## 4 Kinematic Model
### Inverse Kinematics of 3(Rc)PU PM
In the fixed coordinate system \(O_{n}\)-XYZ, the position vector of the center \(A_{i}\) of the Rc limb can be expressed as  
\[O_{n}A_{1}=[0,\ 0,\ R]^{\mathrm{T}},\ \ O_{n}A_{2}=\left[\sqrt{3}R/2,\ 0,-R/2 \right]^{\mathrm{T}},\ \ O_{n}A_{3}=\left[-\sqrt{3}R/2,\ 0,-R/2\right]^{\mathrm{T}}. \tag{19}\]  
The position vector of the center \(B_{i}\) of each limb's universal joint in the moving coordinate system \(O_{u}\)-\(xyz\) can be expressed as  
\[O_{u}B_{1}=(0,\ 0,\ r_{1})^{\mathrm{T}},O_{u}B_{2}=\left(\sqrt{3}r_{1}/2,\ 0,-r_{1}/2 \right)^{\mathrm{T}},O_{u}B_{3}=\left(-\sqrt{3}r_{1}/2,\ 0,-r_{1}/2 \right)^{\mathrm{T}}\]  
The position vector of the origin \(O_{u}\) of the moving coordinate system \(O_{u}\)-\(xyz\) in the fixed coordinate system \(O_{n}\)-\(XYZ\) is  
\[O_{n}O_{u}=(X,Y,Z)^{\mathrm{T}}\]  
The position vector \(O_{u}B_{i}\) (\(i=1,\ 2,\ 3\)) of the center \(B_{i}\) of each limb's universal joint in the moving coordinate system \(O_{u}\)-\(xyz\) can be converted to the description \(O_{n}B_{i}\) (\(i=1,\ 2,\ 3\)) in the fixed coordinate system \(O_{n}\)-XYZ:  
\[O_{n}B_{1}=[m_{1},\ n_{1},\ d_{1}]^{\mathrm{T}},O_{n}B_{2}=[m_{2},\ n_{2},\ d_{2}]^{\mathrm{T}},O_{n}B_{3}=[m_{3},\ n_{3},\ d_{3}]^{\mathrm{T}}. \tag{20}\]  
where  
\[m_{1}=\left[(c\gamma s\beta c\alpha+s\gamma s\alpha)r+X\right],n_{1}=\left[(s\gamma s\beta c\alpha-c\gamma s\alpha)r+Y\right],d_{1}=\left[c\beta c\alpha r-R+Z\right],\] \[m_{2}=\left[\left(\frac{\sqrt{3}}{2}c\gamma c\beta-\frac{1}{2}c \gamma s\beta c\alpha-\frac{1}{2}s\gamma s\alpha\right)r-\frac{\sqrt{3}R}{2} +X\right],n_{2}=\left[\left(\frac{\sqrt{3}}{2}s\gamma c\beta-\frac{1}{2}s \gamma s\beta c\alpha+\frac{1}{2}c\gamma s\alpha\right)r+Y\right],\] \[d_{2}=\left[\left(-\frac{\sqrt{3}}{2}s\beta-\frac{1}{2}c\beta c \alpha\right)r+\frac{R}{2}+Z\right],m_{3}=\left[\left(-\frac{\sqrt{3}}{2}c \gamma c\beta-\frac{1}{2}c\gamma s\beta c\alpha-\frac{1}{2}s\gamma s\alpha \right)r+\frac{\sqrt{3}R}{2}+X\right],\] \[n_{3}=\left[\left(-\frac{\sqrt{3}}{2}s\gamma c\beta-\frac{1}{2} s\gamma s\beta c\alpha+\frac{1}{2}c\gamma s\alpha\right)r+Y\right],d_{3}=\left[ \left(\frac{\sqrt{3}}{2}s\beta-\frac{1}{2}c\beta c\alpha\right)r+\frac{R}{2} +Z\right].\]  
The inverse kinematic solution of the 3(Rc)PU PM can be obtained from Equation (21):  
\[A_{i}B_{i}+O_{n}A_{i}-O_{n}B_{i}=0,\ (i=1,\ 2,\ 3) \tag{18}\]0  
When all limbs are (Rc)\({}_{1}\)PU limbs, \(\theta_{1}=\theta_{2}=\theta_{3}=0\). When all limbs are (Rc)\({}_{2}\)PU limbs,  
\[ \begin{cases}\theta_{1}=\arccos\frac{|A_{1}B_{1}(0,0)|}{|A_{1}B_{1}|}=\arccos \frac{d_{1}}{\sqrt{m_{1}^{2}+n_{1}^{2}+d_{1}^{2}}},\\ \theta_{2}=\arccos\frac{|A_{2}B_{2}(1,0)|}{|A_{2}B_{2}|}=\arccos\frac{m_{2}}{ \sqrt{m_{2}^{2}+n_{2}^{2}+d_{2}^{2}}},\\ \theta_{3}=\arccos\frac{|A_{3}B_{3}(1,0)|}{|A_{3}B_{3}|}=\arccos\frac{m_{3}}{ \sqrt{m_{3}^{2}+n_{3}^{2}+d_{3}^{2}}}.\end{cases}  \tag{22}\]  
The input parameters of each motion branch of the 3(Rc)PU parallel mechanism are shown in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Serial No.** & **Motion Mode** & **Inverse Solution Input Parameter Composition** \\ \hline
1 & 3T & \(\{l_{1},l_{2},l_{3}\}\) \\
2 & 1R3T & \(\{\theta_{2},l_{1},l_{2},l_{3}\}\) \\
3 & 2R3T & \(\{\theta_{2},\theta_{3},l_{1},l_{2},l_{3}\}\) \\
4 & 3R3T & \(\{\theta_{1},\theta_{2},\theta_{3},l_{1},l_{2},l_{3}\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The inverse solution input parameters in various motion modes.