# Kinematics and Statics of the Gough-Stewart Platform
## 3 Kinematics of a Multi-Rigid System
### Kinematics of Gough-Stewart Platform  
In this section, the inverse kinematics of the Gough-Stewart platform are computed. As depicted in Figure 4, the inverse kinematic equation of the Gough-Stewart platform can be established as:  
\[\textbf{5}_{E}=\textbf{S}_{ABC,i}\boldsymbol{\omega}_{ABC,i},(i=1,2,\cdots,6) \tag{16}\]  
where \(i\) represents the \(i\)-th subchain of the Gough-Stewart platform. For the first kinematic chain \(L_{ABC,1}\), the unit velocity screw equation of the moving platform can be expressed as:  
\[\textbf{5}_{E}=\textbf{S}_{ABC,1}\boldsymbol{\omega}_{ABC,1} \tag{17}\]  
where \(\textbf{S}_{ABC,1}= \begin{bmatrix}{}_{0}\textbf{5}_{1,1}&{}_{1}\textbf{5}_{2, 1}&{}_{2}\textbf{5}_{3,1}&{}_{3}\textbf{5}_{4,1}&{}_{4}\textbf{5}_{5,1}&{}_{5 }\textbf{5}_{6,1}\end{bmatrix} \) is the unit screw matrix of the first kinematic subchain, with velocity screws \({}_{0}\textbf{5}_{1,1}= \begin{bmatrix}{}_{0}\textbf{5}_{1,1}\\ {}_{0}\textbf{7}_{A_{1}}\times{}_{0}\textbf{5}_{1,1}\end{bmatrix} \), \({}_{1}\textbf{5}_{2,1}= \begin{bmatrix}{}_{0}\textbf{2}_{1,1}\\ {}_{0}\textbf{7}_{A_{1}}\times{}_{0}\textbf{5}_{2,1}\end{bmatrix} \), \({}_{2}\textbf{5}_{3,1}= \begin{bmatrix}{}_{0}\textbf{5}_{3,1}\\ {}_{0}\textbf{7}_{A_{1}}\times{}_{0}\textbf{5}_{3,1}\end{bmatrix} \), \({}_{3}\textbf{5}_{4,1}= \begin{bmatrix}{}_{0}\textbf{5}_{4,1}\\ {}_{0}\textbf{7}_{C_{1}}\times{}_{0}\textbf{5}_{1,1}\end{bmatrix} \), \({}_{5}\textbf{5}_{6,1}= \begin{bmatrix}{}_{0}\textbf{6}_{1,1}\\ {}_{0}\textbf{7}_{C_{1}}\times{}_{0}\textbf{5}_{1,1}\end{bmatrix} \), and \(\boldsymbol{\omega}_{ABC,1}= \begin{bmatrix}{}_{0}\textbf{5}_{1,1}&{}_{1}\textbf{ 5}_{2,1}&{}_{2}\textbf{5}_{3,1}&{}_{3}\textbf{5}_{4,1}&{}_{4}\textbf{5}_{5,1}&{}_ {5}\textbf{5}_{6,1}\end{bmatrix} ^{T}\). Equation (17) is the velocity screw equation of the kinematic chain \(L_{ABC,1}\) which can be applied to solve the relative angular velocities and relative linear velocities. Therefore, Equation (17) could be rewritten as:  
\[\textbf{S}_{ABC,1}= \begin{bmatrix}{}_{1,1}&{}_{2,1}&{}_{3,1}&{}_{0}\textbf{3 }_{\times 1}&{}_{0}\textbf{5}_{1,1}&{}_{6,1}\\ {}_{0}\textbf{7}_{A_{1}}\times{}_{0}\textbf{5}_{1,1}&{}_{0}\textbf{7}_{A_{1}} \times{}_{0}\textbf{5}_{2,1}&{}_{0}\textbf{7}_{A_{1}}\times{}_{0}\textbf{5}_{3, 1}&{}_{4,1}&{}_{0}\textbf{7}_{C_{1}}\times{}_{0}\textbf{5}_{1,1}&{}_{0}\textbf{7 }_{C_{1}}\times{}_{0}\textbf{5}_{6,1}\end{bmatrix}  \tag{18}\]  
Figure 4: Kinematics analysis of the Gough–Stewart platform: (**a**) kinematic modeling of Gough–Stewart platform; (**b**) kinematic expressions of a single chain. In order to obtain the unit screws in Equation (18), the rotation transformation matrix is applied to express \(\mathbf{e}_{j,i},\ \mathbf{r}_{j,i}\) of each joint in the local coordinate system; \(i\) represents the \(i\)-th chain and \(j\) represents the \(j\)-th joint in this chain. \[\left\{ \begin{array}{l}\mathbf{e}_{j,i}=\mathbf{R}_{Z,i}\left(\prod\limits_{j=2} \mathbf{R}_{j,i}\right)\mathbf{e}_{j,i}(0)\\ \mathbf{r}_{j,i}=\mathbf{R}_{Z,i}\left(\mathbf{o}^{r}\mathbf{r}_{A,i}(0)+\mathbf{R}_{2 }\mathbf{R}_{3}\mathbf{R}_{5}\mathbf{r}_{n\alpha}^{j}\right)\end{array} ,\left(i=1, 2,\cdots,6;j=1,2,\cdots,6\right)\right. \tag{19}\]  
where \(\mathbf{R}_{Z,i}\) is the rotation transformation matrix around the \(z\)-axis, and it can be expressed as \(\mathbf{R}_{Z,i}=\mathbf{R}_{z}((i-1)30^{\circ})\). \(\mathbf{R}_{j}\) is the rotation transformation matrix from link \(L_{i}\) to link \(L_{i-1}\); when it rotates around the \(x\)-axis, \(\mathbf{R}_{j}=\mathbf{R}_{x}\left({}_{j-1}\theta_{j,i}\right)\); when it rotates around \(y\)-axis, \(\mathbf{R}_{j}=\mathbf{R}_{y}\left({}_{j-1}\theta_{j,i}\right)\); and when it rotates around \(z\)-axis, \(\mathbf{R}_{j}=\mathbf{R}_{z}\left({}_{j-1}\theta_{j,i}\right)\). Suppose the velocity screw of the moving platform is given; the relative angular velocity of each kinematic chain \(L_{ABC,i}\), \((i=1,2,\cdots,6)\) could be calculated using Equation (17).