A Novel 2-DOF Lorentz Force Actuator for the Modular Magnetic Suspension Platform

The modular magnetic suspension platform depends on multi degree of freedom of Lorentz force actuators for large bearing capacity, high precision positioning and structure miniaturization. To achieve the integration of vertical driving force and horizontal driving force, a novel 2- (two degrees-of-freedom) DOF Lorentz force actuator is developed by designing the pose of the windings and permanent magnets (PMs). The structure and the working principle are introduced. The electromagnetic force mathematical model is established by the equivalent magnetic circuit method to analyze the coupling of magnetic flux. The distribution characteristics of magnetic flux density are analyzed by the finite-element method (FEM). It is found that the coupling of the magnetic flux and the large magnetic field gradient severely reduce the uniformity of the air-gap magnetic field. The electromagnetic force characteristic is investigated by FEM and measurement experiments. The difference between FEM and experiment results is within 10%. The reasons of driving force fluctuation are explained based on the distribution of air-gap magnetic field. The actuator performance are compared under the sliding mode control algorithm and PID control algorithm and the positioning accuracy is 20 μm and 15 μm respectively. Compared with the similar configuration, the motion range and force coefficient of the Lorentz force actuator in this paper are larger. It has a certain guiding significance on the structure design of the multi degree of freed Lorentz force actuator.


Introduction
The multi-degree-of-freedom positioning platforms are widely used in aerospace and industrial manufacturing, such as optical imaging systems, lithographic processing of semiconductors and precision machining [1][2][3][4][5]. But there are some disadvantages of the friction, wear creep and contact fatigue in mechanical contact platforms, which limits the improvement of positioning accuracy [6,7]. The magnetic suspension platform with electromagnetic actuators can realize the contactless suspension and overcome disadvantages of mechanical contact platforms [8,9]. According to the driving distance, the magnetically levitated actuators can be divided into the long-stroke actuators [10][11][12][13] and the short-stroke actuators [14,15]. The translational motion range of the former in X direction and Y direction is large but the rotation angle is small. The translational motion range of the latter in X direction and Y direction is small but the rotation angle is large, which is more suitable for larger angle attitude adjustment of high precision equipment on spacecraft. Translation along X-axis direction Horizontal driving forces produced by actuators 1and 3 The structure of the modular magnetic suspension platform is shown in Figure 2. Four 2-DOF Lorentz force actuators are distributed orthogonal to each other. The mapping relationship between the motion direction of the platform and the driving force of the actuators is shown in Table 1. With the distance between the two actuators in the symmetrical position adjusted, the bearing capacity and motion range of the platform will be changed accordingly. So the performance of the modular magnetic suspension platform can be improved for varies applications.
The performance of the modular magnetic suspension platform depends on the 2-DOF Lorentz force actuator. It is necessary to study its characteristics of air-gap magnetic field and electromagnetic force.   Table 1. The relationship between motion direction of the platform and driving forces of the actuators.

Movement of the Platform Driving Force
Translation along X-axis direction Horizontal driving forces produced by actuators 1and 3 Translation along Y-axis direction Horizontal driving forces produced by actuators 2 and 4 Translation motion in Z-axis direction Vertical driving forces produced by actuators 1, 2, 3 and 4 Deflection around X-axis Vertical driving forces produced by actuators 1 and 3 Deflection around Y-axis Vertical driving forces produced by actuators 2 and 4 Rotate around Z-axis Horizontal driving forces produced by actuators 1, 2, 3 and 4

Analysis of Magnetic Circuit
The equivalent magnetic circuit method is used to calculate magnetic field of the air-gap. The U-shaped yoke around the PMs contributes to reduction of the flux leakage, so the flux leakage is ignored for calculation convenience. Because the constant magnetic field is provided by PMs and the thickness of the yoke is large, the saturation phenomenon is ignored. The corresponding equivalent magnet circuit of the 2-DOF Lorentz force actuator can be obtained as shown in Figure 3. Circuit 1 is upper flux path produced by the vertical PMs. Circuit 2 is coupling flux path produced by vertical PMs and left horizontal PMs. Circuit 3 is left lower flux path produced by left horizontal PMs. Circuit 4 is flux path produced by vertical PMs and right horizontal PMs. Circuit 5 is right lower flux path produced by right horizontal PMs. F p1 and F p2 are magnetomotive force of vertical PMs and horizontal PMs, respectively. R 1 and R 4 are respectively internal resistance of PMs and horizontal PMs. R 2 and R 5 are respectively magnetic resistance of air-gap in the upper and the lower flux. R 3 is the resistance of partition region where the upper flux and the lower flux are coupling. Φ 1 -Φ 5 are the magnetic flux corresponding to circuits 1 -5 , respectively.

129
The magnetic circuit equation can be obtained by Kirchhoff law, as described by:  The magnetic circuit equation can be obtained by Kirchhoff law, as described by: Sensors 2020, 20, 4365 5 of 17 Circuits 2 and 4 , 3 and 5 are symmetrical respectively and there are Φ 2 = Φ 4 , Φ 3 = Φ 5 . Equation (1) can be simplified as: F p1 and F p2 can be expressed as: where H c is the permanent magnetic field intensity and h i is the thickness of the PMs in the magnetization. The reluctance R 1 -R 5 can be expressed as: where l i is length of magnetic path at R i region. µ 0 is the permeability of vacuum. µ i is the relative permeability. s i is section of magnetic path at R i region. Substituting Equation (3) and Equation (4) into Equation (2), the magnetic flux density B can be calculated as: where Φ 1 , Φ 2 and Φ 3 are magnetic flux through R 1 , R 2 and R 3 , respectively. B 1 , B 2 and B 3 are magnetic flux density at R 1 , R 2 and R 3 , respectively. A 1 , A 2 and A 3 are effective areas corresponding to Φ 1 , Φ 2 and Φ 3.

Calculation of Electromagnetic Force
The effective length of the vertical coil and the horizontal coil do not change when the mover deflects around the x-axis. When the mover deflects α around the y-axis, the effective length of the horizontal coil remains unchanged, while the effective length of the vertical coil increases. When the mover deflects β around the z-axis, the effective length of the vertical coil remains unchanged, while the effective length of the horizontal coil increases. Horizontal driving force F y and vertical driving force F z can be written as: where N 1 and N 2 are numbers of horizontal windings and vertical windings, respectively . I y and I z are control current in the horizontal windings and vertical windings, respectively . L 1 and L 2 are the effective length of the initial state of the horizontal windings and vertical windings in the magnetic field, respectively.
It can be seen from Equation (6) that both horizontal electromagnetic force F y and vertical electromagnetic force F z are affected by the reluctance R 3 where the upper flux and the lower flux are coupling. Horizontal electromagnetic force F y is affected by both the upper air-gap reluctance R 2 and the lower air-gap reluctance R 5 . The vertical electromagnetic force F z is mainly affected by the lower air-gap reluctance R 5 . So there are obvious coupling between the upper flux and the lower flux in the 2-DOF Lorentz force actuator.

Mathematical Model Analysis of Distribution Characteristics of Magnetic Flux Density
The performance of electromagnetic force of the 2-DOF Lorentz force actuator mainly depends on distribution of the air-gap magnetic field. So the magnetic field characteristics of the actuator are analyzed in this section. Combined with the performance requirements and overall size constraints of the magnetic suspension platform in practical applications, the nonlinear model of the Lorentz force actuator size optimization was calculated by the Optimization Toolbox in MATLAB to reduce weight and energy consumption. The optimized parameters of the Lorentz force actuator are listed in Table 2. The mass of the actuator is 2.59 kg, the translation stroke is ±5 mm, the overall dimension is 117 mm × 106 mm × 118 mm and the deflection angle is ±10 • . The PM material is NdFeB. Magnetic energy product (BH product) is about 358 KJ/m 3 . Residual magnetism is about 1.36 T. Coercive force is about 1026 KA/m and working temperature can reach 120 • C. The material of the yoke is 1J22 with high saturation magnetic induction (about 0.245 T). The flux nephogram at the central section is shown in Figure 4. To reduce magnetic flux leakage and increase magnetic field of air-gap, magnetic flux of the upper PMs and the lower PMs form a closed magnetic circuit through the yoke, respectively. However, there is obvious flux diffusion at the region near the yoke marked with black ellipses in Figure 4, which makes the magnetic flied inhomogeneous.   In the vertical winding region 1 , the magnetic flux density is large in the middle of the air-gap and decreases along both ends in the Y direction. The magnetic flux density is small in the middle of the air-gap and increases along both sides in the X direction, as shown in Figure 5a. Because the vertical yoke leads to a large gradient of magnetic flux intensity in the upper end of the vertical winding, the magnetic flux density near the yoke area is lower than that at the symmetric position. The magnetic permeability of the yoke is greater than that of the air, the magnetic flux generated by the upper edge of the vertical PMs pass through the yoke preferentially. Due to the similar configuration of magnetic flux circuit, the magnetic flux density distribution at the left horizontal winding region 2 is similar to that of vertical winding region 1 , as shown in Figure 5b. The magnetic flux density decreases rapidly near the yoke, resulting in a large difference between the maximum value of 0.372 T and minimum value of 0.249 T. The average of the magnetic flux density is 0.332 T.
It can be seen from Equations (6) and (4)  distribution. The coupling of upper flux and lower flux is strong in the region near the yoke and is weak in the region near the air-gap. So the yoke is helpful to reducing magnetic flux leakage and increasing air-gap magnetic field strength. But the magnetic field gradient is large and the magnetic density uniformity is poor at the region near the yoke.
The Line L 1 and the Line L 2 are respectively on the maximum magnetic density region of the vertical yoke and the horizontal yoke in Figure 4 and their magnetic flux density curves versus the excited current are plotted in Figure 6. The magnetic flux density of the vertical yoke increases linearly with the increase of excited current. That's because the coil wound on the vertical yoke produces a large electromagnetic field and the permanent magnetic field produced by the PM is superimposed in the vertical yoke. When the excited current is 3 A, the maximum flux density in the vertical yoke is 1.694 T, which is less than 2.45 T of saturation flux density. Because the number of turns of the horizontal coil is small and it is not around the yoke, the electromagnetic field generated by the coil is low and the maximum magnetic density in the horizontal yoke is 1.598 T. The maximum magnetic flux density in the vertical yoke and the horizontal yoke is not saturated and the magnetic flux density fluctuation is 3.4% and 0.71%, respectively. The magnetic field distribution in the critical region is analyzed above. The driving 232 performance of the actuator is directly reflected in the characteristics of the electromagnetic force.

233
The driving forces are analyzed by 3D FEM and experimental measurement.

234
It is difficult to directly measure the moment of the actuator, so FEM is used to analyze the 235 driving force when the mover deflects around the x-axis, y-axis and z-axis, respectively. As shown in 236 Fig. 7, the vertical driving force is larger than the horizontal driving force, because magnetic flux 237 density in the vertical winding region is larger than that in the horizontal winding region and turns 238 of the vertical winding are more than that of the horizontal winding. As shown in Fig. 7(a), the 239 driving forces increase when the mover deflects around the x-axis. That is because the windings

Characteristics of Electromagnetic Force
The magnetic field distribution in the critical region is analyzed above. The driving performance of the actuator is directly reflected in the characteristics of the electromagnetic force. The driving forces are analyzed by 3D FEM and experimental measurement.
It is difficult to directly measure the moment of the actuator, so FEM is used to analyze the driving force when the mover deflects around the x-axis, y-axis and z-axis, respectively. As shown in Figure 7, the vertical driving force is larger than the horizontal driving force, because magnetic flux density in the vertical winding region is larger than that in the horizontal winding region and turns of the vertical winding are more than that of the horizontal winding. As shown in Figure 7a, the driving forces increase when the mover deflects around the x-axis. That is because the windings approach the surface of the pole of PMs and the magnetic flux density increases, as shown in Figure 5a. The change rates of the vertical driving force and the horizontal driving force are 1.5% and 4.5%, respectively.

249
As shown in Fig. 7(b), the horizontal driving force increases gradually when the mover deflects

255
As shown in Fig. 7(c), the horizontal driving force rapidly reduces when the mover deflects 256 around the z-axis, because the horizontal winding is close to the surface of yoke and magnetic flux 257 density reduce, as shown in Fig. 5(b). The vertical driving force increases, because the vertical 258 winding is close to the magnetic pole and magnetic flux density increase, as shown in Fig. 5(a). The

264
Driving force variation with control current and displacement are studied by FEM and 265 experimental measurements, respectively. To reduce the manufacturing error and assembly error, 266 differential measurement method is adopted in the experiment, as shown in Fig. 8. To compensate 267 the weight of the mover, the actuator is suspended through the thread. The three-axis displacement 268 table is adjusted so that the winding is at the zero position, as shown in Fig. 8(a). The driving force in 269 vertical direction is measured as shown in Fig. 8(b). As shown in Figure 7b, the horizontal driving force increases gradually when the mover deflects around the y-axis, because the horizontal winding is close to the surface of the magnetic pole and magnetic flux density increases. The vertical driving force gradually reduces, because the effective length of the vertical coils is just a small amount of increase and the magnetic flux density rapidly reduces. The change rates of the vertical driving force and the horizontal driving force are 4.1% and 2.3%, respectively.
As shown in Figure 7c, the horizontal driving force rapidly reduces when the mover deflects around the z-axis, because the horizontal winding is close to the surface of yoke and magnetic flux density reduce, as shown in Figure 5b. The vertical driving force increases, because the vertical winding is close to the magnetic pole and magnetic flux density increase, as shown in Figure 5a. The change rates of the horizontal driving force and the vertical driving force are 3.8% and 1.6%, respectively.
Based on the above analysis, the change of the effective length of the coil and the inhomogeneity of air-gap magnetic flux density make the driving force fluctuate obvious. The linearity of the driving force is more affected by the uneven of the magnetic flux density.
Driving force variation with control current and displacement are studied by FEM and experimental measurements, respectively. To reduce the manufacturing error and assembly error, differential measurement method is adopted in the experiment, as shown in Figure 8. To compensate the weight of the mover, the actuator is suspended through the thread. The three-axis displacement table is adjusted so that the winding is at the zero position, as shown in Figure 8a. The driving force in vertical direction is measured as shown in Figure 8b. Driving force variation with control current is shown in Figure 9. With the increase of current, the electromagnetic force increases linearly. Due to ignorance of the magnetic flux leakage in the calculation process, EMCM results of horizontal current stiffness and vertical current stiffness, which are 4.761 N/A and 5.972 N/A, both are larger than FEM results and experimental results, respectively. The error between FEM results and experimental results increases as the control current increases. There are some threaded holes on the yoke for processing and assembly, which inevitably increases the magnetic resistance and reduces the magnetic flux density in air-gap. The magnetic resistance of threaded holes is ignored by FEM. So the FEM results are larger than the experimental results. FEM results of the vertical force coefficient and the horizontal force coefficient are 4.472 N/A and 5.549 N/A, respectively. Compared to experimental results of 4.08 N/A and 5.11 N/A, the error is 9.95% and 8.59%, respectively. With the control current of 1 A, driving force variation with displacement is analyzed by FEM and experimental measurement. The results of FEM and measurement are shown in Figure 10. FEM results are slightly higher the experimental results, because the magnetic resistance of threaded holes is ignored by FEM. The variation trend of driving force is consistent with that of magnetic field. The driving force decreases faster at the region near the yoke, because the magnetic field gradient near the yoke is large and the magnetic flux density decreases rapidly. The maximum of driving force of FEM is 5.76 N, which is smaller than 5.85 N of experiment results. The minimum of the vertical driving force of FEM is 4.86 N, which is larger than 4.17 N of experiment results. FEM value of average vertical driving force is 5.42 N. Compared to the experiment value of 5.19 N, the error is 4.2%. The maximum horizontal value of FEM is 5.03 N, which is larger than 4.44 N of experiment results. The minimum horizontal current stiffness of FEM is 3.92 N, which is larger than 3.66 N of experiment results. FEM value of average horizontal driving force is 4.48 N. Compared to the experiment value of 4.08 N, the error is 8.9%. With the control current of 1 A, driving force variation with displacement is analyzed by FEM and experimental measurement. The results of FEM and measurement are shown in Figure 10. FEM results are slightly higher the experimental results, because the magnetic resistance of threaded holes is ignored by FEM. The variation trend of driving force is consistent with that of magnetic field. The driving force decreases faster at the region near the yoke, because the magnetic field gradient near the yoke is large and the magnetic flux density decreases rapidly. The maximum of driving force of FEM is 5.76 N, which is smaller than 5.85 N of experiment results. The minimum of the vertical driving force of FEM is 4.86 N, which is larger than 4.17 N of experiment results. FEM value of average vertical driving force is 5.42 N. Compared to the experiment value of 5.19 N, the error is 4.2%. The maximum horizontal value of FEM is 5.03 N, which is larger than 4.44 N of experiment results. The minimum horizontal current stiffness of FEM is 3.92 N, which is larger than 3.66 N of experiment results. FEM value of average horizontal driving force is 4.48 N. Compared to the experiment value of 4.08 N, the error is 8.9%.   The control system of the actuator is established through the Simulink Real-Time hardware in 316 the loop simulation system, as shown in Fig. 12(a). The control program is established under the 317 MATLAB/ Simulink environment of the host computer. The control model is shown in Fig. 12(b).

318
The channel of the PCI-6229 DA is used to output the control voltage of the RMDS-405 driver. The 319 channel of the PCI-6229 AD collects the analog voltage value output by the laser displacement 320 sensor. The test system is shown in Fig. 12(c).

301
With the control current of 1 A, driving force variation with displacement in full range of 302 motion is analyzed through experimental measurement, as shown in Fig. 11. The maximum and   The control system of the actuator is established through the Simulink Real-Time hardware in 316 the loop simulation system, as shown in Fig. 12(a). The control program is established under the 317 MATLAB/ Simulink environment of the host computer. The control model is shown in Fig. 12(b).

318
The channel of the PCI-6229 DA is used to output the control voltage of the RMDS-405 driver. The 319 channel of the PCI-6229 AD collects the analog voltage value output by the laser displacement 320 sensor. The test system is shown in Fig. 12(c). The position sensors are laser displacement sensor 321 ILD2300 with a resolution of 1 μm. The command signals are step, sine and continuous step to verify 322 the suspension performance of the actuator. Large horizontal driving force and vertical driving force output is implemented simultaneously by the 2-DOF Lorentz force actuator. The distribution law of air-gap magnetic field is similar to that of driving force. So the linearity of driving force can be improved by heightening the uniformity of air-gap magnetic field.

Levitation Performance
The control system of the actuator is established through the Simulink Real-Time hardware in the loop simulation system, as shown in Figure 12a. The control program is established under the MATLAB/Simulink environment of the host computer. The control model is shown in Figure 12b.
The channel of the PCI-6229 DA is used to output the control voltage of the RMDS-405 driver. The channel of the PCI-6229 AD collects the analog voltage value output by the laser displacement sensor. The test system is shown in Figure 12c. The position sensors are laser displacement sensor ILD2300 with a resolution of 1 µm. The command signals are step, sine and continuous step to verify the suspension performance of the actuator.

330
The step response is set at 1 mm and the sampling frequency is 2 KHz. As shown in Fig. 13, it 331 can be seen that the system can reach the command signal within a limited time range under the two 332 333 Figure 13. Test results with the step response

334
Test results with the sinusoidal signal are shown in Fig. 14  The step response is set at 1 mm and the sampling frequency is 2 KHz. As shown in Figure 13, it can be seen that the system can reach the command signal within a limited time range under the two control algorithms. The system adjustment time under PID control algorithm is 0.38s and the overshoot is 10.8%. Under the sliding mode control algorithm, the system adjustment time is 0.58s and the system has no overshoot.
Test results with the sinusoidal signal are shown in Figure 14. When the input signal frequency is 0.5 Hz, it can be seen that good tracking performance is achieved with two different control algorithm. The closed-loop control system with PID control algorithm has faster response speed and achieves stable tracking in a short time. The system with sliding mode control algorithm takes more time for stable tracking.  The step response is set at 1 mm and the sampling frequency is 2 KHz. As shown in Fig. 13, it 1 can be seen that the system can reach the command signal within a limited time range under the two   As shown in Figure 15, the tracking error roughly takes on the shape of a sine by PID control algorithm. The error under the sliding mode control algorithm floats around 0 when the system is stable and the maximum error is less than that under PID control algorithm. The tracking accuracy of the sliding mode control algorithm is better than that of the PID control algorithm.
The positioning accuracy of the actuator is tested with a 0.02 mm continuous step command signal. The positioning accuracy not only depends on the air gap magnetic field characteristics but also on the sensor detection accuracy, resolution, motion control board, control algorithm and other factors. The data acquisition board is NI PCI −6229 and the sampling frequency is 2 KHz. As shown in Figure 16, the positioning accuracy of the actuator with sliding mode control algorithm and PID control algorithm are about 20 µm and 15 µm, respectively. error roughly takes on the shape of a sine. The error under the sliding mode control algorithm floats 3 around 0 when the system is stable and the maximum error is less than that under PID control 4 algorithm. The tracking accuracy of the sliding mode control algorithm is better than that of the PID 5 control algorithm.   The 2-DOF Lorentz force actuator developed in this paper is similar to that in literature [35]. The   The 2-DOF Lorentz force actuator developed in this paper is similar to that in Reference [35]. The structure parameters and performance are compared, as shown in the Table 3. The motion range of the Lorentz force actuator in this paper is larger, especially the deflection angle is up to ±10 • , which is 2.5 times of the reference value. The larger air gap makes the uniformity of magnetic density poor and the positioning accuracy is only 15 µm, which is lower than 4.4 nm in the reference. Due to the large size of the PM and air gap, the air gap flux density of 0.33 T is close to that of 0.34 T in the reference. Because of the large size of the coil in the magnetic field, vertical force coefficient of 5.549 N/A and horizontal force coefficient of 4.027 N/A are larger than those of 1.8 N/A and 0.31 N/A in the reference, respectively. So Lorentz force actuator in this paper effectively increases the deflection angle and force coefficient at the cost of localization accuracy.

Conclusions
Considering the pose of the windings and the PMs, a novel 2-DOF Lorentz force actuator is developed for the modular magnetic suspension platform. The mass of the actuator is 2.59 kg, the translation stroke is ±5 mm, the overall dimension is 117 mm × 106 mm × 118 mm and the deflection angle is ± 10 • . The mathematical model of electromagnetic force is established by equivalent magnetic circuit method. The characteristics of magnetic field and the driving force are analyzed through the FEM and the experimental measurement. The horizontal force coefficient and the vertical force coefficient are 4.08 N/A and 5.11 N/A, which means large force output of the actuator is realized. The sliding control algorithm and PID control algorithm are respectively applied to the actuator control system. The experimental results show that the position precision under the PID control is 15 µm, which is better than 20 µm of sliding model control. The tracking accuracy of the sliding mode control algorithm is better than that of the PID control algorithm. Compared with the conventional similar 2-DOF Lorentz force actuator, the motion range and force coefficient of the actuator in this paper are larger. Due to the increase of the air gap size, the uniformity of air gap flux density and positioning accuracy are not as good as those of small sized actuator. The results of the paper provide important support for the design of the high performance 2-DOF Lorentz force actuator. The design method of magnetic flux decoupling will be studied in the future.