# Computational Dynamics of Multi-Rigid-Body System in Screw Coordinate
## 4 Kinematics and Dynamics of the Gough-Stewart Platform
### Dynamics Analysis of the Gough-Stewart Platform  
Inverse dynamics equations for the fixed leg \(L_{AB}^{p},(p=1,2\cdots 6)\)  
The fixed leg \(L_{AB}^{p},(p=1,2\cdots 6)\) consists of one spherical joint and one prismatic joint, which is shown in Figure 5b. At spherical joint \(j_{A}^{p},(p=1,2\cdots 6)\), there are three forces \(\mathbf{F}_{A}\) without torque. At prismatic joint \(j_{B}^{p},(p=1,2\cdots 6)\), there are three forces \(\mathbf{F}_{B}\) and three torques \(\mathbf{T}_{B}\). The dynamics equations for the fixed leg \(L_{AB}^{p},(p=1,2\cdots 6)\) can be derived:  
\[ \begin{bmatrix}_{A}\mathbf{r}_{D}\times F_{A}-_{D}\mathbf{r}_{B}\times F_{B}-\mathbf{T}_{ B}\\ \mathbf{F}_{A}-\mathbf{F}_{B}\end{bmatrix} = \begin{bmatrix}\mathbf{J}_{D}&0\\ 0&m_{D}\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{\beta}_{D}\\ \mathbf{a}_{D}\end{bmatrix} + \begin{bmatrix}\mathbf{\omega}_{D}\times\mathbf{J}_{D}\mathbf{ \omega}_{D}\\ m_{D}\mathbf{g}\end{bmatrix}  \tag{31}\]  
Inverse dynamics equations for the moving leg \(L_{BC}^{p},(p=1,2\cdots 6)\)  
The moving leg \(L_{BC}^{p},(p=1,2\cdots 6)\) consists of one universal joint and one prismatic joint which is shown in Figure 5c. At universal joint \(j_{C}^{p},(p=1,2\cdots 6)\), there are three forces \(\mathbf{F}_{C}\) and one torque \(\mathbf{T}_{C}\), the direction of the torque \(\mathbf{T}_{C}\) at joint \(j_{C}^{p},(p=1,2\cdots 6)\) is along the leg. At prismatic joint \(j_{B}^{p},(p=1,2\cdots 6)\), there are three forces \(\mathbf{F}_{B}\) and three torques \(\mathbf{T}_{B}\). The dynamics equations for the fixed leg \(L_{BC}^{p},(p=1,2\cdots 6)\) can be expressed by  
\[ \begin{bmatrix}_{B}\mathbf{r}_{E}\times\mathbf{F}_{B}-_{E}\mathbf{r}_{C}\times\mathbf{F}_{C}+ \mathbf{T}_{B}-\mathbf{T}_{C}\\ \mathbf{F}_{B}-\mathbf{F}_{C}\end{bmatrix} = \begin{bmatrix}\mathbf{J}_{E}&0\\ 0&m_{E}\mathbf{I}\end{bmatrix}\begin{bmatrix}\mathbf{\beta}_{E}\\ \mathbf{a}_{E}\end{bmatrix} + \begin{bmatrix}\mathbf{\omega}_{E}\times\mathbf{J}_{E}\mathbf{ \omega}_{E}\\ m_{E}\mathbf{g}\end{bmatrix}  \tag{32}\]  
Figure 5: Dynamic analysis of the Gough–Stewart platform: (**a**) dynamics of the first kinematic chain; (**b**) dynamics of the fixed leg \(L_{AB}^{p},(p=1,2\cdots 6)\); (**c**) dynamics of the moving leg \(L_{BC}^{p},(p=1,2\cdots 6)\). _Inverse dynamics equations for the moving platform \(L_{P}\)_  
The moving platform \(L_{P}\) consists of six universal joints \(j_{C}^{p}\), \((p=1,2\cdots 6)\), which are shown in Figure 5c. At the universal joint \(j_{C}^{p}\), \((p=1,2\cdots 6)\), there are three forces \(\mathbf{F}_{C}\) and one torque \(\mathbf{T}_{C}\). Suppose that there is an external force system (if any) acting on the moving platform, which consists of a force \(\mathbf{F}_{P}\) and a torque \(\mathbf{T}_{P}\) in the absolute coordinate frame. The dynamics equations for the moving platform \(L_{P}\) are expressed by  
\[\left[ \begin{matrix}\sum\limits_{p=1}^{6}\left(\mathbf{0}\mathbf{r}_{C}\times\mathbf{F}_{ C}\right)+\sum\limits_{p=1}^{6}\mathbf{T}_{C}\\ \sum\limits_{p=1}^{6}\mathbf{F}_{C}\end{matrix} \right]=\left[ \begin{matrix}\mathbf{I}_ {P}&0\\ 0&m_{P}\mathbf{I}\end{matrix} \right]\left[ \begin{matrix}\mathbf{\beta}_{P}\\ \mathbf{a}_{P}\end{matrix} \right]+\left[ \begin{matrix}\mathbf{\omega}_{P}\times\mathbf{J} _{P}\mathbf{\omega}_{P}\\ m_{P}\mathbf{g}\end{matrix} \right]+\left[ \begin{matrix}\mathbf{T}_{P}\\ \mathbf{F}_{P}\end{matrix} \right] \tag{33}\]  
_Inverse dynamics equations for Gough-Stewart platform_  
Suppose,  
\[\mathbf{W}^{\mathcal{S}}=\left[\mathbf{F}_{A}\quad\mathbf{F}_{B}\quad\mathbf{F}_{C}\quad\mathbf{T}_ {B}\quad\mathbf{T}_{C}\right] \tag{34}\]  
where \(\mathbf{F}_{A}=\left[\mathbf{F}_{A}^{1}\quad\mathbf{F}_{A}^{2}\quad\cdots\quad\bm  
Suppose,  
\[\mathbf{W}^{E}= \begin{bmatrix}\mathbf{T}_{P}\\ \mathbf{F}_{P}\end{bmatrix} = \begin{bmatrix}\mathbf{r}_{P}\times\mathbf{F}_{P}\\ \mathbf{F}_{P}\end{bmatrix} = \begin{bmatrix}0\mathbf{D}_{P}(4:6)\times\mathbf{F}_{P}\\ \mathbf{F}_{P}\end{bmatrix}