## 3 Magnetic Levitation Control System  
### System Modeling  
* **EM Model** According to previous studies [21; 22], the attractive force generated by the EM can be written as \[f_{n}=\frac{\mu_{0}N^{2}A}{4}\left(\frac{i_{n}}{c_{n}}\right)^{2}\] (1) where \(\mu_{0}=4\pi\times 10^{-7}\) H/m is the vacuum permeability, \(i\) is the applied current, and \(c\) is the airgap. The values of the number of turns \(N\) and pole area of EM \(A\) that are used are summarized in Table 2.
* **Dynamic Equation** The dynamic equation of the magnetic levitation wafer transfer system can be described by [PERSON]'s equation for translation and [PERSON]'s equation for rotation. [PERSON]'s equation represents the relationship between the force acting on the center of mass of a rigid body and its acceleration, while [PERSON]'s equation represents the relationship between the moments acting on the center of mass of a rigid body and the time derivative of the angular momentum. \[ \begin{array}{l}\sum\mathbf{F}=m\mathbf{a}\\ \sum\mathbf{M_{G}}=\mathbf{\hat{H}}_{G}\end{array} \] (2) The developed magnetic levitation wafer transfer system is mechanically symmetric about the \(x\)- and \(y\)-axes, so the coordinate system used is the principal axes of inertia, and the products of inertia can be neglected: \(I_{xy}=I_{xz}=I_{yz}=0\). Therefore, the angular momentum is expressed as follows: \(\mathbf{H_{G}}=I_{x}\omega_{x}i+I_{y}\omega_{y}j+I_{z}\omega_{z}\hat{k}\). By substituting the time derivative of the angular momentum and the attractive force generated by EMs into Equation (2), the dynamic equation of the magnetic levitation wafer transfer system can be summarized. \[ \begin{array}{l}m\dot{x}=f_{5}-f_{6}+f_{7}-f_{8}+F_{dx}\\ m\dot{z}=f_{1}+f_{2}+f_{3}+f_{4}+F_{dz}\\ I_{x}\dot{g}_{x}-(I_{y}-I_{z})\dot{\theta}_{y}\dot{\theta}_{z}=d(f_{1}+f_{2}-f_ {3}-f_{4})+M_{dx}\\ I_{y}\dot{\theta}_{y}-(I_{z}-I_{x})\dot{\theta}_{z}\dot{\theta}_{x}=w(f_{1}-f_ {2}+f_{3}-f_{4})+h(-f_{5}+f_{6}-f_{7}+f_{8})+M_{dy}\\ I_{z}\dot{\theta}_{z}-(I_{x}-I_{y})\dot{\theta}_{x}\dot{\theta}_{y}=d(-f_{5}+f_ {6}+f_{7}-f_{8})+M_{dz}\end{array} \] (3) The \(y\)-axis motion of the magnetic levitation wafer transfer system is controlled by an LSM, and the 5-DOF motion excluding the \(y\)-axis motion is precisely controlled by the magnetic levitation units attached to the four corners of the magnetic levitation wafer transfer system. To design a linear controller, the dynamic equation Equation (3) is linearized. The simulation results show that the nonlinear terms are relatively small compared to the linear terms (\(I_{x}\ddot{\theta}_{x}\gg(I_{y}-I_{z})\dot{\theta}_{y}\dot{\theta}_{z}\)). Thus, the linearized dynamic equation is sufficient \begin{table}
\begin{tabular}{c l c c} \hline \hline
**Parameters** & **Description** & \(x\)**-axis** & \(z\)**-axis** \\ \hline \(c_{0}\) & Nominal airgap & 1 mm & 1 mm \\ \(f_{0}\) & Nominal force & 0 N & 524 N \\ \(f_{Max}\) & Maximum force & 110 N & 1100 N \\ \(N\) & Number of turns & 266 turn & 396 turn \\ \(A\) & Pole area & 1440 mm\({}^{2}\) & 352 mm\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Major design parameters of the magnetic levitation units. for obtaining meaningful interpretation results. The linearized motion equation can be expressed in the following matrix form: \[\vec{\textbf{q}} =\textbf{M}^{-1}\textbf{B}\textbf{f}+\textbf{M}^{-1}\textbf{d}\] \[=\textbf{M}^{-1}\textbf{u}+\textbf{M}^{-1}\textbf{d}\] where \[\textbf{q} =\left[ \begin{array}{cccccccc}x&z&\theta_{x}&\theta_{y}&\theta_{ z}\end{array} \right]^{T}\] \[\textbf{f} =\left[ \begin{array}{cccccccc}f_{1}&f_{2}&f_{3}&f_{4}&f_{5}&f_{6 }&f_{7}&f_{8}\\ \end{array} \right]^{T}\] \[\textbf{d} =\left[ \begin{array}{cccccccc}F_{dx}&F_{dz}&M_{dx}&M_{dy}&M_{dz }\end{array} \right]^{T}\] \[\textbf{M} =\text{diag}\left( \begin{array}{cccccccc}m&m&I_{x}&I_{y}&I_{z} \end{array} \right)\] \[\textbf{B} =\left[ \begin{array}{cccccccccccc}1&1&1&1&1\\ d&d&-d&-d&&&&&\\ w&-w&w&-w&-h&h&-h&h\\ &&-d&d&d&-d\end{array} \right]\] \[\textbf{u} =\textbf{B}\,\textbf{f}=\left[