# The Mobility Analysis of a 3-CCR Parallel Manipulator with Three Screw-Type Terminal Constraints
## 2 Problem Description  
The terminal constraint refers to the constraints exerted on the end-effector of a manipulator. In screw theory, one independent terminal constraint \(\mathbf{\xi}^{r}\) at the end-effector of a mechanism is represented as following [1]:  
\[\mathbf{\xi}^{r}=\left\{ \begin{array}{ll}\left[\begin{array}{cc}\mathbf{s}^{T}& \left(\mathbf{r}\times\mathbf{s}\right)^{T}\end{array}\right]&\text{constraint force}\\ \left[\begin{array}{cc}\mathbf{0}^{T}&\mathbf{s}^{T}\end{array}\right]&\text{constraint torque}\\ \left[\begin{array}{cc}\mathbf{s}^{T}&\left(\mathbf{r}\times\mathbf{s}+\mathbf{h}\mathbf{s} \right)^{T}\end{array}\right]&\text{constraint screw}\end{array} \right. \tag{1}\]  
where \(\mathbf{s}\) represents the direction vector of \(\mathbf{\xi}^{r}\), \(\mathbf{r}\) denotes the position vector from the reference point to any point on the axis of \(\mathbf{\xi}^{r}\), and \(\mathbf{h}\) is a scalar known as the pitch of the screw. For the existing lower-mobility PMs, almost all terminal constraints are constraint forces or constraint torques. Recent research has shown that there are some PMs [38] with the terminal constraint is one screw pitch in each limb. For PMs with only constraint forces/torque constraints, the DOFs can be visually distinguished based on the geometric relationship of the terminal constraints in each limb [8]. However, for PMs with screw-type terminal constraints, the same geometric relationships of the terminal constraints may still correspond to multiple different types of DOFs. It is not possible to obtain DOFs simply through the traditional process. The mobility problem of this class of PMs is difficult and has not been solved yet. This paper takes a 3-CCR PM as an example to discuss the DOFs problem of this class of PMs. The 3-CCR PM is shown in Figure 1. This PM consists of a fixed pyramid with \(O\) as its center, a moving platform, and three CCR limbs connecting the two platforms. The moving platform is an equilateral triangle with three vertexes, which are \(A_{i}(i=1,2,3)\). The \(i\)-th CCR limb connects the fixed pyramid with the moving platform by two cylindrical joints C\({}_{i1}\) and C\({}_{i2}\), one linkages \(l_{i}\) between C\({}_{i1}\) and C\({}_{i2}\), and there is one revolute joint R at point \(A_{i}\). The joint C\({}_{ij}\) is equivalent to a P joint P\({}_{ij}\), and an R joint R\({}_{ij}\). P\({}_{i1}(i=1,2,3)\) intersects at point \(O\). R\({}_{i3}\) are symmetrically distributed on the moving platform and intersects at a point outside the moving platform. Let \(P_{ij}\), \(R_{ij}(j=1,2,...\,. )\) represent the \(j\)-th P or R joint in the \(i\)-th limbs. Let \({}^{\omega}\,|\,|\,^{\omega},\,\bot\,^{\omega}\) and \({}^{\omega}\,|\,^{\omega}\) represent the parallel, perpendicular, and collinear relations, respectively. The geometric relations among the joints in the 3-CCR can be expressed as follows:  
\[P_{11}\bot P_{21},P_{21}\bot P_{31},P_{11}\bot P_{31},R_{i1}\,1\,P_{i1},R_{i1} \bot R_{i2},R_{i1}\bot l_{i},R_{i2}\bot l_{i},R_{i2}\,1\,P_{i2},R_{i2}\,\bot R _{i3} \tag{2}\]  
The terminal constraint in each CCR limb of this PM is a screw constraint [38]. The three constraint screws imposed on the moving platform form a constraint screw system. In reference [39], the screw systems have been divided into 50 categories. For this PM, there are eight types of possible constraint screw systems (as shown in Table 1) corresponding to the 7 th to 14 th types in reference. The eight constraint screw systems belong to the three-order screw systems. Table 1 shows the relationships between the pitch of the principal screws \(h_{\alpha},h_{\beta}\), and \(h_{\gamma}\) of the constraint screw system and the DOFs corresponding to various situations, as well as the corresponding atlas of screw systems. In Table 1, yellow lines indicate helical DOFs, blue lines indicate rotational DOFs, green lines indicate screw-type constraints, and red lines indicate force constraints.