# The Mobility Analysis of a 3-CCR Parallel Manipulator with Three Screw-Type Terminal Constraints
## 1 Introduction  
Mobility analysis is one of the most fundamental issues in type synthesis, kinematic, and dynamic studies of parallel manipulators [1]. Mobility analysis includes the number of degrees of freedom and motion property of the end-effector. The 3-CCR PM with three DOFs has attracted much attention from scholars. [PERSON] and [PERSON] [2] analyzed the assembly conditions and singular configurations of non-overconstrained 3-CCR PMs with three spherical DOFs based on the Lie group method. [PERSON] et al. [3] proposed a 3-RCC PM with three translational DOFs and investigated its dynamic and position/force hybrid control. [PERSON] et al. [4] investigated the impact of clearance and manufacturing errors on the accuracy of a 3-CCR PM with three spherical DOFs. [PERSON] et al. [5] analyzed the instantaneous mobility of the 3-RCC PM with three translational DOFs using screw theory. [PERSON] et al. [6] studied the motion characteristics of 3-RCC mechanisms with three spherical DOFs using the position and orientation characteristics method. [PERSON] et al. [7] applied the 3-RCC PM with three translational DOFs to the vehicle seat suspension and effectively reduced its vibrations. The existing research on3-CCR PMs mainly focused on the PMs with constraint forces and torques, while studies of the 3-CCR PMs having screw-type constraints have not been attempted. Studying the mobility of the 3-CCR PM with three screw-type constraints is beneficial for gaining a deeper understanding of the types of DOFs in PMs under the influence of multiple screw-type constraints. This promotes the application of PMs with multiple independent helical DOFs, such as in minimally invasive surgical robots and biomimetic joints. During the past 150 years, scholars have made intensive efforts in the mobility analysis of mechanisms. The corresponding studies include screw theory-based methods [8; 9; 10; 11], graph methods [12; 13], Lie group-based methods [14; 15], linear transformation-based methods [16], position and orientation characteristics (POC) methods [17], geometric algebra-based methods [18; 19], and computer-aided methods [20; 21; 22]. Screw theory-based methods establish a general approach for analyzing the DOFs of mechanisms using the theory of the screw systems and reciprocal relationships [8; 9; 10; 11], which have been widely applied to the mobility analysis of lower-mobility PMs, metamorphic PMs, and hybrid mechanisms. Graph-based methods provide a visual representation of screw systems using numerical tools to map various geometric relationships between constraints and DOFs [12; 13]. This method establishes a one-to-one correspondence between the properties of DOFs, motion screw systems, and constraint screw systems through graphical representations. Lie group-based methods employ Euclidean subgroups to describe the fundamental motions of kinematic joints and end-effectors using multiplication and intersection operations between subgroups to calculate the DOFs [14; 15]. The linear transformation-based method requires the establishment of a Jacobian matrix that maps the joint velocity space to the end-effector output velocity. The DOFs are determined by the rank of this Jacobian matrix [16]. Other methods, such as geometric algebra and Clifford Algebra, are also used to analyze the DOFs of PMs [18; 19]. Furthermore, with the rapid development of computer technology, automated mobility analysis methods based on CAD technology and machine learning-based DOFs prediction methods have also begun to attract attention [20; 21; 22]. Up to now, these DOFs determination methods have primarily focused on PMs with only forces/torque constraints. It is not easy to obtain DOFs simply through the traditional process for PMs that have screw-type terminal constraints. When using screw theory-based methods and atlas methods for mobility analysis, it is necessary to determine the type of DOFs based on the geometric relationships between constraints.