# The Mobility Analysis of a 3-CCR Parallel Manipulator with Three Screw-Type Terminal Constraints
## 5 Conclusions  
The contribution of this paper lies in proposing an approach for identifying the DOFs of PMs with screw-type terminal constraints and revealing a special phenomenon that the 3R DOFs and 3H DOFs coexist in the 3-CCR PM.  
For the 3-CCR PM with three screw-type terminal constraints, based on the reciprocal relationship between the DOFs and terminal constraints, the analytical expression of the terminal constraints of the 3-CCR PM is obtained by solving the symbolic equation system. The results show that the terminal constraint in each limb of the PM is a screw-type constraint. Based on the theory of quadratic curve decomposition, the principal  
Figure 5: The axode and pose of 3-CCR PM with 3R DOFs. (**a**) pose 1, (**b**) pose 2, (**c**) pose 3, and (**d**) pose 4.  
screws and the corresponding pitches of the constraint system were solved, and the DOFs determination approach was given, considering the relations of the motion and constraint systems. An analytical expression for the joint vectors and six-dimensional terminal pose parameters of the 3-CCR PM was established using the vector method. The DOFs distribution situations in the workspace of the 3-CCR PM were plotted, revealing the special phenomenon that three rotation and three helical DOFs coexist in the workspace. The distribution surfaces corresponding to the helical DOFs with specific pitches under three poses were drawn. The results showed that helical DOFs with the same pitches were distributed in the same hyperboloid of one sheet, while the pitches in different hyperboloid of one sheet were different. The study provides an effective approach for determining the DOFs of the PMs with screw-type terminal constraints.  
This study helps explain the special phenomenon where PMs with three screw-based constraints exhibit three rotational DOFs. It provides a theoretical basis for developing three rotational DOFs PMs with screw-type constraints, helping to overcome the limitations of traditional three rotational DOFs PMs that can only use the force constraint limbs. This study facilitates the development of parallel biomimetic joints with three helical DOFs. For example, the center of rotation of the human ankle joint changes during actual movement. Using three helical DOFs PMs allows for designing biomimetic joints that mimic these motion characteristics, aiding humanoid robots in achieving more coordinated and natural ankle movements.  
Conceptualization, B.H. and P.B.; methodology, B.H. and P.B.; validation, B.H. and P.B.; investigation, P.B.; resources, B.H.; writing--original draft preparation, B.H. and P.B.; writing--review and editing, B.H. and P.B.; visualization, P.B.; supervision, B.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (grant number 52275033).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} PM & parallel manipulator \\ DOFs & degrees of freedom \\ \end{tabular}