# Kinematics Model Optimization Algorithm for Six Degrees of Freedom Parallel Platform
## 2 Materials and Methods
### Parallel Platform Jacobian Matrix  
The Jacobian matrix is of great significance in the parallel platform system, it shows the relationship between the speed change between the moving platform and each group of manipulators in the parallel platform system [22; 23]. Based on the space geometry model of the parallel platform, the Jacobian matrix of the parallel platform can be deduced. Note that the position vector of the center of mass, \(P\), of the moving platform relative to the center of mass, \(O\), of the static platform is \(\mathbf{p}\), and the velocity vector corresponding to the translation of the center of mass of the moving platform is \(\mathbf{v}_{p}\). Note that the angular velocity of the moving platform around the center of mass is \(\mathbf{w}\). Note that the position vector between the centroid \(O\) of the static platform and the connection point, \(A_{i}\), at the lower end of each robot arm is \(\mathbf{p}_{A_{i}}\). The angular velocity of each mechanical arm around point \(A_{i}\) is \(\mathbf{w}_{i}\). Note that the position vector between the centroid \(O\) of the static platform and the connection point \(B_{i}\) at the upper end of the manipulator arm is \(\mathbf{p}_{B_{i}}\). When the mechanical arm moves, the velocity vector corresponding to \(B_{i}\) is \(\mathbf{v}_{B_{i}}\). According to the spatial geometric model of the parallel platform shown in Figure 1, the length of each manipulator arm can be expressed as Equation (3):  
\[l_{i}\mathbf{n}_{i}=\mathbf{p}_{B_{i}}-\mathbf{p}_{A_{i}} \tag{3}\]  
By deriving both sides of Equation (3) with respect to time simultaneously, we can obtain Equation (4):  
\[\dot{l}_{i}\mathbf{n}_{i}+l_{i}(\mathbf{w}_{i}\times\mathbf{n}_{i})=\mathbf{v}_{B_{i}} \tag{4}\]  
Since the angular velocity, \(\mathbf{w}_{i}\), of the manipulator around \(A_{i}\) is always orthogonal to the unit direction vector, \(\mathbf{n}_{i}\), of the manipulator, multiplying both sides of Equation (4) by \(\mathbf{n}_{i}\) at the same time can eliminate the second half of the left side of the equation. We can obtain the relationship between the length-change speed of the mechanical arm and the speed (\(\mathbf{v}_{B_{i}}\)) at point \(B_{i}\), as shown in Equation (5):  
\[\dot{l}_{i}=\mathbf{v}_{B_{i}}^{T}\mathbf{n}_{i} \tag{5}\]  
According to the spatial geometric model in Figure 1, the motion velocity of point \(B_{i}\) on the moving platform can be expressed by the translation of the center of mass of the moving platform and the rotation around the center of mass, as shown in Equation (6):  
\[\mathbf{v}_{B_{i}}=\mathbf{v}_{p}+\mathbf{w}\times\mathbf{p}_{B_{i}} \tag{6}\]  
Substitute Equation (6) into Equation (5). The relationship between the length change speed of each mechanical arm and the center-of-mass velocity and angular velocity of the moving platform can be obtained, as shown in Equation (7):  
\[\dot{l}_{i}=\mathbf{v}_{p}+\mathbf{w}\times\mathbf{p}_{B_{i}}^{T}\mathbf{n}_{i} \tag{7}\]  
According to Equation (7), the same equations are listed for the six groups of manipulators and written in matrix form. The relationship expressions between the length change rate of each manipulator and the center-of-mass velocity and angular velocity of the moving platform can be obtained, as shown in Equation (8):  
\[\begin{pmatrix}i_{1}\\ \vdots\\ i_{6}\end{pmatrix}=\begin{pmatrix}\mathbf{n}_{1}^{T}&\begin{pmatrix}\mathbf{p}_{\mathbf{B}_{ 1}}\times\mathbf{n}_{1}\\ \vdots&\vdots\\ \mathbf{n}_{6}^{T}&\begin{pmatrix}\mathbf{p}_{\mathbf{B}_{6}}\times\mathbf{n}_{6}\end{pmatrix} ^{T}\end{pmatrix}\begin{pmatrix}\mathbf{v}_{p}\\ \mathbf{w}\end{pmatrix} \tag{8}\]  
Write Equation (8) as a simplified expression between the length change speed of each mechanical arm and the center-of-mass velocity and angular velocity of the moving platform, and Equation (9) can be obtained:  
\[\dot{\mathbf{L}}=J\dot{\mathbf{X}} \tag{9}\]  
In Equation (9), \(J\) is the Jacobian matrix between the moving platform and the manipulator.