# 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 |

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

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

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

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

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

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**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