# 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
### Mobility Identifying Approach  
For the obtained values of \(h_{j}\), they are categorized based on their numerical magnitudes and denoted as \(h_{\alpha}\), \(h_{\beta}\), and \(h_{\gamma}\). \[h_{\alpha}=\max(h_{1},h_{2},h_{3}),\ h_{\gamma}=\min(h_{1},h_{2},h_{3}) \tag{19}\]  
Then, the specific types of DOFs of the PM can be identified based on \(h_{\alpha}\), \(h_{\beta}\) and \(h_{\gamma}\), which are categorized into 8 cases as shown in Table 1.  
When \(h_{\alpha}=h_{\beta}=0>h_{\gamma}\) or \(h_{\alpha}>h_{\beta}=h_{\gamma}=0\), the constraint screw system of the PM is classified as type 7. When \(h_{\alpha}\), \(h_{\beta}\), and \(h_{\gamma}\) are all distinct, and one pitch has a different sign from the other two, the constraint screw system is classified as type 8. When \(h_{\alpha}>h_{\beta}=0>h_{\gamma}\), the constraint screw system is classified as type 9. When \(h_{\alpha}>h_{\beta}>h_{\gamma}=0\) or \(h_{\alpha}=0>h_{\beta}>h_{\gamma}\), the constraint screw system is classified as type 10. When \(h_{\alpha}\), \(h_{\beta}\), and \(h_{\gamma}\) are all distinct and have the same sign, the constraint screw system is classified as type 11. 11. When \(h_{\alpha}>h_{\beta}=h_{\gamma}=0\) or \(h_{\alpha}>h_{\beta}=h_{\gamma}=0\), the constraint screw system is classified as type 12. When \(h_{\alpha}=h_{\beta}>h_{\gamma}=0\) or \(h_{\alpha}=h_{\beta}>h_{\gamma}=0\), the constraint screw system is classified as type 13. When \(h_{\alpha},h_{\beta}\), and \(h_{\gamma}\) have the same sign, and only two of them are equal, the end-effector constraint screw system is classified as type 14. Each type of third-order constraint system uniquely corresponds to one type of third-order motion system [39]. As shown in Table 1, the green lines represent the screw-type constraints, the red lines represent the constraint forces, the blue lines represent the motion screws, and the yellow lines represent the rotations. The above motion screw systems all contain independent helical DOFs. The axes of these helical DOFs are distributed in the cluster straight generatrices of an infinite number of one-sheeted hyperboloids, with the principal generator and the plane of the circular cross-section of each one-sheeted hyperboloid coincide, respectively. For the 7 th and 8 th motion three-systems, except motion screws, there also exists an infinite number of rotational DOFs, which are distributed in the same cluster of straight generatrices of a one-sheeted hyperboloid. For the 9 th motion system, there similarly exists an infinite number of rotational DOFs, which form two planar pencils of lines intersecting at two common points, respectively. For the 12 th motion system, there exists an infinite number of rotational DOFs, which form a planar pencil intersecting at a common point. For the 10 th and 13 th motion systems, only one rotational DOFs exists. For the 11 th and 14 th motion systems, no rotational DOFs exist. Based on the theory of Grassmann line geometry, the dimension of the independent DOFs of the motion systems is 3. Due to the coexistence of multiple rotational and helical DOFs in the above-mentioned motion screw systems, in order to clarify the corresponding DOFs type for each motion screw system, this paper provides a criterion for the DOFs determination of this class of PMs:  
For an \(n\)-order motion screw system, the independent rotation and translation DOFs should be determined first. If the independent rotation and translation DOFs is \(a\), then the independent helical DOFs is \(n\)-\(a\). Thus, among these 8 cases, the 7 th, 8 th, and 9 th motion systems have three linearly independent rotational DOFs. The 12 th motion system has two linearly independent rotational DOFs and one helical DOF. The 10 th and 13 th motion systems have one rotational DOFs and two linearly independent helical DOFs. Only the 11 th and 14 th motion screw systems have three linearly independent pure helical DOFs.