# Optimal Controller Design for Ultra-Precision Fast-Actuation Cutting Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

#### 2.2. Experimental Determination of the System Model

#### 2.3. Modelling of Disturbances and Weighting Functions

^{−7}.

^{8}) to indicate that the motor power is enough for static positioning.

#### 2.4. Modelling of Following Errors

_{1}, P

_{2}, and P

_{3}are the PSD of each error source, and υ is the frequency). Since these disturbances are assumed to be mutually uncorrelated, their powers can be combined to reflect the total error power [20]. The synthesized tool position’s PSD is:

## 3. Results

#### 3.1. Closed-Loop Response with Optimal Control

#### 3.2. Study on the Influences of Structural Parameters on Positioning Following Error

#### 3.2.1. Influence of Moving Mass

#### 3.2.2. Influence of Flexure Bearing Stiffness and Damping

## 4. Conclusions

- The positioning error was reduced from 1.19 nm RMS to 0.68 nm RMS with the new controller, showing the benefits of a deterministic controller design approach;
- Under the given disturbances, there exist optimal bearing stiffness and damping coefficients that result in minimal following errors. The optimal bearing stiffness and damping coefficients are $1.1\times {10}^{5}\text{}\mathrm{N}/\mathrm{m}$ and $237.7\text{}\mathrm{N}/\left(\mathrm{m}\xb7{\mathrm{s}}^{-1}\right)$, respectively;
- It was found that increasing moving mass helps to reduce following errors, but the optimal bandwidth will be smaller.

## 5. Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Fang, F.; Zhang, X.; Weckenmann, A.; Zhang, G.; Evans, C. Manufacturing and measurement of freeform optics. CIRP Ann. Manuf. Technol.
**2013**, 62, 823–846. [Google Scholar] [CrossRef] - Qiao, Z.; Wu, Y.; Wang, B.; Liu, Y.; Qu, D.; Zhang, P. The average effect of multi-divisions cutting method on thermal error in cutting horizontal grooves array on roller mold. Proc. Inst. Mech. Eng. Part B J. Eng. Manuf.
**2018**, 233, 1907–1913. [Google Scholar] [CrossRef] - Liu, Y.; Li, D.; Ding, F.; Wu, Y.; Xue, J.; Qiao, Z.; Wang, B. Reduction of pitch error of the micro-prism array in brightness enhancement film by compensating z-axis positioning accuracy. Appl. Opt.
**2021**, 60, 5278–5284. [Google Scholar] [CrossRef] [PubMed] - Yuan, W.; Cheung, C.-F. Characterization of Surface Topography Variation in the Ultra-Precision Tool Servo-Based Diamond Cutting of 3D Microstructured Surfaces. Micromachines
**2021**, 12, 1448. [Google Scholar] [CrossRef] - Padilla-Garcia, E.A.; Rodriguez-Angeles, A.; Resendiz, J.R.; Cruz-Villar, C.A. Concurrent Optimization for Selection and Control of AC Servomotors on the Powertrain of Industrial Robots. IEEE Access
**2018**, 6, 27923–27938. [Google Scholar] [CrossRef] - Alter, D.M.; Tsao, T.-C. Control of Linear Motors for Machine Tool Feed Drives: Design and Implementation of H∞ Optimal Feedback Control. J. Dyn. Syst. Meas. Control
**1996**, 118, 649–656. [Google Scholar] [CrossRef] - Alter, D.M.; Tsao, T.-C. Control of Linear Motors for Machine Tool Feed Drives: Experimental Investigation of Optimal Feedforward Tracking Control. J. Dyn. Syst. Meas. Control
**1998**, 120, 137–142. [Google Scholar] [CrossRef] - Alter, D.; Tsao, T.-C. Optimal feedforward tracking control of linear motors for machine tool drives. In Proceedings of the 1995 American Control Conference-ACC’95, Seattle, WA, USA, 21–23 June 1995; Volume 1, pp. 210–214. [Google Scholar]
- Dumanli, A.; Sencer, B. Optimal high-bandwidth control of ball-screw drives with acceleration and jerk feedback. Precis. Eng.
**2018**, 54, 254–268. [Google Scholar] [CrossRef] - Huang, W.-W.; Li, L.; Li, Z.-L.; Zhu, Z.; Zhu, L.-M. Robust high-bandwidth control of nano-positioning stages with Kalman filter based extended state observer and H∞ control. Rev. Sci. Instrum.
**2021**, 92, 065003. [Google Scholar] [CrossRef] [PubMed] - Zhu, F.; Wang, H.; Tian, Y. Optimal law based improved reset PID control and application to HDD head-positioning systems. In Proceedings of the 2017 32nd Youth Academic Annual Conference of Chinese Association of Automation (YAC), Hefei, China, 19–21 May 2017; pp. 258–262. [Google Scholar]
- Mendoza-Mondragon, F.; Hernandez-Guzman, V.M.; Rodriguez-Resendiz, J. Robust Speed Control of Permanent Magnet Synchronous Motors Using Two-Degrees-of-Freedom Control. IEEE Trans. Ind. Electron.
**2018**, 65, 6099–6108. [Google Scholar] [CrossRef] - Zheng, C.; Su, Y.; Mercorelli, P. A simple nonlinear PD control for faster and high-precision positioning of servomechanisms with actuator saturation. Mech. Syst. Signal Process.
**2019**, 121, 215–226. [Google Scholar] [CrossRef] - Li, Z.; Guan, C.; Dai, Y.; Xue, S.; Yin, L. Comprehensive Design Method of a High-Frequency-Response Fast Tool Servo System Based on a Full-Frequency Error Control Algorithm. Micromachines
**2021**, 12, 1354. [Google Scholar] [CrossRef] [PubMed] - Hama, T.; Sato, K. High-speed and high-precision tracking control of ultrahigh-acceleration moving-permanent-magnet linear synchronous motor. Precis. Eng.
**2015**, 40, 151–159. [Google Scholar] [CrossRef] - Ramesh, R.; Mannan, M.A.; Poo, A.N. Error compensation in machine tools—A review: Part I: Geometric, cut-ting-force induced and fixture-dependent errors. Int. J. Mach. Tools Manuf.
**2000**, 40, 1235–1256. [Google Scholar] [CrossRef] - Ramesh, R.; Mannan, M.; Poo, A. Error compensation in machine tools—A review: Part II: Thermal errors. Int. J. Mach. Tools Manuf.
**2000**, 40, 1257–1284. [Google Scholar] [CrossRef] - Zhou, K.; Doyle, J.C. Essentials of Robust Control; Prentice Hall: Upper Saddle River, NJ, USA, 1998; Volume 104. [Google Scholar]
- Sperilă, A.; Ciubotaru, B.D.; Oară, C. The optimal H
_{2}controller for generalized discrete-time sys-tems. Automatica**2021**, 133, 109889. [Google Scholar] [CrossRef] - Ding, F.; Luo, X.; Zhong, W.; Chang, W. Design of a new fast tool positioning system and systematic study on its positioning stability. Int. J. Mach. Tools Manuf.
**2019**, 142, 54–65. [Google Scholar] [CrossRef]

**Figure 5.**Measured (

**a**) sensor noise (

**b**), current loop noise, and (

**c**) environmental vibrations and the modelled PSD functions.

**Figure 6.**(

**a**) Calculated optimal controller and transfer functions; (

**b**) controller and transfer functions with PID control.

**Figure 10.**(

**a**) Noise transfer functions; (

**b**) CAS plots for each error source. Performance with increased moving mass (grey: before; colored: after).

**Figure 11.**(

**a**) Errors with variable optimal bandwidth; (

**b**) errors with constant bandwidth. Minimum achievable positioning error decreases with larger moving masses.

**Figure 12.**(

**a**) Noise transfer functions; (

**b**) CAS plots for each error source. Performance with increased flexure stiffness ${k}_{2}$ (grey: before; colored: after).

**Figure 13.**(

**a**) Noise transfer functions; (

**b**) CAS plots for each error source. Performance with increased flexure damping ${c}_{2}$ (grey: before; colored: after).

Parameters | Values |
---|---|

${m}_{1}+{m}_{2}$ | $0.074\text{}\mathrm{Kg}$ |

${k}_{2}$ | $\mathrm{21,965}\text{}\mathrm{N}/\mathrm{m}$ |

${c}_{2}$ | $2.28\text{}\mathrm{N}/\left(\mathrm{m}/\mathrm{s}\right)$ |

${m}_{4}$ | $0.008\text{}\mathrm{Kg}$ |

${k}_{4}$ | $\mathrm{56,285}\text{}\mathrm{N}/\mathrm{m}$ |

${c}_{4}$ | $6.18\text{}\mathrm{N}/\left(\mathrm{m}/\mathrm{s}\right)$ |

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

© 2021 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**

Ding, F.; Luo, X.; Li, D.; Qiao, Z.; Wang, B.
Optimal Controller Design for Ultra-Precision Fast-Actuation Cutting Systems. *Micromachines* **2022**, *13*, 33.
https://doi.org/10.3390/mi13010033

**AMA Style**

Ding F, Luo X, Li D, Qiao Z, Wang B.
Optimal Controller Design for Ultra-Precision Fast-Actuation Cutting Systems. *Micromachines*. 2022; 13(1):33.
https://doi.org/10.3390/mi13010033

**Chicago/Turabian Style**

Ding, Fei, Xichun Luo, Duo Li, Zheng Qiao, and Bo Wang.
2022. "Optimal Controller Design for Ultra-Precision Fast-Actuation Cutting Systems" *Micromachines* 13, no. 1: 33.
https://doi.org/10.3390/mi13010033