# Research on Space Maglev Vibration Isolation Control System Modeling and Simulation
## 3 Maglev Vibration Isolation System Modeling and Simulation
### Maglev Mathematical Modeling  
The maglev system adopts a four-bar actuator layout, with a single actuator having a bidirectional force output function. The actuators are mounted in two directions, according \begin{table}
\begin{tabular}{c c c} \hline
**Wire Diameter** & **Rated Current** & **Current Density** \\ \hline
0.2 mm & 0.4 A & 12.73 A/mm\({}^{2}\) \\ \hline
0.3 mm & 0.8 A & 9.95 A/mm\({}^{2}\) \\ \hline
0.4 mm & 1.2 A & 9.55 A/mm\({}^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of enameled wire. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Wire Di-** & **Outer Diameter of** & **Coil** & **Number of** & **Rated** & **Ampere** \\
**ameter** & **Enameled Wire** & **Cross-Section** & **Turns** & **Current** & **Turn** \\ \hline
0.4 & 0.439 & 5 mm*10 mm & 238 & 1.2 & 285.6 \\ \hline \end{tabular}
\end{table} 
Table 3: Coil parameters. to the decoupling control principle of \"rotation control does not affect translation control, and translation control does not affect rotation control\" [16; 17; 18]. The two degrees of freedom (output force) of each actuator are indicated by the purple arrows in Figure 8.  
If the command control force is \(F_{cmd}\), and the command control torque is \(T_{cmd}\), then  
\[\left\{ \begin{aligned} F_{cmd}&=\begin{bmatrix}F_{x} &F_{y}&F_{z}\end{bmatrix}\\ T_{cmd}&=\begin{bmatrix}T_{x}&T_{y}&T_{z}\end{bmatrix}\end{aligned} \right. \tag{2}\]  
where \(Fi\) and \(Ti\) (\(i=x,y,z\)) are the three-axis control forces and torque [1]. One actuator has two directions of force and torque outputs; four actuators have eight vectors. Set the output force of four actuators as:  
\[F= \begin{bmatrix}A_{1x}&A_{1z}&A_{2x}&A_{2z}&A_{3y}&A_{3z}&A_{4y}&A_{4z} \end{bmatrix} ^{T} \tag{3}\]  
Set the projection of the position vectors of the four actuators as \(L\), relative to the centroid of the payload module in the mechanical coordinate system, represented by subscripts \(A_{1},A_{2},A_{3}\), and \(A_{4}\). Based on the layout of the four actuators, the two-dimensional force output direction, and the formula for torque \(T=F\,^{*}\,L\), it can be obtained that  
\[\left\{ \begin{aligned} F_{x}&=A_{1x}+A_{2x}\\ F_{y}&=A_{3y}+A_{4y}\\ F_{z}&=A_{1z}+A_{2z}+A_{3z}+A_{4z}\\ T_{x}&=-A_{1z}L_{A1}(y)+A_{2z}L_{A2}(y)-A_{3z}L_{A3}(y)+A_{4z}L_{A4}(y )+A_{3y}L_{A3}(z)+A_{4y}L_{A4}(z)\\ T_{y}&=-A_{1x}L_{A1}(z)-A_{2x}L_{A2}(z)+A_{1z}L_{A1}(x)-A_{2z}L_{A2}( x)-A_{3z}L_{A3}(x)+A_{4z}L_{A4}(x)\\ T_{z}&=A_{1x}L_{A1}(y)-A_{2x}L_{A2}(y)+A_{3y}L_{A3}(x)-A_{4y}L_{A4}(x )\\ \end{aligned} \right. \tag{4}\]  
According to Equations (3) and (4), rewrite in matrix form, with  
\[M\cdot F= \begin{bmatrix}F_{cmd}&T_{cmd}\end{bmatrix} ^{T} \tag{5}\]  
\(M\) is the allocation matrix, which distributes the six-dimensional command force and torque to four 8-way maglev propulsion. Figure 8: Maglev modeling. \[M= \begin{bmatrix}1&0&1&0&0&0&0&0\\ 0&0&0&0&1&0&1&0\\ 0&1&0&1&0&1&0&1\\ 0&-L_{A1}(y)&0&L_{A2}(y)&L_{A3}(z)&-L_{A3}(y)&L_{A4}(z)&L_{A4}(y)\\ -L_{A1}(z)&L_{A1}(x)&-L_{A2}(z)&-L_{A2}(x)&0&-L_{A3}(x)&0&L_{A4}(x)\\ L_{A1}(y)&0&-L_{A2}(y)&0&L_{A3}(x)&0&-L_{A4}(x)&0\end{bmatrix}  \tag{6}\]  
According to the pseudoinverse algorithm, the pseudoinverse solution can be obtained as follows, thereby obtaining 8-way command output forces:  
\[F=M^{T}\cdot\left(M\cdot M^{T}\right)^{-1}\cdot \begin{bmatrix}F_{cmd}&T_{cmd} \end{bmatrix} ^{T} \tag{7}\]  
The maglev system adopts a four-bar actuator layout, and each direction has a redundant design, such as \(A_{1x}\) is a redundancy of \(A_{2x}\). Based on the above deduction, when there are one or two faults of the actuator, the output force and torque are as follows (Table 4). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \(\mathbf{A_{1x}}\) & \(\mathbf{A_{1z}}\) & \(\mathbf{A_{2x}}\) & \(\mathbf{A_{2x}}\) & \(\mathbf{A_{3y}}\) & \(\mathbf{A_{3z  
Therefore, when there is only a one-way failure in the actuator, it does not affect the payload module's motion of 6-DOF (degrees of freedom).