# Kinematics and Statics of the Gough-Stewart Platform
## 3 Kinematics of a Multi-Rigid System
### Kinematics of a Closed Kinematic Chain  
A single closed kinematic chain is illustrated in Figure 3. Based on the linear combination, the kinematics of the end joint \(J_{i+1}\) can be expressed as:  
\[\left\{\begin{array}{c}{}_{0}\boldsymbol{\omega}_{i+1}={}_{0}\boldsymbol{ \omega}_{1}+{}_{1}\boldsymbol{\omega}_{2}+{}_{2}\boldsymbol{\omega}_{3}+ \cdots+{}_{i-2}\boldsymbol{\omega}_{i-1}+{}_{i-1}\boldsymbol{\omega}_{i}+{}_ {i}\boldsymbol{\omega}_{i+1}\\ {}_{0}\boldsymbol{v}_{i+1}={}_{0}\boldsymbol{v}_{1}+{}_{1}\boldsymbol{v}_{2 }+{}_{2}\boldsymbol{v}_{3}+\cdots+{}_{i-2}\boldsymbol{v}_{i-1}+{}_{i-1} \boldsymbol{v}_{i}+{}_{i}\boldsymbol{v}_{i+1}\end{array}\right. \tag{11}\]  
The absolute angular velocity and absolute linear velocity of joint \(J_{i+1}\) on the expanded rigid body of link \(L_{i}\) that is superimposed with the origin of the coordinate system are 0. Therefore, Equation (11) can be rewritten as:  
\[\left\{\begin{array}{c}0={}_{0}\boldsymbol{\omega}_{1}+{}_{1}\boldsymbol{ \omega}_{2}+{}_{2}\boldsymbol{\omega}_{3}+\cdots+{}_{i-2}\boldsymbol{\omega}_ {i-1}+{}_{i-1}\boldsymbol{\omega}_{i}+{}_{i}\boldsymbol{\omega}_{i+1}\\ 0={}_{0}\boldsymbol{v}_{1}+{}_{1}\boldsymbol{v}_{2}+{}_{2}\boldsymbol{v}_{3}+ \cdots+{}_{i-2}\boldsymbol{v}_{i-1}+{}_{i-1}\boldsymbol{v}_{i}+{}_{i} \boldsymbol{v}_{i+1}\end{array}\right. \tag{12}\]  
With Equations (6) and (12), the forward kinematics of a single closed kinematic chain can be derived as:  
\[{}_{0}\textbf{ \Large$\boldsymbol{\xi}$}_{i+1}=\sum_{i=0}^{n}{}_{i}\textbf{ \Large$\boldsymbol{\xi}$}_{i+1}=0 \tag{13}\]  
When the kinematics of any link \(L_{n}\) in the closed serial kinematic chain shown in Figure 3 are given in the absolute coordinate system, that is to say, the screw matrix \({}_{0}\textbf{ \Large$\boldsymbol{\xi}$}_{n}\) of link \(L_{n}\) is known, the inverse kinematics of the single closed kinematic chain could be solved based on Equation (6):  
\[\begin{array}{c}{}_{0}\textbf{ \Large$\boldsymbol{\xi}$}_{n}={}_{0}\textbf{ \Large$\boldsymbol{\xi}$}_{1}+{}_{1}\textbf{ \Large$\boldsymbol{\xi}$}_{2}+\cdots+{}_{n-2}\textbf{ \Large$\boldsymbol{\xi}$}_{n-1}+{}_{n-1}\textbf{ \Large$\boldsymbol{\xi}$}_{n}\\ -{}_{0}\textbf{ \Large$\boldsymbol{\xi}$}_{n}={}_{n}\textbf{ \Large$\boldsymbol{\xi}$}_{n+1}+{}_{n+1}\textbf{ \Large$\boldsymbol{\xi}$}_{n+2}+\cdots+{}_{i-2}\textbf{ \Large$\boldsymbol{\xi}$}_{i-1}+{}_{i-1}\textbf{\Large$\boldsymbol{\xi}$}_{i} \end{array} \tag{14}\]  
where only \({}_{0}\textbf{ \Large$\boldsymbol{\xi}$}_{n}\) is known.  
Figure 3: Closed kinematic chain.  
When the kinematics of all joint except the link \(L_{n}\) are prescribed, the forward kinematics of the single closed kinematic chain could be obtained based on Equation (6) through  
\[\begin{array}{c}{}_{n-1}\textbf{5}_{n}={}_{0}\textbf{5}_{1}+{}_{1}\textbf{5}_{2 }+\cdots+{}_{n-2}\textbf{5}_{n-1}+{}_{n}\textbf{5}_{n+1}+\cdots+{}_{i-2}\textbf{5 }_{i-1}+{}_{i-1}\textbf{5}_{i}\\ {}_{0}\textbf{5}_{n}={}_{0}\textbf{5}_{1}+{}_{1}\textbf{5}_{2}+\cdots+{}_{n-2} \textbf{5}_{n-1}+{}_{n-1}\textbf{5}_{n}\end{array} \tag{15}\]  
where \({}_{n-1}\textbf{5}_{n}\) is the relative velocity screw, and \({}_{0}\textbf{5}_{n}\) is the absolute velocity screw.