Abstract

A measurement system with three degrees of freedom (3 DOF) that compensates for errors caused by incident beam drift is proposed. The system’s measurement model (i.e. its mathematical foundation) is analyzed, and a measurement module (i.e. the designed orientation measurement unit) is developed and adopted to measure simultaneously straightness errors and the incident beam direction; thus, the errors due to incident beam drift can be compensated. The experimental results show that the proposed system has a deviation of 1 μm in the range of 200 mm for distance measurements, and a deviation of 1.3 μm in the range of 2 mm for straightness error measurements.

© 2015 Optical Society of America

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References

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  1. J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).
  2. K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
    [Crossref]
  3. F. Qibo, Z. Bin, C. Cunxing, K. Cuifang, Z. Yusheng, and Y. Fenglin, “Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide,” Opt. Express 21(22), 25805–25819 (2013).
    [Crossref] [PubMed]
  4. W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
    [Crossref]
  5. G. Zamiela and M. Dobosz, “Corner cube reflector lateral displacement evaluation simultaneously with interferometer length measurement,” Opt. Laser Technol. 50, 118–124 (2013).
    [Crossref]
  6. P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machine,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
    [Crossref]
  7. C. Chou, L. Y. Chou, C. K. Peng, Y. C. Huang, and K. C. Fan, “CCD-based CMM Geometrical error measurement using fourier phase shift algorithm,” Int. J. Mach. Tools Manuf. 37(5), 579–590 (1997).
    [Crossref]
  8. Q. Feng, B. Zhang, and C. Kuang, “Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift,” Int. J. Prec. Eng. Manufact. 9, 26–31 (2008).
  9. K. C. Fan and M. J. Chen, “6-Degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
    [Crossref]
  10. J. A. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom displacement measurement systems using cooperative targets,” Precis. Eng. 26(1), 99–104 (2002).
    [Crossref]
  11. J. S. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71(8), 3214–3219 (2000).
    [Crossref]
  12. Automated Precision Inc, “XD LASER,” http://www.apisensor.com/index.php/products-en/machine-tool-calibration-en/xd-laser-en .
  13. J. S. Chen, T. W. Kou, and S. H. Chiou, “Geometric error calibration of multi-axis machines using an auto-alignment laser interferometer,” Precis. Eng. 26(1), 99–104 (2002).
  14. C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sensor. Actuat. A 125(1), 100–108 (2005).
  15. Renishaw plc Laser System Manual, Reference Section (2007), version 2.1.
  16. Renishaw plc Laser System Manual, Linear measurement (2007), version 2.1.
  17. Agilent Technologies and Optics and Laser Heads for Laser-Interferometer Positioning Systems,” 5964–6190 (2000).
  18. P. L. Teoh, B. Shirinzadeh, C. W. Foong, and G. Alici, “The measurement uncertainties in the laser interferometry-based sensing and tracking technique,” Measurement 32(2), 135–150 (2002).
    [Crossref]
  19. Agilent Laser and Optics User’s Manual; 2007, Volume I.

2013 (2)

G. Zamiela and M. Dobosz, “Corner cube reflector lateral displacement evaluation simultaneously with interferometer length measurement,” Opt. Laser Technol. 50, 118–124 (2013).
[Crossref]

F. Qibo, Z. Bin, C. Cunxing, K. Cuifang, Z. Yusheng, and Y. Fenglin, “Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide,” Opt. Express 21(22), 25805–25819 (2013).
[Crossref] [PubMed]

2008 (1)

Q. Feng, B. Zhang, and C. Kuang, “Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift,” Int. J. Prec. Eng. Manufact. 9, 26–31 (2008).

2006 (1)

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[Crossref]

2005 (1)

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sensor. Actuat. A 125(1), 100–108 (2005).

2002 (3)

P. L. Teoh, B. Shirinzadeh, C. W. Foong, and G. Alici, “The measurement uncertainties in the laser interferometry-based sensing and tracking technique,” Measurement 32(2), 135–150 (2002).
[Crossref]

J. S. Chen, T. W. Kou, and S. H. Chiou, “Geometric error calibration of multi-axis machines using an auto-alignment laser interferometer,” Precis. Eng. 26(1), 99–104 (2002).

J. A. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom displacement measurement systems using cooperative targets,” Precis. Eng. 26(1), 99–104 (2002).
[Crossref]

2000 (2)

J. S. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71(8), 3214–3219 (2000).
[Crossref]

K. C. Fan and M. J. Chen, “6-Degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

1998 (1)

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

1997 (1)

C. Chou, L. Y. Chou, C. K. Peng, Y. C. Huang, and K. C. Fan, “CCD-based CMM Geometrical error measurement using fourier phase shift algorithm,” Int. J. Mach. Tools Manuf. 37(5), 579–590 (1997).
[Crossref]

1995 (1)

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machine,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

1992 (1)

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).

Agilent Technologies,

Agilent Technologies and Optics and Laser Heads for Laser-Interferometer Positioning Systems,” 5964–6190 (2000).

Alici, G.

P. L. Teoh, B. Shirinzadeh, C. W. Foong, and G. Alici, “The measurement uncertainties in the laser interferometry-based sensing and tracking technique,” Measurement 32(2), 135–150 (2002).
[Crossref]

Arai, Y.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[Crossref]

Bae, E. W.

J. A. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom displacement measurement systems using cooperative targets,” Precis. Eng. 26(1), 99–104 (2002).
[Crossref]

J. S. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71(8), 3214–3219 (2000).
[Crossref]

Bin, Z.

Chen, J. S.

J. S. Chen, T. W. Kou, and S. H. Chiou, “Geometric error calibration of multi-axis machines using an auto-alignment laser interferometer,” Precis. Eng. 26(1), 99–104 (2002).

Chen, M. J.

K. C. Fan and M. J. Chen, “6-Degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

Chen, S.

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sensor. Actuat. A 125(1), 100–108 (2005).

Chiou, S. H.

J. S. Chen, T. W. Kou, and S. H. Chiou, “Geometric error calibration of multi-axis machines using an auto-alignment laser interferometer,” Precis. Eng. 26(1), 99–104 (2002).

Chou, C.

C. Chou, L. Y. Chou, C. K. Peng, Y. C. Huang, and K. C. Fan, “CCD-based CMM Geometrical error measurement using fourier phase shift algorithm,” Int. J. Mach. Tools Manuf. 37(5), 579–590 (1997).
[Crossref]

Chou, L. Y.

C. Chou, L. Y. Chou, C. K. Peng, Y. C. Huang, and K. C. Fan, “CCD-based CMM Geometrical error measurement using fourier phase shift algorithm,” Int. J. Mach. Tools Manuf. 37(5), 579–590 (1997).
[Crossref]

Cuifang, K.

Cunxing, C.

Dobosz, M.

G. Zamiela and M. Dobosz, “Corner cube reflector lateral displacement evaluation simultaneously with interferometer length measurement,” Opt. Laser Technol. 50, 118–124 (2013).
[Crossref]

Fan, K. C.

K. C. Fan and M. J. Chen, “6-Degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

C. Chou, L. Y. Chou, C. K. Peng, Y. C. Huang, and K. C. Fan, “CCD-based CMM Geometrical error measurement using fourier phase shift algorithm,” Int. J. Mach. Tools Manuf. 37(5), 579–590 (1997).
[Crossref]

Feng, Q.

Q. Feng, B. Zhang, and C. Kuang, “Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift,” Int. J. Prec. Eng. Manufact. 9, 26–31 (2008).

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sensor. Actuat. A 125(1), 100–108 (2005).

Fenglin, Y.

Foong, C. W.

P. L. Teoh, B. Shirinzadeh, C. W. Foong, and G. Alici, “The measurement uncertainties in the laser interferometry-based sensing and tracking technique,” Measurement 32(2), 135–150 (2002).
[Crossref]

Gao, W.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[Crossref]

Huang, P. S.

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machine,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).

Huang, W. M.

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

Huang, Y. C.

C. Chou, L. Y. Chou, C. K. Peng, Y. C. Huang, and K. C. Fan, “CCD-based CMM Geometrical error measurement using fourier phase shift algorithm,” Int. J. Mach. Tools Manuf. 37(5), 579–590 (1997).
[Crossref]

Kim, J. A.

J. A. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom displacement measurement systems using cooperative targets,” Precis. Eng. 26(1), 99–104 (2002).
[Crossref]

Kim, J. S.

J. S. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71(8), 3214–3219 (2000).
[Crossref]

Kim, K. C.

J. S. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71(8), 3214–3219 (2000).
[Crossref]

Kim, S. H.

J. A. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom displacement measurement systems using cooperative targets,” Precis. Eng. 26(1), 99–104 (2002).
[Crossref]

J. S. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71(8), 3214–3219 (2000).
[Crossref]

Kiyono, S.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[Crossref]

Kou, T. W.

J. S. Chen, T. W. Kou, and S. H. Chiou, “Geometric error calibration of multi-axis machines using an auto-alignment laser interferometer,” Precis. Eng. 26(1), 99–104 (2002).

Kuang, C.

Q. Feng, B. Zhang, and C. Kuang, “Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift,” Int. J. Prec. Eng. Manufact. 9, 26–31 (2008).

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sensor. Actuat. A 125(1), 100–108 (2005).

Kwak, Y. K.

J. A. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom displacement measurement systems using cooperative targets,” Precis. Eng. 26(1), 99–104 (2002).
[Crossref]

J. S. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71(8), 3214–3219 (2000).
[Crossref]

Liu, B.

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sensor. Actuat. A 125(1), 100–108 (2005).

Ni, J.

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machine,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).

Park, C. H.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[Crossref]

Peng, C. K.

C. Chou, L. Y. Chou, C. K. Peng, Y. C. Huang, and K. C. Fan, “CCD-based CMM Geometrical error measurement using fourier phase shift algorithm,” Int. J. Mach. Tools Manuf. 37(5), 579–590 (1997).
[Crossref]

Qibo, F.

Shibuya, A.

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[Crossref]

Shirinzadeh, B.

P. L. Teoh, B. Shirinzadeh, C. W. Foong, and G. Alici, “The measurement uncertainties in the laser interferometry-based sensing and tracking technique,” Measurement 32(2), 135–150 (2002).
[Crossref]

Teoh, P. L.

P. L. Teoh, B. Shirinzadeh, C. W. Foong, and G. Alici, “The measurement uncertainties in the laser interferometry-based sensing and tracking technique,” Measurement 32(2), 135–150 (2002).
[Crossref]

Wu, S. M.

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).

Yusheng, Z.

Zamiela, G.

G. Zamiela and M. Dobosz, “Corner cube reflector lateral displacement evaluation simultaneously with interferometer length measurement,” Opt. Laser Technol. 50, 118–124 (2013).
[Crossref]

Zhang, B.

Q. Feng, B. Zhang, and C. Kuang, “Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift,” Int. J. Prec. Eng. Manufact. 9, 26–31 (2008).

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sensor. Actuat. A 125(1), 100–108 (2005).

Zhang, Z.

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sensor. Actuat. A 125(1), 100–108 (2005).

Int. J. Mach. Tools Manuf. (3)

K. C. Fan, M. J. Chen, and W. M. Huang, “A six-degree-of-freedom measurement system for the motion accuracy of linear stages,” Int. J. Mach. Tools Manuf. 38(3), 155–164 (1998).
[Crossref]

P. S. Huang and J. Ni, “On-line error compensation of coordinate measuring machine,” Int. J. Mach. Tools Manuf. 35(5), 725–738 (1995).
[Crossref]

C. Chou, L. Y. Chou, C. K. Peng, Y. C. Huang, and K. C. Fan, “CCD-based CMM Geometrical error measurement using fourier phase shift algorithm,” Int. J. Mach. Tools Manuf. 37(5), 579–590 (1997).
[Crossref]

Int. J. Prec. Eng. Manufact. (1)

Q. Feng, B. Zhang, and C. Kuang, “Four degree-of-freedom geometric measurement system with common-path compensation for laser beam drift,” Int. J. Prec. Eng. Manufact. 9, 26–31 (2008).

J. Eng. Ind. (1)

J. Ni, P. S. Huang, and S. M. Wu, “A multi-degree-of-freedom measurement system for CMM geometric errors,” J. Eng. Ind. 114, 362–389 (1992).

Measurement (1)

P. L. Teoh, B. Shirinzadeh, C. W. Foong, and G. Alici, “The measurement uncertainties in the laser interferometry-based sensing and tracking technique,” Measurement 32(2), 135–150 (2002).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

G. Zamiela and M. Dobosz, “Corner cube reflector lateral displacement evaluation simultaneously with interferometer length measurement,” Opt. Laser Technol. 50, 118–124 (2013).
[Crossref]

Precis. Eng. (4)

W. Gao, Y. Arai, A. Shibuya, S. Kiyono, and C. H. Park, “Measurement of multi-degree-of-freedom error motions of a precision linear air-bearing stage,” Precis. Eng. 30(1), 96–103 (2006).
[Crossref]

K. C. Fan and M. J. Chen, “6-Degree-of-freedom measurement system for the accuracy of X-Y stages,” Precis. Eng. 24(1), 15–23 (2000).
[Crossref]

J. A. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Design methods for six-degree-of-freedom displacement measurement systems using cooperative targets,” Precis. Eng. 26(1), 99–104 (2002).
[Crossref]

J. S. Chen, T. W. Kou, and S. H. Chiou, “Geometric error calibration of multi-axis machines using an auto-alignment laser interferometer,” Precis. Eng. 26(1), 99–104 (2002).

Rev. Sci. Instrum. (1)

J. S. Kim, K. C. Kim, E. W. Bae, S. H. Kim, and Y. K. Kwak, “Six-degree-of-freedom displacement measurement system using a diffraction grating,” Rev. Sci. Instrum. 71(8), 3214–3219 (2000).
[Crossref]

Sensor. Actuat. A (1)

C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, “A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module,” Sensor. Actuat. A 125(1), 100–108 (2005).

Other (5)

Renishaw plc Laser System Manual, Reference Section (2007), version 2.1.

Renishaw plc Laser System Manual, Linear measurement (2007), version 2.1.

Agilent Technologies and Optics and Laser Heads for Laser-Interferometer Positioning Systems,” 5964–6190 (2000).

Agilent Laser and Optics User’s Manual; 2007, Volume I.

Automated Precision Inc, “XD LASER,” http://www.apisensor.com/index.php/products-en/machine-tool-calibration-en/xd-laser-en .

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Figures (12)

Fig. 1
Fig. 1 Schematic diagram of a straightness error measurement. PSD: position-sensitive detector.
Fig. 2
Fig. 2 Schematic diagrams of the dual position-sensitive detector (PSD)-based orientation measurement unit and its improved pattern: (a) original pattern and (b) improved pattern.
Fig. 3
Fig. 3 Configuration of the modified dual position-sensitive detector (PSD)-based unit.
Fig. 4
Fig. 4 Schematic diagram of the fiber-coupled laser being used to correct the three parameters E, F, and P2 . PSD: position-sensitive detector.
Fig. 5
Fig. 5 Schematic diagram of the dual-frequency, three-degrees-of-freedom (3 DOF) measurement system. PBS: polarizing beam splitter. RCC: reference cube corner. PM: plane mirror.
Fig. 6
Fig. 6 Placement of the retro-reflectors: (a) measurement along the X-axis; (b) measurement along the Y-axis; (c) measurement along the Z-axis.
Fig. 7
Fig. 7 The raw data of measured displacements (blue square points). Deviations of the displacement along the X-axis before (a) and after (b) laser beam direction variation, respectively (black circle points).
Fig. 8
Fig. 8 The raw data of measured straightness errors along the Y-axis (blue square points). Deviations of the straightness errors along the Y-axis before (a) and after (b) laser beam direction variation, respectively (black circle points).
Fig. 9
Fig. 9 The raw data of measured straightness errors along the Z-axis (blue square points). Deviations of the straightness errors along the Z-axis before (a) and after (b) laser beam direction variation, respectively (black circle points).
Fig. 10
Fig. 10 (a) Straightness errors along the Y-axis. (b) Straightness errors along the Z-axis.
Fig. 11
Fig. 11 (a) Straightness errors with time elapsing along the Y-axis. (b) Straightness errors with time elapsing along the Z-axis.
Fig. 12
Fig. 12 (a) Beam drifts along the Y-axis for the three runs. (b) Beam drifts along the Z-axis for the three runs.

Equations (11)

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

x ( x i ' + x c ) m i = y ( y i ' + y c ) n i = z ( z i ' + z c ) q i ,
x i ' = ( 2 m i 2 1 ) ( x i x c ) + 2 m i n i ( y i y c ) + 2 m i q i ( z i z c ) y i ' = 2 m i n i ( x i x c ) + ( 2 n i 2 1 ) ( y i y c ) + 2 n i q i ( z i z c ) z i ' = 2 m i q i ( x i x c ) + 2 n i q i ( y i y c ) + ( 2 q i 2 1 ) ( z i z c ) .
x y o z = 0 y y o z = n i m i ( x i ' + x c ) + y i ' + y c z y o z = q i m i ( x i ' + x c ) + z i ' + z c .
y c h y c f = ( y y o z h y y o z f ) 2 + n i m i ( x c h x c f ) z c h z c f = ( z y o z h z y o z f ) 2 + q i m i ( x c h x c f ) .
y c h y c f = ( y y o z h y y o z f ) 2 + L M n i z c h z c f = ( z y o z h z y o z f ) 2 + L M q i .
x 1 = x 0 + n L 2 y 0 m y 1 = L 2 z 1 = z 0 + q L 2 y 0 m .
x 2 = x 0 " n L 3 + y 0 " m y 2 = L 3 z 2 = z 0 " + q L 3 + y 0 " m ,
x 0 " = m y 0 ' + n x 0 ' + m L 1 n + m y 0 " = m y 0 ' n x 0 ' + n L 1 n + m z 0 " = ( x 0 ' y 0 ' + L 1 ) q n + m + z 0 ' .
x 0 ' = ( 2 m 2 1 ) x 0 + 2 m n y 0 + 2 m q z 0 y 0 ' = 2 m n x 0 + ( 2 n 2 1 ) y 0 + 2 n q z 0 z 0 ' = 2 m q x 0 + 2 n q y 0 + ( 2 q 2 1 ) z 0 .
m = 2 L 1 + L 3 L 2 Δ n = 2 L 1 x 1 x 2 Δ q = z 1 + z 2 Δ ,
m = E Δ n = M 1 F Δ q = M 2 + P 2 Δ ,

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