## 3 Kinematics Analysis
### Inverse Kinematics  
The inverse kinematics aims to find the drive displacements of driving parts given the pose of the mobile platform. As shown in Figure 7, based on the global coordinate system, we established a base platform local coordinate system \(O_{0i}-X_{0i}Y_{0i}Z_{0i}\) (\(i=1,\ldots,6\)), where the point \(O_{0i}\) (coincidence point \(B_{i}\) at the zero position) is the origin. Based on the mobile platform's local coordinate system, the mobile platform local coordinate system \(O_{6i}-X_{6i}Y_{6i}Z_{6i}\) (\(i=1,\ldots,6\)) is established with the point \(O_{6i}\) (coincident with point \(P_{i}\)) as its origin. Because of the presence of the offset joint, it is necessary to create five additional local coordinate systems on each leg, which are denoted by systems \(O_{1i}-X_{1i}Y_{1i}Z_{1i}\) (\(i=1,\ldots,6\)) to \(O_{5i}-X_{5i}Y_{5i}Z_{5i}\) (\(i=1,\ldots,6\)). Each leg of the proposed parallel platform can be equivalent to the P-RR-R-RR series mechanism. According to the D-H parameter theory used for series robots, the D-H \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(R_{P}\) & 0.13 m \\ \(R_{B}\) & 0.222 m \\ \(\theta_{P}\) & 30\({}^{\circ}\) \\ \(\theta_{B}\) & 90\({}^{\circ}\) \\ \(U_{P}\) & 0.01 m \\ \(U_{B}\) & 0.01 m \\ \(H\) & 0.1 m \\ \(H_{P}\) & 0.066 m \\ \(H_{B}\) & 0.128 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parallel platform configuration parameters. Figure 8: Layouts of local coordinate systems: (**a**) At the base platform; (**b**) At the mobile platform. parameter of the \(i_{th}\) leg can be obtained as shown in Table 2. Here, \(S_{1i}\) represents the displacement of the lower joint point \(B_{i}\) along the inclined plane. Based on the D-H parameter method, the kinematics transformation matrix for the \(i_{th}\) leg kinematic chain of the parallel platform can be derived as:  
\[{}^{j}\mathbf{T}_{6(i)}=\mathbf{A}_{j+1(i)}\cdots\mathbf{A}_{6(i)}\ (j=0,\ 1,\cdots,5) \tag{2}\]  
where \(\mathbf{A}_{j(i)}\) is the transformation matrix between two adjacent coordinate systems. The distribution diagrams of the local coordinate systems that were defined on the base and mobile platforms are shown in Figure 8. The position of the origin \(O_{0i}\) of the local coordinate system in \(O_{B}-X_{B}Y_{B}Z_{B}\) is expressed as:  
\[{}^{O_{B}}\mathbf{O}_{0i}= \begin{bmatrix}R_{B}\cdot c\theta_{O_{0i}}&R_{B}\cdot s \theta_{O_{0i}}&H_{B}\end{bmatrix} ^{T} \tag{3}\]  
The position of the origin \(O_{6i}\) of the local coordinate system in \(O_{P}-X_{P}Y_{P}Z_{P}\) is then expressed as:  
\[{}^{O_{P}}\mathbf{O}_{6i}= \begin{bmatrix}R_{P}\cdot c\theta_{O_{6i}}&R_{P}\cdot s \theta_{O_{6i}}&-H_{P}\end{bmatrix} ^{T} \tag{4}\]  
The static coordinate system \(O_{B}-X_{B}Y_{B}Z_{B}\) first rotates at the angle \(\theta_{Z_{B}(i)}\) around the axis \(Z_{B}\) and then rotates at the angle \(\theta_{\chi^{\prime}{}_{B}(i)}\) around the new axis \(Y^{\prime}{}_{B}\) so that the newly obtained coordinate system has the same orientation as \(O_{0i}-X_{0i}Y_{0i}Z_{0i}\). In the coordinate system \(O_{0i}-X_{0i}Y_{0i}Z_{0i}\), the definitions of \(X_{0i}\) and \(Z_{0i}\) are as shown in Figure 7, and the definitions of \(Y_{0i}\) are as shown in Figure 8. The expression for the transfer matrix \({}^{O_{B}}\mathbf{T}_{O_{0i}}\) of \(O_{0i}-X_{0i}Y_{0i}Z_{0i}\) to \(O_{B}-X_{B}Y_{B}Z_{B}\) is given as follows:  
\[{}^{O_{B}}\mathbf{T}_{O_{0i}}= \begin{bmatrix}{}^{O_{B}}\mathbf{R}_{O_{0i}}&{}^{O_{B}} \mathbf{O}_{0i}\\ 0&1\end{bmatrix}  \tag{5}\]  
where \({}^{O_{B}}\mathbf{R}_{O_{0i}}\) denotes the rotation matrix of \(O_{B}-X_{B}Y_{B}Z_{B}\) to \({}^{O_{B}}-X^{\prime\prime}{}_{B}Y^{\prime\prime}{}_{B}Z^{\prime\prime}{}_{B}\). Similarly, the transfer matrix \({}^{O_{P}}\mathbf{T}_{O_{0i}}\) from \(O_{6i}-X_{6i}Y_{6i}Z_{6i}\) to \(O_{P}-X_{P}Y_{P}Z_{P}\) is expressed as follows:  
\[{}^{O_{P}}\mathbf{T}_{O_{0i}}= \begin{bmatrix}{}^{O_{P}}\mathbf{R}_{O_{0i}}&{}^{O_{P}} \mathbf{O}_{6i}\\ 0&1\end{bmatrix}