# The Mobility Analysis of a 3-CCR Parallel Manipulator with Three Screw-Type Terminal Constraints
## 4 Mobility Analysis of 3-CCR PM
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{3}{*}{**No**} & \multicolumn{3}{c}{**Principal Screws of the**} & \multirow{3}{*}{**Pitch of the**} & \multirow{3}{*}{**Axode**} & \multirow{3}{*}{**No**} & \multicolumn{3}{c}{**Principal Screws of the**} & \multirow{3}{*}{**Pitch of the**} & \multirow{3}{*}{**Axode**} \\  & & \multicolumn{1}{c}{**Constraint Screw System**} & & & & & & **Constraint Screw System** & \\ \cline{2-2} \cline{6-8}  & \(\mathbf{\xi}^{r}_{p1}\) & \(\mathbf{\xi}^{r}_{p2}\) & \(\mathbf{\xi}^{r}_{p3}\) & & & & & \(\mathbf{\xi}^{r}_{p1}\) & \(\mathbf{\xi}^{r}_{p2}\) & \(\mathbf{\xi}^{r}_{p3}\) & \\ \hline \multirow{8}{*}{1} & \(\{0.5483\}\) & \(-0.6458\) & \(-0.5313\) & \multirow{8}{*}{\(h_{1}=79.39\)} & \multirow{8}{*}{Figure 4a} & \(\{0.3232\}\) & \(-0.3086\) & \(0.8946\) & \multirow{8}{*}{\(h_{1}=83.07\)} & \multirow{8}{*}{Figure 4b} \\  & \(0.6288\) & \(-0.1004\) & \(0.7710\) & & & & \(0.5468\) & \(0.8324\) & \(0.0896\) & \\  & \(0.5513\) & \(0.7569\) & \(-0.3511\) & & \(h_{1}=79.39\) & & & \(0.7723\) & \(-0.4602\) & \(-0.4378\) & \\  & \(13.6980\) & \(-152.2053\) & \(-154.0919\) & & \(h_{2}=261.01\), & & & \(36.4089\) & \(-102.7470\) & \(112.0896\) & \\  & \(61.5057\) & \(-93.4742\) & \(131.8786\) & & & & \(19.6546\) & \(153.2852\) & \(69.1029\) & \\  & \(60.2177\) & \(202.5824\) & \(-31.6441\) & & & & \(78.4009\) & \(-25.4499\) & \(-95.8437\) & \\  & \(0.6149\) & \(0.1769\) & \(-0.76858\) & & & & \(0.7708\) & \(-0.2128\) & \(0.6005\) & \\  & \(0.6649\) & \(0.4078\) & \(0.6258\) & & & & \(0.2154\) & \(0.9741\) & \(0.6688\) & \\  & \(0.4241\) & \(-0.8958\) & \(0.1331\) & & \(h_{1}=87.01\), & & & \(0.5996\) & \(-0.0763\) & \\  & \(31.2971\) & \(23.8994\) & \(-198.1323\) & & & & \(74.4607\) & \(-29.15886\) & \(85.9335\) & \\  & \(55.3421\) & \(155.5982\) & \(108.0792\) & & & & \(140.4947\) & \(604.4967\) & \(92.5120\) & \\  & \(73.0361\) & \(-190.5953\) & \(69.9514\) & & & & \(-42.6582\) & \(-49.5566\) & \(-291.0721\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The principal screw information of 3-CCR PM with 3H DOFs in some poses. The method from reference [26] was utilized to generate the axodes of helical DOFs with specific pitches within the motion screw systems at the four poses. To verify whether there are any rotational DOFs at the selected poses, we specifically plotted the axodes corresponding to a pitch of \(h^{m}=0\) (representing rotational DOF) using a yellow surface. Additionally, the axodes corresponding to three different pitches in \([\min(h^{m}_{i})\), \(\max(h^{m}_{i})]\) were plotted and represented by blue surfaces, as shown in Figure 4. The red labeled numbers in Figure 4 indicate the pitch corresponding to the DOFs within the axodes. The results indicate that no yellow surfaces were generated under the four poses listed in Table 2, which implies that no rotational DOFs exist in the four poses. And the helical DOFs with the same pitch are distributed on the same hyperboloid of one sheet. Under the aforementioned four poses, the motion screw system of the 3-CCR mechanism contains only helical DOFs with negative pitches. For comparison, four other poses of the 3-CCR mechanism with 3R DOFs are chosen, as presented in Table 3.  
The first position parameters are given as \(X_{o}\) = 30.25 mm, \(Y_{o}\) = 43.75 mm, and \(Z_{o}\) = 73 mm. The active limbs are solved as \(d_{11}\) = 29.59 mm, \(d_{21}\) = 48.07 mm, and \(d_{31}\) = 77.31 mm. The second position parameters are given as \(X_{o}\) = 28 mm, \(Y_{o}\) = 84.25 mm, and \(Z_{o}\) = 66.25 mm. The active limbs are solved as \(d_{11}\) = 35.53 mm, \(d_{21}\) = 94.36 mm, and \(d_{31}\) = 70.26 mm. The third position parameters are given as \(X_{o}\) = 30.25 mm, \(Y_{o}\) = 46 mm, and \(Z_{o}\) = 43.75 mm. The active limbs are solved as \(d_{11}\) = 27.33 mm, \(d_{21}\) = 46.30 mm, and \(d_{31}\) = 40.74 mm. The fourth position parameters are given as \(X_{o}\) = 37 mm, \(Y_{o}\) = 50.50 mm, and \(Z_{o}\) = 39.25 mm.