# Path Planning Based on NURBS for Hyper-Redundant Manipulator Used in Narrow Space
## 2 Kinematics Modeling and Workspace Analysis of UCR Parallel Mechanism
### Kinematics Modeling of UCR Parallel Mechanism  
(1)The driven model of UCR parallel mechanism  
As shown in Figure 1, \(O_{1}\) and \(O_{2}\) are the center of the fixed platform and the moving platform, respectively. The connection points of the three chains to the fixed platform and the moving platform are \(P_{1,i}\), \(Q_{1,i}\) (\(i=1,2,3\)). The radius of the fixed platform and moving platform is r. A fixed coordinate system \(O-XYZ\) is established on the fixed platform. The two orthogonal axes of the universal joint are used as the \(X\) and \(Y\)-axes, where the \(X\)-axis points from \(O_{1}\) to \(P_{1,1}\), and the \(Z\) axis is the fixed platform normal. A moving coordinate system \(O_{2}-UVW\) is established on the moving platform, where the U axis is from \(O_{2}\) to \(Q_{1,1}\), and the \(W\) axis is the normal of the moving platform. The transformation matrix of moving coordinate system \(O_{2}-UVW\) relative to fixed coordinate system \(O_{1}-XYZ\) is  
\[T=\left[ \begin{array}{cccc}n_{x}&o_{x}&a_{x}&X_{c}\\ n_{y}&o_{y}&a_{y}&Y_{c}\\ n_{z}&o_{z}&a_{z}&Z_{c}\\ 0&0&0&1\end{array} \right] \tag{1}\]  
The coordinates of \(P_{1,1}\), \(P_{1,2}\), and \(P_{1,3}\) in the coordinate system \(O_{1}-XYZ\) are:  
\[P_{1,1}=\left[ \begin{array}{c}r\\ 0\\ 0\end{array} \right]\quad P_{1,2}=\left[ \begin{array}{c}-\frac{r}{2}\\ \frac{\sqrt{3}r}{2}\\ 0\end{array} \right]\quad P_{1,3}=\left[ \begin{array}{c}-\frac{r}{2}\\ -\frac{\sqrt{3}r}{2}\\ 0\end{array} \right] \tag{2}\]  
The coordinates of \(Q_{1,1}\), \(Q_{1,2}\), and \(Q_{1,3}\) in the coordinate system \(O_{2}-UVW\) are:  
\[Q_{1,1}=\left[ \begin{array}{c}r\\ 0\\ 0\end{array} \right]\quad Q_{1,2}=\left[ \begin{array}{c}-\frac{r}{2}\\ \frac{\sqrt{3}r}{2}\\ 0\end{array} \right]\quad Q_{1,3}=\left[ \begin{array}{c}-\frac{r}{2}\\ -\frac{\sqrt{3}r}{2}\\ 0\end{array} \right] \tag{3}\]  
Because of \(\left[ \begin{array}{c}Q_{i}\\ 1\end{array} \right]=T\left[ \begin{array}{c}Q_{1,i}\\ 1\end{array} \right](i=1,2,3)\), so  
Figure 1: Structure of UCR parallel mechanism. \[Q_{1}=\left[ \begin{array}{c}Q_{1_{x}}\\ Q_{1_{y}}\\ Q_{1_{z}}\end{array} \right]=\left[ \begin{array}{c}rn_{x}+X_{c}\\ rn_{y}+Y_{c}\\ rn_{z}+Z_{c}\end{array} \right] \tag{4}\]  
\[Q_{2}=\left[ \begin{array}{c}Q_{2_{x}}\\ Q_{2_{y}}\\ Q_{2_{z}}\end{array} \right]=\left[ \begin{array}{c}-\frac{r}{2}n_{x}+\frac{ \sqrt{3}r}{2}o_{x}+X_{c}\\ -\frac{r}{2}n_{y}+\frac{\sqrt{3}r}{\sqrt{3}}o_{y}+Y_{c}\\ -\frac{r}{2}n_{z}+\frac{\sqrt{3}r}{2}o_{z}+Z_{c}\end{array} \right] \tag{5}\]  
\[Q_{3}=\left[ \begin{array}{c}Q_{3_{x}}\\ Q_{3_{y}}\\ Q_{3_{z}}\end{array} \right]=\left[ \begin{array}{c}-\frac{r}{2}n_{x}-\frac{ \sqrt{3}r}{2}O_{x}+X_{c}\\ -\frac{r}{2}n_{y}-\frac{\sqrt{3}r}{2}O_{y}+Y_{c}\\ -\frac{r}{2}n_{z}-\frac{\sqrt{3}r}{2}O_{z}+Z_{c}\end{array} \right] \tag{6}\]  
The driving functions is obtained as follows:  
\[l_{1\_j}^{2}=\left\|Q_{1\_i}-P_{1\_i}\right\|^{2}=\left[RQ_{1\_j}+c-P_{1\_i} \right]^{T}[RQ_{1\_i}+c-P_{1\_i}](i=1,2,3) \tag{7}\]  
where \(l_{1\_i}\) represents the length of chain. (2) The constraint model of UCR parallel mechanism  
The chains of the UCR parallel mechanism are symmetric. Take one of the chains and analyze its motion spiral and constraint spiral. The chain is shown in Figure 2.  
According to the screw theory, the screw system of the chain is:  
\[\left\{\begin{array}{c}\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\Figure 2.0\Figure 2.1\Figure 2.2\Figure 2.3\Figure 2.4\Figure 2.5\Figure 2.6\Figure 2.7\Figure 2.8\Figure 2.9\$\$0\$\$1\$\$2\$\$3\$\$4\$  
3. The parasitic motion of UCR parallel mechanism Attitude description based on \(Z-X-Z\) Euler Angle: \[R=\left[ \begin{array}{ccc}Ca^{2}+Sa^{2}C\beta&CaS\alpha(1-C\beta)&SaS\beta\\ Sa\alpha Ca(1-C\beta)&Sa^{2}+Ca^{2}C\beta&-CaS\beta\\ -SaS\beta&S\beta Ca&C\beta\end{array} \right]\] (11)  
The rotation matrix is equivalent to rotate the \(\beta\) degrees around the axis that is \(\alpha\) degrees from the \(X\)-axis.