# High Precision Magnetic Levitation Actuator for Micro-EDM

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## Abstract

**:**

## 1. Introduction

## 2. 5-DOF Controlled Magnetic Levitation Actuator

#### 2.1. The Structure of Magnetic Levitation Actuator

#### 2.2. The Principle of Magnetic Levitation Actuator

#### 2.3. Mathematical Model of 5-DOF Magnetic Levitation Actuator

_{z}is the air gap stiffness in the Z direction; k

_{x}is the air gap stiffness in the X direction; k

_{θ}is the air gap stiffness in the θ direction; F

_{z}is the electromagnetic force in the Z direction; F

_{x}is the electromagnetic force in the X direction; M

_{θ}is the torque generated in the θ direction; l is the distance from the center of the kinematic to the force point. Meanwhile, the forces generated between the coil and the permanent magnet in the Z and X directions are F

_{z}and F

_{x}, and the torque generated in the θ direction is also M

_{θ}.

_{i}is the current stiffness factor of the coil and i

_{k}is the drive current supplied to each coil set. In addition, in this actuator, the control currents of the eight groups of coils for controlling vertical motion in the Z direction are the same and set to i

_{z}, the control currents of the four sets of coils for controlling horizontal motion in the X direction are the same and set to i

_{x}, and the control currents of the four sets of coils for controlling rotational motion in the θ direction are the same and set to i

_{θ}. The spindle’s equations of motion in the Z, X, and θ directions are

_{L}is the inductor voltage drop; k

_{v}is the reverse electric potential coefficient; U

_{R}is the resistance voltage drop; E is the reverse electric potential; L is the coil inductance; R is the coil resistance. The electrical and mechanical models are coupled, and the Rasch transform can sort out the transfer function of the magnetic levitation actuator.

#### 2.4. Experimental Magnetic Levitation Actuator

^{2}, the cross-sectional area of the copper wire with a diameter of 0.7 mm is 0.38 mm

^{2}, and the safe ampacity is within 3.08 A. From Equation (7) in Section 2.3, it can be obtained that the load of the magnetic levitation actuator is 12.93 N. To improve the accuracy of the displacement sensor displacement detection, the detection material of both sides and the upper end of the spindle is made of stainless steel (SUS304). Considering the remanent magnetism, coercivity, maximum magnetic energy product, and economy, the permanent magnet ring is made of NdFeB–the third-generation permanent magnet material. The spindle displacement in the direction of five degrees of freedom is measured by five eddy current displacement sensors (PU-09, AEC Corp., Dallas, TX, USA), and the actuator is measured by a digital signal processor (DSP; DS1103 PPC Controller Board, dSPACE Corp., Paderborn, Germany) with a sampling rate of 10 kHz.

## 3. Magnetic Field Characteristic Analysis

## 4. 5-DOF Magnetic Levitation Actuator Controller Design and Positioning Performance

#### 4.1. Actuator Motion Control System

_{1}and a

_{0}are the denominator parameters of the regulator, and b

_{2}, b

_{1}, and b

_{0}are the numerator parameters of the regulator.

_{dz}, and the controller design for the other directions is the same as for the Z-direction. Table 1 shows the model parameters, and Table 2 shows the control parameters of the actuator, which were determined by experimental results and numerical simulations.

#### 4.2. Composition of the Experimental System

#### 4.3. Performance Test

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The structure diagram of the 5-DOF controlled magnetic levitation actuator. (

**a**) Actuator; (

**b**) Permanent magnets; (

**c**) Air-core coils.

**Figure 13.**Structure of the experimental system. (

**a**) Step response in the X direction; (

**b**) Step response in the Y direction; (

**c**) Step response in the Z direction; (

**d**) Step response in the Φ direction; (

**e**) Step response in the θ direction.

**Figure 14.**Positioning resolutions. (

**a**) Positioning resolution in the X direction; (

**b**) Positioning resolution in the Y direction; (

**c**) Positioning resolution in the Z direction; (

**d**) Positioning resolution in the Φ direction; (

**e**) Positioning resolution in the θ direction.

**Figure 15.**Positioning resolutions. (

**a**) The stroke in the X direction; (

**b**) The stroke in the Y direction; (

**c**) The stroke in the Z direction; (

**d**) The stroke in the Φ direction; (

**e**) The stroke in the θ direction.

**Figure 16.**Frequency responses. (

**a**) The bandwidth in the X direction; (

**b**) The bandwidth in the Y direction; (

**c**) The bandwidth in the Z direction; (

**d**) The bandwidth in the Φ direction; (

**e**) The bandwidth in the θ direction.

Variable Name | X (Y) Z θ (Φ) Direction | Unit | |
---|---|---|---|

mass of the spindle | m | 0.80 | kg |

coil inductance | L | 35.4 | mH |

coil resistance | R | 2.6 | Ω |

torque | l | 25 | mm |

rotational inertia | J_{θ} | 2 | kg·m^{2} |

first-order delay system time constants | T_{d} | 3.9 × 10^{−3} | / |

current stiffness | k_{i} | 4.2 | N·A^{−1} |

air gap stiffness in the X direction | k_{x} | 367.57 | N·m^{−1} |

air gap stiffness in the Z direction | k_{z} | 170.7 | N·m^{−1} |

air gap stiffness in the θ direction | k_{θ} | 9.19 | N·rad^{−1} |

damping coefficient | c | 1 | N·s·m^{−1} |

X (Y) Z θ (Φ) Direction Controller | |
---|---|

δ_{x} | 256.45 |

δ_{z} | 256.50 |

δ_{θ} | 3.07 × 10^{4} |

a_{0x} | 3.07 × 10^{5} |

a_{0z} | 1.49 × 10^{5} |

a_{0θ} | 4.53 × 10^{5} |

a_{1x} | 873.31 |

a_{1z} | 1.93 × 10^{3} |

a_{1θ} | 819.73 |

b_{0x} | 1.30 × 10^{8} |

b_{0z} | 6.78 × 10^{9} |

b_{0θ} | 1.79 × 10^{6} |

b_{1x} | 3.52 × 10^{6} |

b_{1z} | 8.50 × 10^{7} |

b_{1θ} | 3.09 × 10^{4} |

b_{2x} | 4.01 × 10^{4} |

b_{2z} | 2.16 × 10^{5} |

b_{2θ} | 93.14 |

α_{x} | 35 |

α_{z} | 35 |

α_{θ} | 35 |

ε_{x} | 2565 |

ε_{z} | 2565 |

ε_{θ} | 2565 |

Response Time | Stroke | Positioning Resolution | Bandwidth | |
---|---|---|---|---|

X direction | 6.7 ms | 4 mm | 1 μm | 101 Hz |

Y direction | 6.8 ms | 4 mm | 1 μm | 101 Hz |

Z direction | 26.3 ms | 4 mm | 1 μm | 51 Hz |

Φ direction | 39.9 ms | 70 mrad | 25 μrad | 42 Hz |

θ direction | 16.2 ms | 70 mrad | 20 μrad | 45 Hz |

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**MDPI and ACS Style**

Luan, B.; Zhang, X.; Xu, F.; Yang, G.; Jin, J.; Xu, C.; Sun, F.; Oka, K.
High Precision Magnetic Levitation Actuator for Micro-EDM. *Actuators* **2022**, *11*, 361.
https://doi.org/10.3390/act11120361

**AMA Style**

Luan B, Zhang X, Xu F, Yang G, Jin J, Xu C, Sun F, Oka K.
High Precision Magnetic Levitation Actuator for Micro-EDM. *Actuators*. 2022; 11(12):361.
https://doi.org/10.3390/act11120361

**Chicago/Turabian Style**

Luan, Boran, Xiaoyou Zhang, Fangchao Xu, Guang Yang, Junjie Jin, Chengcheng Xu, Feng Sun, and Koichi Oka.
2022. "High Precision Magnetic Levitation Actuator for Micro-EDM" *Actuators* 11, no. 12: 361.
https://doi.org/10.3390/act11120361