# Parasitic Motions of 3-PRS Parallel Mechanisms with Two Different Branch Chain Arrangements
## 3 3-PRS PM II
### Kinematics Analysis  
The joint position vectors and motion direction vectors are obtained, as shown in Table 1. According to Figures 13 and 14, the vectors of the kinematic pairs of the three branches in the \(\{O_{\text{B}}\}\) coordinate system are as follows.  
The position vectors of \(B_{1}\), \(B_{2}\), and \(B_{3}\) in the \(\{O_{\text{B}}\}\) coordinate system are \(\mathbf{a_{1}}\), \(\mathbf{a_{2}}\), and \(\mathbf{a_{3}}\):  
\[\begin{array}{l}\mathbf{a_{1}}=\left[\begin{array}{ccc}-a&0&0\\ \end{array}\right]^{T}\\ \mathbf{a_{2}}=\left[\begin{array}{ccc}a&0&0\\ \end{array}\right]^{T}\\ \mathbf{a_{3}}=\left[\begin{array}{ccc}0&b&0\\ \end{array}\right]^{T}\end{array} \tag{11}\]  
The position vectors of the prismatic joint \(P_{i}\) (\(i\) = 1, 2, 3) in the \(\{O_{\text{B}}\}\) coordinate system are:  
\[\begin{array}{l}\mathbf{P_{1}}=\left[\begin{array}{ccc}-a&0&d_{1}\\ \end{array}\right]^{\text{T}}\\ \mathbf{P_{2}}=\left[\begin{array}{ccc}a&0&d_{2}\\ \end{array}\right]^{\text{T}}\\ \mathbf{P_{3}}=\left[\begin{array}{ccc}0&b&d_{3}\\ \end{array}\right]^{\text{T}}\end{array} \tag{12}\]  
The position vectors of the spherical pairs \(S_{1}\), \(S_{2}\), and \(S_{3}\) in the \(\{O_{\text{m}}\}\) coordinate system are \(\mathbf{S_{1}}^{\prime}\), \(\mathbf{S_{2}}^{\prime}\), and \(\mathbf{S_{3}}^{\prime}\):  
\[\begin{array}{l}\mathbf{S^{\prime}}_{1}=\left[\begin{array}{ccc}-c&0&0\\ \end{array}\right]^{\text{T}}\\ \mathbf{S^{\prime}}_{2}=\left[\begin{array}{ccc}c&0&0\\ \end{array}\right]^{\text{T}}\\ \mathbf{S^{\prime}}_{3}=\left[\begin{array}{ccc}0&e&0\\ \end{array}\right]^{\text{T}}\end{array} \tag{13}\]  
The position vectors of spherical pairs \(S_{1}\), \(S_{2}\), and \(S_{3}\) are represented as \(I_{i}\) (\(i\) = 1,2,3) in the \(\{O_{\text{B}}\}\) coordinate system \(\{B\}\) by \(\mathbf{S_{1}}^{\prime}\), \(\mathbf{S_{2}}^{\prime}\), and \(\mathbf{S_{3}}^{\prime}\) using a closed loop vector method:  
\[\mathbf{I_{i}}=\mathbf{S_{i}}+\mathbf{p}=\mathbf{T}\cdot\mathbf{S^{{}^{\prime}}}_{i}+\mathbf{p} \tag{14}\]  
Substituting all the vectors aforementioned into Equation (14) yields:  
\[\begin{array}{l}l_{1x}=x+T_{11}\mathbf{S^{\prime}}_{1x}\\ l_{1y}=y+T_{21}\mathbf{S^{\prime}}_{1x}\end{array} \tag{15a}\] \[\begin{array}{l}l_{2x}=x+T_{11}\mathbf{S^{\prime}}_{2x}\\ l_{2y}=y+T_{21}\mathbf{S^{\prime}}_{2x}\end{array}\] (15b) \[\begin{array}{l}l_{3x}=x+T_{12}\mathbf{S^{\prime}}_{3y}\\ l_{3y}=y+T_{22}\mathbf{S^{\prime}}_{3y}\end{array} \tag{15c}\]  
According to the constraints of the revolute pairs of the mechanism structure, each branch chain of the PM only moves in the corresponding plane as in Figure 15. Constrained  
Figure 14: Vector diagram of 3-PRS PM II.  
by the structural conditions of the mechanism, each branch chain can obtain three constraint equations through Equations (15a)-(15c), as follows:  
\[l_{\text{1y}} =0 \tag{16a}\] \[l_{\text{2y}} =0\] (16b) \[l_{\text{3x}} =0 \tag{16c}\]  
The solution of vector \(\mathbf{p}\) is obtained by combining Equation (16):  
\[x=-e\sin\psi\sin\theta,y=0,\ \phi=0 \tag{17}\]  
According to Equation (17), it is known that the moving platform \(S_{1}S_{2}S_{3}\) produces one translation movement \(x\) of the moving platform center point along the axis \(x\). The parasitic motion \(x\) is related to the structural parameter \(e\) and kinematic parameters \(\psi\) and \(\theta\). When the kinematic parameters \(\psi\) and \(\theta\) and structural parameter \(e\) of the moving platform \(S_{1}S_{2}S_{3}\) are given, the parasitic motions \(x\) can be obtained.