuated parallel mechanism, the prismatic pairs of branched chains 1 and 2 and the cylindrical pair of branched chain 4 are the driving pairs of the mechanism. According to the calculation, it can be concluded that in the 2T1R motion mode, the workspace performance index of the redundantly actuated parallel mechanism is 0.9619, and the workspace performance index of the non-redundantly actuated parallel mechanism is 0.9619. After introducing branched chains with actuation redundancy, the size and shape of the reachable workspace of the mechanism have not changed, as shown in Figure 7.  
Figure 6: The three-dimensional distribution map and corresponding contour lines of rotational dexterity index at \(Z=300\) mm: (**a**) redundantly actuated parallel mechanism and (**b**) non-redundantly actuated parallel mechanism.  
The region with a low load capacity performance is represented as a blue region. By comparing Figure 7a,b, it can be concluded that after introducing branched chains with actuation redundancy, the load capacity performance index represented by the blue region of the redundantly actuated parallel mechanism is significantly higher than that of the non-redundantly actuated parallel mechanism, and the load capacity performance of the mechanism is clearly improved. The region with a high load capacity performance is represented as a yellow region. With an increase in the \(\alpha\) value, the load capacity performance of the redundantly actuated parallel mechanism increases first and then decreases, which is reflected in the transition from green to yellow, green and blue.  
Under the initial parameters when \(\alpha=0^{\circ}\), the load capacity performance distribution of the redundantly actuated parallel mechanism and non-redundantly actuated parallel mechanism is obtained, as shown in Figure 8. When comparing Figure 8a,b, it can be concluded that the load capacity performance of the redundantly actuated parallel mechanism is significantly higher than that of the non-redundantly actuated parallel mechanism. From Figure 8a, it can be concluded that the load capacity performance of the redundantly actuated parallel mechanism is symmetrical when \(X=0\) mm. As the \(Z\) value increases, the load capacity performance gradually increases. From Figure 8b, it can be concluded that with an increase in the \(X\) value and \(Z\) value, the load capacity performance of the non-redundantly actuated parallel mechanism increases gradually.  
Figure 8: The three-dimensional distribution map and corresponding contour lines of load capacity performance index at \(\alpha=0^{\circ}\): (**a**) redundantly actuated parallel mechanism and (**b**) non-redundantly actuated parallel mechanism.  
Figure 7: The load capacity performance index distribution in the workspace: (**a**) a redundantly actuated parallel mechanism and (**b**) a non-redundantly actuated parallel mechanism.