Paper:

# Square Layout Four-Point Method for Two-Dimensional Profile Measurement and Self-Calibration Method of Zero-Adjustment Error

## Hiroki Shimizu^{†}, Ryousuke Yamashita, Takuya Hashiguchi, Tasuku Miyata, and Yuuma Tamaru

Kyushu Institute of Technology

1-1 Sensui-cho, Tobata-ku, Kitakyushu-shi, Fukuoka 804-8550, Japan

^{†}Corresponding author

An on-machine measurement method, called the square-layout four-point (SLFP) method with angle compensation, for evaluating two-dimensional (2-D) profiles of flat machined surfaces is proposed. In this method, four displacement sensors are arranged in a square and mounted to the scanning table of a 2-D stage. For measuring the 2-D profile of a target plane, height data corresponding to all measuring points are acquired by means of the raster scanning motion. At the same time, pitching data of the first primary scan line and rolling data of the first subsidiary scan line are monitored by means of two auto-collimators to compensate for major profile errors that arise out of the posture error. Use of the SLFP method facilitates connection of the results of straightness-measurements results obtained for each scanning line by using two additional sensors and rolling data of the first subsidiary scan line. Specifically, the height of a measuring point is calculated by means of a recurrence equation using three predetermined height data for adjacent points in conjunction with data acquired by the four displacement sensors. Results of the numerical simulation performed in this study demonstrate higher efficiency of the SLFP method with angle compensation. During actual measurement, however, it is difficult to perfectly align inline the origin height of each displacement sensor. With regard to the SLFP method, zero-adjustment error is defined as the relative height of a sensor’s origin with respect to the plane comprising origins of the other three sensors. This error accumulates in proportion to number of times the recurrence equation is applied. Simulation results containing the zero-adjustment error demonstrate that accumulation of the said error results in unignorable distortion of measurement results. Therefore, a new self-calibration method for the zero-adjustment error has been proposed. During 2-D profile measurement, two different calculation paths – the raster scan path and orthogonal path – can be used to determine the height of a measurement point. Although heights determined through use of the two paths must ideally be equal, they are observed to be different because accumulated zero-adjustment errors for the two paths are different. In view of this result, the zero-adjustment error can be calculated backwards and calibrated. Validity of the calibration method has been confirmed via simulations and experiments.

*Int. J. Automation Technol.*, Vol.12, No.5, pp. 707-713, 2018.

- [1] W. R. Moore, “Foundations of mechanical accuracy,” The Moore Special Company, 1979.
- [2] A. E. Ennos and M. S. Virdee “High Accuracy Profile Measurement of Quasi-conical Mirror Surfaces by Laser Autocollimation,” Precision Engineering, Vol.4, No.1, pp. 5-8, 1982.
- [3] M. S. Virdee, “Non-contacting straightness measurement to nanometre accuracy,” Int. J. of Machine Tools and Manufacture, Vol.35, No.2, pp. 157-164, 1995.
- [4] S. Kiyono, “Ultra Precision Measurement,” 2007 (in Japanese).
- [5] D. J. Whitehouse, “Some theoretical aspects of error separation techniques in surface metrology,” J. of Physics E: Scientific Instruments, Vol.9, pp. 531-536, 1976.
- [6] C. J. Evans, R. J. Hocken, and W. T. Estler, “Self-Calibration: Reversal, Redundancy, Error Separation, and ‘Absolute Testing’,” CIRP Annals, Vol.45, No.2, pp. 617-634, 1996.
- [7] H. Tanaka and H. Sato, “Basic characteristics of straightness measurement by sequential two point method,” J. of JSME Series C, Vol.48, No.436, pp. 1930-1937, 1982 (in Japanese).
- [8] M. Obi and S. Furukawa, “A Study of the Measuring for Straightness Using a Sequential Point Method (1st Report, The Functions of Sequential Point Methods and Theoretical Analysis of the Error),” J. of JSME Series C, Vol.57, No.542, pp. 3197-3201, 1991 (in Japanese).
- [9] K. Tozawa, H. Sato, and M. O-hori, “A new method for the measurement of the straightness of machine tools and machined work,” J. Mech. Des., Vol.104, No.3, pp. 587-592, 1982.
- [10] S. Kiyono and W. Gao, “Estimation and improvement of accidental and systematic errors in profile measurements using software datum,” J. of JSME Series C, Vol.58, No.551, pp. 2262-2267, 1992 (in Japanese).
- [11] E. Okuyama and H. Ishikawa, “Generalized two-point method using inverse filtering for surface profile measurement – theoretical analysis and experimental results for error propagation –,” Int. J. Automation Technol., Vol.8, No.1, pp. 43-48, 2014.
- [12] E. Okuyama, “Multi-probe method for straightness profile measurement based on least uncertainty propagation (1st report): Two-point method considering cross-axis translational motion and sensor’s random error,” Precision Engineering, Vol.34, No.1, pp. 49-54, 2010.
- [13] E. Okuyama, K. Konda, and H. Ishikawa, “Surface profile measurement based on the concept of multi-step division of length,” Int. J. Automation Technol., Vol.11, No.5, pp. 716-720, 2017.
- [14] W. Gao and S. Kiyono, “On-machine profile measurement of machined surface using the combined three-point method,” JSME Int. J. Series C, Vol.40, No.2, pp. 253-259, 1997.
- [15] P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, “Multi-probe scanning system comprising three laser interferometers and one autocollimator for measuring flat bar mirror profile with nanometer accuracy,” Precision Engineering, Vol.35, No.4, pp. 686-692, 2011.
- [16] I. Fujimoto, T. Takatsuji, K. Nishimura, and Y. S. Pyun, “An uncertainty analysis of displacement sensors with the three-point method,” Measurement Science and Technology, Vol.23, No.11, p. 115102, 2012.
- [17] I. Weingärtner and C. Elster, “System of four distance sensors for high-accuracy measurement of topography,” Precision Engineering, Vol.28, No.2, pp. 164-170, 2004.
- [18] W. Gao, J. Yokoyama, H. Kojima, and S. Kiyono, “Precision measurement of cylinder straightness using a scanning multi-probe system,” Precision Engineering, Vol.26, No.3, pp. 279-288, 2002.
- [19] S. Itoh, T. Narikiyo, Y. Satoh, and Y. Okada, “Measurement of flat form error by a 2 D least square serial two point method,” J. of the Japan Society for Precision Engineering, Vol.57, No.10, pp. 1844-1849, 1991 (in Japanese).
- [20] Z. Ge, W. Gao, and S. Kiyono, “Basic study on measurement of 2-D surface profile. (2nd report: measurement error analysis),” JSME Int. J. Series C, Vol.40, No.3, pp. 439-446, 1997.