# The Mobility Analysis of a 3-CCR Parallel Manipulator with Three Screw-Type Terminal Constraints
## 3 Terminal Constraint and DOFs Determination Based on Principal Screws Analysis
### Calculating the Principal Screws of the Constraint Screw System  
The constraint screw \(\mathbf{\xi}_{i}^{r}\) of each limb form a constraint screw system, and any constraint screw in this system can be expressed by the linear combination of \(\mathbf{\xi}_{i}^{r}\) as follows:  
\[\mathbf{\xi}^{r}=\begin{bmatrix}\mathbf{s}^{r}\\ \mathbf{s}_{0}^{r}\end{bmatrix}=\begin{bmatrix}\mathbf{s}_{1}^{r}&\mathbf{s}_{2}^{r}&\mathbf{s }_{3}^{r}\\ \mathbf{s}_{10}^{r}&\mathbf{s}_{20}^{r}&\mathbf{s}_{30}^{r}\end{bmatrix}\begin{bmatrix}k_{1} \\ k_{2}\\ k_{3}\end{bmatrix} \tag{8}\]  
where \(k_{1}\), \(k_{2}\), and \(k_{3}\) are three coefficients.  
The three solutions \(h_{1}\), \(h_{2}\), and \(h_{3}\) of Equation (14) are the pitch values of the principal screws, which can be expressed as follows:  
\[[h_{1},\ h_{2},\ h_{3}]=\left\{\begin{array}{l}\left[-\frac{c_{3}}{c_{2}},\ -\frac{c_{3}}{c_{2}},\ -\frac{c_{3}}{c_{2}}\right],\ A=0,\ B=0\\ \left[-\frac{c_{2}}{c_{1}}+K,\ -\frac{K}{K},\ -\frac{K}{K}\right],\ B^{2}-4 AC=0,\ A\
eq 0,\ B\
eq 0\\ \left[\frac{-c_{2}-2\sqrt{A}\cos(\theta/3)}{3x_{1}},\ -\frac{c_{2}+\sqrt{A}M}{3x_{1}},\ -\frac{c_{2}+\sqrt{A}N}{3x_{1}}\right],\ B^{2}-4 AC<0 \end{array}\right. \tag{15}\]  
where  
\[\begin{array}{l}A=c_{2}^{2}-3c_{1}c_{3},\ B=c_{2}c_{3}-9c_{1}c_{4},\ C=c_{3}^ {2}-3c_{2}c_{4}K=\frac{c_{2}c_{3}-9c_{1}c_{4}}{c_{2}^{2}-3c_{1}c_{3}},\ M=\cos \frac{\theta}{3}+\sqrt{3}\sin\frac{\theta}{3},\\ N=\cos\frac{\theta}{3}-\sqrt{3}\sin\frac{\theta}{3},\ T=\frac{2 Ac_{2}-3 Ac_{1}}{2 \sqrt{A^{3}}},\ \theta=\arccos T\end{array}\]  
By substituting \(h_{j}=1\), 2, 3 in Equation (15) into Equation (12), the corresponding values \(a_{11,j}\), \(a_{12,j}\), \(a_{13,j}\), \(a_{22,j}\), \(a_{23,j}\), and \(a_{33j}\) can be obtained. Then, by substituting these values into Equation (11), a bivariate quadratic equation in terms of \(u_{j}\), \(v_{j}\) is derived. The solutions of this equation are shown as the following [40]:  
\[u_{j}=\frac{a_{22,j}a_{13,j}-a_{12,j}a_{23,j}}{a_{12,j}^{2}-a_{11,j}a_{22,j}}, \ v_{j}=-\frac{a_{23,j}+a_{12,j}u_{j}}{a_{22,j}} \tag{16}\]  
By considering \(u=k_{1}/k_{3}\), \(w=k_{2}/k_{3}\) and by substituting Equation (16) into Equation (8), the three principal screws \(\mathbf{\xi}_{pj}^{r}\)(\(j=1\),\(2\),\(3\)) corresponding to the terminal constraint system can be obtained:  
\[\mathbf{\xi}_{pj}^{r}=\left[\left(u_{j}\mathbf{\xi}_{1}^{r}+v_{j}\mathbf{\xi}_{2}^{r}+\bm {\xi}_{3}^{r}\right)^{T}\ \ \ \ \left(u_{j}\mathbf{\xi}_{10}^{r}+v_{j}\mathbf{\xi}_{20}^{r}+\mathbf{\xi}_{30}^{r}\right)^{T} \right],\ j=1,2,3 \tag{17}\]  
Equation (17) is the analytical expression of \(\mathbf{\xi}_{pj}^{r}\) with respect to \(\mathbf{\xi}_{i}^{r}\), and Equation (15) is the analytical expression of \(h_{j}\) with respect to \(\mathbf{\xi}_{i}^{r}\). By substituting Equations (4) and (5) into Equation (15), \(h_{i}\) can obtained, and then by substituting the obtained \(h_{i}\) into Equation (17), the analytical expression of \(\mathbf{\xi}_{pj}^{r}\) can be obtained.  
To verify whether the solution of the principal screw is correct, one can check if the elements of \(\mathbf{\xi}_{pj}^{r}\) satisfy the perpendicular and intersecting conditions. The specific verification conditions are listed as follows:  
\[\begin{array}{l}\mathbf{s}_{p1}^{r}\cdot\mathbf{s}_{p2}^{r}=0,\ \mathbf{s}_{p1}^{r}\cdot\mathbf{s}_{p3}^{r}=0,\ \mathbf{s}_{p2}^{r}\cdot\mathbf{s}_{p3}^{r}=0,\\ \mathbf{s}_{p1}^{r}\cdot\mathbf{s}_{p20}^{r}+\mathbf{s}_{p2}^{r}\cdot\mathbf{s}_{p10}^{r}=0,\ \mathbf{s}_{p1}^{r}\cdot\mathbf{s}_{p30}^{r}+\mathbf{s}_{p3}^{r}\cdot\mathbf{s}_{p10}^{r}=0, \ \mathbf{s}_{p2}^{r}\cdot\mathbf{s}_{p30}^{r}+\mathbf{s}_{p3}^{r}\cdot\mathbf{s}_{p20}^{r}=0 \end{array} \tag{18}\]