Abstract

A multidirectional interferometer system is developed to determine the position and orientation of a stage moved in a two-dimensional (2-D) space. In this system four corner-cube prisms are mounted on the moving stage, and four laser beams are incident on the corner cubes in different directions. Moving distances in the observed directions are measured by laser interferometers. The position and orientation of the stage are calculated from the moving distances of the corner cubes. Some experiments are done on the 2-D moving stage with four interferometers, and measurement errors are estimated from redundant data. The estimated accuracy is higher than 0.2 μm for translation and 0.3 × 10-3 deg for rotation for a measurement range of 0.5 mm and 0.5 deg.

© 2003 Optical Society of America

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  1. E. H. Bokelberg, H. J. Sommer, M. W. Trethewey, “A six-degree-of-freedom laser vibrometer,” J. Sound Vib. 178, 643–667 (1994).
    [CrossRef]
  2. I. J. Busch-Vishniac, A. B. Buckman, W. Wang, D. Qian, V. Mancevski, “Noncontact position measurement systems using optical sensor,” U.S. patent5,367,373 (22November1994).
  3. N. K. S. Lee, Y. Cai, A. Joneja, “High-resolution multidimensional displacement monitoring system,” Opt. Eng. 36, 2287–2293 (1997).
    [CrossRef]
  4. J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71, 3214–3219 (2000).
    [CrossRef]
  5. J. A. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Application of sensitivity analysis for the design of six-degree-of-freedom measurement,” Opt. Eng. 40, 2837–2844 (2001).
    [CrossRef]
  6. W. S. Park, H. S. Cho, “Measurement of fine 6-degree-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation,” Opt. Eng. 41, 860–871 (2002).
    [CrossRef]
  7. K. C. Fan, M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24, 15–23 (2000).
    [CrossRef]
  8. O. Nakamura, M. Goto, “Laser interferometric calibration of microscan mechanisms by using three laser beams,” Precis. Eng. 15, 39–43 (1993).
    [CrossRef]
  9. O. Nakamura, M. Goto, K. Toyoda, N. Takai, T. Kurosawa, T. Nakamuta, “A laser tracking robot-performance calibration system using ball-seated bearing mechanisms and a spherically shaped cat’s-eye retroreflector,” Rev. Sci. Instrum. 65, 1006–1011 (1994).
    [CrossRef]
  10. T. Takatsuji, M. Goto, T. Kurosawa, T. Tanimura, Y. Koseki, “The first measurement of a three-dimensional coordinate by use of a laser tracking interferometer system based on trilateration,” Meas. Sci. Technol. 9, 38–41 (1998).
    [CrossRef]
  11. J. Zhang, K. Iwata, H. Kikuta, C. S. Park, “An interferometric system for measuring position and orientation of a positioning stage,” in Seventh International Symposium on Laser Metrology Applied to Science, Industry, and Everyday Life, Y. V. Chugui, S. N. Bagayev, A. Weckenmann, P. Herbert Osanna, eds., Proc. SPIE4900, 304–311 (2002).
    [CrossRef]

2002 (1)

W. S. Park, H. S. Cho, “Measurement of fine 6-degree-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation,” Opt. Eng. 41, 860–871 (2002).
[CrossRef]

2001 (1)

J. A. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Application of sensitivity analysis for the design of six-degree-of-freedom measurement,” Opt. Eng. 40, 2837–2844 (2001).
[CrossRef]

2000 (2)

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71, 3214–3219 (2000).
[CrossRef]

K. C. Fan, M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24, 15–23 (2000).
[CrossRef]

1998 (1)

T. Takatsuji, M. Goto, T. Kurosawa, T. Tanimura, Y. Koseki, “The first measurement of a three-dimensional coordinate by use of a laser tracking interferometer system based on trilateration,” Meas. Sci. Technol. 9, 38–41 (1998).
[CrossRef]

1997 (1)

N. K. S. Lee, Y. Cai, A. Joneja, “High-resolution multidimensional displacement monitoring system,” Opt. Eng. 36, 2287–2293 (1997).
[CrossRef]

1994 (2)

E. H. Bokelberg, H. J. Sommer, M. W. Trethewey, “A six-degree-of-freedom laser vibrometer,” J. Sound Vib. 178, 643–667 (1994).
[CrossRef]

O. Nakamura, M. Goto, K. Toyoda, N. Takai, T. Kurosawa, T. Nakamuta, “A laser tracking robot-performance calibration system using ball-seated bearing mechanisms and a spherically shaped cat’s-eye retroreflector,” Rev. Sci. Instrum. 65, 1006–1011 (1994).
[CrossRef]

1993 (1)

O. Nakamura, M. Goto, “Laser interferometric calibration of microscan mechanisms by using three laser beams,” Precis. Eng. 15, 39–43 (1993).
[CrossRef]

Bae, E. W.

J. A. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Application of sensitivity analysis for the design of six-degree-of-freedom measurement,” Opt. Eng. 40, 2837–2844 (2001).
[CrossRef]

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71, 3214–3219 (2000).
[CrossRef]

Bokelberg, E. H.

E. H. Bokelberg, H. J. Sommer, M. W. Trethewey, “A six-degree-of-freedom laser vibrometer,” J. Sound Vib. 178, 643–667 (1994).
[CrossRef]

Buckman, A. B.

I. J. Busch-Vishniac, A. B. Buckman, W. Wang, D. Qian, V. Mancevski, “Noncontact position measurement systems using optical sensor,” U.S. patent5,367,373 (22November1994).

Busch-Vishniac, I. J.

I. J. Busch-Vishniac, A. B. Buckman, W. Wang, D. Qian, V. Mancevski, “Noncontact position measurement systems using optical sensor,” U.S. patent5,367,373 (22November1994).

Cai, Y.

N. K. S. Lee, Y. Cai, A. Joneja, “High-resolution multidimensional displacement monitoring system,” Opt. Eng. 36, 2287–2293 (1997).
[CrossRef]

Chen, M. J.

K. C. Fan, M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24, 15–23 (2000).
[CrossRef]

Cho, H. S.

W. S. Park, H. S. Cho, “Measurement of fine 6-degree-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation,” Opt. Eng. 41, 860–871 (2002).
[CrossRef]

Fan, K. C.

K. C. Fan, M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24, 15–23 (2000).
[CrossRef]

Goto, M.

T. Takatsuji, M. Goto, T. Kurosawa, T. Tanimura, Y. Koseki, “The first measurement of a three-dimensional coordinate by use of a laser tracking interferometer system based on trilateration,” Meas. Sci. Technol. 9, 38–41 (1998).
[CrossRef]

O. Nakamura, M. Goto, K. Toyoda, N. Takai, T. Kurosawa, T. Nakamuta, “A laser tracking robot-performance calibration system using ball-seated bearing mechanisms and a spherically shaped cat’s-eye retroreflector,” Rev. Sci. Instrum. 65, 1006–1011 (1994).
[CrossRef]

O. Nakamura, M. Goto, “Laser interferometric calibration of microscan mechanisms by using three laser beams,” Precis. Eng. 15, 39–43 (1993).
[CrossRef]

Iwata, K.

J. Zhang, K. Iwata, H. Kikuta, C. S. Park, “An interferometric system for measuring position and orientation of a positioning stage,” in Seventh International Symposium on Laser Metrology Applied to Science, Industry, and Everyday Life, Y. V. Chugui, S. N. Bagayev, A. Weckenmann, P. Herbert Osanna, eds., Proc. SPIE4900, 304–311 (2002).
[CrossRef]

Joneja, A.

N. K. S. Lee, Y. Cai, A. Joneja, “High-resolution multidimensional displacement monitoring system,” Opt. Eng. 36, 2287–2293 (1997).
[CrossRef]

Kikuta, H.

J. Zhang, K. Iwata, H. Kikuta, C. S. Park, “An interferometric system for measuring position and orientation of a positioning stage,” in Seventh International Symposium on Laser Metrology Applied to Science, Industry, and Everyday Life, Y. V. Chugui, S. N. Bagayev, A. Weckenmann, P. Herbert Osanna, eds., Proc. SPIE4900, 304–311 (2002).
[CrossRef]

Kim, J. A.

J. A. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Application of sensitivity analysis for the design of six-degree-of-freedom measurement,” Opt. Eng. 40, 2837–2844 (2001).
[CrossRef]

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71, 3214–3219 (2000).
[CrossRef]

Kim, K. C.

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71, 3214–3219 (2000).
[CrossRef]

Kim, S. H.

J. A. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Application of sensitivity analysis for the design of six-degree-of-freedom measurement,” Opt. Eng. 40, 2837–2844 (2001).
[CrossRef]

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71, 3214–3219 (2000).
[CrossRef]

Koseki, Y.

T. Takatsuji, M. Goto, T. Kurosawa, T. Tanimura, Y. Koseki, “The first measurement of a three-dimensional coordinate by use of a laser tracking interferometer system based on trilateration,” Meas. Sci. Technol. 9, 38–41 (1998).
[CrossRef]

Kurosawa, T.

T. Takatsuji, M. Goto, T. Kurosawa, T. Tanimura, Y. Koseki, “The first measurement of a three-dimensional coordinate by use of a laser tracking interferometer system based on trilateration,” Meas. Sci. Technol. 9, 38–41 (1998).
[CrossRef]

O. Nakamura, M. Goto, K. Toyoda, N. Takai, T. Kurosawa, T. Nakamuta, “A laser tracking robot-performance calibration system using ball-seated bearing mechanisms and a spherically shaped cat’s-eye retroreflector,” Rev. Sci. Instrum. 65, 1006–1011 (1994).
[CrossRef]

Kwak, Y. K.

J. A. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Application of sensitivity analysis for the design of six-degree-of-freedom measurement,” Opt. Eng. 40, 2837–2844 (2001).
[CrossRef]

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71, 3214–3219 (2000).
[CrossRef]

Lee, N. K. S.

N. K. S. Lee, Y. Cai, A. Joneja, “High-resolution multidimensional displacement monitoring system,” Opt. Eng. 36, 2287–2293 (1997).
[CrossRef]

Mancevski, V.

I. J. Busch-Vishniac, A. B. Buckman, W. Wang, D. Qian, V. Mancevski, “Noncontact position measurement systems using optical sensor,” U.S. patent5,367,373 (22November1994).

Nakamura, O.

O. Nakamura, M. Goto, K. Toyoda, N. Takai, T. Kurosawa, T. Nakamuta, “A laser tracking robot-performance calibration system using ball-seated bearing mechanisms and a spherically shaped cat’s-eye retroreflector,” Rev. Sci. Instrum. 65, 1006–1011 (1994).
[CrossRef]

O. Nakamura, M. Goto, “Laser interferometric calibration of microscan mechanisms by using three laser beams,” Precis. Eng. 15, 39–43 (1993).
[CrossRef]

Nakamuta, T.

O. Nakamura, M. Goto, K. Toyoda, N. Takai, T. Kurosawa, T. Nakamuta, “A laser tracking robot-performance calibration system using ball-seated bearing mechanisms and a spherically shaped cat’s-eye retroreflector,” Rev. Sci. Instrum. 65, 1006–1011 (1994).
[CrossRef]

Park, C. S.

J. Zhang, K. Iwata, H. Kikuta, C. S. Park, “An interferometric system for measuring position and orientation of a positioning stage,” in Seventh International Symposium on Laser Metrology Applied to Science, Industry, and Everyday Life, Y. V. Chugui, S. N. Bagayev, A. Weckenmann, P. Herbert Osanna, eds., Proc. SPIE4900, 304–311 (2002).
[CrossRef]

Park, W. S.

W. S. Park, H. S. Cho, “Measurement of fine 6-degree-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation,” Opt. Eng. 41, 860–871 (2002).
[CrossRef]

Qian, D.

I. J. Busch-Vishniac, A. B. Buckman, W. Wang, D. Qian, V. Mancevski, “Noncontact position measurement systems using optical sensor,” U.S. patent5,367,373 (22November1994).

Sommer, H. J.

E. H. Bokelberg, H. J. Sommer, M. W. Trethewey, “A six-degree-of-freedom laser vibrometer,” J. Sound Vib. 178, 643–667 (1994).
[CrossRef]

Takai, N.

O. Nakamura, M. Goto, K. Toyoda, N. Takai, T. Kurosawa, T. Nakamuta, “A laser tracking robot-performance calibration system using ball-seated bearing mechanisms and a spherically shaped cat’s-eye retroreflector,” Rev. Sci. Instrum. 65, 1006–1011 (1994).
[CrossRef]

Takatsuji, T.

T. Takatsuji, M. Goto, T. Kurosawa, T. Tanimura, Y. Koseki, “The first measurement of a three-dimensional coordinate by use of a laser tracking interferometer system based on trilateration,” Meas. Sci. Technol. 9, 38–41 (1998).
[CrossRef]

Tanimura, T.

T. Takatsuji, M. Goto, T. Kurosawa, T. Tanimura, Y. Koseki, “The first measurement of a three-dimensional coordinate by use of a laser tracking interferometer system based on trilateration,” Meas. Sci. Technol. 9, 38–41 (1998).
[CrossRef]

Toyoda, K.

O. Nakamura, M. Goto, K. Toyoda, N. Takai, T. Kurosawa, T. Nakamuta, “A laser tracking robot-performance calibration system using ball-seated bearing mechanisms and a spherically shaped cat’s-eye retroreflector,” Rev. Sci. Instrum. 65, 1006–1011 (1994).
[CrossRef]

Trethewey, M. W.

E. H. Bokelberg, H. J. Sommer, M. W. Trethewey, “A six-degree-of-freedom laser vibrometer,” J. Sound Vib. 178, 643–667 (1994).
[CrossRef]

Wang, W.

I. J. Busch-Vishniac, A. B. Buckman, W. Wang, D. Qian, V. Mancevski, “Noncontact position measurement systems using optical sensor,” U.S. patent5,367,373 (22November1994).

Zhang, J.

J. Zhang, K. Iwata, H. Kikuta, C. S. Park, “An interferometric system for measuring position and orientation of a positioning stage,” in Seventh International Symposium on Laser Metrology Applied to Science, Industry, and Everyday Life, Y. V. Chugui, S. N. Bagayev, A. Weckenmann, P. Herbert Osanna, eds., Proc. SPIE4900, 304–311 (2002).
[CrossRef]

J. Sound Vib. (1)

E. H. Bokelberg, H. J. Sommer, M. W. Trethewey, “A six-degree-of-freedom laser vibrometer,” J. Sound Vib. 178, 643–667 (1994).
[CrossRef]

Meas. Sci. Technol. (1)

T. Takatsuji, M. Goto, T. Kurosawa, T. Tanimura, Y. Koseki, “The first measurement of a three-dimensional coordinate by use of a laser tracking interferometer system based on trilateration,” Meas. Sci. Technol. 9, 38–41 (1998).
[CrossRef]

Opt. Eng. (3)

N. K. S. Lee, Y. Cai, A. Joneja, “High-resolution multidimensional displacement monitoring system,” Opt. Eng. 36, 2287–2293 (1997).
[CrossRef]

J. A. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Application of sensitivity analysis for the design of six-degree-of-freedom measurement,” Opt. Eng. 40, 2837–2844 (2001).
[CrossRef]

W. S. Park, H. S. Cho, “Measurement of fine 6-degree-of-freedom displacement of rigid bodies through splitting a laser beam: experimental investigation,” Opt. Eng. 41, 860–871 (2002).
[CrossRef]

Precis. Eng. (2)

K. C. Fan, M. J. Chen, “A 6-degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24, 15–23 (2000).
[CrossRef]

O. Nakamura, M. Goto, “Laser interferometric calibration of microscan mechanisms by using three laser beams,” Precis. Eng. 15, 39–43 (1993).
[CrossRef]

Rev. Sci. Instrum. (2)

O. Nakamura, M. Goto, K. Toyoda, N. Takai, T. Kurosawa, T. Nakamuta, “A laser tracking robot-performance calibration system using ball-seated bearing mechanisms and a spherically shaped cat’s-eye retroreflector,” Rev. Sci. Instrum. 65, 1006–1011 (1994).
[CrossRef]

J. A. Kim, K. C. Kim, E. W. Bae, S. H. Kim, Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71, 3214–3219 (2000).
[CrossRef]

Other (2)

I. J. Busch-Vishniac, A. B. Buckman, W. Wang, D. Qian, V. Mancevski, “Noncontact position measurement systems using optical sensor,” U.S. patent5,367,373 (22November1994).

J. Zhang, K. Iwata, H. Kikuta, C. S. Park, “An interferometric system for measuring position and orientation of a positioning stage,” in Seventh International Symposium on Laser Metrology Applied to Science, Industry, and Everyday Life, Y. V. Chugui, S. N. Bagayev, A. Weckenmann, P. Herbert Osanna, eds., Proc. SPIE4900, 304–311 (2002).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Measurement principle of the position and orientation with multidirectional interferometers: C k (k = 1, 2, …, K), corner-cube prisms mounted on the moving stage. The directions of the laser beams are expressed by unit vectors S k . Only four corner cubes are shown.

Fig. 2
Fig. 2

System configuration for measuring the position and orientation for a 2-D moving stage. The system consists of four distance-measuring interferometers with the fringe-counting method. Four corner-cube prisms are mounted on a moving stage.

Fig. 3
Fig. 3

Layout of the four corner-cube prisms and laser beams: O, origin of the moving stage.

Fig. 4
Fig. 4

Four corner-cube prisms.

Fig. 5
Fig. 5

Schematic of the systematic errors of Δθ k , Δr k , and Δψ k .

Fig. 6
Fig. 6

Measurement results of standard errors σ x , σ y , and r 0σϕ calculated from the measurement data in Table 2 based on Eq. (10 ).

Fig. 7
Fig. 7

Simulation results of standard errors σ x , σ y , r 0σϕ and standard deviations σ x ′, σ y ′, r 0σϕ′. In this simulation the errors of |δL k | ≤ ±0.05 μm, |Δθ k | ≤ ±0.05 deg, |Δr k | ≤ ±0.05 mm, and |Δψ k | ≤ ±3 deg are uniform random errors. Standard errors are the average of the standard errors in 50 simulations, and the standard deviations are calculated from 50 simulations.

Fig. 8
Fig. 8

Schematic of the optical path length measured with a corner cube.

Fig. 9
Fig. 9

Additional OPD caused by the ordinary corner-cube prism when its orientation changes.

Tables (2)

Tables Icon

Table 1 Measured System Parameters of the Experimental System

Tables Icon

Table 2 Measurement Results with the Four Interferometers

Equations (17)

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

Lk=Sk · U+R-E · rkk=1, 2, , K,
R · R*=E,
V=k=1KLk-Sk · U+R-E · rk2.
Lk=cos θkUx+sin θkUy+rk cosθk-ψk-ϕ-rk cosθk-ψk.
Ux=Ux+ΔUx, Uy=Uy+ΔUy, ϕ=ϕ+Δϕ,
A11ΔUx+A12ΔUy+A13r0Δϕ=b1, A21ΔUx+A22ΔUy+A23r0Δϕ=b2, A31ΔUx+A32ΔUy+A33r0Δϕ=b3.
A11=k=1Kcos2 θk,A12=A21=k=1Kcos θk sin θk,A13=A31=1r0k=1K rk cos θk sinθk-ψk-ϕ,A22=k=1Ksin2 θk,A23=A32=1r0k=1K rk sin θk sinθk-ψk-ϕ,A33=1r02k=1K rk2 sin2θk-ψk-ϕ,b1=k=1KLk-Lkcos θk,b2=k=1KLk-Lksin θk,b3=k=1KLk-Lksinθk-ψk-ϕ,
Δθ1δL1/δx.
r2=δL2/δϕ.
ΔLk=rk1-cos ϕΔψk.
σx=m=14Uxm-Ux231/2,
σy=m=14Uym-Uy231/2,
r0σϕ=m=14 r02ϕm-ϕ231/2,
s2=s1-2s1 · e1e1,s3=s2-2s2 · e2e2,s4=s3-2s3 · e3e3.
d=r2-r1 · s2+r3-r2 · s3+r4-r3 · s4.
d=-2|r|.
OPD=2n2d1cos i2-1+2n1d1-cosi1-i2cos i2.

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