# 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  
### Analytical Expression of the Terminal Constraint  
Establish the fixed coordinate system \(O\)-\(XYZ\) at point \(O\) with \(Z\bot P_{31}\), \(X\mid I\)\(P_{21}\), \(Y\mid I_{11}\) are satisfied. Establish the moving coordinate system \(o\)-\(xyz\) at point \(o\) with \(x\mid I_{3}A_{1}\) and \(y\mid\mid\)\(oA_{2}\) are satisfied, \(z\) axis can be determined by \(x\) and \(y\) (see Figure 1). Let \(\mathbf{R}_{ij}\), \(\mathbf{P}_{ij}\) denote the unit vectors of the \(j\)-th R joint in \(i\)-th limbs, \(\mathbf{r}_{i}\) denotes the position vector from a reference point to an arbitrary point on the axis of the R joint. Let \(\mathbf{\mathfrak{F}}_{i}^{r}\) be the terminal constraint of \(i\)-th limbs, \(\mathbf{s}_{i}^{r}\) and \(\mathbf{s}_{i0}^{r}\) be the direction and moment components of \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No**} & \multirow{2}{*}{**Relations Among \(\mathbf{h_{\alpha}}\), \(\mathbf{h_{\beta}}\), and \(\mathbf{h_{\gamma}}\)**} & **DOFs** & \begin{tabular}{c} **Type of the** \\ **Three-Order** \\ **Screw System** \\ \end{tabular} & \
\begin{tabular}{c} **Sketch of Motion** \\ **Screw System** \\ \end{tabular} &
\begin{tabular}{c} **Sketch of Constraint** \\ **Screw System** \\ \end{tabular} \\ \hline
1 & \(h_{\alpha}=h_{\beta}\), =0 \(>h_{\gamma}\), \(h_{\alpha}>h_{\beta}\), =\(h_{\gamma}=0\) & 3R & 7 th type & \\
2 & \(h_{\alpha}\), \(h_{\beta}\), \(h_{\gamma}\) are mutually unequal and of opposite signs & 3R & 8 th type & \\
3 & \(h_{\alpha}>h_{\beta}=0>h_{\gamma}\) & 3R & 9 th type & \\
4 & \(h_{\alpha}>h_{\beta}>h_{\gamma}=0\), \(h_{\alpha}=0>h_{\beta}>h_{\gamma}\) & 3R & 10 th type & \\
5 & \(h_{\alpha}\), \(h_{\beta}\), \(h_{\gamma}\) are mutually unequal and of opposite signs & 3H & 11 th type & \\
6 & \(h_{\alpha}>h_{\beta}=h_{\gamma}=0\), \(h_{\alpha}=h_{\beta}=0>h_{\gamma}\) & 2R1H & 12 th type & \\
7 & \(h_{\alpha}=h_{\beta}>h_{\gamma}=0\), \(h_{\alpha}=0>h_{\beta}>h_{\gamma}\) & 1R2H & 13 th type & \\
8 & \(h_{\alpha}=h_{\beta}>h_{\gamma}>0\), \(0>h_{\alpha}>h_{\beta}=h_{\gamma}\) & 3H & 14 th type & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The relationships between the pitch of the principal screws, the DOFs, and the atlas of the screw system. \(\mathbf{\xi}_{i}^{r}\), respectively. From the reciprocal relation between the motion and the constraint screw system, it leads to the following:  
\[ \begin{bmatrix}\mathbf{R}_{i1}^{T}&(r_{i1}\times\mathbf{R}_{i1})^{T}\\ \mathbf{R}_{i2}^{T}&(r_{i2}\times\mathbf{R}_{i2})^{T}\\ \mathbf{R}_{i3}^{T}&(r_{i3}\times\mathbf{R}_{i3})^{T}\\ \mathbf{0}^{T}&\mathbf{P}_{i1}^{T}\\ \mathbf{0}^{T}&\mathbf{P}_{i2}^{T}\end{bmatrix}\begin{bmatrix}\mathbf{s}_{i0}^{r}\\ \mathbf{s}_{i}^{r}\end{bmatrix} =0 \tag{3}\]  
From Equation (3), \(\mathbf{s}_{i}^{r}\) can be solved as follows:  
\[\mathbf{s}_{i}^{r}=\mathbf{P}_{i1}\times\mathbf{P}_{i2} \tag{4}\]  
In the general case, \(\mathbf{R}_{i1}\), \(\mathbf{R}_{i2}\), and \(\mathbf{R}_{i3}\) are linearly independent. Substituting Equation (4) into Equation (3), \(\mathbf{s}_{i0}^{r}\) can be solved as follows:  
\[\mathbf{s}_{i0}^{r}=\frac{-[(\mathbf{r}_{i1}\times\mathbf{R}_{i1})\cdot\mathbf{s}_{i}^{r}]\mathbf{ R}_{i2}\times\mathbf{R}_{i3}-[(\mathbf{r}_{i2}\times\mathbf{R}_{i2})\cdot\mathbf{s}_{i}^{r}]\mathbf{ R}_{i3}\times\mathbf{R}_{i1}-[(\mathbf{r}_{i3}\times\mathbf{R}_{i3})\cdot\mathbf{s}_{i}^{r}]\mathbf{ R}_{i1}\times\mathbf{R}_{i2}}{\mathbf{R}_{i1}\times\mathbf{R}_{i2}\cdot\mathbf{R}_{i3}} \tag{5}\]  
From Equation (2), it can be concluded that some joint vectors of the 3-CCR PM satisfy the following relationship:  
\[