# A Real-Time Numerical Decoupling Method for Multi-DoF Magnetic Levitation Rotary Table
## 2 Magnetic Levitation Rotary Table and Modeling Method
### Working Principle of Maglev Actuator
#### 2.2.2 Permanent Magnet Selection Law for Magnetic Force Computation  
Based on superposition theory, magnetic flux density \(\mathbf{B}\) in Equation (3) contains the results from all magnets in the rotor, where \(j\) is from 0 to 47. However, if all magnets are taken into account, the computation burden becomes huge, so real-time computation is hard to achieve. Because the magnetic field intensity produced by the permanent magnet at a certain point is inversely proportioned to the square of the distance between this point and the permanent magnet center, the magnets are not considered in the wrench model if they produce a magnetic field intensity less than 3% of the magnetic field from the nearest magnet to this point. Under this circumstance, the deviation angle of the current-carrying region, given as \(\theta_{e}\) and related to the rotary table is used as the standard for selecting the permanent magnets. The deviation angle of each long side of a coil is solved by  
\[\theta_{e}\left(i\right)=-{}^{s}\gamma_{r}+\mathbf{floor}\left(\frac{i}{2} \right)\cdot\frac{\pi}{2}-\Delta d\div R, \tag{10}\]  
where \(R\) is the radius given in Figure 1, \(\mathbf{floor}\left(\frac{i}{2}\right)\cdot\frac{\pi}{2}\) represents the rotation angles of each coil coordinate system \(\left\{c_{i}\right\}\) referring to the \(\left\{s\right\}\) coordinate system, and \(\Delta d\) is the distance between original point \({}^{r}o\) and the central line of each long side. In order to choose the suitable magnets in the magnetic field computation, Figure 2 is introduced for the analysis. As illustrated in Figure 2a, \(\Delta d\) is \(-{}^{s}y_{r}+1.5\cdot r_{c}+w_{c}\) if the right long side of **Coil5** is considered.  
Meanwhile, the intensity of the magnetic flux density produced by the red magnet in Figure 2a is explored to decide the reasonable magnet amount used in the wrench model. The testing points are located 2 mm below the bottom surface of the permanent magnet and distributed at the red circumference in this figure. Results are shown in Figure 2b, where the space between each testing point is the pitch angle \(\frac{\pi}{24}\). Observing the results, the magnetic intensity at the fourth point referring to the 0 degree point is less independent of the remanence direction of the permanent magnets. Therefore, the 7 nearest magnets to the current-carrying region, indicated by \(j-3,j-2,j-1,j,j+1,j+2,j+3\), are selected for force computation in this work. The permanent magnet index \(n\) is solved by Equation (11):  
\[j=\left\{\begin{array}{lcl}\mathbf{mod}\left(\theta_{e}\left(i\right),\frac {\pi}{24}\right)&\mathbf{if}&\mathbf{floor}\left(\theta_{e}\left(i\right), \frac{\pi}{24}\right)\leq\frac{\pi}{48}\\ \mathbf{mod}\left(\theta_{e}\left(i\right),\frac{\pi}{24}\right)+1&\mathbf{if} &\mathbf{floor}\left(\theta_{e}\left(i\right),\frac{\pi}{24}\right)>\frac{ \pi}{48}\end{array}\right., \tag{11}\]  
where \(\mathbf{mod}\)\(()\) and \(\mathbf{floor}\)\(()\) are the modulo function and round-down function, as mentioned before. With such a selecting law, the permanent magnets required to compute the force vector for a certain coil are decided due to Equation (11) after the POV \({}^{s}\mathbf{p}\) is obtained from the sensing system in the real prototype.