# Design of a Measurement System for Six-Degree-of-Freedom Geometric Errors of a Linear Guide of a Machine Tool

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## Abstract

**:**

## 1. Introduction

## 2. Structure Layout and Measuring Principle

## 3. Numerical Simulation and Mathematical Model

_{PSD1}= F

_{X}

_{1}(δ

_{x}, δ

_{z}, ε

_{x}, ε

_{y}, ε

_{z}),

_{PSD1}= F

_{Y}

_{1}(δ

_{x}, δ

_{z}, ε

_{x}, ε

_{y}, ε

_{z}),

_{PSD2}= F

_{X2}(δ

_{x}, δ

_{z}, ε

_{x}, ε

_{y}, ε

_{z}),

_{PSD2}= F

_{Y2}(δ

_{x}, δ

_{z}, ε

_{x}, ε

_{y}, ε

_{z}),

_{PSD3}= F

_{X3}(δ

_{x}, δ

_{z}, ε

_{x}, ε

_{y}, ε

_{z}),

_{PSD3}= F

_{Y3}(δ

_{x}, δ

_{z}, ε

_{x}, ε

_{y}, ε

_{z})

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

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

_{PSD1}= Aεy+Bεz+C,

_{PSD1}= Dεx+E,

_{PSD2}= Fεz+G,

_{PSD2}= Hεx+I,

_{PSD3}= Jδx+Kεy+Lεz+M,

_{PSD3}= Nδz+Oεy+Pεx+Q,

## 4. Experimental characterization

## 5. Conclusions

## 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] - 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] - Renishaw plc, “XL-80,” Renishaw. Available online: https://www.renishaw.com.tw/tw/xl-80-laser- system--8268 (accessed on 14 November 2018).
- Okafor, A.C.; Ertekin, Y.M. Vertical machining center accuracy characterization using laser interferometer, part one: Linear positional errors. J. Mater. Process. Technol.
**2000**, 105, 394–406. [Google Scholar] [CrossRef] - Okafor, A.C.; Ertekin, Y.M. Vertical machining center accuracy characterization using laser interferometer, part two: Angular errors. J. Mater. Process. Technol.
**2000**, 105, 407–420. [Google Scholar] [CrossRef] - 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] - 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] - Sun, Y.T.; Hu, J.C.; Chen, L.M. Study on fast and precise measurement of three-dimensional displacement using hall sensors. Adv. Mater. Res.
**2013**, 694–697, 1034–1038. [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 dof 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] - Fan, K.C.; Chen, M.J.; Huang, W.M. A six-degree-of-freedom measurement system for the motion accuracy of linear stages. Int. J. Mach. Tools Manuf.
**1998**, 38, 155–164. [Google Scholar] [CrossRef] - 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] - 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] - 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, 97–103. [Google Scholar] [CrossRef] - Gao, W.; Saito, Y.; Muto, H.; Arai, Y.; Shimizu, Y. A three-axis autocollimator for detection of angular error motions of a precision stage. CIRP Ann.-Manuf. Tech.
**2011**, 60, 515–518. [Google Scholar] [CrossRef] - Kuang, C.F.; Hong, H.; Ni, J. A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques. AIP Rev. Sci. Instrum.
**2007**, 78, 095105. [Google Scholar] [CrossRef] [PubMed] - Feng, Q.; Zhang, B.; Kuang, C. Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift. Int. J. Precis. Eng. Manuf.
**2008**, 9, 26–31. [Google Scholar] - 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] - Zhao, Y.; Zhang, B.; Feng, Q. Measurement system and model for simultaneously measuring 6DOF geometric errors. Opt. Express
**2017**, 25, 20993–201007. [Google Scholar] [CrossRef] - 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] - 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] - Liu, C.S.; Lin, P.D. Determination of linear equations of position sensing detectors in small motion measurement systems. J. Opt. Soc. Am. A-Opt. Image Sci. Vis.
**2010**, 27, 2480–2487. [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] - Lin, P.D. New Computation Methods for Geometrical Optics; Springer: Singapore, 2013. [Google Scholar]
- Chang, Y.H.; Liu, C.S.; Chen, C.C. Design and characterization of a fast steering mirror compensation system based on double Porro prisms by a screw-ray tracing method. Sensors
**2018**, 18, 4046. [Google Scholar] [CrossRef] [PubMed] - 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.; 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]

**Figure 7.**Simulation results showing variation of light spot positions on PSDs with geometric errors.

**Figure 9.**Verification of the mathematical model: (

**a**) pitch, (

**b**) roll, (

**c**) yaw, (

**d**) horizontal straightness, and (

**e**) vertical straightness, respectively.

**Figure 12.**Measurement results for variation of geometric error with position: (

**a**) pitch, (

**b**) yaw, (

**c**) horizontal straightness, (

**d**) vertical straightness, (

**e**) roll, and (

**f**) positioning errors, respectively.

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**MDPI and ACS Style**

Liu, C.-S.; Lai, J.-J.; Luo, Y.-T.
Design of a Measurement System for Six-Degree-of-Freedom Geometric Errors of a Linear Guide of a Machine Tool. *Sensors* **2019**, *19*, 5.
https://doi.org/10.3390/s19010005

**AMA Style**

Liu C-S, Lai J-J, Luo Y-T.
Design of a Measurement System for Six-Degree-of-Freedom Geometric Errors of a Linear Guide of a Machine Tool. *Sensors*. 2019; 19(1):5.
https://doi.org/10.3390/s19010005

**Chicago/Turabian Style**

Liu, Chien-Sheng, Jia-Jun Lai, and Yong-Tai Luo.
2019. "Design of a Measurement System for Six-Degree-of-Freedom Geometric Errors of a Linear Guide of a Machine Tool" *Sensors* 19, no. 1: 5.
https://doi.org/10.3390/s19010005