# The Mobility Analysis of a 3-CCR Parallel Manipulator with Three Screw-Type Terminal Constraints
## 4 Mobility Analysis of 3-CCR PM  
This section analyzes the mobility of the 3-CCR mechanism according to the steps for mobility analysis of PMs with three screw-type constraints. The terminal constraint and DOFs properties of the 3-CCR PM may vary in different poses, so this section establishes a position solution model for the 3-CCR PM. Based on this, combined with the terminal constraint and DOFs determination method obtained in the previous section, the terminal constraints and DOFs of this PM at different poses in the workspace are determined. The position vector \(\mathbf{A}_{i}\) of point \(A_{i}\) in \(\{n_{0}\}\) can be expressed as follows:  
\[\mathbf{A}_{i}= \begin{subarray}{c}n_{0}\\ n_{1}\end{subarray} \mathbf{R}^{n_{1}}\mathbf{A}_{i}+\mathbf{o}, \begin{subarray}{c}n_{1} \end{subarray} \mathbf{A}_{i}=e_{2}\left[ \begin{array}{c}c_{\theta}\\ s_{\theta}\\ 0\end{array} \right], \begin{subarray}{c}n_{0}\\ n_{1}\end{subarray} \mathbf{R}=\left[ \begin{array}{ccc}x_{1}&y_{1}&z_{l}\\ x_{m}&y_{m}&z_{m}\\ x_{n}&y_{n}&z_{n}\end{array} \right], \begin{array}{c}\mathbf{o}=\left[\begin{array} []{c}X_{o}\\ Y_{o}\\ Z_{o}\end{array}\right],\\ \theta=\pi/6+(2\pi/3)(i-1),\;i=1,2,3\end{array}  \tag{20}\]  
where \(\mathbf{A}_{i}\) and \( \begin{subarray}{c}n_{1}\end{subarray} \mathbf{A}_{i}\) are the position vectors of \(A_{i}\) in O-XYZ and \(o\)-xyz, respectively. \(\mathbf{o}\) is the position vector of \(o\) in \(o\)-xyz. \( \begin{subarray}{c}n_{0}\\ n_{1}\end{subarray} \mathbf{R}\) is the rotation matrix, and \((x_{l},x_{m},x_{n},y_{l},y_{m},y_{n},z_{l},z_{m},z_{n})\) are the rotation elements. \(e_{1}\) denotes the distances from \(o\) to \(A_{i}\). From Equation (20), \( \begin{subarray}{c}n_{1}\end{subarray} \mathbf{A}_{i2}\) and \(\mathbf{A}_{ij}\) can be expressed as follows:  
\[ \begin{array}{l}\begin{subarray}{c}n_{1}\end{subarray}\mathbf{A}_{1}=\frac{\alpha }{2}\left[\begin{array}{c}q\\ -1\\ 0\end{array}\right],\begin{subarray}{c}n_{1}\end{subarray}\mathbf{A}_{2}=\left[ \begin{array}{c}0\\ e_{2}\\ 0\end{array}\right],\begin{subarray}{c}n_{1}\end{subarray}\mathbf{A}_{3}=\frac{ \alpha}{2}\left[\begin{array}{c}-q\\ -1\\ 0\end{array}\right],\begin{subarray}{c}q=\sqrt{3},\\ \end{array} \\ \begin{array}{c}A_{11}=\frac{1}{2}\left[\begin{array}{c}q_{2}x_{-1}-e_{2} y_{l}+2X_{c1}\\ q_{2}x_{m}-e_{2}y_{m}+2Y_{c1}\\ q_{2}x_{n}-e_{2}y_{n}+2Z_{c1}\end{array}\right],\begin{subarray}{c}\mathbf{A}_{2}= \left[\begin{array}{c}e_{2}y_{l}+X_{c1}\\ e_{2}y_{m}+Y_{c1}\\ e_{2}y_{n}+Z_{c1}\end{array}\right],\begin{subarray}{c}\mathbf{A}_{3}=\frac{1}{2} \left[\begin{array}{c}-q_{2}x_{l}-e_{2}y_{l}+2X_{c1}\\ -q_{2}x_{m}-e_{2}y_{m}+2Y_{c1}\\ -q_{2}x_{n}-e_{2}y_{n}+2Z_{c1}\end{array}\right]\end{array}  \tag{21}\]  
The direction vectors of Pi1 and Ri1 in \(\{n0\}\) can be expressed as follows:  
\[\mathbf{P}_{11}=\mathbf{R}_{11}= \begin{bmatrix}0\\ 1\\ 0\end{bmatrix} ,\begin{array}{c}\mathbf{P}_{21}=\mathbf{R}_{21}= \begin{bmatrix}1\\ 0\\ 0\end{bmatrix} , \begin{array}{c}\mathbf{P}_{31}=\mathbf{R}_{31}=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix}\end{array}  \tag{22}\]  
According to Equation (2), the direction vectors of Pi2 and Ri2 in \(\{n0\}\) can be expressed as follows:  
\[\mathbf{P}_{i2}=\mathbf{R}_{i2}=\frac{\mathbf{R}_{i3}\times\mathbf{R}_{i1}}{\sqrt{(\mathbf{R}_{i3} \times\mathbf{R}_{i1})\cdot(\mathbf{R}_{i3}\times\mathbf{R}_{i1})}} \tag{23}\]  
The direction vector of \(R_{i3}\) in \(\{n_{0}\}\) can be expressed as follows:  
\[\mathbf{R}_{i3}= \begin{subarray}{c}n_{0}\\ n_{1}\end{subarray} \mathbf{R}^{n_{1}}\mathbf{R}_{i3}, \begin{subarray}{c}n_{1}\end{subarray}  \mathbf{R}_{i3}= \begin{bmatrix}\sin(\pi/4)\cos(\theta)\\ \sin(\pi/4)sin(\theta)\\ \cos(\pi/4)\end{bmatrix}  \tag{24}\]  
Equations (21)-(24) are the joint vector expressions with respect to the pose parameters of the moving platform. From Equation (2), the geometric constraint equations of this PM can be expressed as follows:  
\[\mathbf{r}_{i}\cdot\mathbf{P}_{i1}=0, \begin{array}{c}\mathbf{r}_{i}\cdot\mathbf{P}_{i2}=0, \;i=1,2,3\end{array}