# Kinematics and Performance Analysis of the 2T1R/1T2R Reconfigurable Decoupled Parallel Mechanism
## 2 Analysis of Reconfigurable Parallel Mechanism  
### Configuration Analysis  
The reconfigurable parallel mechanism consists of a static platform, a moving platform, and three serial chains connecting the moving and static platforms, as shown in Figure 1. The static platform is a square with the area \(a\times a\), and the moving platform is a square with the area \(b\times b\). For the convenience of expression, the kinematic pairs of the three serial chains are represented by K\({}_{kl}\) (\(k\) = 1, 2, 3; 1 = 1, 2, 3, 4, 5), where the subscript \(k\) represents the \(k\) serial chain and the subscript \(l\) represents the \(l\) kinematic pair of the \(k\) serial chain.  
Therefore, serial chain 1 can be represented as a P\({}_{11}\)R\({}_{12}\)R\({}_{13}\)R\({}_{14}\) serial chain, and the axis of P\({}_{11}\) is perpendicular to the axis of the R\({}_{12}\) and R\({}_{13}\) pairs. Serial chain 2 is a U\({}_{21}\)R\({}_{22}\)C\({}_{23}\), and the axis of R\({}_{22}\) is parallel to the axis of C\({}_{23}\). Serial chain 3 is a reconfigurable serial chain P\({}_{31}\)R\({}_{32}\)R\({}_{33}\)P\({}_{34}\)P\({}_{35}\), where the R\({}_{32}\) pair is a reconfigurable kinematic pair, specifically named rR pair. The three kinematic pairs of the serial chain are perpendicular in axial direction, and the serial chain can be converted between modes P\({}_{31}\)R\({}_{32}\)R\({}_{33}\)P\({}_{34}\) and R\({}^{\prime}_{31}\)R\({}^{\prime}_{32}\)P\({}^{\prime}_{33}\)P\({}^{\prime}_{34}\) by locking the P\({}_{31}\) pair and P\({}_{35}\) pair, respectively.  
The kinematic pair rR is a revolute pair with a variable axis. The method proposed in [32] is adopted to realize the change of the rotation axis by using a worm gear system. The rotation bar is mounted to the worm wheel in which the worm wheel can be rotated by the worm. This motion creates the alteration of the (rR) pair axes which is driven by a motor attached to the worm. Figure 2 shows two modes of reconfigurable motion on rR:  
This paper realizes the locking of the prismatic pair through the function of the brake. When the lead screw needs to be locked, the brake automatically starts and clamps the lead screw so that it stops moving without artificial intervention.  
When the mechanism is in 2T1R motion mode and serial chain 3 is in P\({}_{31}\)R\({}_{32}\)R\({}_{33}\)P\({}_{34}\) mode, the translational pair P\({}_{35}\) is locked. The axis of P\({}_{31}\) is parallel to the axis of P\({}_{11}\) in serial chain 1, the axis of rR is perpendicular to the axis of R\({}_{12}\) in serial chain 1, and the axis of R\({}_{33}\) is parallel to the axis of rR. P\({}_{34}\) is a passive prismatic pair, and the axis of P\({}_{34}\) is perpendicular to the plane of the moving platform.  
When the mechanism is in 1T2R motion mode and serial chain 3 is in R\({}^{\prime}_{31}\)R\({}^{\prime}_{32}\)P\({}^{\prime}_{33}\)P\({}^{\prime}_{34}\) mode, the translational pair P\({}_{31}\) is locked. The axis of rR is parallel to the axis of R\({}_{12}\) in serial chain 1, the axis of R\({}_{33}\) is perpendicular to the axis of rR, and the axis of P\({}_{34}\) is perpendicular to the plane of the moving platform. The P\({}_{35}\) is a passive prismatic pair, located in the plane of the moving platform, with its axis perpendicular to the axis of R\({}_{14}\) in serial chain 1.