# Parasitic Motions of 3-PRS Parallel Mechanisms with Two Different Branch Chain Arrangements
## 2 3-PRS PM I
### Kinematics Analysis  
The aim of kinematics analysis is to obtain the constraint equations according to the geometric characteristics of the PMs. In this paper, a \(y\)-\(x\)-\(z\) Euler angle is introduced to describe the posture of the moving platform in the \(\{O_{\text{B}}\}\) coordinate system; that is, let the rotation angle around the \(y\) axis be first \(\theta\), then the rotation angle around the \(x\) axis can be \(\psi\), and, finally, the rotation angle around the \(z\) axis can be \(\phi\). The rotation matrix \(\mathbf{T}\) is:  
\[ \begin{split}&\mathbf{T}=\mathbf{T}_{yxz}=\mathbf{R}_{y}(\theta)\mathbf{R}_{x}( \psi)\mathbf{R}_{z}(\varphi)\\ &=\left[\begin{array}{ccc}\text{c}\theta\text{c}\varphi+\text{ s}\psi\text{s}\theta\text{s}\varphi&-\text{c}\theta\text{s}\varphi+\text{s} \psi\text{s}\theta\text{c}\varphi&\text{c}\psi\theta\\ \text{c}\psi\text{s}\varphi&\text{c}\psi\text{c}\varphi&-\text{s}\psi\\ -\text{s}\theta\text{c}\varphi+\text{s}\psi\text{s}\theta\text{s}\varphi& \text{s}\theta\text{s}\varphi+\text{s}\psi\text{c}\theta\text{c}\varphi& \text{c}\psi\text{c}\theta\end{array}\right]=\left[\begin{array}{ccc}\text {T}_{11}&\text{T}_{12}&\text{T}_{13}\\ \text{T}_{21}&\text{T}_{22}&\text{T}_{23}\\ \text{T}_{31}&\text{T}_{32}&\text{T}_{33}\end{array}\right]\end{split}  \tag{1}\]  
where c represents the cosine and s represents the sine. \[\mathbf{R}_{y}(\theta)=\left[ \begin{array}{ccc}\cos\theta&0&\sin\theta\\ 0&1&0\\ -\sin\theta&0&\cos\theta\end{array} \right]\mathbf{R}_{x}(\psi)=\left[ \begin{array} []{ccc}1&0&0\\ 0&\cos\psi&-\sin\psi\\ 0&\sin\psi&\cos\psi\end{array} \right]\mathbf{R}_{z}(\phi)=\left[ \begin{array}{ cccc}\cos\varphi&\sin\varphi&0\\ -\sin\varphi&\cos\varphi&0\\ 0&0&1\end{array} \right]\]  
According to Figures 2 and 3, the vectors of the joint position and motion direction are obtained, as shown in Table 1.  
In the \(\{O_{\text{B}}\}\) coordinate system, the vectors of the kinematic pairs of the three branch chains are as follows. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Expression**} & \multicolumn{2}{c}{**Physical Meaning (\(i=1,2,3\))**} \\ \hline \(\mathbf{p}=\left[x\right.\) & \(y\) & \(z\right]^{\text{T}}\) & In the \(\{O_{\text{B}}\}\) coordinate system, the vector from the origin \(O_{\text{B}}\) to the origin \(O_{\text{m}}\). \\ \(\mathbf{a}_{i}=\left[a_{ix}\right.\) & \(a_{iy}\) & \(a_{iz}\right]^{\text{T}  
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 \(\mathbf{a}_{3}\):  
\[ \begin{array}{l}\mathbf{a}_{1}=\left[\begin{array}{cc}-\frac{\sqrt{3}R}{2}&- \frac{R}{2}&0\end{array}\right]^{\text{T}}\\ \mathbf{a}_{2}=\left[\begin{array}{cc}\frac{\sqrt{3}R}{2}&-\frac{R}{2}&0\end{array} \right]^{\text{T}}\\ \mathbf{a}_{3}=\left[\begin{array}{cc}0&R&0\end{array}\right]^{\text{T}}\end{array}  \tag{2}\]  
The position vectors of the prismatic joint \(P_{i}\) (\(i=1\), \(2\), \(3\)) in the \(\{O_{\text{B}}\}\) coordinate system are:  
\[ \begin{array}{l}\mathbf{P}_{1}=\left[\begin{array}{cc}-\frac{\sqrt{3}R}{2}&- \frac{R}{2}&d_{1}\end{array}\right]^{\text{T}}\\ \mathbf{P}_{2}=\left[\begin{array}{cc}\frac{\sqrt{3}R}{2}&-\frac{R}{2}&d_{2} \end{array}\right]^{\text{T}}\\ \mathbf{P}_{3}=\left[\begin{array}{cc}0&R&d_{3}\end{array}\right]^{\text{T}} \end{array}  \tag{3}\]  
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}_{1}^{\prime}=\left[\begin{array}{cc}-\frac{\sqrt{3 }}{2}&-\frac{r}{2}&0\end{array}\right]^{\text{T}}\\ \mathbf{S}_{2}^{\prime}=\left[\begin{array}{cc}\frac{\sqrt{3}r}{2}&-\frac{r}{2}&0 \end{array}\right]^{\text{T}}\\ \mathbf{S}_{3}^{\prime}=\left[\begin{array}{cc}0&r&0\end{array}\right]^{\text{T} }\end{array}