# The Mobility Analysis of a 3-CCR Parallel Manipulator with Three Screw-Type Terminal Constraints
## 1 Introduction
However, when multiple screw-type constraints exist at the end-effector, not only do the geometric relationships between screw-type constraints affect the type of DOF, but the pitch values of screw-type constraints also influence the identification of DOFs (detailed analysis is provided in Section 3.1). Therefore, these two types of methods are not suitable for analyzing the DOFs in PMs with three screw-type constraints. Currently, determining the principal screws of the motion screw system is a viable approach to analyzing the mobility properties of PMs [23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37]. Principal screws are defined as a set of mutually orthogonal and intersecting screws. The number of principal screws equals the order of the screw system. Furthermore, any screw within the screw system can be represented as a linear combination of these principal screws. The methods for solving principal screws can be divided into two main categories. The first type is based on the geometric properties of principal screws being orthogonal and intersecting with each other. [PERSON] [23] proposed a method for constructing principal screws from given screws, which is applicable to second and third-order screw systems. Based on this, [PERSON] [24] further investigated geometric construction methods and classified screw systems of different orders. [PERSON] et al. [25] studied special bases for second- and third-order screw systems, providing geometric conditions and classification methods. [PERSON] et al. [26; 27] utilized the theory of quadric decomposition to identify the principal screws of a third-order screw system and applied this method to analyze the principal screws of the 3-RPS PM. The second category is based on the eigenvalue and eigenvector analysis, which typically relies on numerical computations. [PERSON] and [PERSON] [28; 29] introduced an algebraic method based on dual vectors to identify principal screws and introduced the concept of DOFs partitioning. [PERSON] [30] proposed a method for solving principal screws applicable to two- to five-order screw systems. [PERSON] and [PERSON] [31; 32] proposed a generalized eigenvalue method based on dual-number algebra to determine principal screws, providing examples for solving the principal screws of serial, parallel, and hybrid mechanisms. [PERSON] et al. [33; 34] developed an algorithm for solving principal screws based on the decomposition of input and output spaces. [PERSON] and [PERSON] [35] proposed a method for solving principal screws using dual quaternions. [PERSON] et al. [36] introduced a general-special decomposition method to determine the standard bases of screw systems. [PERSON] and [PERSON] [37] introduced a numerical algorithm based on singular value decomposition for solving principal screws. However, when using the principal screw solution algorithm to analyze mobility, the solving process is numerical, which prevents the establishment of analytical relationships between the structure of the mechanism and the types of DOFs. This approach is not conducive to subsequent research on how changes in structural parameters affect the types of DOF. This paper addresses the mobility analysis problem of a 3-CCR PM with three screw-type constraints. By solving for the principal screws of the constraint screw system, the special DOFs phenomena of the 3-CCR PM are investigated. The remainder of this paper is organized as follows: Section 2 provides a detailed description of the differences between the mobility analysis of the 3-CCR PM having screw-type terminal constraints and traditional PMs. Section 3 introduces the analytical expressions of the terminal screw-type constraints and the corresponding principal screws and establishes the DOFs identifying approach for the 3-CCR PM. Section 4 gives the numerical example for the DOFs determination of the 3-CCR PM. Finally, some concluding remarks are given in Section 5.