## 3 Kinematic Analysis  
The unified expression of the three plunger groove axes \(m_{1},m_{2},m_{3}\) in coordinate system \(O^{\prime}X^{\prime}Y^{\prime}Z^{\prime}\) is:  
\[\left\{ \begin{array}{l}X^{\prime}=r\cos\bigl{[}\varphi_{1}+(j-1)\frac{2}{3} \pi\bigr{]}\\ Y^{\prime}=r\sin\bigl{[}\varphi_{1}+(j-1)\frac{2}{3}\pi\bigr{]}\end{array}  \right.\qquad(j=1,2,3) \tag{1}\]  
This utilizes the following coordinate transformations:  
\[ \begin{bmatrix}X^{\prime}\\ Y^{\prime}\\ Z^{\prime}\end{bmatrix} = \begin{bmatrix}\cos\beta&0&-\sin\beta\\ 0&1&0\\ \sin\beta&0&\cos\beta\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}  \tag{2}\]  
These are transformed into coordinate system \(oxyz\)  
\[\left\{ \begin{array}{l}x\cos\beta-z\sin\beta=r\cos\bigl{[}\varphi_{1}+(j-1) \frac{2}{3}\pi\bigr{]}\\ y=r\sin\bigl{[}\varphi_{1}+(j-1)\frac{2}{3}\pi\bigr{]}\end{array} \right. \qquad(j=1,2,3) \tag{3}\]  
The expression of the axes \(n_{1},n_{2},n_{3}\) of the three-rod in the moving coordinate system \(O^{\prime}X^{\prime\prime}Y^{\prime\prime}Z^{\prime\prime}\), in the fixed coordinate system \(oxyz\), can be referred to as shown in Figure 4.  
\[\left\{ \begin{array}{l}Y^{\prime\prime}=X^{\prime\prime}tg\bigl{[}\varphi_{ 0}+(j-1)\frac{2}{3}\pi\bigr{]}\\ Z^{\prime\prime}=0\end{array} \right.\qquad(j=1,2,3) \tag{4}\]  
The coordinates of the origin \(O^{\prime\prime}\) of the moving coordinate system \(O^{\prime\prime}X^{\prime\prime}Y^{\prime\prime}Z^{\prime\prime}\) in the fixed coordinate system \(oxyz\) can be determined by referencing Figure 4.  
\[\left\{ \begin{array}{l}X^{(O^{\prime\prime})}=\rho\cos\theta\sin 3\varphi_{0} \\ Y^{(O^{\prime\prime})}=\rho\cos\theta\cos 3\varphi_{0}\\ Z^{(O^{\prime\prime})}=\rho\sin\theta\end{array} \right. \tag{5}\]  
Since the \(O^{\prime\prime}Y^{\prime\prime}\) axis of the moving coordinate system \(O^{\prime\prime}X^{\prime\prime}Y^{\prime\prime}Z^{\prime\prime}\) is always perpendicular to the \(ox\) axis of the fixed coordinate system, it can be considered by, as shown in  
Figure 4: Rotation the angle (\(\varphi_{0}\)) of the axis (\(O^{\prime}Z^{\prime\prime}\)) of the three-pronged rod in the moving coordinate system (\(O^{\prime}X^{\prime\prime}Y^{\prime\prime}Z^{\prime\prime}\)). Figure 2, first rotating the \(oxyz\) coordinate system about the \(ox\) axis by an angle \(\theta_{x}\) to align the \(oy\) axis parallel to the \(O^{\prime\prime}Y^{\prime\prime}\) axis, then rotating about the new \(oy\) axis by an angle \(\theta_{y}\) to align the \(x\) axis parallel to \(O^{\prime\prime}X^{\prime\prime}\), while also making the \(oz\) axis parallel to the \(O^{\prime\prime}Z^{\prime\prime}\) axis. Due to the transformation relationship between the fixed coordinate system \(oxyz\) and the moving coordinate system \(O^{\prime\prime}X^{\prime\prime}Y^{\prime\prime}Z^{\prime\prime}\),  
\[ \begin{bmatrix}x\\ y\\ z\end{bmatrix} = \begin{bmatrix}\rho\cos\theta\sin 3\varphi_{0}\\ \rho\cos\theta\cos 3\varphi_{0}\\ \rho\sin\theta\end{bmatrix} + \begin{bmatrix}1&0&0\\ 0&\cos\theta_{x}&-\sin\theta_{x}\\ 0&\sin\theta_{x}&\cos\theta_{x}\end{bmatrix}\begin{bmatrix}\cos\theta_{y}&0& \sin\theta_{y}\\ 0&1&0\\ -\sin\theta_{y}&0&\cos\theta_{y}\end{bmatrix}\begin{bmatrix}X^{\prime\prime }\\ Y^{\prime\prime}\\ Z^{\prime\prime}\end{bmatrix}  \tag{6}\]  
This is based on the properties of matrix transformations  
\[ \begin{bmatrix}X^{\prime\prime}\\ Y^{\prime\prime}\\ Z^{\prime\prime}\end{bmatrix} = \begin{bmatrix}\cos\theta_{y}&\sin\theta_{x} \sin\theta_{y}&-\cos\theta_{x}\sin\theta_{y}\\ 0&\cos\theta_{x}&\sin\theta_{x}\\ \sin\theta_{y}&-\sin\theta_{x}\cos\theta_{y}&\cos\theta_{x}\cos\theta_{y} \end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix} - \begin{bmatrix}\rho\cos\theta\sin 3\varphi_{0}\\ \rho\cos\theta\cos 3\varphi_{0}\\ \rho\sin\theta\end{bmatrix}  \tag{7}\]  
After expanding,  
\[\left\{