# Development of Magnetically Levitated Rotary Table for Repetitive Trajectory Tracking

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Model Description of the MLRT

## 3. LPIDDC Approach

#### 3.1. PID Term

#### 3.2. Disturbance Compensation Term

#### 3.3. Iterative Learning Term

**Theorem**

**1.**

**Proof of Theorem**

**1.**

#### 3.4. Summary of LPIDDC Control Strategy

## 4. Experimental Studies

#### 4.1. Hardware Setup

- (1)
- Track1: the MLRT is controlled to track the sinusoidal trajectory in ${}^{s}z$-axis below with the unit being $\mathrm{mm}$,$${z}_{d}=3+0.5sin\left(\pi t\right),$$
- (2)
- Track2: the MLRT is controlled to track the sinusoidal trajectory in ${}^{s}\gamma $-axis below with unit being $\mathrm{rad}$,$${\gamma}_{d}=0.1sin\left(\pi t\right),$$

- (1)
- ${e}_{\mathrm{RMS}}=\sqrt{\frac{1}{T}{\int}_{0}^{T}{\left|{x}_{d}\left(t\right)-{x}_{k}\left(t\right)\right|}^{2}dt}$, the root-mean-square value of the trajectory tracking error, where T is the period of tracking trajectory.
- (2)
- ${e}_{\mathrm{M}}=max\left\{\left|{x}_{d}\left(t\right)-{x}_{k}\left(t\right)\right|\right\}$, the maximal absolute value of the trajectory tracking error.

#### 4.2. Trajectory Tracking without External Disturbance

#### 4.3. Trajectory Tracking with External Disturbance

#### 4.3.1. Step Disturbance

#### 4.3.2. Complex Disturbance

#### 4.3.3. Disturbance Caused by Polyfoam

#### 4.3.4. Circle Trajectory Tracking

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MLRT | Magnetically levitated rotary table |

PID | Proportion-integral-derivative |

ILC | Iterative learning control |

DC | Disturbance compensation |

LPIDDC | Iterative learning PID control strategy with disturbance compensation |

PM | Permanent magnet |

PIDDC | PID with Disturbance compensation |

LPID | Iterative learning feed-forwad PID |

## References

- Xu, T.; Yu, J.; Yan, X.; Choi, H.; Zhang, L. Magnetic Actuation Based Motion Control for Microrobots: An Overview. Micromachines
**2015**, 6, 1346–1364. [Google Scholar] [CrossRef] - Kumar, P.; Malik, S.; Toyserkani, E.; Khamesee, M.B. Development of an Electromagnetic Micromanipulator Levitation System for Metal Additive Manufacturing Applications. Micromachines
**2022**, 13, 585. [Google Scholar] [CrossRef] [PubMed] - Li, Z.; Wu, Q.; Liu, B.; Gong, Z. Optimal Design of Magneto-Force-Thermal Parameters for Electromagnetic Actuators with Halbach Array. Actuators
**2021**, 10, 231. [Google Scholar] [CrossRef] - Dyck, M.; Lu, X.; Altintas, Y. Magnetically Levitated Rotary Table with Six Degrees of Freedom. IEEE/ASME Trans. Mechatronics
**2016**, 22, 530–540. [Google Scholar] [CrossRef] - Xu, X.; Zheng, C.; Xu, F. A Real-Time Numerical Decoupling Method for Multi-DoF Magnetic Levitation Rotary Table. Appl. Sci.
**2019**, 9, 3263. [Google Scholar] [CrossRef] [Green Version] - Kou, B.; Xing, F.; Zhang, L.; Zhang, C.; Zhou, Y. A Real-Time Computation Model of the Electromagnetic Force and Torque for a Maglev Planar Motor with the Concentric Winding. Appl. Sci.
**2017**, 7, 98. [Google Scholar] [CrossRef] [Green Version] - Lu, X. 6D Direct-drive Technology for Planar Motion Stages. CIRP Ann.
**2012**, 61, 359–362. [Google Scholar] [CrossRef] - Li, J.H.; Chiou, J.S. GSA-Tuning IPD Control of a Field-Sensed Magnetic Suspension System. Sensors
**2015**, 15, 31781–31793. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kim, W.J.; Verma, S.; Shakir, H. Design and Precision Construction of Novel Magnetic-levitation-based Multi-axis Nanoscale Positioning Systems. Precis. Eng.
**2007**, 31, 337–350. [Google Scholar] [CrossRef] - Silva-Rivas, J.C.; Kim, W.J. Multivariable Control and Optimization of a Compact 6-DOF Precision Positioner With Hybrid and Digital Filtering. IEEE Trans. Control Syst. Technol.
**2012**, 21, 1641–1651. [Google Scholar] [CrossRef] - Fallaha, C.; Kanaan, H.; Saad, M. Real Time Implementation of a Sliding Mode Regulator for Current-Controlled Magnetic Levitation System. In Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation Intelligent Control, Limassol, Cyprus, 27–29 June 2005; pp. 696–701. [Google Scholar] [CrossRef]
- Zhang, Z.; Li, X. Real-Time Adaptive Control of a Magnetic Levitation System with a Large Range of Load Disturbance. Sensors
**2018**, 18, 1512. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, M.Y.; Tsai, C.F.; Fu, L.C. A Novel Design and Control to Improve Positioning Precision and Robustness for a Planar Maglev System. IEEE Trans. Ind. Electron.
**2019**, 66, 4860–4869. [Google Scholar] [CrossRef] - Basovich, S.; Arogeti, S.A.; Menaker, Y.; Brand, Z. Magnetically Levitated Six-DOF Precision Positioning Stage with Uncertain Payload. IEEE/ASME Trans. Mechatronics
**2016**, 21, 660–673. [Google Scholar] [CrossRef] - de Jesús Rubio, J.; Zhang, L.; Lughofer, E.; Cruz, P.; Alsaedi, A.; Hayat, T. Modeling and Control with Neural Networks for a Magnetic Levitation System. Neurocomputing
**2017**, 227, 113–121. [Google Scholar] [CrossRef] - Liang, S.; Xi, R.; Xiao, X.; Yang, Z. Adaptive Sliding Mode Disturbance Observer and Deep Reinforcement Learning Based Motion Control for Micropositioners. Micromachines
**2022**, 13, 458. [Google Scholar] [CrossRef] - Kazemzadeh Heris, P.; Khamesee, M.B. Design and Fabrication of a Magnetic Actuator for Torque and Force Control Estimated by the ANN/SA Algorithm. Micromachines
**2022**, 13, 327. [Google Scholar] [CrossRef] [PubMed] - Zhang, Y.; Huang, Y.; Wang, Y. Research on Compound PID Control Strategy Based on Input Feedforward and Dynamic Compensation Applied in Noncircular Turning. Micromachines
**2022**, 13, 341. [Google Scholar] [CrossRef] [PubMed] - Mishra, S.; Tomizuka, M. Projection-based iterative learning control for wafer scanner systems. IEEE/ASME Trans. Mechatronics
**2009**, 14, 388–393. [Google Scholar] [CrossRef] - Mao, W.L. Indirect fuzzy contour tracking for X–Y PMSM actuated motion system applications. IET Electr. Power Appl.
**2018**, 12, 12–24. [Google Scholar] [CrossRef] - Yu, D.; Zhu, Y.; Yang, K.; Hu, C.; Li, M. A Time-varying Q-filter Design for Iterative Learning Control with Application to an Ultra-precision Dual-stage Actuated Wafer Stage. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng.
**2014**, 228, 658–667. [Google Scholar] [CrossRef] - Bai, L.; Feng, Y.W.; Li, N.; Xue, X.F. Robust Model-Free Adaptive Iterative Learning Control for Vibration Suppression Based on Evidential Reasoning. Micromachines
**2019**, 10, 196. [Google Scholar] [CrossRef] [Green Version] - Peng, J.; Huang, J.; Wang, J.; Meng, F.; Gong, H.; Ping, B. The Driving Waveform Design Method of Power-Law Fluid Piezoelectric Printing Based on Iterative Learning Control. Sensors
**2022**, 22, 935. [Google Scholar] [CrossRef] [PubMed] - Chen, W.H.; Yang, J.; Guo, L.; Li, S. Disturbance-observer-based Control and Related Methods—An Overview. IEEE Trans. Ind. Electron.
**2016**, 63, 1083–1095. [Google Scholar] [CrossRef] [Green Version] - Lu, X.; Zheng, T.; Xu, F.; Xu, X. Semi-Analytical Solution of Magnetic Force and Torque for a Novel Magnetically Levitated Actuator in Rotary Table. IEEE Trans. Magn.
**2019**, 55, 1–8. [Google Scholar] [CrossRef]

**Figure 4.**Tracking errors of the MLRT without external disturbance. (

**a**) Tracking error of Track1. (

**b**) Tracking error of Track2.

**Figure 5.**Tracking results of Track1 and Track2 with step disturbance. (

**a**) Tracking error in ${}^{s}z$-axis. (

**b**) Tracking error in ${}^{s}\gamma $-axis.

**Figure 6.**Tracking results of Track1 with complex disturbance. (

**a**) Tracking error in ${}^{s}z$-axis. (

**b**) Disturbance quantity and its estimated value. (

**c**) Convergence rate of ${e}_{\mathrm{RMS}}$ in ${}^{s}z$-axis.

**Figure 7.**Tracking results of Track2 with complex disturbance. (

**a**) Tracking error in ${}^{s}\gamma $-axis. (

**b**) Disturbance quantity and its estimated value. (

**c**) Convergence rate of ${e}_{\mathrm{RMS}}$ in ${}^{s}\gamma $-axis.

**Figure 8.**Photo of the MLRT with polyfoam producing unknown disturbance. Experimental video is found in the Supplementary Video S1.

**Figure 9.**Tracking results of Track1 with polyfoam disturbance. (

**a**) Tracking error in ${}^{s}z$-axis. (

**b**) Convergence rate of ${e}_{\mathrm{RMS}}$ in ${}^{s}z$-axis.

Trajectory | Track1 | Track1 | Track2 | Track2 |
---|---|---|---|---|

Index | ${\mathit{e}}_{\mathbf{RMS}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\mu}\mathbf{m}\right)$ | ${\mathit{e}}_{\mathbf{M}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\mu}\mathbf{m}\right)$ | ${\mathit{e}}_{\mathrm{RMS}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mrad}\right)$ | ${\mathit{e}}_{\mathbf{M}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mrad}\right)$ |

$\mathrm{M}1$ | $3.083$ | $6.996$ | $2.159$ | $4.999$ |

$\mathrm{M}2$ | $3.076$ | $6.984$ | $2.161$ | $3.611$ |

$\mathrm{M}3$ | $1.917$ | $4.450$ | $1.412$ | $2.285$ |

$\mathrm{M}4$ | $1.876$ | $4.332$ | $1.409$ | $2.191$ |

Trajectory | Track1 | Track1 | Track2 | Track2 |
---|---|---|---|---|

Index | ${\mathit{e}}_{\mathbf{RMS}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\mu}\mathbf{m}\right)$ | ${\mathit{e}}_{\mathbf{M}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\mu}\mathbf{m}\right)$ | ${\mathit{e}}_{\mathrm{RMS}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mrad}\right)$ | ${\mathit{e}}_{\mathbf{M}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mrad}\right)$ |

$\mathrm{M}1$ | $34.058$ | $59.915$ | $2.918$ | $5.418$ |

$\mathrm{M}2$ | $11.759$ | $21.527$ | $2.035$ | $2.883$ |

$\mathrm{M}3$ | $2.238$ | $9.373$ | $1.819$ | $2.379$ |

$\mathrm{M}4$ | $2.001$ | $7.445$ | $1.587$ | $1.907$ |

Index | ${\mathit{e}}_{\mathbf{RMS}}\left(\mathsf{\mu}\mathbf{m}\right)$ | ${\mathit{e}}_{\mathbf{M}}\left(\mathsf{\mu}\mathbf{m}\right)$ |
---|---|---|

$\mathrm{M}1$ | $52.034$ | $97.302$ |

$\mathrm{M}2$ | $13.496$ | $36.248$ |

$\mathrm{M}3$ | $10.896$ | $31.637$ |

$\mathrm{M}4$ | $3.749$ | $17.547$ |

Index | ${\mathit{e}}_{\mathbf{RMS}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\mu}\mathbf{m}\right)$ | ${\mathit{e}}_{\mathbf{M}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathsf{\mu}\mathbf{m}\right)$ |
---|---|---|

$\mathrm{M}1$ | $59.869$ | $96.935$ |

$\mathrm{M}2$ | $35.142$ | $83.218$ |

$\mathrm{M}3$ | $29.511$ | $94.900$ |

$\mathrm{M}4$ | $20.135$ | $62.428$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xu, F.; Zhang, K.; Xu, X.
Development of Magnetically Levitated Rotary Table for Repetitive Trajectory Tracking. *Sensors* **2022**, *22*, 4270.
https://doi.org/10.3390/s22114270

**AMA Style**

Xu F, Zhang K, Xu X.
Development of Magnetically Levitated Rotary Table for Repetitive Trajectory Tracking. *Sensors*. 2022; 22(11):4270.
https://doi.org/10.3390/s22114270

**Chicago/Turabian Style**

Xu, Fengqiu, Kaiyang Zhang, and Xianze Xu.
2022. "Development of Magnetically Levitated Rotary Table for Repetitive Trajectory Tracking" *Sensors* 22, no. 11: 4270.
https://doi.org/10.3390/s22114270