## 3 Implementation of Raising Power Loss Equalizing Degree (PLED) by Convex Optimization Commutation
### Implementation of Raising PLED by Convex Quadratic Optimization Commutation  
The commutated coil currents depend upon the desired electromagnetic forces and torques determined by the motion profile of MLPM. A most likely motion profile is that the coil array stage is levitated above the permanent magnet array stage and moves straight from one point to another in the horizontal direction. Supposing the coil array stage is levitated above the permanent magnet array stage at the air gap of 1 mm (the working air gap of the MLPM under investigation), and moves at the acceleration of 1g in \(x\)-direction ranging from \(-10.61\) mm to \(10.61\) mm (\(y=10.61\) mm), the desired electromagnetic force/torque vector \(\mathbf{W}_{\text{d}}\) can be calculated as:  
\[\mathbf{W}_{\text{d}}=\left(F_{\text{xd}},F_{\text{yd}},F_{\text{zd}},T_{\text{xd }},T_{\text{yd}},T_{\text{zd}}\right)^{T}=\left(10\text{N},\ 0,\ 10\text{N},\ 0,\ 0,\ 0\right)^{\text{T}} \tag{25}\]  
While the coil currents are calculated according to Equation (9), i.e., the coils are commutated according to the optimization commutation as shown in Equation (24), without any boundary constraints, the commutated currents of four forcers are as shown in Figure 4. From this figure, it can be observed that the commutated currents are seriously imbalanced. The current amplitudes of forcer I and forcer III are relatively higher, the current amplitudes of forcer II and forcer IV are relatively lower, and the maximum and minimum amplitudes of commutated currents are 3.0 A and 0.18 A, respectively. Due to these unbalanced current amplitudes, the power loss distribution among the four forcers as well as among all coils will be seriously imbalanced. While the coils are commutated by the optimization commutation with boundary constraint I\({}_{\text{m}}=2.0\) A, the commutated currents of four forcers are as shown in Figure 5. It can be observed from this figure that the maximum current amplitudes of four forcers are limited within 2.0 A and all commutated currents of the four forcers are inclined towards a more equalizing trend. Figure 4: Coil currents of (**a**) Forcer I, (**b**) Forcer II, (**c**) Forcer III, and (**d**) Forcer IV by optimization commutation without boundary constraint I\({}_{\text{m}}\). When the maximum current amplitude is limited by the optimization commutation, the maximum local power loss is reduced and thus, the PLED is raised. Figure 6 shows the PLED curves when commutations are optimized without boundary constraint \(\rm I_{m}\) and with boundary constraint \(\rm I_{m}=2.6\) A, 2.4 A, 2.2 A, 2.0 A, respectively. It can be seen from Figure 6 that PLED is raised in whole \(x\)-direction ranging from -10.61 mm to 10.61 mm, along with decrease of boundary constraint \(\rm I_{m}\). When the commutations are optimized without boundary constraint \(\rm I_{m}\) and with boundary constraint \(\rm I_{m}=2.6\) A, 2.4 A, 2.2 A, 2.0 A, respectively, the local power loss distributions in position \(x=-6.896\) mm and \(x=-4.774\) mm, which are the maximum and minimum PLED positions, are as shown in Figure 7, respectively. It can be seen from Figure 7 that the maximum local power loss is gradually  
Figure 5: Coil currents of (**a**) Forcer I, (**b**) Forcer II, (**c**) Forcer III, and (**d**) Forcer IV by optimization commutation with boundary constraint \(\rm I_{m}=2.0\) A.  
Figure 6: Power loss equalizing degree when the commutations are optimized without boundary constraint \(\rm I_{m}\), with boundary constraint \(\rm I_{m}=2.6\) A, 2.4 A, 2.2 A, and 2.0 A, respectively. reduced and the local power loss distribution becomes more and more uniform, along with decrease of boundary constraint \(\rm I_{m}\). When MPLM runs in real time, all the six-DOF movements of the mover have to be controlled completely. Therefore, efforts taken to control the six-DOF movements should be taken into account.