# Kinematics and Statics of the Gough-Stewart Platform
## 4 Statics of the Gough-Stewart Platform in Plucker Coordinate  
### Statics of a Series Kinematic Chain  
The equilibrium of an ideal object without considering its size is stated by [PERSON]'s first law. Force and couple are two basic physical quantities of statics. In reality, the parameters of forces and torques, such as magnitude and acting position, are taken into account in the equilibrium of a rigid body. We therefore discuss the action of forces and couples on a rigid body together.  
Suppose that there is an external force \(\mathbf{F}_{n}\) exerted on rigid link \(L_{n}\) as shown in Figure 5. Therefore, a force \(\mathbf{F}_{i}\) and a torque \(\mathbf{M}_{i}\) should be applied to joint \(J_{i},\ (i=1,2,\cdots)\) to maintain the equilibrium of the chain.  
Figure 5: Statics of kinematic chain.  
Associating the force \(\textbf{F}_{i}\) with the couple of \({}_{0}\textbf{r}_{i}\times\textbf{F}_{i}\), where \({}_{0}\textbf{r}_{i}\) is the position vector from the origin to joint \(J_{i}\), the force screw can be expressed as:  
\[\textbf{\xi}_{i}^{W}=\begin{bmatrix}\textbf{F}_{i}\\ {}_{0}\textbf{r}_{i}\times\textbf{F}_{i}\end{bmatrix}=\begin{bmatrix}\textbf{ F}_{i}\\ {}_{0}\textbf{r}_{i}\end{bmatrix} \tag{21}\]  
In Equation (21), \(\textbf{F}_{i}\) is called the force vector of the force screw and \(\textbf{M}_{i}={}_{0}\textbf{r}_{i}\times\textbf{F}_{i}\) represents the moment of the force about the origin.  
Suppose \(\textbf{e}_{i}\) presents the unit direction vector of force \(\textbf{F}_{i}\), namely \(\|\textbf{e}_{i}\|=1\), where \(\|\cdot\|\) indicates the norm of vector \({}^{\prime}\cdot^{\prime}\). Equation (21) can be rewritten as:  
\[\textbf{\xi}_{i}^{W}=\textbf{F}_{i}\textbf{\xi}_{iu} \tag{22}\]  
with \(\textbf{\xi}_{iu}=\begin{bmatrix}\textbf{e}_{i}\\ {}_{0}\textbf{r}_{i}\times\textbf{e}_{i}\end{bmatrix}\).  
The equilibrium condition of the rigid object, whose external loads are all expressed in screw coordinates, should be that the resultant force screw must be zero:  
\[\sum_{i=1}^{n}\textbf{\xi}_{i}^{W}=0 \tag{23}\]  
When there is an external load exerted on the mechanism, which can be expressed as \(\textbf{\xi}_{n}^{W}\), Equation (23) is rewritten as:  
\[\textbf{S}^{W}\textbf{F}=\textbf{\xi}_{n}^{W} \tag{24}\]  
where \(\textbf{S}^{W}=\begin{bmatrix}\textbf{\xi}_{1u}&\textbf{\xi}_{2u}&\cdots& \textbf{\xi}_{iu}\end{bmatrix}\) is the unit force screw matrix composed of unit force screws of each joint, and the force vector \(\textbf{F}=\begin{bmatrix}F_{1}&F_{2}&\cdots&F_{i}\end{bmatrix}^{T}\) contains all force information of each joint. Therefore, the inverse statics of each joint \(J_{i}\) could be carried out through Equation (24):  
\[\left(\textbf{S}^{W}\right)^{T}\textbf{S}^{W}\textbf{F}=\left(\textbf{S}^{W} \right)^{T}\textbf{\xi}_{n}^{W} \tag{25}\]  
When \(\left|\left(\textbf{S}^{W}\right)^{T}\textbf{S}_{n}^{W}\right|=0\), the kinematic chain is a redundantly actuated mechanism, or it is in a singular configuration. Otherwise, Equation (25) could be further derived as:  
\[\textbf{F}=\left[\left(\textbf{S}^{W}\right)^{T}\textbf{S}_{n}^{W}\right]^{-1 }\left(\textbf{S}^{W}\right)^{T}\textbf{\xi}_{n}^{W} \tag{26}\]  
Based on the definition of velocity screw and force screw, the kinematic and static modeling could be associated with each other through unit screws. Through Equations (6), (7) and (10), expressed in Section 3, the unit screw \(\textbf{\xi}_{iu}\) of each joint in the absolute coordinate system, which consists of position \({}_{0}\textbf{r}_{i}\) and posture \(\textbf{e}_{i}\), could be obtained via an iterative procedure. And the solved kinematic parameters, which are expressed in screw form, can be directly employed in establishing the static equations based on Equation (26). The relationship and unified procedure of the kinematic and static modeling of a kinematic chain are given in Figure 6.