## 3 Magnetic Levitation Control System
### System Modeling
\begin{array}{cccccccc}u_{x}&u_{ z}&u_{\theta x}&u_{\theta y}&u_{\theta z}\end{array} \right]^{T}.\] The **q** matrix represents the movements of the 5-DOF of the magnetic levitation wafer transfer system, the **f** matrix represents the attractive forces generated by the EMs, the **d** matrix represents the disturbances, the **M** matrix is the inertia matrix, the **B** matrix represents the geometric installation information of the EMs, and the **u** matrix is the decoupled control input that is used to achieve a desired output. * **Kinematics** The 5-DOF motion of the magnetic levitation wafer transfer system is measured by the gap sensors installed at the four corners. To control the system, a process of reconstructing the 5-DOF motion based on the measured sensor signals is carried out. The motion of the electromagnet module 1 (EM1) can be expressed as \(\vec{\textbf{r}}_{\text{EM1}}\) based on the inertial reference coordinate. And, it can also be expressed as the sum of the center of gravity motion \(\vec{\textbf{r}}_{\text{CG}}\) and the relative motion with respect to center of gravity \(\vec{\textbf{\theta}}\times\vec{\textbf{r}}_{\text{EM1/CG}}\)[23]. The relationship between the 5-DOF motion of the magnetic levitation wafer transfer system and the gap sensor measurements installed at the four corners is determined by the following kinematics. \[\vec{\textbf{r}}_{\text{EM1}}=\vec{\textbf{r}}_{\text{CG}}+\vec{\textbf{\theta }}\times\vec{\textbf{r}}_{\text{EM1/CG}}\] (5) For example, the relationship between the 5-DOF motion of the magnetic levitation wafer transfer system and the gap sensor measurements in EM1 is \[ \begin{array}{ll}\vec{\textbf{r}}_{\text{EM1}}&=(x\hat{i}+y_{j}^{\dagger}+z \hat{k})+(\theta_{x}\hat{i}+\theta_{y}\hat{j}+\theta_{z}\hat{k})\times(-w\hat{i }+d\hat{j}-h\hat{k})\\ &=(x-h\theta_{y}-d\theta_{z})\hat{j}+(y+h\theta_{x}-w\theta_{z})\hat{j}+(z+d \theta_{x}+w\theta_{y})\hat{k}\end{array} \] (6) As shown in Figure 3, the \(z\)-axis motion measured by Gap Sensor 1 in EM1 can be expressed as follows: \(c_{1}=c_{1}-c_{0z}=z+d\theta_{x}+w\theta_{y}\). Similarly, the \(x\)-axis motion measured by Gap Sensor 5 in EM1 can be expressed as follows: \(c_{5}=c_{5}-c_{0x}=-x+h\theta_{y}+d\theta_{z}\). By repeating the same process for EM2, EM3, and EM4, the relationship between the 5-DOF motion of the magnetic levitation wafer transfer system and the gap sensor measurements installed at the four corners can be obtained as follows: \[\textbf{\Delta c}=\textbf{Tq}\] (7)where \[\mathbf{\Delta}\mathbf{c}=\left[ \begin{array}{cccccc}\Delta c_{1}&\Delta c_{2}&\Delta c _{3}&\Delta c_{4}&\Delta c_{5}&\Delta c_{6}&\Delta c_{7}&\Delta c_{8}\end{array}  \right]^{T}\] \[\mathbf{T}=\left[ \begin{array}{cccccc}1&d&w&\\ 1&d&-w&\\ 1&-d&w&\\ 1&-d&-w&\\ -1&&h&d\\ 1&-h&-d\\ -1&&h&-d\\ 1&-h&d\end{array} \right]\] The \(\mathbf{T}\) matrix represents the geometric installation information of the gap sensors. Conversely, the 5-DOF motion of the magnetic levitation wafer transfer system can be reconstructed from the measured gap sensor signals using the pseudo inverse. \[\mathbf{q}=\mathbf{T}^{+}\mathbf{\Delta}\mathbf{c}\] (8)
* **Block diagram** Figure 5 shows the block diagram of the feedback control system of the magnetic levitation wafer transfer system. The input of the dynamic equation is the decoupled control input \(\mathbf{u}\) and disturbance \(\mathbf{d}\), and the output is the 5-DOF motion \(\mathbf{q}\). The 5-DOF motion of the magnetic levitation wafer transfer system is measured by the gap sensors \(\mathbf{\Delta}\mathbf{c}\) installed at the four corners, and the sensor noise is included in the measurement process. The measured sensor signal is reconstructed into the 5-DOF motion, and the difference \(\mathbf{e}\) between the reference and the measured 5-DOF motion is calculated and utilized as the input of the feedback controller. The output of the feedback controller is the required force and momentum \(\mathbf{u_{ref}}\) that are used to achieve the desired 5-DOF motion.