# Kinematics Model Optimization Algorithm for Six Degrees of Freedom Parallel Platform
## 2 Materials and Methods  
### Kinematic Modeling  
To study the kinematics of the parallel platform, it is necessary to establish a mathematical model of the parallel platform. Mathematically abstract the physical system of the parallel platform and use three-dimensional space vectors to describe each component. The six-degree-of-freedom platform studied in this paper is based on the Stewart-Gough platform, so it is referred to as Stewart-Gough in the following [20]. The Stewart-Gough parallel platform is mainly composed of three parts: the bottom static platform, six sets of parallel mechanical arms, and the top dynamic platform. The mechanical arm is connected to the dynamic and static platforms through the structure of universal joints. The six-degree-of-freedom attitude of the parallel platform in the working space refers to the \(xyz\) coordinates of the center of mass of the top moving platform in three-dimensional space. Moreover, the pitch angle (\(\alpha\)) around the \(x\)-axis, the roll angle (\(\beta\)) around the \(y\)-axis, and the yaw angle (\(\gamma\)) around the \(z\)-axis are three rotation angles, expressed as \([x,y,z,\alpha,\beta,\gamma]^{T}\) in vector. In this study, the RPY angle method was used to represent the rotational motion of the moving platform around the axis. The rotation operators \(R_{X}(\alpha)\), \(R_{Y}(\beta)\), and \(R_{Z}(\gamma)\) were used to represent the rotation of the moving platform by an angle of \(\alpha\) around the \(x\)-axis, the angle of \(\beta\) around the \(y\)-axis, and the angle of \(\gamma\) around the \(z\)-axis. The rotation operator of the parallel platform satisfies Equation (1) [21]:  
\[\left\{ \begin{aligned} R_{X}(\alpha)&=\begin{bmatrix} 1&0&0\\ 0&\cos\alpha&-\sin\alpha\\ 0&\sin\alpha&\cos\alpha\end{bmatrix}\\ R_{Y}(\beta)&=\begin{bmatrix}\cos\beta&0&\sin\beta\\ 0&1&0\\ -\sin\beta&0&\cos\beta\end{bmatrix}\\ R_{Z}(\gamma)&=\begin{bmatrix}\cos\gamma&-\sin\gamma&0\\ \sin\gamma&\cos\gamma&0\\ 0&0&1\end{bmatrix}\end{aligned} \right. \tag{1}\]  
When the moving platform rotates around multiple axes simultaneously in the workspace, matrix multiplication is performed in the order of \(R_{Z}(\gamma)\), \(R_{Y}(\beta)\), and \(R_{X}(\alpha)\)Constitute the rotation matrix \(R_{XYZ}(\alpha,\beta,\gamma)\). If \(\sin\) is abbreviated as s and \(\cos\) is abbreviated as \(c\), then we can obtain Equation (2):  
\[R_{XYZ}(\alpha,\beta,\gamma)= \begin{bmatrix}c\gamma c\beta&c\gamma s\beta s \alpha&-s\gamma c\alpha&c\gamma s\beta c\alpha+s\gamma s\alpha\\ s\gamma c\beta&s\gamma s\beta s\alpha+c\gamma c\alpha&s\gamma s\beta c\alpha-c \gamma s\alpha\\ -s\beta&c\beta s\alpha&c\beta c\alpha\end{bmatrix}  \tag{2}\]  
In order to study the kinematics relationship of the parallel platform, the mechanical structure of the parallel platform is abstracted mathematically, and the spatial geometric model is established. With the center of mass, \(O\), of the static platform at the bottom of the parallel platform as the origin, a space Cartesian coordinate system, \(Oxyz\), is established. With the center of mass, \(P\), of the top moving platform as the origin, a space rectangular coordinate system, \(Px^{\prime}y^{\prime}z^{\prime}\), is established. Simplify the mechanical structure of each mechanical arm of the parallel platform and the upper and lower platforms. Establish the parallel platform space geometric model in MATLAB, as shown in Figure 1.  
A1 to A6 in Figure 1 are the connection points between the bottom static platform and each parallel manipulator arm. B1 to B6 are the connection points between the top moving platform and each group of parallel mechanical arms. The six connection points B1 to B6 of the top moving platform and the mechanical arm are on the same circumscribed circle. The six connection points A1 to A6 of the bottom static platform are on the same circumscribed circle. In the initial zero state, the six robotic arms are of equal length, and the centers of the circumscribed circles of the moving platform and the static platform coincide.