## 3 Implementation of Raising Power Loss Equalizing Degree (PLED) by Convex Optimization Commutation
### Verification Schem and Measurement Results  
As the motion of MLPM under investigation has not been controlled in real time, the static measurement scheme is adopted to verify the proposed convex quadratic optimization commutation method. This scheme has five steps: 1) Using the adjustable mechanism, ensure the gap between two stages to be \(z\) = 1 mm; 2) using the _X-Y_ plane motion mechanism, adjust and ensure the relative position of two stages in the horizontal direction; 3) using the Matlab routine quadprog on the host computer, calculate the coil currents that can generate the desired electromagnetic forces/torques in present position; 4) using the D/A converter and power amplifiers, give the just-calculated currents to the coils  
Figure 11: Photograph of comprehensive measurement platform (system). Figure 12: Configuration of comprehensive measurement platform (system). and ensure the coils are actuated; 5) using the multi-dimension force/torque sensor, the generated electromagnetic forces and torques are measured in present position and under present coil currents. Sampling frequency of measurement data of the sensor is at 250 Hz and the measurement time of each point is 2 seconds at least. Figure 13 shows the measured results of one point electromagnetic forces and torques. It can be observed that the force precision is about \(\pm\) 0.5 N and torque precision is less than \(\pm\)60 N\(\cdot\)mm. Table 2 shows the coil currents solved by the convex quadratic optimization commutation method when the desired electromagnetic forces and torques are \(W_{\text{d}}\) = (\(F_{\text{xd}}\), \(F_{\text{yd}}\), \(F_{\text{zd}}\), \(T_{\text{xd}}\), \(T_{\text{yd}}\), \(T_{\text{zd}}\))T = (10N, 0, 10N, 0, 0, 0)T, supposing the air gap \(z\) = 1 mm and the coil array stage, respectively, locates in 17 positions in the \(x\)-direction, ranging from \(-\)10.61 mm to 10.61 mm (\(y\) = 10.61 mm). These coil currents are solved on the host computer by the Matlab routine quadprog, where the current boundary constraints are given as I\({}_{\text{m}}\) = 2.0 A. Figure 14 shows the measured results of electromagnetic forces and torques of 17 different points in the \(x\)-direction, ranging from \(-\)10.61 mm to 10.61 mm (\(y\) = 10.61 mm) when the coils are actuated according to the currents as shown in Table 2. It can be observed from Figure 14 that the desired electromagnetic forces and torques are generated when the coils are actuated by the currents calculated by the proposed convex quadratic optimization commutation algorithm. The force ripples are within \(\pm\) 0.5 N and the torque ripples are about \(\pm\) 50 N\(\cdot\)mm. These ripples result from measurement errors of  
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Position**} & \multicolumn{1}{c}{**\(-\)10.608**} & \multicolumn{1}{c}{**\(-\)9.202**} & \multicolumn{1}{c}{**\(-\)7.956**} & \multicolumn{1}{c}{**\(-\)6.630**} & \multicolumn{1}{c}{**\(-\)5.304**} & \multicolumn{1}{c}{**\(-\)3.978**} & \multicolumn{1}{c}{**\(-\)2.652**} & \multicolumn{1}{c}{**\(-\)1.326**} & \multicolumn{1}{c}{**\(-\)0.000**} & \multicolumn{1}{c}{**\(\cdot\)1.326**} & \multicolumn{1}{c}{**\(\cdot\)2.652**} & \multicolumn{1}{c}{**\(\cdot\)3.978**} & \multicolumn{1}{c}{**\(\cdot\)5.304**} & \multicolumn{1}{c}{**\(\cdot\)6.830**} & \multicolumn{1}{c}{**\(\cdot\)7.956**} & \multicolumn{1}{c}{**\(\cdot\)9.282**} & \multicolumn  
the multi-dimension force/torque sensor, the current errors, and the position errors between the coil array and the permanent magnet array. More measurements like the ones above are taken in other positions, wherein the coil array stage is located in different \(y\)-direction positions relative to the permanent magnet array stage at the \(X\)-\(Y\) plane. These measured results indicate that the proposed convex quadratic optimization commutation method is feasible and effective.