Abstract

Our work deals with point length standards, which can be practically realized by two precise spheres or parts of spheres connected by a bar or array of spherical surfaces. The distance between the centers of spheres precisely determines the length. Two methods (mechanical and optical) are shown for determination of the centers of spheres of the length standard. The proposed optical method is based on the interference of light. General formulas are derived that make it possible to calculate accurately the center position of the spherical surface, which is used for the length standards. An analysis of the proposed method is described based on the third-order aberration theory. The proposed technique can be used for calibration and checking of computer numerical controlled machines.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2010 (3)

2009 (3)

A. Mikš, Applied Optics (Czech Technical University, 2009).
[PubMed]

T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications (CRC, 2009).
[CrossRef]

K. H. Grote, E. K. Antonsson, Springer Handbook of Mechanical Engineering (Springer, 2009).
[CrossRef]

2008 (2)

F. M. Santoyo, Handbook of Optical Metrology (CRC, 2008).

H. F. F. Castro, “Uncertainty analysis of a laser calibration system for evaluating the position accuracy of a numerically controlled axis of coordinate measuring machines and machines tools,” Precis. Eng. 32, 106–113 (2008).
[CrossRef]

2007 (2)

D. Malacara, Optical Shop Testing (Wiley & Sons, 2007).
[CrossRef]

N. Suga, Metrology Handbook: The Science of Measurement (Mitutoyo, 2007).

2006 (1)

C. Dotson, Fundamentals of Dimensional Metrology, 5th ed. (Delmar Learning, 2006).

2005 (2)

K. Umetsu, R. Furutnani, S. Osawa, T. Takatsuji, and T. Kurosawa, “Geometric calibration of a coordinate measuring machine using a laser tracking system,” Meas. Sci. Technol. 16, 2466–2472 (2005).
[CrossRef]

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

2003 (1)

H. F. F. Castro and M. Burdekin, “Dynamic calibration of the positioning accuracy of machine tools and coordinate measuring machines using a laser interferometer,” Int. J. Mach. Tools Manuf. 43, 947–954 (2003).
[CrossRef]

2002 (2)

G. T. Smith, Industrial Metrology (Springer, 2002).

T. Pfeifer, Production Metrology (Oldenbourg Wissenschaftsverlag, 2002).

2001 (2)

I. Kovac and A. Frank, “Testing and calibration of coordinate measuring arms,” Precis. Eng. 25, 90–99 (2001).
[CrossRef]

H. Schwenke, F. Wäldele, C. Weiskirch, and H. Kunzmann, “Opto-tactile sensor for 2D and 3D measurement of small structures on coordinate measuring machines,” CIRP Annals Manuf. Technol. 50, 361–364 (2001).
[CrossRef]

1999 (1)

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

1996 (1)

1995 (3)

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[CrossRef] [PubMed]

J. F. Ouyang and I. S. Jawahir, “Ball array calibration on a coordinate measuring machine using a gage block,” Measurement 16, 219–229 (1995).
[CrossRef]

J. A. Bosch, Coordinate Measuring Machines and Systems (CRC, 1995).

1993 (1)

D. C. Williams, Optical Methods in Engineering Metrology (Chapman & Hall, 1993).
[CrossRef]

1992 (1)

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering (Academic, 1992), Vol.  XI, Chap. 1.

1991 (1)

G. X. Zhang and Y. F. Zang, “A method for machine geometry calibration using 1-D ball array,” CIRP Annals Manuf. Technol. 40, 519–522 (1991).
[CrossRef]

1988 (2)

K. Kinnstaetter, A. W. Lohmann, J. Schwider, and N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef] [PubMed]

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1988), Vol.  XXVI, pp. 349–393.
[CrossRef]

1987 (1)

1985 (2)

K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24, 3101–3105 (1985).
[CrossRef] [PubMed]

K. Busch, H. Kunzmann, and F. Wäldele, “Calibration of coordinate measuring machines,” Precis. Eng. 7, 139–144 (1985).
[CrossRef]

1983 (1)

1982 (1)

1974 (2)

1950 (1)

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).

Antonsson, E. K.

K. H. Grote, E. K. Antonsson, Springer Handbook of Mechanical Engineering (Springer, 2009).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Bosch, J. A.

J. A. Bosch, Coordinate Measuring Machines and Systems (CRC, 1995).

Brangaccio, D. J.

Bruning, J. H.

Burdekin, M.

H. F. F. Castro and M. Burdekin, “Dynamic calibration of the positioning accuracy of machine tools and coordinate measuring machines using a laser interferometer,” Int. J. Mach. Tools Manuf. 43, 947–954 (2003).
[CrossRef]

Burow, R.

Busch, K.

K. Busch, H. Kunzmann, and F. Wäldele, “Calibration of coordinate measuring machines,” Precis. Eng. 7, 139–144 (1985).
[CrossRef]

Castro, H. F. F.

H. F. F. Castro, “Uncertainty analysis of a laser calibration system for evaluating the position accuracy of a numerically controlled axis of coordinate measuring machines and machines tools,” Precis. Eng. 32, 106–113 (2008).
[CrossRef]

H. F. F. Castro and M. Burdekin, “Dynamic calibration of the positioning accuracy of machine tools and coordinate measuring machines using a laser interferometer,” Int. J. Mach. Tools Manuf. 43, 947–954 (2003).
[CrossRef]

Creath, K.

J. Schmit and K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35, 5642–5649(1996).
[CrossRef] [PubMed]

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[CrossRef] [PubMed]

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering (Academic, 1992), Vol.  XI, Chap. 1.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1988), Vol.  XXVI, pp. 349–393.
[CrossRef]

Dobosz, M.

O. Iwasinska-Kowalska and M. Dobosz, “A new method of non-contact gauge block calibration using a fringe-counting technique I, II,” Opt. Laser Technol. 42, 149–155 (2010).
[CrossRef]

Dotson, C.

C. Dotson, Fundamentals of Dimensional Metrology, 5th ed. (Delmar Learning, 2006).

Elssner, K.-E.

Frank, A.

I. Kovac and A. Frank, “Testing and calibration of coordinate measuring arms,” Precis. Eng. 25, 90–99 (2001).
[CrossRef]

Furutnani, R.

K. Umetsu, R. Furutnani, S. Osawa, T. Takatsuji, and T. Kurosawa, “Geometric calibration of a coordinate measuring machine using a laser tracking system,” Meas. Sci. Technol. 16, 2466–2472 (2005).
[CrossRef]

Gallagher, J. E.

Grote, K. H.

K. H. Grote, E. K. Antonsson, Springer Handbook of Mechanical Engineering (Springer, 2009).
[CrossRef]

Grzanna, J.

Herriott, D. R.

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).

Ina, H.

Iwasinska-Kowalska, O.

O. Iwasinska-Kowalska and M. Dobosz, “A new method of non-contact gauge block calibration using a fringe-counting technique I, II,” Opt. Laser Technol. 42, 149–155 (2010).
[CrossRef]

Jawahir, I. S.

J. F. Ouyang and I. S. Jawahir, “Ball array calibration on a coordinate measuring machine using a gage block,” Measurement 16, 219–229 (1995).
[CrossRef]

Kinnstaetter, K.

Kobayashi, S.

Kovac, I.

I. Kovac and A. Frank, “Testing and calibration of coordinate measuring arms,” Precis. Eng. 25, 90–99 (2001).
[CrossRef]

Kunzmann, H.

H. Schwenke, F. Wäldele, C. Weiskirch, and H. Kunzmann, “Opto-tactile sensor for 2D and 3D measurement of small structures on coordinate measuring machines,” CIRP Annals Manuf. Technol. 50, 361–364 (2001).
[CrossRef]

K. Busch, H. Kunzmann, and F. Wäldele, “Calibration of coordinate measuring machines,” Precis. Eng. 7, 139–144 (1985).
[CrossRef]

Kurosawa, T.

K. Umetsu, R. Furutnani, S. Osawa, T. Takatsuji, and T. Kurosawa, “Geometric calibration of a coordinate measuring machine using a laser tracking system,” Meas. Sci. Technol. 16, 2466–2472 (2005).
[CrossRef]

Lohmann, A. W.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley & Sons, 2007).
[CrossRef]

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

Merkel, K.

Mikš, A.

Novak, J.

Novak, P.

Nugent, K. A.

Osawa, S.

K. Umetsu, R. Furutnani, S. Osawa, T. Takatsuji, and T. Kurosawa, “Geometric calibration of a coordinate measuring machine using a laser tracking system,” Meas. Sci. Technol. 16, 2466–2472 (2005).
[CrossRef]

Ouyang, J. F.

J. F. Ouyang and I. S. Jawahir, “Ball array calibration on a coordinate measuring machine using a gage block,” Measurement 16, 219–229 (1995).
[CrossRef]

Pfeifer, T.

T. Pfeifer, Production Metrology (Oldenbourg Wissenschaftsverlag, 2002).

Roddier, C.

Roddier, F.

Rosenfeld, D. P.

Santoyo, F. M.

F. M. Santoyo, Handbook of Optical Metrology (CRC, 2008).

Schmit, J.

Schwenke, H.

H. Schwenke, F. Wäldele, C. Weiskirch, and H. Kunzmann, “Opto-tactile sensor for 2D and 3D measurement of small structures on coordinate measuring machines,” CIRP Annals Manuf. Technol. 50, 361–364 (2001).
[CrossRef]

Schwider, J.

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

Smith, G. T.

G. T. Smith, Industrial Metrology (Springer, 2002).

Spolaczyk, R.

Streibl, N.

Suga, N.

N. Suga, Metrology Handbook: The Science of Measurement (Mitutoyo, 2007).

Takatsuji, T.

K. Umetsu, R. Furutnani, S. Osawa, T. Takatsuji, and T. Kurosawa, “Geometric calibration of a coordinate measuring machine using a laser tracking system,” Meas. Sci. Technol. 16, 2466–2472 (2005).
[CrossRef]

Takeda, M.

Umetsu, K.

K. Umetsu, R. Furutnani, S. Osawa, T. Takatsuji, and T. Kurosawa, “Geometric calibration of a coordinate measuring machine using a laser tracking system,” Meas. Sci. Technol. 16, 2466–2472 (2005).
[CrossRef]

Wäldele, F.

H. Schwenke, F. Wäldele, C. Weiskirch, and H. Kunzmann, “Opto-tactile sensor for 2D and 3D measurement of small structures on coordinate measuring machines,” CIRP Annals Manuf. Technol. 50, 361–364 (2001).
[CrossRef]

K. Busch, H. Kunzmann, and F. Wäldele, “Calibration of coordinate measuring machines,” Precis. Eng. 7, 139–144 (1985).
[CrossRef]

Weiskirch, C.

H. Schwenke, F. Wäldele, C. Weiskirch, and H. Kunzmann, “Opto-tactile sensor for 2D and 3D measurement of small structures on coordinate measuring machines,” CIRP Annals Manuf. Technol. 50, 361–364 (2001).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).

White, A. D.

Williams, D. C.

D. C. Williams, Optical Methods in Engineering Metrology (Chapman & Hall, 1993).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Wyant, J. C.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering (Academic, 1992), Vol.  XI, Chap. 1.

Yoshizawa, T.

T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications (CRC, 2009).
[CrossRef]

Zang, Y. F.

G. X. Zhang and Y. F. Zang, “A method for machine geometry calibration using 1-D ball array,” CIRP Annals Manuf. Technol. 40, 519–522 (1991).
[CrossRef]

Zhang, G. X.

G. X. Zhang and Y. F. Zang, “A method for machine geometry calibration using 1-D ball array,” CIRP Annals Manuf. Technol. 40, 519–522 (1991).
[CrossRef]

Appl. Opt. (7)

CIRP Annals Manuf. Technol. (2)

H. Schwenke, F. Wäldele, C. Weiskirch, and H. Kunzmann, “Opto-tactile sensor for 2D and 3D measurement of small structures on coordinate measuring machines,” CIRP Annals Manuf. Technol. 50, 361–364 (2001).
[CrossRef]

G. X. Zhang and Y. F. Zang, “A method for machine geometry calibration using 1-D ball array,” CIRP Annals Manuf. Technol. 40, 519–522 (1991).
[CrossRef]

Int. J. Mach. Tools Manuf. (1)

H. F. F. Castro and M. Burdekin, “Dynamic calibration of the positioning accuracy of machine tools and coordinate measuring machines using a laser interferometer,” Int. J. Mach. Tools Manuf. 43, 947–954 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

Meas. Sci. Technol. (1)

K. Umetsu, R. Furutnani, S. Osawa, T. Takatsuji, and T. Kurosawa, “Geometric calibration of a coordinate measuring machine using a laser tracking system,” Meas. Sci. Technol. 16, 2466–2472 (2005).
[CrossRef]

Measurement (1)

J. F. Ouyang and I. S. Jawahir, “Ball array calibration on a coordinate measuring machine using a gage block,” Measurement 16, 219–229 (1995).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

O. Iwasinska-Kowalska and M. Dobosz, “A new method of non-contact gauge block calibration using a fringe-counting technique I, II,” Opt. Laser Technol. 42, 149–155 (2010).
[CrossRef]

Opt. Lett. (1)

Precis. Eng. (3)

H. F. F. Castro, “Uncertainty analysis of a laser calibration system for evaluating the position accuracy of a numerically controlled axis of coordinate measuring machines and machines tools,” Precis. Eng. 32, 106–113 (2008).
[CrossRef]

I. Kovac and A. Frank, “Testing and calibration of coordinate measuring arms,” Precis. Eng. 25, 90–99 (2001).
[CrossRef]

K. Busch, H. Kunzmann, and F. Wäldele, “Calibration of coordinate measuring machines,” Precis. Eng. 7, 139–144 (1985).
[CrossRef]

Other (17)

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005).
[CrossRef]

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering (Academic, 1992), Vol.  XI, Chap. 1.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E.Wolf, ed. (Elsevier Science, 1988), Vol.  XXVI, pp. 349–393.
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

A. Mikš, Applied Optics (Czech Technical University, 2009).
[PubMed]

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).

D. Malacara, Optical Shop Testing (Wiley & Sons, 2007).
[CrossRef]

G. T. Smith, Industrial Metrology (Springer, 2002).

C. Dotson, Fundamentals of Dimensional Metrology, 5th ed. (Delmar Learning, 2006).

N. Suga, Metrology Handbook: The Science of Measurement (Mitutoyo, 2007).

D. C. Williams, Optical Methods in Engineering Metrology (Chapman & Hall, 1993).
[CrossRef]

F. M. Santoyo, Handbook of Optical Metrology (CRC, 2008).

T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications (CRC, 2009).
[CrossRef]

J. A. Bosch, Coordinate Measuring Machines and Systems (CRC, 1995).

K. H. Grote, E. K. Antonsson, Springer Handbook of Mechanical Engineering (Springer, 2009).
[CrossRef]

T. Pfeifer, Production Metrology (Oldenbourg Wissenschaftsverlag, 2002).

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Figures (6)

Fig. 1
Fig. 1

Scheme of the ball bar defined by two spheres.

Fig. 2
Fig. 2

Determination of the sphere center.

Fig. 3
Fig. 3

Sensitivity of the sphere center determination.

Fig. 4
Fig. 4

Interferogram ( F = 1 , λ = 0.633 μm , δ X = 0.01 mm , δ Y = 0.01 mm , and δ Z = 0.03 mm ).

Fig. 5
Fig. 5

Proposed ball bars using plano–convex or plano–concave lenses.

Fig. 6
Fig. 6

Principle of measurement using the 3D coordinate measuring machine.

Tables (1)

Tables Icon

Table 1 ZYGO Measurement of the Spherical Surface

Equations (39)

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

d = ( x c 1 x c 2 ) 2 + ( y c 1 y c 2 ) 2 + ( z c 1 z c 2 ) 2 .
( r r c ) T . ( r r c ) = R 2 ,
( r i r c ) T . ( r i r c ) = R 2 , ( r j r c ) T . ( r j r c ) = R 2 .
r c T . ( r i r j ) = 1 2 ( r i T . r i r j T . r j ) .
A r c = b ,
A = ( x 1 x 2 y 1 y 2 z 1 z 2 x 1 x 3 y 1 y 3 z 1 z 3 x 1 x 4 y 1 y 4 z 1 z 4 ) , b = 1 2 ( r 1 2 r 2 2 r 1 2 r 3 2 r 1 2 r 4 2 ) , r c = ( x c y c z c ) , r k 2 = x k 2 + y k 2 + z k 2 , k = 1 , 2 , 3 , 4.
A r c = b ,
A = ( x 1 x 2 y 1 y 2 z 1 z 2 x 1 x N y 1 y N z 1 z N ) , b = 1 2 ( r 1 2 r 2 2 r 1 2 r N 2 ) , r c = ( x c y c z c ) , r k 2 = x k 2 + y k 2 + z k 2 , k = 1 , 2 , 3 , , N .
( r Q r 1 ) · ( r Q r 1 ) = R 1 2 ,
( r B r 2 ) · ( r B r 2 ) = R 2 2 ,
r B = r Q + δ r Q , r 2 = r 1 + δ r C .
2 ( r Q r 1 ) . δ r Q 2 r Q . δ r C + 2 r 1 . δ r C + δ r Q 2 + δ r C 2 2 δ r Q . δ r C = R 2 2 R 1 2 .
R 1 2 + 2 r 1 . δ r C + δ r C 2 = R 2 2 .
2 ( r Q r 1 ) . δ r Q 2 r Q . δ r C + δ r Q 2 2 δ r Q . δ r C = 0.
s = ( r 1 r Q ) / R 1 .
s . δ r Q = r Q . δ r C / R 1 .
δ W = n ( s . δ r Q ) = n R 1 r Q . δ r C ,
δ r C = ( δ X , δ Y , δ Z ) ,
r Q = ( X Q , Y Q , Z Q ) = ( R 1 sin U sin φ , R 1 sin U cos φ , R 1 ( 1 cos U ) ) ,
δ W = n [ ( δ X sin φ + δ Y cos φ ) sin U + δ Z ( 1 cos U ) ] .
δ r C = ( 0 , 0 , δ Z ) .
δ r C = ( 0 , δ Y , 0 ) .
r Q = ( 0 , Y Q , Z Q ) .
δ W L = n δ Z ( 1 cos U ) = n δ Z sin 2 U / ( 1 + cos U ) 1 2 n δ Z sin 2 U .
δ W T = n δ Y ( Y Q / R 1 ) = n δ Y sin U .
δ Z = 2 n ( δ W L ) max n 2 sin 2 U = 8 n F 2 ( δ W L ) max , δ Y = ( δ W T ) max n sin U max = 2 F ( δ W T ) max ,
F = 1 2 n sin U max ,
W = W 1 x x + W 1 y y + W 20 ( x 2 + y 2 ) + W 40 ( x 2 + y 2 ) 2 + W 31 ( x 2 + y 2 ) y + ,
δ y = y P ( y P 2 + x P 2 ) 2 n K σ K ( s 1 s ¯ 1 ) 3 σ 1 3 S I ,
S I = i = 1 i = K h i ( Δ σ i Δ ( 1 / n i ) ) 2 Δ ( σ i n i ) ,
n i σ i = n i σ i + h i ( n i n i ) / r i , h i + 1 = h i d i σ i , σ i + 1 = σ i , n i + 1 = n i , i = 1 , 2 , , K , n i / s i n i / s i = ( n i n i ) / r i , s i + 1 = s i d i , n i + 1 = n i , i = 1 , 2 , , K ,
S I = 1 4 h ( σ σ ) 2 ( σ + σ ) = 2 h 4 r ( 1 r 1 s ) 2 ,
s = r + δ Z .
S I = 2 h 4 r ( 1 r 1 s ) 2 = 2 h 4 r [ 1 r 1 r + δ Z ] 2 2 h 4 r 5 δ Z 2 .
δ y y P 3 r 4 δ Z 2 = δ Z 2 r sin 3 U , and δ s y P 2 r 3 δ Z 2 = δ Z 2 r sin 2 U ,
δ s δ Z 2 r sin 2 U = δ Z 2 4 F 2 r ( sin U sin U m ) 2 = 1 4 F 2 ( δ Z 2 r ) y 2 ,
W 40 = δ s m 16 F 2 = 1 64 F 4 ( δ Z 2 r ) ,
W 20 = δ Z 8 F 2 .
W 40 W 20 = δ Z 8 F 2 r .

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