# Kinematics and Performance Analysis of the 2T1R/1T2R Reconfigurable Decoupled Parallel Mechanism
## 2 Analysis of Reconfigurable Parallel Mechanism
### Analysis of Degrees of Freedom
Therefore, when serial chain 3 is in \(\text{P}_{31}\text{R}_{32}\text{R}_{33}\text{P}_{34}\)mode, the motion mode of the mechanism is \(2\text{T}1\text{R}\), and the mobility is full-cycle. When serial chain 3 is in \(\text{R}_{31}^{\prime}\text{R}_{32}^{\prime}\text{P}_{33}^{\prime}\text{P}_{34}\) mode, the kinematic screw of each kinematic pair on serial chain 3 is  
\[ \begin{split}\boldsymbol{\xi}_{31}^{\prime}&=\left( \begin{array}{cccccc}0&1&0&;&-Z_{A_{3}^{\prime}}&0&X_{A_{3}^{\prime}}\\ \end{array}\right)\\ \boldsymbol{\xi}_{32}^{\prime}&=\left(\begin{array}{cccccc}1&0&0&;&0&Z_{B_{3 }^{\prime}}&-Y_{B_{3}^{\prime}}\\ \end{array}\right)\\ \boldsymbol{\xi}_{33}^{\prime}&=\left(\begin{array}{cccccc}0&0&0&;&0&0&1\\ \end{array}\right)\\ \boldsymbol{\xi}_{34}^{\prime}&=\left(\begin{array}{cccccc}0&0&0&;&0&1&0\\ \end{array}\right)\end{split}  \tag{8}\]where \(X_{A_{3}^{\prime}},Z_{A_{3}^{\prime}},Y_{B_{3}^{\prime}},Z_{B_{3}^{\prime}}\) are the coordinate parameters of \(A_{3}^{\prime}\), \(B_{3}^{\prime}\), and \(A_{3}^{\prime}\), \(B_{3}^{\prime}\) are the center points of the axes of R\({}_{31}^{\prime}\), R\({}_{32}^{\prime}\), respectively. It can be concluded that the reciprocal screw is  
\[ \begin{array}{l}\mathbf{\xi}_{31}^{\prime\prime}=\left(\begin{array}{cccccc}0&0& 0&;&0&0&1\\ 1&0&0&;&0&Z_{A_{3}^{\prime}}&0\\ \end{array}\right)\\ \end{array}  \tag{9}\]  
Serial chain 3 provides a constraint couple parallel to the \(Z\)-axis and a constraint force parallel to the \(X\)-axis, as shown in Figure 3b. Based on the above analysis, it can be concluded that the moving platform of the mechanism is subjected to constraints parallel to the \(Y\)-axis and the \(X\)-axis, as well as a constraint couple parallel to the \(Z\)-axis. Each serial chain provides a constraint couple parallel to the \(Z\)-axis, which is equivalent to an independent constraint couple. The mechanism has a common constraint; therefore, the order of the mechanism is \(d=6\)\(-1=5\). Using the modified GK formula to calculate the degree of freedom of the mechanism, the result are as follows:  
\[M_{2}=d(n-g-1)+\sum_{i=1}^{g}f_{i}+\
u-\zeta=5\times(9-10-1)+13=3 \tag{10}\]  
Thus, when the serial chain 3 is in R\({}_{31}^{\prime}\) R\({}_{32}^{\prime}\)P\({}_{33}^{\prime}\)P\({}_{34}^{\prime}\) mode, the degree of freedom of the mechanism is three, and the moving platform is subject to two independent constraint forces and one independent constraint couple. An independent constraint force parallel to the \(Y\)-axis constrains the movement of the moving platform along the \(Y\)-axis; an independent constraint force parallel to the \(X\)-axis constrains the movement of the moving platform along the \(X\)-axis; one independent constraint couple is parallel to the \(Z\)-axis, constraining the rotation of the moving platform around the \(Z\)-axis. In the process of motion, the position relationship between the two constraint forces and the one constraint couple remains unchanged, and the constraint properties of the moving platform do not change. Therefore, when serial chain 3 is in R\({}_{31}^{\prime}\)R\({}_{32}^{\prime}\)P\({}_{33}^{\prime}\)P\({}_{34}^{\prime}\) mode, the motion mode of the mechanism is 1T2R, and the mobility is full-cycle. When the mechanism is in 2T1R motion mode, the driving pair in serial chain 1 is set to R\({}_{12}\), the driving pair of serial chain 2 is set to rotate parallel to the axis of P\({}_{11}\) in the U pair connected to the static platform, and the driving pair in serial chain 3 is set to P\({}_{31}\). When the mechanism is in 1T2R motion mode, the driving pair in serial chain 1 is set to R\({}_{12}\), the driving pair of serial chain 2 is set to rotate parallel to the axis of P\({}_{11}\) in the U pair connected to the static platform, and the driving pair in serial chain 3 is set to R\({}_{31}^{\prime}\). Based on the above analysis, when serial chain 3 is in P\({}_{31}\)R\({}_{32}\)R\({}_{33}\)P\({}_{34}\) mode, the mechanism is in 2T1R motion mode, and the degree of freedom has full circumference.