# Kinematics and Performance Analysis of the 2T1R/1T2R Reconfigurable Decoupled Parallel Mechanism
## 2 Analysis of Reconfigurable Parallel Mechanism
### Analysis of Degrees of Freedom  
\(O\) and \(O_{1}\) are the geometric centers of the static and moving platforms, respectively. The static coordinate system \(O\)-\(XYZ\) and the moving coordinate system \(O_{1}\)-\(X_{1}Y_{1}Z_{1}\) are established with \(O\) and \(O_{1}\) as the origins, respectively, as shown in Figure 3. Figure 3a,b shows two motion modes, 2T1R and 1T2R, respectively. In 2T1R motion mode, the distance between the axis of P\({}_{31}\) and the \(X\)-axis is \(a\), the distance between the intersection point of the axis of U\({}_{21}\) and the \(Y\)-axis is \(a\), and the distance between the axis of C\({}_{23}\) and the \(Y_{1}\) -axis is \(b\). In 1T2R motion mode, the distance between the axis center of R\({}^{\prime}_{31}\) and the \(X\)-axis and  
Figure 1: Reconfigurable parallel mechanism. Figure 2: Two modes of reconfigurable kinematic pair rR: (**a**) the axis of the kinematic pair rR coincides with the \(X_{R}\)-axis and (**b**) the axis of the kinematic pair rR coincides with the \(Y_{R}\)-axis. The serial chain 2 provides a constraint couple parallel to the \(Z\)-axis. When serial chain 3 is in \(\text{P}_{31}\text{R}_{32}\text{R}_{33}\text{P}_{34}\) mode, the kinematic screw of serial chain 3 is  
\[ \begin{split}\boldsymbol{\xi}_{31}&=\left(\begin{array} []{cccccc}0&0&0&;&1&0&0\\ \boldsymbol{\xi}_{32}&=\left(\begin{array}{cccccc}1&0&0&;&0&0&0\\ \end{array}\right)\\ \boldsymbol{\xi}_{33}&=\left(\begin{array}{cccccc}1&0&0&;&0&Z_{B3}&-Y_{B3} \\ \end{array}\right)\\ \boldsymbol{\xi}_{34}&=\left(\begin{array}{cccccc}0&0&0&;&0&0&1\\ \end{array}\right)\end{split}  \tag{5}\]  
where \(Y_{B3}\) and \(Z_{B3}\) are the coordinate parameters of \(B_{3}\), and \(B_{3}\) is the center point of the \(\text{R}_{33}\) axis. The constraint screw can be obtained as follows:  
\[ \begin{split}\boldsymbol{\xi}_{31}&=\left(\begin{array}{cccccc} 0&0&0&;&0&1&0\\ \boldsymbol{\xi}_{32}^{*}&=\left(\begin{array}{cccccc}0&0&0&;&0&0&1\\ \end{array}\right)\end{array}\right)\\ \boldsymbol{\xi}_{32}^{*}&=\left(\begin{array}{cccccc}0&0&0&;&0&0&1\\ \end{array}\right)\end{split}  \tag{6}\]  
Serial chain 3 provides a constraint couple parallel to the \(Z\)-axis and a constraint couple parallel to the \(Y\)-axis. From the analysis presented, it can be concluded that the moving platform of the mechanism is subject to constraint forces parallel to the \(Y\)-axis, as well as constraint couples parallel to both the \(Y\)-axis and the \(Z\)-axis. Each serial chain provides a constraint couple parallel to the \(Z\)-axis, which is equivalent to a single independent constraint couple. Therefore, the mechanism has one common constraint, and thus, the order of the mechanism is calculated as \(d=6-1=5\). By using the modified GK formula to calculate the degree of freedom of the mechanism, the following result can be obtained:  
\[M_{1}=d(n-g-1)+\sum_{i=1}^{g}f_{i}+\
u-\zeta=5\times(9-10-1)+13=3 \tag{7}\]  
where \(n\) is the number of components of the mechanism, \(g\) is the number of kinematic pairs of the mechanism, \(f_{i}\) is the degree of freedom of the kinematic pairs of the mechanism, \(\
u\) is the number of parallel redundant constraints, and \(\zeta\) is the local degree of freedom. It can be concluded that when serial chain 3 is in \(\text{P}_{31}\text{R}_{32}\text{R}_{33}\text{P}_{34}\) mode, the degree of freedom of the mechanism is three, and the moving platform is subjected to one independent constraint force and two independent constraint couples. An independent constraint force parallel to the \(Y\)-axis constrains the movement of the moving platform along the \(Y\)-axis; two independent constraint couples are parallel to the \(Y\)-axis and the \(Z\)-axis, respectively, constraining the rotation of the moving platform around the \(Y\)-axis and the \(Z\)-axis. In the process of motion, the position relationship between one independent constraint force and two independent constraint couples remains unchanged, and the constraint properties of the moving platform do not change.