# The Mobility Analysis of a 3-CCR Parallel Manipulator with Three Screw-Type Terminal Constraints
## 4 Mobility Analysis of 3-CCR PM
 \tag{25}\]  
Figure 2: The flowchart of mobility analysis of PMs with three screw-type constraints. \[e[3y_{l}^{3}+4y_{n}x_{m}y_{l}^{2}+(-5x_{l}^{2}+12x_{m}x_{n}x_{l}-4y_{m} y_{n}x_{l}+3y_{m}^{2}-3x_{m}^{2}-4)y_{l}-12x_{l}^{2}x_{n}y_{m}-2x_{m}y_{m}x_{l}-\] \[2x_{n}y_{m}+6y_{n}x_{m}]\,X_{o}+4[\sqrt{3}(4x_{l}x_{m}y_{n}-4x_{m} x_{n}y_{l}+2x_{m}y_{m})-4x_{n}y_{l}y_{m}+4x_{l}y_{m}y_{n}-3y_{m}^{2}-x_{m}^{2} +4]Y_{o}^{2}\] \[+[-4x_{n}y_{l}^{2}+(32x_{m}x_{n}y_{m}+8x_{l}y_{m}+8y_{n}y_{l})y_{ l}+4x_{l}^{2}x_{m}+(-32x_{n}y_{m}^{2}+24x_{n})x_{l}+4x_{n}y_{m}^{2}-\] \[(4(-x_{m}^{2}+16))x_{m}]+4e[4x_{n}y_{l}^{3}+(-4x_{l}y_{n}+3y_{m})y _{l}^{2}+(12x_{m}^{2}x_{n}+4x_{n}y_{m}^{2}-2x_{l}x_{m}-2x_{n})y_{l}-\] \[3y_{m}x_{l}^{2}+(-12x_{m}x_{n}y_{m}-4y_{m}^{2}y_{n}-6y_{n})x_{l}+ 3y_{m}^{3}+(-5x_{m}^{2}-4)y_{m}]\sqrt{3}x_{o}+\] \[(4e^{2[(-3x_{m}x_{n}-x_{l})y_{l}^{3}+y_{m}y_{l}^{2}(3x_{l}x_{n}-x _{m})+(x_{l}^{3}-3x_{l}^{2}x_{m}x_{n}+x_{m}^{2}x_{l}-y_{m}^{2}x_{l}+2x_{l}-3x_ {m}^{3}x_{n}+ \tag{32}\] \[3y_{m}^{2}x_{n}x_{m}-6x_{n}x_{m})y_{l}+y_{m}(3x_{n}^{3}x_{n}+x_{ l}^{2}x_{m}x_{n}x_{l}+3y_{m}^{2}x_{n}x_{l}+2x_{n}x_{l}+x_{m}^{3}-y_{m}^{2}x_{m} +2x_{m})]+\] \[8L^{2}[y_{l}(2x_{m}x_{n}-x_{l})-\sqrt{3}(2x_{n}y_{m}x_{l}+x_{m} y_{m})]+(-3y_{l}^{4}-4(x_{m}y_{n}+x_{n}y_{m})y_{l}^{3}+\] \[[2x_{l}^{2}+(-36x_{m}x_{n}+8y_{m}y_{n})x_{l}-6y_{m}^{2}+10]y_{l}^ {2}+4y_{l}[9x_{l}^{2}x_{n}y_{m}+x_{m}y_{m}x_{l}-x_{n}y_{m}^{3}+\] \[e[3y_{l}^{3}+4y_{n}x_{m}y_{l}^{2}+(-5x_{l}^{2}+12x_{m}x_{n}x_{l}-4y_{m} y_{n}x_{l}+3y_{m}^{2}-3x_{m}^{2}-4)y_{l}-12x_{l}^{2}x_{n}y_{m}-2x_{m}y_{m}x_{l}-\]0 \[e[3y_{l}^{3}+4y_{n}x_{m}y_{l}^{2}+(-5x_{l}^{2}+12x_{m}x_{n}x_{l}-4y_{m} y_{n}x_{l}+3y_{m}^{2}-3x_{m}^{2}-4)y_{l}-12x_{l}^{2}x_{n}y_{m}-2x_{m}y_{m}x_{l}-\]1  
Using \(ZYX\) Euler rotations with \(\alpha\), \(\beta\), and \(\gamma\) are three Euler angles, the rotation matrix of this PM can be expressed as follows:  
\[ \begin{array}{ccc}n_{0}\\ n_{1}\end{array} \mathbf{R}= \begin{bmatrix}\cos(\beta)\cos(\gamma)&\sin(\alpha) \sin(\beta)\cos(\gamma)-\cos(\alpha)\sin(\gamma)&\cos(\alpha)\sin(\beta)\cos( \gamma)+\sin(\alpha)\sin(\gamma)\\ \cos(\beta)\sin(\gamma)&\sin(\alpha)\sin(\beta)\sin(\gamma)+\cos(\alpha)\cos( \gamma)&\cos(\alpha)\sin(\beta)\sin(\gamma)-\sin(\alpha)\cos(\gamma)\\ -\sin(\beta)&\cos(\beta)\sin(\alpha)&\cos(\beta)\cos(\alpha)\end{bmatrix}  \tag{33}\]  
Substituting Equation (33) into Equations (30)-(32) yields three constraint equations containing the terminal pose parameters \(X_{o}\), \(Y_{o}\), \(Z_{o}\), \(\alpha\), \(\beta\), and \(\gamma\). The position solution of this PM is relatively complex and requires solving nonlinear equations. In this paper, the numerical method is used to solve the position solution, and then the DOFs at different points in the workspace are identified. The specific operation is as follows:  
Set \(X_{o}\), \(Y_{o}\), \(Z_{o}\)\(\in\) [10 mm, 100 mm], and set the constraint conditions as \(d_{11}\), \(d_{i2}\)\(\in\) [10 mm, 100 mm]. For the given \(X_{o}\), \(Y_{o}\), \(Z_{o}\), solve \(\alpha\), \(\beta\), \(\gamma\) that satisfy Equations (30)-(32) using a numerical method. Substitute the pose parameters \(X_{o}\), \(Y_{o}\), \(Z_{o}\), \(\alpha\), \(\beta\), \(\gamma\) into Equation (27), calculate \(d_{i1}\), \(d_{i2}\), and verify the values according to the constraint conditions. If the given points satisfy the constraint conditions, substitute the pose parameters into Equations (21)-(24), obtain the joint vectors at this pose, and then solve the principal screws and the pitches of the constraint screw system using Equations (4), (5), (15) and (17). Finally, determine the DOFs considering the relationship between the principal screws of the constraint screw system and DOFs types. The points with different DOFs are drawn in the workspace, as shown in Figure 3.  
Figure 3: **(a)** The workspace with 3R and 3H DOFs. **(b)** The workspace with 3H DOFs.