# Design of a Measurement System for Simultaneously Measuring Six-Degree-Of-Freedom Geometric Errors of a Long Linear Stage

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{x}, horizontal straightness error δ

_{y}, and vertical straightness error δ

_{z}) and three angular errors (pitch error ε

_{y}, yaw error ε

_{z}, and roll error ε

_{x}) [8,9]. As a result, in order to improve the accuracy and repeatability of the multi-axis machine tools, 6DOF geometric motion errors of the linear precision stage should be accurately identified and effectively compensated [10,11].

_{x}for a long traveling range (>200 mm) with high accuracy and repeatability due to its inherent limits.

_{x}for a long traveling range (> 200 mm) with high accuracy and repeatability due to its inherent limits. Therefore, Fang et al. proposed a measurement system to simultaneously measure 6DOF geometric errors. The measurement method is based on a combination of laser interferometry and laser fiber collimation. Positioning error measurement was achieved by laser interferometry, and other five-degree-of-freedom (5DOF) geometric motion errors were obtained by fiber collimation measurement [28,29,30,31]. However, to the best of the current authors’ knowledge, these techniques are very few.

## 2. Structure Layout and Measuring Principle

_{y}, vertical straightness error δ

_{z}, pitch error ε

_{y}, yaw error ε

_{z}, and roll error ε

_{x}, respectively.

_{4}and PSD

_{5}) are used to measure linear fluctuations (δ

_{ly}and δ

_{lz}) and angular fluctuations (ε

_{ly}and ε

_{lz}) of the laser source in this study. As stated in Section 3, a forward light ray tracing method is used to follow the laser beam. From these data, the effects of 6DOF geometric errors of the linear stage on the light spot positions on the PSDs are determined, and a reverse derivation is applied to find the 6DOF geometric errors of the linear stage from the light spot position information [7]. Subsequently the measurement accuracy of the proposed measurement system can be improved by compensating the measured linear and angular fluctuations of the laser source. As a result, the 6DOF geometric errors of the linear stage can be obtained by analyzing the position information of the light spots on the PSDs and the optical paths in the proposed measurement system.

## 3. Numerical Simulation and Mathematical Model

_{i}is the vector from the source P

_{i − 1}to the destination point P

_{i}, then λ

_{i}is as follows:

_{PSD1}= F

_{X}

_{1}(δ

_{x}, δ

_{y}, δ

_{z}, ε

_{y}, ε

_{z}, ε

_{x}, δ

_{ly}, δ

_{lz}, ε

_{ly}, ε

_{lz}),

_{PSD1}= F

_{Y}

_{1}(δ

_{x}, δ

_{y}, δ

_{z}, ε

_{y}, ε

_{z}, ε

_{x}, δ

_{ly}, δ

_{lz}, ε

_{ly}, ε

_{lz}),

_{PSD2}= F

_{X2}(δ

_{x}, δ

_{y}, δ

_{z}, ε

_{y}, ε

_{z}, ε

_{x}, δ

_{ly}, δ

_{lz}, ε

_{ly}, ε

_{lz}),

_{PSD2}= F

_{Y2}(δ

_{x}, δ

_{y}, δ

_{z}, ε

_{y}, ε

_{z}, ε

_{x}, δ

_{ly}, δ

_{lz}, ε

_{ly}, ε

_{lz}),

_{PSD3}= F

_{X3}(δ

_{x}, δ

_{y}, δ

_{z}, ε

_{y}, ε

_{z}, ε

_{x}, δ

_{ly}, δ

_{lz}, ε

_{ly}, ε

_{lz}),

_{PSD3}= F

_{Y3}(δ

_{x}, δ

_{y}, δ

_{z}, ε

_{y}, ε

_{z}, ε

_{x}, δ

_{ly}, δ

_{lz}, ε

_{ly}, ε

_{lz}),

_{PSD4}= F

_{X4}(δ

_{ly}, δ

_{lz}, ε

_{ly}, ε

_{lz}),

_{PSD4}= F

_{Y4}(δ

_{ly}, δ

_{lz}, ε

_{ly}, ε

_{lz}),

_{PSD5}= F

_{X5}(δ

_{ly}, δ

_{lz}, ε

_{ly}, ε

_{lz}),

_{PSD5}= F

_{Y5}(δ

_{ly}, δ

_{lz}, ε

_{ly}, ε

_{lz}),

_{PSDi}(i = 1, 2, 3, 4, and 5) and Y

_{PSDi}(i = 1, 2, 3, 4, and 5) are the image centroid coordinates of the light spot on PSDi in the X-direction and Y-direction, respectively.

_{y}= G

_{δy}(X

_{PSD1}, Y

_{PSD1}, X

_{PSD2}, Y

_{PSD2}, X

_{PSD3}, Y

_{PSD3}, X

_{PSD4}, Y

_{PSD4}, X

_{PSD5}, Y

_{PSD5}),

_{z}= G

_{δz}(X

_{PSD1}, Y

_{PSD1}, X

_{PSD2}, Y

_{PSD2}, X

_{PSD3}, Y

_{PSD3}, X

_{PSD4}, Y

_{PSD4}, X

_{PSD5}, Y

_{PSD5}),

_{y}= G

_{εy}(X

_{PSD1}, Y

_{PSD1}, X

_{PSD2}, Y

_{PSD2}, X

_{PSD3}, Y

_{PSD3}, X

_{PSD4}, Y

_{PSD4}, X

_{PSD5}, Y

_{PSD5}),

_{z}= G

_{εz}(X

_{PSD1}, Y

_{PSD1}, X

_{PSD2}, Y

_{PSD2}, X

_{PSD3}, Y

_{PSD3}, X

_{PSD4}, Y

_{PSD4}, X

_{PSD5}, Y

_{PSD5}),

_{x}= G

_{εx}(X

_{PSD1}, Y

_{PSD1}, X

_{PSD2}, Y

_{PSD2}, X

_{PSD3}, Y

_{PSD3}, X

_{PSD4}, Y

_{PSD4}, X

_{PSD5}, Y

_{PSD5}),

## 4. Experimental characterization

## 5. Conclusions

_{x}for a long traveling range (> 200 mm). In contrast to conventional laser interferometers using only the interferometer method, the proposed measurement system can simultaneously measure six-degree-of-freedom geometric motion errors of a long linear stage with lower cost and faster operational time. The performance of the proposed measurement system has been evaluated using a laboratory-built prototype. The experimental results have shown that the proposed measurement system can simultaneously measure 6-DOF geometric motion errors of a linear stage for a long traveling range of 250 mm, and the measured accuracy of the proposed measurement system for yaw errors is about 4 µm when comparing to that of the laser interferometer.

## 6. Patents

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Lee, C.B.; Kim, G.H.; Lee, S.K. Uncertainty investigation of grating interferometry in six degree-of-freedom motion error measurements. Int. J. Precis. Eng. Manuf.
**2012**, 13, 1509–1515. [Google Scholar] [CrossRef] - Lee, C.B.; Lee, S.K. Multi-degree-of-freedom motion error measurement in an ultraprecision machine using laser encoder – Review. J. Mech. Sci. Technol.
**2013**, 27, 141–152. [Google Scholar] [CrossRef] - Gao, W. Precision Nanometrology: Sensors and Measuring Systems for Nanomanufacturing; Springer: London, UK, 2010. [Google Scholar]
- Liu, C.H.; Jywe, W.Y.; Jeng, Y.R.; Hsu, T.H.; Wang, M.S.; Deng, S.Y. Development of a straightness measuring system and compensation technique using multiple corner cubes for precision stages. Proc. IMechE Part B J. Eng. Manuf.
**2010**, 224, 483–492. [Google Scholar] [CrossRef] - Jywe, W.Y.; Chou, C.T.; Chen, C.J.; Yang, T.Y.; Jwo, H.H. Development of a three-dimensional contouring measuring system and error compensation method for a CNC machine tool. Proc. IMechE Part B J. Eng. Manufact.
**2007**, 221, 1755–1761. [Google Scholar] [CrossRef] - Liu, C.H.; Jywe, W.Y.; Hsu, C.C.; Hsu, T.H. Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage. Rev. Sci. Instrum.
**2005**, 76, 055110. [Google Scholar] [CrossRef] - 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] - Fan, K.C.; Chen, M.J. A 6-degree-of-freedom measurement system for the accuracy of X-Y stages. Precis. Eng.
**2000**, 24, 15–23. [Google Scholar] [CrossRef] - Cui, C.; Feng, Q.; Zhang, B.; Zhao, Y. System for simultaneously measuring 6DOF geometric motion errors using a polarization maintaining fiber-coupled dual-frequency laser. Opt. Express
**2016**, 24, 6735–6748. [Google Scholar] [CrossRef] [PubMed] - Lee, S.W.; Mayor, R.; Ni, J. Development of a six-degree-of-freedom geometric error measurement system for a meso-scale machine tool. J. Manuf. Sci. Eng.-Trans. ASME
**2005**, 127, 857–865. [Google Scholar] [CrossRef] - Feng, Q.; Zhang, B.; Cui, C.; Kuang, C.; Zhai, Y.; You, F. Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide. Opt. Express
**2013**, 21, 25805–25819. [Google Scholar] - Wang, W.; Kweon, S.H.; Hwang, C.S.; Kang, N.C.; Kim, Y.S.; Yang, S.H. Development of an optical measuring system for integrated geometric errors of a three-axis miniaturized machine tool. Int. J. Adv. Manuf. Technol.
**2009**, 43, 701–709. [Google Scholar] [CrossRef] - Allred, C.J.; Jolly, M.R.; Buckner, G.D. Real-time estimation of helicopter blade kinematics using integrated linear displacement sensors. Aerosp. Sci. Technol.
**2015**, 42, 274–286. [Google Scholar] [CrossRef] - Mura, A. Six d.o.f. displacement measuring device based on a modified Stewart platform. Mechatronics
**2011**, 21, 1309–1316. [Google Scholar] [CrossRef] - Mura, A. Multi-dofs MEMS displacement sensors based on the Stewart platform theory. Microsyst. Technol.
**2012**, 18, 575–579. [Google Scholar] [CrossRef] - Mura, A. Sensitivity analysis of a six degrees of freedom displacement measuring device. Proc. Inst. Mech. Eng. C
**2014**, 228, 158–168. [Google Scholar] [CrossRef] - 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] [Green Version] - Lee, H.W.; Liu, C.H. High precision optical sensors for real-time on-line measurement of straightness and angular errors for smart manufacturing. Smart Sci.
**2016**, 4, 134–141. [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] - Kuang, C.; Feng, Q.; Zhang, B.; Liu, B.; Chen, S.; Zhang, Z. A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module. Sens. Actuators A
**2005**, 125, 100–108. [Google Scholar] [CrossRef] - Kuang, C.; Hong, E.; Ni, J. A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques. Rev. Sci. Instrum.
**2007**, 78, 095105. [Google Scholar] [CrossRef] [PubMed] - Gao, W.; Arai, Y.; Shibuya, A.; Kiyono, S.; Park, C.H. Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage. Precis. Eng.
**2006**, 30, 96–103. [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] - Chen, B.; Xu, B.; Yan, L.; Zhang, E.; Liu, Y. Laser straightness interferometer system with rotational error compensation and simultaneous measurement of six degrees of freedom error parameters. Opt. Express
**2015**, 23, 9052–9073. [Google Scholar] [CrossRef] [PubMed] - Huang, H.L.; Liu, C.H.; Jywe, W.Y.; Wang, M.S.; Fan, T.H. Development of a three-degree-of-freedom laser linear encoder for error measurement of a high precision stage. Rev. Sci. Instrum.
**2007**, 78, 066103. [Google Scholar] [CrossRef] [PubMed] - Liu, C.H.; Huang, H.L.; Lee, H.W. Five-degrees-of-freedom diffractive laser encoder. Appl. Opt.
**2009**, 48, 2767–2777. [Google Scholar] [CrossRef] - Kimura, A.; Gao, W.; Lijiang, Z. Position and out-of-straightness measurement of a precision linear air-bearing stage by using a two-degree-of-freedom linear encoder. Meas. Sci. Technol.
**2010**, 21, 054005. [Google Scholar] [CrossRef] - Cui, C.; Feng, Q.; Zhang, B. Compensation for straightness measurement systematic errors in six degree-of-freedom motion error simultaneous measurement system. Appl. Opt.
**2015**, 54, 3122–3131. [Google Scholar] [CrossRef] [PubMed] - Gao, S.; Zhang, B.; Feng, Q.; Cui, C.; Chen, S.; Zhao, Y. Errors crosstalk analysis and compensation in the simultaneous measuring system for five-degree-of-freedom geometric error. Appl. Opt.
**2015**, 54, 458–466. [Google Scholar] [CrossRef] - Lou, Y.; Yan, L.; Chen, B.; Zhang, S. Laser homodyne straightness interferometer with simultaneous measurement of six degrees of freedom motion errors for precision linear stage metrology. Opt. Express
**2017**, 25, 6805–6821. [Google Scholar] [CrossRef] [PubMed] - Zhao, Y.; Zhang, B.; Feng, Q. Measurement system and model for simultaneously measuring 6DOF geometric errors. Opt. Express
**2017**, 25, 20993–21007. [Google Scholar] [CrossRef] [PubMed] - Liu, C.S.; Jiang, S.H. A novel laser displacement sensor with improved robustness toward geometrical fluctuations of the laser beam. Meas. Sci. Technol.
**2013**, 24, 105101. [Google Scholar] [CrossRef] - Liu, C.S.; Jiang, S.H. Precise autofocusing microscope with rapid response. Opt. Lasers Eng.
**2015**, 66, 294–300. [Google Scholar] [CrossRef] - Liu, C.S.; Lin, Y.C.; Hu, P.H. Design and characterization of precise laser-based autofocusing microscope with reduced geometrical fluctuations. Microsyst. Technol.
**2015**, 19, 1717–1724. [Google Scholar] [CrossRef] - Liu, C.S.; Lin, K.W. Numerical and experimental characterization of reducing geometrical fluctuations of laser beam based on rotating optical diffuser. Opt. Eng.
**2014**, 53, 122408. [Google Scholar] [CrossRef] - Liu, C.S.; Lin, P.D. Jacobian and Hessian matrices of optical path length for computing the wave front shape, irradiance, and caustics in optical systems. J. Opt. Soc. Am. A-Opt. Image Sci. Vis.
**2012**, 29, 2272–2280. [Google Scholar] - Chen, Y.T.; Huang, Y.S.; Liu, C.S. An optical sensor for measuring the position and slanting direction of flat surfaces. Sensors
**2016**, 16, 1061. [Google Scholar] [CrossRef] [PubMed] - Tsai, C.Y. Free-form surface design method for a collimator TIR lens. J. Opt. Soc. Am. A-Opt. Image Sci. Vis.
**2016**, 33, 785–792. [Google Scholar] [CrossRef] [PubMed] - Lin, P.D. New Computation Methods for Geometrical Optics; Springer: Singapore, 2013. [Google Scholar]
- Rodríguez-Navarro, D.; Lázaro-Galilea, J.L.; Bravo-Muñoz, I.; Gardel-Vicente, A.; Tsirigotis, G. Analysis and calibration of sources of electronic error in PSD sensor response. Sensors
**2016**, 16, 619. [Google Scholar] [CrossRef] [PubMed]

**Figure 2.**Motion error-induced changes in positions of light spots on position sensitive detectors (PSDs). (

**a**) Linear error along the Y-axis (δ

_{y}) or Z-axis (δ

_{z}). (

**b**) Angular error along the Z-axis (ε

_{z}) or Y-axis (ε

_{y}). (

**c**) Angular error along the X-axis (δ

_{x}).

**Figure 3.**Fluctuation-induced changes in positions of light spots on PSDs. (

**a**) Linear fluctuation along the Y-axis or Z-axis. (

**b**) Angular fluctuation along the Y-axis or Z-axis.

**Figure 9.**Experimental results for variation of geometric motion error with position: (

**a**) horizontal straightness, (

**b**) vertical straightness, (

**c**) pitch, (

**d**) yaw, (

**e**) roll, and (

**f**) positioning errors, respectively.

**Figure 10.**Comparative test results of proposed measurement system and Renishaw laser interferometer.

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/s18113875

**AMA Style**

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(11):3875.
https://doi.org/10.3390/s18113875

**Chicago/Turabian Style**

Liu, Chien-Sheng, Yu-Fan Pu, Yu-Ta Chen, and Yong-Tai Luo.
2018. "Design of a Measurement System for Simultaneously Measuring Six-Degree-Of-Freedom Geometric Errors of a Long Linear Stage" *Sensors* 18, no. 11: 3875.
https://doi.org/10.3390/s18113875