# Computational Dynamics of Multi-Rigid-Body System in Screw Coordinate
## 4 Kinematics and Dynamics of the Gough-Stewart Platform
### Kinematics Analysis of the Gough-Stewart Platform  
This section gives the inverse kinematics analysis of the Gough-Stewart platform. As shown in Figure 4, the inverse kinematics of the Gough-Stewart platform can be expressed as:  
\[\mathbf{S}\mathbf{\omega}=\mathbf{V}_{\text{C}} \tag{26}\]  
where \(\mathbf{S}=\textit{diag}\big{[}\mathbf{S}^{1}\quad\mathbf{S}^{2}\quad\mathbf{S}^{3}\quad\mathbf{S} ^{4}\quad\mathbf{S}^{5}\quad\mathbf{S}^{6}\big{]}\) in which \(\mathbf{S}^{p}=\big{[}\mathbf{\hat{s}}_{1}^{p}\quad\mathbf{\hat{s}}_{2}^{p}\quad\mathbf{\hat{ s}}_{3}^{p}\quad\mathbf{\hat{s}}_{4}^{p}\quad\mathbf{\hat{s}}_{5}^{p}\quad\mathbf{\hat{s}}_{6}^{p}\big{]}\), and \(\mathbf{\hat{s}}_{1}^{p}=\begin{bmatrix}\mathbf{e}_{1}^{p}\\ \mathbf{o}_{A}^{p}\times\mathbf{e}_{1}^{p}\end{bmatrix}\), \(\mathbf{\hat{s}}_{2}^{p}=\begin{bmatrix}\mathbf{e}_{2}^{p}\\ \mathbf{o}_{A}^{p}\times\mathbf{e}_{2}^{p}\end{bmatrix}\), \(\mathbf{\hat{s}}_{3}^{p}=\begin{bmatrix}\mathbf{e}_{3}^{p}\\ \mathbf{o}_{A}^{p}\times\mathbf{e}_{3}^{p}\end{bmatrix}\), \(\mathbf{\hat{s}}_{4}^{p}=\begin{bmatrix}\mathbf{0}\\ \mathbf{e}_{4}^{p}\end{bmatrix}\), \(\mathbf{\hat{s}}_{5}^{p}=\begin{bmatrix}\mathbf{e}_{5}^{p}\\ \mathbf{o}_{\text{C}}^{p}\times\mathbf{e}_{5}^{p}\end{bmatrix}\), \(\mathbf{\hat{s}}_{6}^{p}=\begin{bmatrix}\mathbf{e}_{6}^{p}\\ \mathbf{o}_{\text{C}}^{p}\times\mathbf{e}_{6}^{p}\end{bmatrix}\), \(\mathbf{p}=1,2,\cdots,6\), \(\mathbf{\omega}=\begin{bmatrix}\mathbf{\omega}^{1}\quad\mathbf{\omega}^{2}\quad\mathbf{\omega}^{3 }\quad\mathbf{\omega}^{4}\quad\mathbf{\omega}^{5}\quad\mathbf{\omega}^{6}\end{bmatrix}^{T}\) and where \(\mathbf{\omega}^{1}=\begin{bmatrix}\mathbf{o}_{1}^{1}&\mathbf{1}\mathbf{\omega}_{2}^{1}&\mathbf{2 }\mathbf{\omega}_{3}^{1}&\mathbf{\hat{s}}_{4}^{1}&\mathbf{4}\mathbf{\omega}_{5}^{1}&\mathbf{5}\bm {\omega}_{6}^{1}\end{bmatrix}^{T}\).  
To get the unit screws in Equation (26), the rotation matrix is applied to express \(\mathbf{e}_{n}^{p},\mathbf{r}_{n}^{p}\) of each joint in the local coordinate frame, \(p\) represents the \(pth\) chain, and \(n\) represents the \(nth\) joint in a chain.  
\[\left\{\begin{array}{l}\mathbf{e}_{n}^{p}=\mathbf{R}_{Z}^{p}\bigg{(}\prod\limits_{n =2}\mathbf{R}_{n}^{p}\bigg{)}\mathbf{e}_{n}^{p}(0)\\ \mathbf{r}_{n}^{p}=\mathbf{R}_{Z}^{p}\bigg{(}\mathbf{o}_{A}^{p}(0)+\mathbf{R}_{Z}^{p}\mathbf{R}_{3}^ {p}\mathbf{R}_{5}^{p}\mathbf{r}_{AC}^{p}\bigg{)}\end{array}\right.\cdot\begin{cases}(n =1,2,\cdots,6)\\ (p=1,2,\cdots,6)\end{cases} \tag{27}\]  
where \(\mathbf{R}_{Z}^{p}\) is the rotation matrix around z axis, which can be expressed as: \(\mathbf{R}_{Z}^{p}=\mathbf{R}_{z}\big{(}\frac{\pi}{6}(p-1)\big{)}\). \(\mathbf{R}_{n}^{p}\) is the rotation matrix from link \(L_{n}\) to link \(L_{n-1}\), when it rotates around \(x\)-axis, \(\mathbf{R}_{n}^{p}=\mathbf{R}_{x}\Big{(}{}_{n-1}\theta_{n}^{p}\Big{)}\), when it rotates around \(y\)-axis, \(\mathbf{R}_{n}^{p}=\mathbf{R}_{y}\Big{(}{}_{n-1}\theta_{n}^{p}\Big{)}\), and when it rotates around \(z\)-axis, \(\mathbf{R}_{n}^{p}=\mathbf{R}_{z}\Big{(}{}_{n-1}\theta_{n}^{p}\Big{)}\). The initial condition parameters are illustrated in Table 1, and \(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\where \(\mathbf{V}_{P}\) is the velocity screw of the reference point \(P\) on the moving platform, and it is given for the inverse kinematics. \({}_{C}\mathbf{V}_{P}^{p},(p=1,2\cdots 6)\) is the relative kinematics screw from joint \(j_{C}^{p},(p=1,2\cdots 6)\) to the reference point \(P\) on the moving platform. It can be expressed as:  
\[{}_{C}\mathbf{V}_{P}^{p}=\begin{bmatrix}0\\ _{0}\mathbf{\omega}_{C}^{p}\times_{0}\mathbf{r}_{P}^{p}\end{bmatrix} \tag{30}\]  
where \({}_{0}\mathbf{\omega}_{C}^{p},(p=1,2\cdots 6)\) is the absolute angular velocity of the moving platform, and \({}_{0}\mathbf{r}_{P}^{p},(p=1,2\cdots 6)\) is the absolute position vector of the reference point \(P\) on the moving platform.