## 3 Motion Parameter Analysis of Propulsion Mechanism  
Because the propeller installation platform includes many motion parameters and there are motion couplings among them, the decoupling of motion parameters should be carried out first. For the convenience of attitude description of the \(M\) coordinate, the attitude uses Euler angle to express. The \(M\) coordinate coincides with the \(S\) coordinate at initial, and rotates in the following order:  
1. Rotating angle \(\alpha\) around \(z\)-axis of \(M\) coordinate;
2. Rotating angle \(\beta\) around \(y\)-axis of \(M\) coordinate;
3. Rotating angle \(\gamma\) around \(z\)-axis of \(M\) coordinate. The rotation matrix is as follows:  
\[\mathbf{T}_{SM}=\left[ \begin{array}{ccc}\cos\alpha\cos\beta\cos\gamma-\sin\alpha \sin\gamma&-\cos\alpha\cos\beta\sin\gamma-\sin\alpha\cos\gamma&\cos\alpha\sin \beta\\ \sin\alpha\cos\beta\cos\gamma+\cos\alpha\sin\gamma&-\sin\alpha\cos\beta\sin \gamma+\cos\alpha\cos\gamma&\sin\alpha\sin\beta\\ -\sin\beta\cos\gamma&\sin\beta\sin\gamma&\cos\beta\end{array} \right], \tag{1}\]  
In the matrix, \(\alpha\) is the precession angle, which is the angle between the \(z\)-axis projection of the M-coordinate in \(XOY\) plane of the \(S\) coordinate and \(x\)-axis of the \(S\) coordinate, meaning the azimuth angle of propeller installation platform swinging relative to the \(S\) coordinate. \(\beta\) is the nutation angle, which is the angle between the \(z\)-axis of the \(M\) coordinate and \(z\)-axis of the \(S\) coordinate, meaning the inclination angle of propeller installation platform relative to the \(S\) coordinate on the \(\alpha\) direction. \(\gamma\) is the self-rotation angle, which is the angle of propeller installation platform spinning around the \(z\)-axis. Because it rotates \(\alpha\) and \(\gamma\) around the \(z\)-axis twice before and after, the rotation angle of the installation platform around the \(z\)-axis is \(\varphi=\alpha+\gamma\). Figure 1: 2-DOF (degree-of-freedom) parallel vector propulsion mechanism. If the coordinate origin of \(M\) coordinate is \((x_{o},y_{o},z_{o})^{\text{T}}\), then the coordinates of three spherical joints are respectively shown as follows:  
\[\textbf{P}_{Si}^{S}=\textbf{T}_{SM}\cdot\textbf{P}_{Si}^{\text{M}}+[x_{o},y_{o},z_{o}]^{\text{T}}, \tag{2}\]  
\[\textbf{P}_{S1}^{S}=\left[ \begin{array}{c}r(\cos\alpha\cos\beta\cos\gamma- \sin\alpha\sin\gamma)+x_{o}\\ r(\sin\alpha\cos\beta\cos\gamma+\cos\alpha\sin\gamma)+y_{o}\\ -r\sin\beta\cos\gamma+z_{o}\end{array} \right]\]  
\[\textbf{P}_{S2}^{S}=\left[ \begin{array}{c}-(r/2)(\cos\alpha\cos\beta\cos \gamma-\sin\alpha\sin\gamma)+\left(\sqrt{3}r/2\right)(-\cos\alpha\cos\beta\sin \gamma-\sin\alpha\cos\gamma)+x_{o}\\ -(r/2)(\sin\alpha\cos\beta\cos\gamma+\cos\alpha\sin\gamma)+\left(\sqrt{3}r/2 \right)(-\sin\alpha\cos\beta\sin\gamma+\cos\alpha\cos\gamma)+y_{o}\\ (r/2)\sin\beta\cos\gamma+\left(\sqrt{3}r/2\right)\sin\beta\sin\gamma+z_{o} \end{array} \right]\]  
\[\textbf{P}_{S3}^{S}=\left[ \begin{array}{c}-(r/2)(\cos\alpha\cos\beta\cos \gamma-\sin\alpha\sin\gamma)-\left(\sqrt{3}r/2\right)(-\cos\alpha\cos\beta \sin\gamma-\sin\alpha\cos\gamma)+x_{o}\\ -(r/2)(\sin\alpha\cos\beta\cos\gamma+\cos\alpha\sin\gamma)-\left(\sqrt{3}r/2 \right)(-\sin\alpha\cos\beta\sin\gamma+\cos\alpha\cos\gamma)+y_{o}\\ (r/2)\sin\beta\cos\gamma-\left(\sqrt{3}r/2\right)\sin\beta\sin\gamma+z_{o} \end{array} \right]\]  
Constrained by revolution joints \(R_{1}\), \(R_{2}\), \(R_{3}\), the spherical joints move in three plane of the S coordinate, \(Y=0\), \(Y=\pm\sqrt{3}X.\) Therefore,  
\[\textbf{P}_{S1}^{S}(2)=0, \tag{3}\]  
\[\textbf{P}_{S2}^{S}(2)=-\sqrt{3}\textbf{P}_{S2}^{S}(1), \tag{4}\]  
\[\textbf{P}_{S3}^{S}(2)=\sqrt{3}\textbf{P}_{S3}^{S}(1), \tag{5}\]  
Let Equation (4) reduce Equation (5),  
\[2x_{o}=r(\alpha c\beta c\gamma-s\alpha s\gamma+s\alpha c\beta s\gamma-\alpha c \gamma), \tag{6}\]  
where \(c=\cos\), \(s=\sin\)(same below). Let Equation (4) plus Equation (5).