# Kinematics and Performance Analysis of the 2T1R/1T2R Reconfigurable Decoupled Parallel Mechanism
## 3 Kinematic Analysis  
### Inverse Position  
We established kinematic models of the mechanism in two modes, 2T1R and 1T2R, using the DH parameter method [33]. Firstly, we established the coordinate system for serial chain 1 and serial chain 2, as shown in Figure 4, and obtained the DH parameters for serial chain 1 and serial chain 2, as shown in Tables 2 and 3. The coordinate transformation of the serial chain coordinate system adopts the prepositive serial chain model. According to the basic homogeneous rotation and translation formula, the general formula for the transformation matrix between two adjacent coordinate systems \(\{i\}\) and \(\{i-1\}\) is  
\[\overset{i-1}{i}\mathbf{T}= \begin{bmatrix}\mathrm{c}\theta_{i}&-\mathrm{s} \theta_{i}&0&a_{i-1}\\ \mathrm{s}\theta_{\mathrm{j}}\mathrm{c}\alpha_{i-1}&\mathrm{c}\theta_{\mathrm{ j}}\mathrm{c}\alpha_{i-1}&-\mathrm{s}\alpha_{i-1}&-d_{\mathrm{j}}\mathrm{s}\alpha_{i-1}\\ \mathrm{s}\theta_{\mathrm{j}}\mathrm{s}\alpha_{i-1}&\mathrm{c}\theta_{i} \mathrm{s}\alpha_{i-1}&\mathrm{c}\alpha_{i-1}&d_{\mathrm{j}}\mathrm{c}\alpha_{ i-1}\\ 0&0&0&1\end{bmatrix}  \tag{11}\]  
Figure 4: DH coordinate system of serial chains 1 and 2: (**a**) coordinate system of serial chain 1 and (**b**) coordinate system of serial chain 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{i}\) & \(\mathbf{\alpha_{i-1}}\)/\({}^{\mathrm{(c)}}\) & \(\mathbf{a_{i-1}}\)/mm & \(\mathbf{\theta_{i}}\)/\({}^{\mathrm{(c)}}\) & \(\mathbf{d_{i}}\)/mm \\ \hline
1 & 0 & 0 & 0 & \(d_{11}\) \\
2 & 0 & \(H_{11}\) & \(\theta_{12}\) & \\
3 & 90 & \(l_{11}\) & \(\theta_{13}\) & 0 \\
4 & 0 & \(l_{12}\) & \(\theta_{14}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: DH parameters of serial chain 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{i}\) & \(\mathbf{\alpha_{i-1}}\)/\({}^{\mathrm{(c)}}\) & \(\mathbf{a_{i-1}}\)/mm & \(\mathbf{\theta_{i}}\)/\({}^{\mathrm{(c)}}\) & \(\mathbf{d_{i}}\)/mm \\ \hline
1 & 0 & 0 & \(\theta_{21}\) & 0 \\
2 & 90 & 0 & \(\theta_{22}\) & 0 \\
3 & 0 & \(l_{23}\) & \(\theta_{23}\) & 0 \\
4 & 0 & \(l_{24}\) & \(\theta_{24}\) & 0 \\
5 & 0 & 0 & 0 & \(d_{25}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: DH parameters of serial chain 2. For serial chain 1, the transformation matrix from the moving platform to the static platform can be obtained as follows:  
\[ \begin{array}{cccc}\overset{O}{O}\textbf{T}=\overset{O}{0}\textbf{T}_{01}^{01} \textbf{T}_{02}^{1}\textbf{T}_{03}^{1}\textbf{T}_{1}^{2}\textbf{T}_{2}^{2} \textbf{T}_{3}^{3}\textbf{T}_{4}^{4}\textbf{T}_{O_{11}}^{O_{11}}\textbf{T}_{O_{1 2}}^{O_{12}}\textbf{T}_{O_{13}}^{O_{13}}\textbf{T}=\begin{bmatrix}M_{11}&M_{12}& M_{13}&p_{X}\\ M_{21}&M_{22}&M_{23}&p_{Y}\\ M_{31}&M_{32}&M_{33}&p_{Z}\\ 0&0&0&1\end{bmatrix}\end{array}  \tag{12}\]  
where \(M_{11}=\text{cos}(\theta_{12}+\theta_{13})\); \(M_{12}=\text{sin}(\theta_{12}+\theta_{13})\text{sin}\theta_{14}\); \(M_{13}=\text{sin}(\theta_{12}+\theta_{13})\text{cos}\theta_{14}\); \(M_{21}=0\); \(M_{22}=\text{cos}\theta_{14}\); \(M_{23}=-\text{sin}\theta_{14}\); \(M_{31}=-\text{sin}(\theta_{12}+\theta_{13})\); \(M_{32}=\text{cos}(\theta_{12}+\theta_{13})\text{sin}\theta_{14}\); \(Y=0\); \(M_{33}=\text{cos}(\theta_{12}+\theta_{13})\text{cos}\theta_{14}\); \(X=d_{11}+l_{12}\text{sin}(\theta_{12}+\theta_{13})+l_{11}\text{sin}\theta_{12}\); \(Z=H+l_{12}\text{cos}(\theta_{12}+\theta_{13})+l_{11}\text{cos}\theta_{12}\)  
The rotation matrix of serial chain 1 adopts \(Y\)-\(X\) type Euler angles, and the Euler angle pose transformation matrix \(\textbf{R}_{1}\) of serial chain 1 is  
\[\textbf{R}_{1}= \begin{bmatrix}\cos\beta&\sin\beta\sin\alpha&\sin\beta\cos \alpha&x\\ 0&\cos\alpha&-\sin\alpha&y\\ -\sin\beta&\cos\beta\sin\alpha&\cos\alpha\cos\beta&z\\ 0&0&0&1\end{bmatrix}  \tag{13}\]  
where \(\alpha\) and \(\beta\) are the rotation angles of the moving platform around \(X\) and \(Y\), respectively. When the mechanism is in 2T1R mode, \(\beta=0\).