Abstract

This paper presents and evaluates an active dual-sensor autofocusing system that combines an optical vision sensor and a tactile probe for autofocusing on arrays of small holes on freeform surfaces. The system has been tested on a two-axis test rig and then integrated onto a three-axis computer numerical control (CNC) milling machine, where the aim is to rapidly and controllably measure the hole position errors while the part is still on the machine. The principle of operation is for the tactile probe to locate the nominal positions of holes, and the optical vision sensor follows to focus and capture the images of the holes. The images are then processed to provide hole position measurement. In this paper, the autofocusing deviations are analyzed. First, the deviations caused by the geometric errors of the axes on which the dual-sensor unit is deployed are estimated to be 11 μm when deployed on a test rig and 7 μm on the CNC machine tool. Subsequently, the autofocusing deviations caused by the interaction of the tactile probe, surface, and small hole are mathematically analyzed and evaluated. The deviations are a result of the tactile probe radius, the curvatures at the positions where small holes are drilled on the freeform surface, and the effect of the position error of the hole on focusing. An example case study is provided for the measurement of a pattern of small holes on an elliptical cylinder on the two machines. The absolute sum of the autofocusing deviations is 118 μm on the test rig and 144 μm on the machine tool. This is much less than the 500 μm depth of field of the optical microscope. Therefore, the method is capable of capturing a group of clear images of the small holes on this workpiece for either implementation.

© 2014 Optical Society of America

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References

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  1. A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
    [CrossRef]
  2. http://www.werth.de/en/user-angebot/products/sensors.html .
  3. H. Cao, X. Chen, K. T. V. Grattan, and Y. Sun, “Automatic micro dimension measurement using image processing methods,” Measurement 31, 71–76 (2002).
    [CrossRef]
  4. Z. G. Fei, J. J. Guo, and C. S. Li, “A new method for detecting the diameter and spatial location of tiny through-hole,” Adv. Mater. Res. 189–193, 4186–4190 (2011).
  5. Y. Luo, D. Hu, and T. Gu, “Research on real-time error measurement in curve grinding process based on machine vision,” in IEEE International Technology and Innovation Conference (2006), pp. 656–661.
  6. H. P. Syahputra and I. J. Ko, “Application of image processes to micro-milling process for surface texturing,” Int. J. Precis. Eng. Manuf. 14, 1507–1512 (2013).
    [CrossRef]
  7. A. Zhou, J. Guo, W. Shao, and J. Yang, “Multipose measurement of surface defects on rotary metal parts with a combined laser-and-camera sensor,” Opt. Eng. 52, 104104 (2013).
    [CrossRef]
  8. R. Yusvana, D. Headon, and G. H. Markx, “Isolation and measurement of the features of arrays of cel aggregates formed by dielectrophoresis using the user-specified multi regions masking (MRM) technique,” J. Phys.: Conf. Ser. 183, 012022 (2009).
    [CrossRef]
  9. T. Shu, Y. Zheng, and Z. Shi, “Image processing-based wheel steer angle detection,” J. Electron. Imaging 22, 043005 (2013).
    [CrossRef]
  10. F. Shen, L. Hodgson, and K. Hahn, “Digital autofocus methods for automated microscopy,” Methods Enzymol. 414, 620–632 (2006).
    [CrossRef]
  11. M. Yang and L. Luo, “A rapid auto-focus method in automatic microscope,” in IEEE International Conference on Signal Processing (ICSP) (2008), pp. 502–505.
  12. C. J. Kuo and C. Chiu, “Improved auto-focus search algorithms for CMOS image-sensing module,” J. Inf. Sci. Eng. 27, 1377–1393 (2011).
  13. X. Qiu and Z. Yang, “Study on entropy function optimization problem in auto-focusing algorithm applied for radar imaging,” in IEEE International Conference on Microwave and Millimeter Wave Technology (ICMMT) (2008), pp. 2055–2058.
  14. A. B. Mario, Á. B. Josué, A. Leonardo, and C. María, “Fast autofocus algorithm for automated microscopes,” Opt. Eng. 44, 063601 (2005).
    [CrossRef]
  15. W. Chen, M. Er, and S. Wu, “Illumination compensation and normalization for robust face recognition using discrete cosine transform in logarithm domain,” IEEE Trans. Syst. Man Cybern., Part B, Cybern. 36, 458–466 (2006).
    [CrossRef]
  16. J. He, R. Zhou, and Z. Hong, “Modified fast climbing search auto-focus algorithm with adaptive step size searching technique for digital camera,” IEEE Trans. Consum. Electron. 49, 257–262 (2003).
  17. http://www.optiv.net/ .
  18. X. Chen, A. Longstaff, S. Parkinson, and A. Myers, “A method for rapid detection and evaluation of position errors of patterns of small holes on complex curved and freeform surfaces,” Int. J. Precis. Eng. Manuf. 15, 209–217 (2014).
    [CrossRef]
  19. , Test code for machine tools—Part 2: Determination of accuracy and repeatability of positioning of numerically controlled axes, ISO (2011).
  20. , Test code for machine tools—Part 1: Geometric accuracy of machines operating under no-load or quasi-static conditions, ISO (2012).
  21. A. Jeffrey, Mathematics for Engineers and Scientists, 6th ed. (Chapman & Hall, 2004).

2014

X. Chen, A. Longstaff, S. Parkinson, and A. Myers, “A method for rapid detection and evaluation of position errors of patterns of small holes on complex curved and freeform surfaces,” Int. J. Precis. Eng. Manuf. 15, 209–217 (2014).
[CrossRef]

2013

H. P. Syahputra and I. J. Ko, “Application of image processes to micro-milling process for surface texturing,” Int. J. Precis. Eng. Manuf. 14, 1507–1512 (2013).
[CrossRef]

A. Zhou, J. Guo, W. Shao, and J. Yang, “Multipose measurement of surface defects on rotary metal parts with a combined laser-and-camera sensor,” Opt. Eng. 52, 104104 (2013).
[CrossRef]

T. Shu, Y. Zheng, and Z. Shi, “Image processing-based wheel steer angle detection,” J. Electron. Imaging 22, 043005 (2013).
[CrossRef]

2011

C. J. Kuo and C. Chiu, “Improved auto-focus search algorithms for CMOS image-sensing module,” J. Inf. Sci. Eng. 27, 1377–1393 (2011).

Z. G. Fei, J. J. Guo, and C. S. Li, “A new method for detecting the diameter and spatial location of tiny through-hole,” Adv. Mater. Res. 189–193, 4186–4190 (2011).

2009

R. Yusvana, D. Headon, and G. H. Markx, “Isolation and measurement of the features of arrays of cel aggregates formed by dielectrophoresis using the user-specified multi regions masking (MRM) technique,” J. Phys.: Conf. Ser. 183, 012022 (2009).
[CrossRef]

2006

A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
[CrossRef]

F. Shen, L. Hodgson, and K. Hahn, “Digital autofocus methods for automated microscopy,” Methods Enzymol. 414, 620–632 (2006).
[CrossRef]

W. Chen, M. Er, and S. Wu, “Illumination compensation and normalization for robust face recognition using discrete cosine transform in logarithm domain,” IEEE Trans. Syst. Man Cybern., Part B, Cybern. 36, 458–466 (2006).
[CrossRef]

2005

A. B. Mario, Á. B. Josué, A. Leonardo, and C. María, “Fast autofocus algorithm for automated microscopes,” Opt. Eng. 44, 063601 (2005).
[CrossRef]

2003

J. He, R. Zhou, and Z. Hong, “Modified fast climbing search auto-focus algorithm with adaptive step size searching technique for digital camera,” IEEE Trans. Consum. Electron. 49, 257–262 (2003).

2002

H. Cao, X. Chen, K. T. V. Grattan, and Y. Sun, “Automatic micro dimension measurement using image processing methods,” Measurement 31, 71–76 (2002).
[CrossRef]

Cao, H.

H. Cao, X. Chen, K. T. V. Grattan, and Y. Sun, “Automatic micro dimension measurement using image processing methods,” Measurement 31, 71–76 (2002).
[CrossRef]

Chen, W.

W. Chen, M. Er, and S. Wu, “Illumination compensation and normalization for robust face recognition using discrete cosine transform in logarithm domain,” IEEE Trans. Syst. Man Cybern., Part B, Cybern. 36, 458–466 (2006).
[CrossRef]

Chen, X.

X. Chen, A. Longstaff, S. Parkinson, and A. Myers, “A method for rapid detection and evaluation of position errors of patterns of small holes on complex curved and freeform surfaces,” Int. J. Precis. Eng. Manuf. 15, 209–217 (2014).
[CrossRef]

H. Cao, X. Chen, K. T. V. Grattan, and Y. Sun, “Automatic micro dimension measurement using image processing methods,” Measurement 31, 71–76 (2002).
[CrossRef]

Chiu, C.

C. J. Kuo and C. Chiu, “Improved auto-focus search algorithms for CMOS image-sensing module,” J. Inf. Sci. Eng. 27, 1377–1393 (2011).

Er, M.

W. Chen, M. Er, and S. Wu, “Illumination compensation and normalization for robust face recognition using discrete cosine transform in logarithm domain,” IEEE Trans. Syst. Man Cybern., Part B, Cybern. 36, 458–466 (2006).
[CrossRef]

Fei, Z. G.

Z. G. Fei, J. J. Guo, and C. S. Li, “A new method for detecting the diameter and spatial location of tiny through-hole,” Adv. Mater. Res. 189–193, 4186–4190 (2011).

Grattan, K. T. V.

H. Cao, X. Chen, K. T. V. Grattan, and Y. Sun, “Automatic micro dimension measurement using image processing methods,” Measurement 31, 71–76 (2002).
[CrossRef]

Gu, T.

Y. Luo, D. Hu, and T. Gu, “Research on real-time error measurement in curve grinding process based on machine vision,” in IEEE International Technology and Innovation Conference (2006), pp. 656–661.

Guo, J.

A. Zhou, J. Guo, W. Shao, and J. Yang, “Multipose measurement of surface defects on rotary metal parts with a combined laser-and-camera sensor,” Opt. Eng. 52, 104104 (2013).
[CrossRef]

Guo, J. J.

Z. G. Fei, J. J. Guo, and C. S. Li, “A new method for detecting the diameter and spatial location of tiny through-hole,” Adv. Mater. Res. 189–193, 4186–4190 (2011).

Hahn, K.

F. Shen, L. Hodgson, and K. Hahn, “Digital autofocus methods for automated microscopy,” Methods Enzymol. 414, 620–632 (2006).
[CrossRef]

He, J.

J. He, R. Zhou, and Z. Hong, “Modified fast climbing search auto-focus algorithm with adaptive step size searching technique for digital camera,” IEEE Trans. Consum. Electron. 49, 257–262 (2003).

Headon, D.

R. Yusvana, D. Headon, and G. H. Markx, “Isolation and measurement of the features of arrays of cel aggregates formed by dielectrophoresis using the user-specified multi regions masking (MRM) technique,” J. Phys.: Conf. Ser. 183, 012022 (2009).
[CrossRef]

Hodgson, L.

F. Shen, L. Hodgson, and K. Hahn, “Digital autofocus methods for automated microscopy,” Methods Enzymol. 414, 620–632 (2006).
[CrossRef]

Hoffmann, J.

A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
[CrossRef]

Hong, Z.

J. He, R. Zhou, and Z. Hong, “Modified fast climbing search auto-focus algorithm with adaptive step size searching technique for digital camera,” IEEE Trans. Consum. Electron. 49, 257–262 (2003).

Hu, D.

Y. Luo, D. Hu, and T. Gu, “Research on real-time error measurement in curve grinding process based on machine vision,” in IEEE International Technology and Innovation Conference (2006), pp. 656–661.

Jeffrey, A.

A. Jeffrey, Mathematics for Engineers and Scientists, 6th ed. (Chapman & Hall, 2004).

Josué, Á. B.

A. B. Mario, Á. B. Josué, A. Leonardo, and C. María, “Fast autofocus algorithm for automated microscopes,” Opt. Eng. 44, 063601 (2005).
[CrossRef]

Ko, I. J.

H. P. Syahputra and I. J. Ko, “Application of image processes to micro-milling process for surface texturing,” Int. J. Precis. Eng. Manuf. 14, 1507–1512 (2013).
[CrossRef]

Kuo, C. J.

C. J. Kuo and C. Chiu, “Improved auto-focus search algorithms for CMOS image-sensing module,” J. Inf. Sci. Eng. 27, 1377–1393 (2011).

Leonardo, A.

A. B. Mario, Á. B. Josué, A. Leonardo, and C. María, “Fast autofocus algorithm for automated microscopes,” Opt. Eng. 44, 063601 (2005).
[CrossRef]

Li, C. S.

Z. G. Fei, J. J. Guo, and C. S. Li, “A new method for detecting the diameter and spatial location of tiny through-hole,” Adv. Mater. Res. 189–193, 4186–4190 (2011).

Longstaff, A.

X. Chen, A. Longstaff, S. Parkinson, and A. Myers, “A method for rapid detection and evaluation of position errors of patterns of small holes on complex curved and freeform surfaces,” Int. J. Precis. Eng. Manuf. 15, 209–217 (2014).
[CrossRef]

Luo, L.

M. Yang and L. Luo, “A rapid auto-focus method in automatic microscope,” in IEEE International Conference on Signal Processing (ICSP) (2008), pp. 502–505.

Luo, Y.

Y. Luo, D. Hu, and T. Gu, “Research on real-time error measurement in curve grinding process based on machine vision,” in IEEE International Technology and Innovation Conference (2006), pp. 656–661.

María, C.

A. B. Mario, Á. B. Josué, A. Leonardo, and C. María, “Fast autofocus algorithm for automated microscopes,” Opt. Eng. 44, 063601 (2005).
[CrossRef]

Mario, A. B.

A. B. Mario, Á. B. Josué, A. Leonardo, and C. María, “Fast autofocus algorithm for automated microscopes,” Opt. Eng. 44, 063601 (2005).
[CrossRef]

Markx, G. H.

R. Yusvana, D. Headon, and G. H. Markx, “Isolation and measurement of the features of arrays of cel aggregates formed by dielectrophoresis using the user-specified multi regions masking (MRM) technique,” J. Phys.: Conf. Ser. 183, 012022 (2009).
[CrossRef]

Myers, A.

X. Chen, A. Longstaff, S. Parkinson, and A. Myers, “A method for rapid detection and evaluation of position errors of patterns of small holes on complex curved and freeform surfaces,” Int. J. Precis. Eng. Manuf. 15, 209–217 (2014).
[CrossRef]

Parkinson, S.

X. Chen, A. Longstaff, S. Parkinson, and A. Myers, “A method for rapid detection and evaluation of position errors of patterns of small holes on complex curved and freeform surfaces,” Int. J. Precis. Eng. Manuf. 15, 209–217 (2014).
[CrossRef]

Peggs, G.

A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
[CrossRef]

Qiu, X.

X. Qiu and Z. Yang, “Study on entropy function optimization problem in auto-focusing algorithm applied for radar imaging,” in IEEE International Conference on Microwave and Millimeter Wave Technology (ICMMT) (2008), pp. 2055–2058.

Shao, W.

A. Zhou, J. Guo, W. Shao, and J. Yang, “Multipose measurement of surface defects on rotary metal parts with a combined laser-and-camera sensor,” Opt. Eng. 52, 104104 (2013).
[CrossRef]

Shen, F.

F. Shen, L. Hodgson, and K. Hahn, “Digital autofocus methods for automated microscopy,” Methods Enzymol. 414, 620–632 (2006).
[CrossRef]

Shi, Z.

T. Shu, Y. Zheng, and Z. Shi, “Image processing-based wheel steer angle detection,” J. Electron. Imaging 22, 043005 (2013).
[CrossRef]

Shu, T.

T. Shu, Y. Zheng, and Z. Shi, “Image processing-based wheel steer angle detection,” J. Electron. Imaging 22, 043005 (2013).
[CrossRef]

Sun, Y.

H. Cao, X. Chen, K. T. V. Grattan, and Y. Sun, “Automatic micro dimension measurement using image processing methods,” Measurement 31, 71–76 (2002).
[CrossRef]

Syahputra, H. P.

H. P. Syahputra and I. J. Ko, “Application of image processes to micro-milling process for surface texturing,” Int. J. Precis. Eng. Manuf. 14, 1507–1512 (2013).
[CrossRef]

Weckenmann, A.

A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
[CrossRef]

Wu, S.

W. Chen, M. Er, and S. Wu, “Illumination compensation and normalization for robust face recognition using discrete cosine transform in logarithm domain,” IEEE Trans. Syst. Man Cybern., Part B, Cybern. 36, 458–466 (2006).
[CrossRef]

Yang, J.

A. Zhou, J. Guo, W. Shao, and J. Yang, “Multipose measurement of surface defects on rotary metal parts with a combined laser-and-camera sensor,” Opt. Eng. 52, 104104 (2013).
[CrossRef]

Yang, M.

M. Yang and L. Luo, “A rapid auto-focus method in automatic microscope,” in IEEE International Conference on Signal Processing (ICSP) (2008), pp. 502–505.

Yang, Z.

X. Qiu and Z. Yang, “Study on entropy function optimization problem in auto-focusing algorithm applied for radar imaging,” in IEEE International Conference on Microwave and Millimeter Wave Technology (ICMMT) (2008), pp. 2055–2058.

Yusvana, R.

R. Yusvana, D. Headon, and G. H. Markx, “Isolation and measurement of the features of arrays of cel aggregates formed by dielectrophoresis using the user-specified multi regions masking (MRM) technique,” J. Phys.: Conf. Ser. 183, 012022 (2009).
[CrossRef]

Zheng, Y.

T. Shu, Y. Zheng, and Z. Shi, “Image processing-based wheel steer angle detection,” J. Electron. Imaging 22, 043005 (2013).
[CrossRef]

Zhou, A.

A. Zhou, J. Guo, W. Shao, and J. Yang, “Multipose measurement of surface defects on rotary metal parts with a combined laser-and-camera sensor,” Opt. Eng. 52, 104104 (2013).
[CrossRef]

Zhou, R.

J. He, R. Zhou, and Z. Hong, “Modified fast climbing search auto-focus algorithm with adaptive step size searching technique for digital camera,” IEEE Trans. Consum. Electron. 49, 257–262 (2003).

Adv. Mater. Res.

Z. G. Fei, J. J. Guo, and C. S. Li, “A new method for detecting the diameter and spatial location of tiny through-hole,” Adv. Mater. Res. 189–193, 4186–4190 (2011).

IEEE Trans. Consum. Electron.

J. He, R. Zhou, and Z. Hong, “Modified fast climbing search auto-focus algorithm with adaptive step size searching technique for digital camera,” IEEE Trans. Consum. Electron. 49, 257–262 (2003).

IEEE Trans. Syst. Man Cybern., Part B, Cybern.

W. Chen, M. Er, and S. Wu, “Illumination compensation and normalization for robust face recognition using discrete cosine transform in logarithm domain,” IEEE Trans. Syst. Man Cybern., Part B, Cybern. 36, 458–466 (2006).
[CrossRef]

Int. J. Precis. Eng. Manuf.

H. P. Syahputra and I. J. Ko, “Application of image processes to micro-milling process for surface texturing,” Int. J. Precis. Eng. Manuf. 14, 1507–1512 (2013).
[CrossRef]

X. Chen, A. Longstaff, S. Parkinson, and A. Myers, “A method for rapid detection and evaluation of position errors of patterns of small holes on complex curved and freeform surfaces,” Int. J. Precis. Eng. Manuf. 15, 209–217 (2014).
[CrossRef]

J. Electron. Imaging

T. Shu, Y. Zheng, and Z. Shi, “Image processing-based wheel steer angle detection,” J. Electron. Imaging 22, 043005 (2013).
[CrossRef]

J. Inf. Sci. Eng.

C. J. Kuo and C. Chiu, “Improved auto-focus search algorithms for CMOS image-sensing module,” J. Inf. Sci. Eng. 27, 1377–1393 (2011).

J. Phys.: Conf. Ser.

R. Yusvana, D. Headon, and G. H. Markx, “Isolation and measurement of the features of arrays of cel aggregates formed by dielectrophoresis using the user-specified multi regions masking (MRM) technique,” J. Phys.: Conf. Ser. 183, 012022 (2009).
[CrossRef]

Meas. Sci. Technol.

A. Weckenmann, G. Peggs, and J. Hoffmann, “Probing systems for dimensional micro- and nano-metrology,” Meas. Sci. Technol. 17, 504–509 (2006).
[CrossRef]

Measurement

H. Cao, X. Chen, K. T. V. Grattan, and Y. Sun, “Automatic micro dimension measurement using image processing methods,” Measurement 31, 71–76 (2002).
[CrossRef]

Methods Enzymol.

F. Shen, L. Hodgson, and K. Hahn, “Digital autofocus methods for automated microscopy,” Methods Enzymol. 414, 620–632 (2006).
[CrossRef]

Opt. Eng.

A. B. Mario, Á. B. Josué, A. Leonardo, and C. María, “Fast autofocus algorithm for automated microscopes,” Opt. Eng. 44, 063601 (2005).
[CrossRef]

A. Zhou, J. Guo, W. Shao, and J. Yang, “Multipose measurement of surface defects on rotary metal parts with a combined laser-and-camera sensor,” Opt. Eng. 52, 104104 (2013).
[CrossRef]

Other

http://www.werth.de/en/user-angebot/products/sensors.html .

Y. Luo, D. Hu, and T. Gu, “Research on real-time error measurement in curve grinding process based on machine vision,” in IEEE International Technology and Innovation Conference (2006), pp. 656–661.

http://www.optiv.net/ .

M. Yang and L. Luo, “A rapid auto-focus method in automatic microscope,” in IEEE International Conference on Signal Processing (ICSP) (2008), pp. 502–505.

X. Qiu and Z. Yang, “Study on entropy function optimization problem in auto-focusing algorithm applied for radar imaging,” in IEEE International Conference on Microwave and Millimeter Wave Technology (ICMMT) (2008), pp. 2055–2058.

, Test code for machine tools—Part 2: Determination of accuracy and repeatability of positioning of numerically controlled axes, ISO (2011).

, Test code for machine tools—Part 1: Geometric accuracy of machines operating under no-load or quasi-static conditions, ISO (2012).

A. Jeffrey, Mathematics for Engineers and Scientists, 6th ed. (Chapman & Hall, 2004).

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Figures (12)

Fig. 1.
Fig. 1.

Ambiguity by only optically focusing the small hole on a turbine blade surface at lens position C1 and C2: corresponding images 1 and 2 are blurred in the upper/lower semicircle and clear in the opposite semicircle.

Fig. 2.
Fig. 2.

Focusing curves based on image discrete cosine transformation (DCT) evaluation function for a hole on an elliptic cylinder shell under different illuminating light intensity (LI) and a hardness indentation on a flat workpiece.

Fig. 3.
Fig. 3.

Dual-sensor autofocusing configurations (a) parallel and (b) perpendicular to each other.

Fig. 4.
Fig. 4.

Dual-sensor autofocusing unit applied on the test rig.

Fig. 5.
Fig. 5.

Dual-sensor autofocusing unit applied on the machine tool.

Fig. 6.
Fig. 6.

Schematic of a tactile probe contacting a small hole on the protruding part of a freeform surface.

Fig. 7.
Fig. 7.

Schematic of autofocusing deviation caused by a tactile probe if it aligns on the freeform surface at (a) the centerline of the hole and (b) offset from the centerline.

Fig. 8.
Fig. 8.

Elliptical cylinder shell.

Fig. 9.
Fig. 9.

Schematic of an inductive sensor head contacting a small hole on an elliptical cylinder.

Fig. 10.
Fig. 10.

(a) Images of a pattern of 12 small holes of 0.5 mm (3.5 times optical magnification) autofocused and captured on the test rig, and (b) their binary images marked with the calculated centroids [18].

Fig. 11.
Fig. 11.

(a) Images of a pattern of 12 small holes of 0.5 mm (3.75 times optical magnification) autofocused and captured on machine tool, and (b) their binary images marked with the calculated centroids.

Fig. 12.
Fig. 12.

Repeatability of detected deviations of hole centroid in directions of (a) circumference (XiXi) (μm) and (b) axis (YiYi) (μm), where dashed lines and solid lines are the results from the test rig and machine tool, respectively.

Tables (3)

Tables Icon

Table 1. Thresholds of Skewed Angles at the Higher Edge of a Small Hole on a Freeform Surface

Tables Icon

Table 2. Average ΔX¯ and Repeatability (σ) of Centroid Deviations in Circumferential Direction (XiXi)

Tables Icon

Table 3. Average ΔY¯ and Repeatability (σ) of Centroid Deviations in Axis Direction (YiYi)

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

uc(f)=[2RZ2+2BZ2+EPX2(d)+EBX2(d)+EMPE2]1/2,
uc(f)=[2RZ2+2BZ2+EZX2(d)+EBX2(d)+ERP2]1/2,
s=RR2r2(R>r).
t=ρρ2r2(ρ>r),
n=(FX(X,Y,Z),FY(X,Y,Z),FZ(X,Y,Z)).
n=(fX(X,Y),fY(X,Y),1).
fE(X,Y)=f(X,Y)+R·n⃗.
cosγ=1/|n⃗|=1/fX2(X,Y)+fY2(X,Y)+1.
fE(X,Y)f(X,Y)+R/cosγ.
ΔZ=fE(X,Y)f(X,Y)RR(1/cosγ1),
tanγ=[fX2(X,Y)+fY2(X,Y)]1/2.
ΔZR·[(tan2γ+1)1/21].
ΔZ=Δl·tanγr·tanγ.
{x=a·cosαz=b·sinα(0α<2π).
{X=xcosθzsinθZ=xsinθ+zcosθ.
tanθ=xz=ab·cosαsinα.
tanγ=dZdX=(dx/dα)·sinθ+(dz/dα)·cosθ(dx/dα)·cosθ(dz/dα)·sinθ.
tanγ=a2b2absin(2α)(0α<π/2).
ΔZR·(1+tan2γ1)=R·[(1+(a2b2)24a2b2·sin22α)121]
ΔZr·tanγ=r·a2b22absin(2α).
{dzdx=bacotαd2zdx2=ba3sin3α
K=(d2zdx2)/(1+(dzdx)2)3/2,=ab/(a2sin2α+b2cos2α)3/2
ρ=|K1|=1ab|(a2sin2α+b2cos2α)3/2|.
{δmin=[s2+t2+uc2(f)]1/2δmax=[(ΔZ+ΔZ)2+uc2(f)]1/2.

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