Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation
Abstract
:1. Introduction
2. Interferometer Measurement Model
2.1. Single-Axis Interferometer Laser Path Model
2.2. 3-DOF Displacement Measurement Model
2.3. 6-DOF Displacement Measurement Model
3. Multi-DOF Displacement Computation
3.1. Problem of Solving Nonlinear Equation System
3.2. Solution Component Uniqueness Theory
Algorithm 1: Solution component uniqueness algorithm. |
is a solution in the solution set of (12) |
9: Compute the uniqueness of each component by Equation (24) |
3.3. Principle of Displacement Computation
Algorithm 2: Newton method with MP inverse. |
satisfy (28) |
7: k = k +1 |
8: end while |
4. Simulation Results
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Yu, X.; Gillmer, S.R.; Woody, S.C.; Ellis, J.D. Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology. Rev. Sci. Instrum. 2016, 87, 065109. [Google Scholar] [CrossRef] [PubMed]
- Chen, Y.T.; Lin, W.C.; Liu, C.S. Design and experimental verification of novel six-degree-of freedom geometric error measurement system for linear stage. Opt. Lasers Eng. 2017, 92, 94–104. [Google Scholar] [CrossRef]
- Zimmermann, N.; Ibaraki, S. Self-calibration of rotary axis and linear axes error motions by an automated on-machine probing test cycle. Int. J. Adv. Manuf. Technol. 2020, 107, 2107–2120. [Google Scholar] [CrossRef]
- Christiansen, A.J.; Naylor, D.A.; Gom, B.G. Multiaxis applications of a cryogenic range-resolved laser interferometer. Photonic Instrum. Eng. X 2023, 12428, 271–282. [Google Scholar]
- Leun, E.V. Highly Sensitive Single-Coordinate Measurement Electric Field Strength by Jet-Drop Optical Measuring Systems with a Pendant Drop. In Proceedings of the Dynamics of Systems, Mechanisms and Machines (Dynamics), Omsk, Russia, 15–17 November 2022; pp. 1–5. [Google Scholar]
- Haitjema, H. Calibration of displacement laser interferometer systems for industrial metrology. Sensors 2019, 19, 4100. [Google Scholar] [CrossRef] [PubMed]
- Gao, Z.; Hu, J.; Zhu, Y.; Duan, G. A new 6-degree-of-freedom measurement method of XY stages based on additional information. Precis. Eng. 2013, 37, 606–620. [Google Scholar] [CrossRef]
- Li, X.; Gao, W.; Muto, H.; Shimizu, Y.; Ito, S.; Dian, S. A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage. Precis. Eng. 2013, 37, 771–781. [Google Scholar] [CrossRef]
- Hsieh, H.L.; Pan, S.W. Development of a grating-based interferometer for six-degree-of-freedom displacement and angle measurements. Opt. Express 2015, 23, 2451–2465. [Google Scholar] [CrossRef] [PubMed]
- Dejima, S.; Gao, W.; Shimizu, H.; Kiyono, S.; Tomita, Y. Precision positioning of a five degree-of-freedom planar motion stage. Mechatronics 2005, 15, 969–987. [Google Scholar] [CrossRef]
- Holmes, M.; Hocken, R.; Trumper, D. The long-range scanning stage: A novel platform for scanned-probe microscopy. Precis. Eng. 2000, 24, 191–209. [Google Scholar] [CrossRef]
- Cai, K.; Tian, Y.; Wang, F.; Zhang, D.; Liu, X.; Shirinzadeh, B. Design and control of a 6-degree-of-freedom precision positioning system. Robot. Comput. Integr. Manuf. 2017, 44, 77–96. [Google Scholar] [CrossRef]
- Zhu, J.; Wang, G.; Wang, S.; Li, X. A reflective-type heterodyne grating interferometer for three-degree-of-freedom subnanometer measurement. IEEE Trans. Instrum. Meas. 2022, 71, 1–9. [Google Scholar] [CrossRef]
- Van Den Brink, M.A.; Straaijer, A. Lithographic Apparatus for Step-and-Scan Imaging of Mask Pattern with Interferometer Mirrors on the Mask and Wafer Holders. U.S. Patent No. 6,084,673, 4 July 2000. [Google Scholar]
- Liu, C.S.; Pu, Y.F.; Chen, Y.T.; Luo, Y.T. Design of a measurement system for simultaneously measuring six-degree-of-freedom geometric errors of a long linear stage. Sensors 2018, 18, 3875. [Google Scholar] [CrossRef] [PubMed]
- Madsen, K.; Nielsen, H.B.; Tingleff, O. The Gauss-Newton Method. In Methods for Non-Linear Least Squares Problems, 2nd ed.; Informatics and Mathematical Modelling, Technical University of Denmark: Kongens Lyngby, Denmark, 2004; pp. 20–23. [Google Scholar]
- Sugihara, T. Solvability-unconcerned inverse kinematics by the Levenberg–Marquardt method. IEEE Trans. Robot. 2011, 27, 984–991. [Google Scholar] [CrossRef]
- Fan, J. The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence. Math. Comput. 2012, 81, 447–466. [Google Scholar] [CrossRef]
- Ueda, K.; Yamashita, N. On a global complexity bound of the Levenberg-Marquardt method. J. Optim. Theory Appl. 2010, 147, 443–453. [Google Scholar] [CrossRef]
- Dan, H.; Yamashita, N.; Fukushima, M. Convergence properties of the inexact Levenberg-Marquardt method under local error bound conditions. Optim. Methods Softw. 2002, 17, 605–626. [Google Scholar] [CrossRef]
- Zeng, Z. A Newton’s iteration converges quadratically to nonisolated solutions too. Math. Comput. 2023, 92, 2795–2824. [Google Scholar] [CrossRef]
- Li, C.; Zhang, W.H.; Jin, X.Q. Convergence and uniqueness properties of Gauss-Newton’s method. Comput. Math. Appl. 2004, 47, 1057–1067. [Google Scholar] [CrossRef]
- Leon, S.J.; De Pillis, L.G. Gaussian Elimination. In Linear Algebra with Applications.; Pearson Education, Inc.: Upper Saddle River, NJ, USA, 2006; pp. 414–418. [Google Scholar]
- Zhang, Z. Introduction to machine learning: K-nearest neighbors. Ann. Transl. Med. 2016, 4, 218. [Google Scholar] [CrossRef] [PubMed]
Laser Number | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
(10−3 mm) | −0.96 | 0.61 | −1.40 | 0.67 | 1.10 | −0.74 |
(10−3 mm) | −0.21 | 0.23 | −1.10 | 2.60 | −0.48 | 0.38 |
(10−3 mrad) | 0.51 | −1.20 | 0.09 | −1.70 | −1.00 | 0.91 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 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
Wen, T.; Hu, J.; Zhu, Y.; Hua, G. Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation. Appl. Sci. 2023, 13, 10887. https://doi.org/10.3390/app131910887
Wen T, Hu J, Zhu Y, Hua G. Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation. Applied Sciences. 2023; 13(19):10887. https://doi.org/10.3390/app131910887
Chicago/Turabian StyleWen, Tingrui, Jinchun Hu, Yu Zhu, and Guojie Hua. 2023. "Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation" Applied Sciences 13, no. 19: 10887. https://doi.org/10.3390/app131910887
APA StyleWen, T., Hu, J., Zhu, Y., & Hua, G. (2023). Multi-Axis Laser Interferometer Not Affected by Installation Errors Based on Nonlinear Computation. Applied Sciences, 13(19), 10887. https://doi.org/10.3390/app131910887