# Kinematics and Statics of the Gough-Stewart Platform
## 5 Numerical Experiment and Discussion  
In order to verify the static modeling method of the Stewart platform, a trajectory of the moving platform is prescribed for the inverse kinematics, which is shown in Figure 8. The Gough-Stewart platform has six DoFs consisting of three translational motions and three rotational motions. Given a translational path of a circular trajectory for the moving platform, the parametric equations of the trajectory in the absolute coordinate system are expressed as:  
\[\left\{ \begin{aligned} & x(t)=-10\cos(t)\\ & y(t)=-10\sin(t)\\ & z(t)=20\cos(t)\end{aligned} \right. \tag{32}\]  
where \(x(t)\), \(y(t)\), and \(z(t)\) are the three motion variables of the center point on the moving platform at time \(t\). From Equation (32), the unit velocity screw of the moving platform can be written as \({}_{0}\boldsymbol{\mathsf{S}}_{in}= \begin{bmatrix}0&0&1&10\sin(n\Delta t)&10\cos(n\Delta t)&0\end{bmatrix} ^{T}\), and the discrete conditions are shown in Table 1. The kinematics analysis of the Gough-Stewart platform is performed through programming the introduced algorithm in MATLAB 2018a. These programs are executed on Windows 11 computers equipped with AMD Ryzen 7 6800 HS with Radeon Graphics 3.20 GHz. The displacement parameters of each joint are obtained through the numerical calculation method. Therefore, using the numerical calculation method introduces differences between the real solution of a practical problem and the approximate solution obtained via numerical calculation. In the iteration simulation, the errors could be minimized through optimizing the step length of the discrete conditions. There are no big differences between the results when the interval of iteration is smaller than 0.001, and it only brings high computation costs. With the initial conditions of displacement \(\theta_{j,i}(0)\) illustrated in Table 2, the initial relative angular velocities \(\omega_{i}(1)\) of the \(i\)-th kinematic chain could be derived using Equation (20). Figure 9: The kinematics of each revolute pair through following the circular path: (**a**) angular displacement; (**b**) angular velocity. Figure 10 depicts the linear displacement (Figure 10a) and linear velocity (Figure 10b) of the linear pairs of each kinematic chain. All diagrams are plotted in MATLAB 2018a. \begin{table}
\begin{tabular}{c c c} \hline
**Discrete Condition** & **Parameters** & **Value** \\ \hline Total time of the path & \(t(s)\) & \(10\) \\ Interval of iteration & \(\Delta t\) & \(0.001\) \\ Steps of iteration & \(k\) & \(10000\) \\ \hline \end{tabular}
\end{table} 
Table 1: Discrete conditions of the simulation. Figure 8: The given circular trajectory of the moving platform and the external loads on the Gough–Stewart platform. Figure 9: The kinematics of each revolute pair through following the circular path: (**a**) angular displacement; (**b**) angular velocity. According to the inverse kinematics that are calculated in the above section, the kinematics of each joint in the absolute coordinate system could be calculated. Suppose there are an external force \(\mathbf{F}_{E}= \begin{bmatrix}0&0&10\end{bmatrix} (kN)\) and a torque \(\mathbf{M}_{E}= \begin{bmatrix}0&0&10\end{bmatrix} (kN\cdot m)\) exerted on the moving platform, which are shown in Figures 7 and 8. And the force screw of the moving platform can be expressed in screw coordinates:  
\[ \begin{bmatrix}\mathbf{F}_{E}(t)\\ \mathbf{M}_{E}(t)\end{bmatrix} = \begin{bmatrix}0&0&10&100\sin(k\Delta t)&-100\cos(k \Delta t)&-200\sin(k\Delta t)+10\end{bmatrix} ^{\mathbf{T}} \tag{33}\]  
In this paper, the mass and inertia properties of the Gough-Stewart platform are ignored. Based on the structure parameters and the external forces and torques added to the moving platform, the driving forces \(\mathbf{F}_{B,i}\) to hold the equilibrium of the mechanism could be calculated through carrying out the proposed statics analysis, and the results are plotted in Figure 11. It is clear that the driving forces are adjusted to maintain equilibrium during the motion process.