# Parasitic Motions of 3-PRS Parallel Mechanisms with Two Different Branch Chain Arrangements
## 2 3-PRS PM I
### Kinematics Analysis
 \tag{4}\]  
The position vectors of the 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 by \(\mathbf{S_{1}}^{\prime}\),\(\mathbf{S_{2}}^{\prime}\), and \(\mathbf{S_{3}}^{\prime}\) using the closed-loop vector method. \[\mathbf{I}_{i}=\mathbf{S}_{i}+\mathbf{p}=\mathbf{T}\cdot\mathbf{S^{{}^{\prime}}}_{i}+\mathbf{p} \tag{5}\]  
Substituting all the vectors aforementioned into Equation (5) yields:  
\[ \begin{array}{l}l_{1x}=x+T_{11}b^{\prime}_{1x}+T_{12}b^{\prime}_{1y}\\ l_{1y}=y+T_{21}b^{\prime}_{1x}+T_{22}b^{\prime}_{1y}\end{array}  \tag{6a}\] \[ \begin{array}{l}l_{2x}=x+T_{11}b^{\prime}_{2x}+T_{12}b^{\prime}_{2y}\\ l_{2y}=y+T_{21}b^{\prime}_{2x}+T_{22}b^{\prime}_{2y}\end{array} \] (6b) \[ \begin{array}{l}l_{3x}=x+T_{12}b^{\prime}_{3y}\\ l_{3y}=y+T_{22}b^{\prime}_{3y}\end{array}  \tag{6c}\]  
According to the constraints of the revolute pairs of the mechanism's structure, each branch chain of the 3-PRS PM I only moves in the corresponding plane in Figure 4. Constrained by the structural conditions of the mechanism, each branch chain can obtain three constraint equations through Equation (6a-c) as follows:  
\[ \begin{array}{l}l_{1y}=\
icefrac{{\sqrt{3}}}{{3}}/\
icefrac{{}}{{3}}l_{1x} \end{array}  \tag{7a}\] \[ \begin{array}{l}l_{2y}=-\
icefrac{{\sqrt{5}}}{{3}}/\
icefrac{{}}{{3}}l_{2x} \end{array} \] (7b) \[ \begin{array}{l}l_{3x}=0\end{array}  \tag{7c}\]The solution of the vector \(\mathbf{p}\) is obtained by combining Equations (6) and (7):  
\[x=r(\cos\theta\sin\varphi-\sin\psi\sin\theta\cos\varphi) \tag{8}\]  
\[y=\frac{r}{2}(\sqrt{3}\cos\psi\sin\varphi+\cos\psi\cos\varphi+\sqrt{3}\cos\theta \sin\varphi-\sqrt{3}\sin\psi\sin\theta\cos\varphi-\cos\theta\cos\varphi-\sin \psi\sin\theta\sin\varphi) \tag{9}\]  
\[\varphi=\arctan\frac{\sin\psi\sin\theta}{\cos\psi+\cos\theta} \tag{10}\]  
According to Equations (8)-(10), it is known that the moving platform \(S_{1}S_{2}S_{3}\) produces one rotation \(\phi\) and two translation movements, \(x\) and \(y\), of the moving platform center point \(O_{\text{m}}\) along the \(x\) and \(y\) axes. The small amplitude motions of the constrained degrees of freedom are called parasitic motions, that is the non-independent degrees of freedom, which are obtained by the constraint analysis of the center point of the moving platform. The existence of parasitic motion will cause the center point of the mechanism moving platform move in a non-direction of the degree of freedom, which brings great difficulties to the dimensional synthesis and trajectory planning of the mechanism. It will cause an effect on the precision of the mechanism and is harmful to the mechanism. The minimization and elimination of the parasitic motion have a significant influence on the practical application. Additionally, the motion parameter \(\phi\) is related to the kinematic parameters \(\psi\) and \(\theta\). The motion parameters \(x\) and \(y\) are related to the structural parameter \(r\) and the kinematic parameters \(\psi\) and \(\theta\).