# Kinematics and Performance Analysis of the 2T1R/1T2R Reconfigurable Decoupled Parallel Mechanism
## 3 Kinematic Analysis
### Inverse Position
The elements at the corresponding positions of the transformation matrix \(\overset{o}{o}\textbf{T}\) and the Euler angle pose matrix \(\textbf{R}_{1}\) are equal, that is  
\[\overset{o}{o}\textbf{T}=\textbf{R}_{1} \tag{14}\]  
From Equations (12) and (13), it can be seen that when the mechanism is in 2T1R and 1T2R modes, the relationship between the rotation angle of serial chain 1 and the rotation angle of the moving platform is  
\[\beta=\theta_{12}+\theta_{13} \tag{15}\]  
\[\alpha=\theta_{14} \tag{16}\]  
The solution for driving \(\theta_{12}\) with serial chain 1 is  
\[\theta_{12}=\text{arccos}\bigg{(}\frac{Z-H-l_{12}\cos\beta}{l_{11}}\bigg{)} \tag{17}\]  
For serial chain 2, the transformation matrix from the moving platform to the static platform can be obtained as follows:  
\[\overset{O}{O}\textbf{T}=\overset{O}{0}\textbf{T}_{1}^{0}\textbf{T}_{2}^{1} \textbf{T}_{3}^{2}\textbf{T}_{4}^{3}\textbf{T}_{4}^{5}\textbf{T}_{O_{1}}^{5} \textbf{T}= \begin{bmatrix}M_{11}^{\prime}&M_{12}^{\prime}&M_{13}^{\prime}&p_{X }\\ M_{21}^{\prime}&M_{22}^{\prime}&M_{23}^{\prime}&p_{Y}\\ M_{31}^{\prime}&M_{32}^{\prime}&M_{33}^{\prime}&p_{Z}\\ 0&0&0&1\end{bmatrix}  \tag{18}\]  
where \(M_{11}^{\prime}=cos(\theta_{22}+\theta_{23}+\theta_{24})\); \(M_{12}^{\prime}=0\); \(M_{13}^{\prime}=sin(\theta_{22}+\theta_{23}+\theta_{24})\); \(M_{21}^{\prime}=M_{13}sin\theta_{21}\); \(M_{22}^{\prime}=cos\theta_{21}\); \(M_{23}^{\prime}=-M_{11}^{\prime}sin\theta_{21}\); \(M_{31}^{\prime}=-M_{13}^{\prime}cos\theta_{21}\); \(M_{32}^{\prime}=sin\theta_{21}\); \(M_{33}^{\prime}=M_{11}^{\prime}cos\theta_{21}\). When serial chain 3 is in \(\text{P}_{31}\text{R}_{32}\text{R}_{33}\text{P}_{34}\) mode, the transformation matrix from the moving platform to the static platform can be obtained as follows:  
\[\overset{O}{O}\mathbf{T}=\overset{O}{0}\mathbf{T}_{1}^{0}\mathbf{T}_{2}^{1} \mathbf{T}_{3}^{2}\mathbf{T}_{4}^{3}\mathbf{T}_{4}^{5}\mathbf{T}_{O_{1}}^{5} \mathbf{T}= \begin{bmatrix}N_{11}&N_{12}&N_{13}&p_{X}\\ N_{21}&N_{22}&N_{23}&p_{Y}\\ N_{31}&N_{32}&N_{33}&p_{Z}\\ 0&0&0&1\end{bmatrix}  \tag{21}\]  
where \(N_{11}=1\); \(N_{12}=0\); \(N_{13}=0\); \(N_{21}=0\); \(N_{22}=\cos(\alpha_{32}+\alpha_{33})\); \(N_{23}=-\sin(\alpha_{32}+\alpha_{33})\); \(N_{31}=0\); \(N_{32}=\sin(\alpha_{32}+\alpha_{33})\); \(N_{33}=\cos(\alpha_{32}+\alpha_{33})\); \(X=a_{31}+a-b\)  
When serial chain 3 is in \(\text{R}_{31}^{\prime}\text{R}_{32}^{\prime}\text{P}_{33}^{\prime}\text{P}_{3 4}^{\prime}\) mode, the transformation matrix from the moving platform to the static platform can be obtained as follows:  
\[\overset{O}{O}\mathbf{T}=\overset{O}{0}\mathbf{T}_{1}^{0}\mathbf{T}_{2}^{1} \mathbf{T}_{3}^{2}\mathbf{T}_{4}^{3}\mathbf{T}_{4}^{5}\mathbf{T}_{5}^{6} \mathbf{T}_{O_{1}}^{6}\mathbf{T}= \begin{bmatrix}N_{11}^{\prime}&N_{12}^{\prime }&N_{13}^{\prime}&p_{X}\\ N_{21}^{\prime}&N_{22}^{\prime}&N_{23}^{\prime}&p_{Y}\\ N_{31}^{\prime}&N_{32}^{\prime}&N_{33}^{\prime}&p_{Z}\\ 0&0&0&1\end{bmatrix}  \tag{22}\]  
where \(N_{11}^{\prime}=\cos\theta_{31}\); \(N_{12}^{\prime}=\sin\theta_{31}\sin\theta_{32}\); \(N_{13}^{\prime}=\sin\theta_{31}\cos\theta_{32}\); \(N_{21}^{\prime}=0\); \(N_{22}^{\prime}=\cos\theta_{32}\); \(N_{23}^{\prime}=-\sin\theta_{32}\); \(N_{31}^{\prime}=-\sin\theta_{31}\); \(N_{32}^{\prime}=\cos\theta_{31}\sin\theta_{32}\); \(N_{33}^{\prime}=\cos\theta_{31}\cos\theta_{32}\). The rotation matrix of serial chain 3 adopts Y-X type Euler angles, and the Euler angle pose transformation matrix of serial chain 3 is  
\[R_{3}^{\prime}= \begin{bmatrix}\cos\beta&\sin\beta\sin\alpha&\sin\beta\cos \alpha&x\\ 0&\cos\alpha&-\sin\alpha&y\\ -\sin\beta&\cos\beta\sin\alpha&\cos\alpha\cos\beta&z\\ 0&0&0&1\end{bmatrix}  \tag{23}\]  
When the mechanism is in 2T1R mode, the rotation angle \(\beta\) of the moving platform around Y is 0. The elements at the corresponding positions of transformation matrix \(\overset{o}{o}\mathbf{T}\) and Euler angle pose matrix \(\mathbf{R}_{3}^{\prime}\) are equal.