Abstract

A novel laser speckle velocimeter with two self-mixing laser diodes (SM-LD’s) for velocity and length measurements of moving surfaces is reported. The mean frequency of the speckle signal obtained in the measurement system depends on the surface path illuminated by the SM-LD. This behavior of the speckle signal in the SM-LD’s is exploited to detect the front and the end edges of a target surface by sampling continuously the number of intensity changes in a speckle signal waveform. Once the edges are determined, the velocity and the length of the surface are calculated easily. The error for length and velocity measurements of a target with a homogeneous rough plane surface of 60-mm length, moving at a velocity of 200 mm/s, can be as low as 2.1% and 1.75%, respectively.

© 1999 Optical Society of America

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References

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  1. T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
    [CrossRef]
  2. R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
    [CrossRef]
  3. K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, Dordrecht, The Netherlands, 1991).
  4. P. J. de Groot, G. M. Gallatin, S. H. Macomber, “Ranging and velocimetry signal generation in a backscatter-modulated laser diode,” Appl. Opt. 27, 4475–4480 (1988).
    [CrossRef] [PubMed]
  5. S. Donati, G. Giuliani, S. Merlo, “Laser diode feedback interferometer for measurements of displacement without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
    [CrossRef]
  6. W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single mode laser for optical sensing applications,” IEEE J. Lightwave Technol. 12, 1577–1587 (1994).
    [CrossRef]
  7. N. Servagent, T. Bosch, M. Lescure, “A laser displacement sensor using the self-mixing effect for modal analysis and defect detection,” IEEE Trans. Instrum. Meas. 46, 847–850 (1997).
    [CrossRef]
  8. S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, M. Sumi, “Compact and high precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
    [CrossRef]
  9. S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeter with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
    [CrossRef]
  10. T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Novel velocity detection of moving object with rough surface vertically illuminated by self-mixing laser diode,” in Proceedings of the Korean Automatic Control Conference (KACC’94), Taejeon, South Korea, 1994 (Korea Association of Automatic Control, Seoul, Korea, 1994), pp. 494–497.
  11. T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. 45, 499–503 (1996).
    [CrossRef]
  12. T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Automatic measurement of velocity and length of moving plate using self-mixing laser diode,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IMTC’96, IMEKO TC-7), Brussels, 1996 (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 78–83.
  13. T. Takasu, Ş. K. Özdemir, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Length measurement by speckle velocimeter using two self-mixing laser diodes,” in Conference on Lasers and Electro-Optics-Europe, Vol. 6 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 200.

1997 (1)

N. Servagent, T. Bosch, M. Lescure, “A laser displacement sensor using the self-mixing effect for modal analysis and defect detection,” IEEE Trans. Instrum. Meas. 46, 847–850 (1997).
[CrossRef]

1996 (1)

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. 45, 499–503 (1996).
[CrossRef]

1995 (1)

S. Donati, G. Giuliani, S. Merlo, “Laser diode feedback interferometer for measurements of displacement without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
[CrossRef]

1994 (1)

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single mode laser for optical sensing applications,” IEEE J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

1992 (1)

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, M. Sumi, “Compact and high precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

1989 (1)

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeter with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
[CrossRef]

1988 (1)

1981 (1)

T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

1980 (1)

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

Asakura, T.

T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

Bosch, T.

N. Servagent, T. Bosch, M. Lescure, “A laser displacement sensor using the self-mixing effect for modal analysis and defect detection,” IEEE Trans. Instrum. Meas. 46, 847–850 (1997).
[CrossRef]

Boyle, W. J. O.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single mode laser for optical sensing applications,” IEEE J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

de Groot, P. J.

Donati, S.

S. Donati, G. Giuliani, S. Merlo, “Laser diode feedback interferometer for measurements of displacement without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
[CrossRef]

Gallatin, G. M.

Giuliani, G.

S. Donati, G. Giuliani, S. Merlo, “Laser diode feedback interferometer for measurements of displacement without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
[CrossRef]

Grattan, K. T. V.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single mode laser for optical sensing applications,” IEEE J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

Ikeda, H.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. 45, 499–503 (1996).
[CrossRef]

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, M. Sumi, “Compact and high precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeter with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
[CrossRef]

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Automatic measurement of velocity and length of moving plate using self-mixing laser diode,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IMTC’96, IMEKO TC-7), Brussels, 1996 (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 78–83.

T. Takasu, Ş. K. Özdemir, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Length measurement by speckle velocimeter using two self-mixing laser diodes,” in Conference on Lasers and Electro-Optics-Europe, Vol. 6 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 200.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Novel velocity detection of moving object with rough surface vertically illuminated by self-mixing laser diode,” in Proceedings of the Korean Automatic Control Conference (KACC’94), Taejeon, South Korea, 1994 (Korea Association of Automatic Control, Seoul, Korea, 1994), pp. 494–497.

Kobayashi, K.

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

Lang, R.

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

Lescure, M.

N. Servagent, T. Bosch, M. Lescure, “A laser displacement sensor using the self-mixing effect for modal analysis and defect detection,” IEEE Trans. Instrum. Meas. 46, 847–850 (1997).
[CrossRef]

Macomber, S. H.

Merlo, S.

S. Donati, G. Giuliani, S. Merlo, “Laser diode feedback interferometer for measurements of displacement without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
[CrossRef]

Naito, H.

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeter with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
[CrossRef]

Nishide, K.

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, M. Sumi, “Compact and high precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

Özdemir, S. K.

T. Takasu, Ş. K. Özdemir, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Length measurement by speckle velocimeter using two self-mixing laser diodes,” in Conference on Lasers and Electro-Optics-Europe, Vol. 6 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 200.

Palmer, A. W.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single mode laser for optical sensing applications,” IEEE J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

Petermann, K.

K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, Dordrecht, The Netherlands, 1991).

Sawaki, T.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. 45, 499–503 (1996).
[CrossRef]

Servagent, N.

N. Servagent, T. Bosch, M. Lescure, “A laser displacement sensor using the self-mixing effect for modal analysis and defect detection,” IEEE Trans. Instrum. Meas. 46, 847–850 (1997).
[CrossRef]

Shibata, T.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. 45, 499–503 (1996).
[CrossRef]

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Novel velocity detection of moving object with rough surface vertically illuminated by self-mixing laser diode,” in Proceedings of the Korean Automatic Control Conference (KACC’94), Taejeon, South Korea, 1994 (Korea Association of Automatic Control, Seoul, Korea, 1994), pp. 494–497.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Automatic measurement of velocity and length of moving plate using self-mixing laser diode,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IMTC’96, IMEKO TC-7), Brussels, 1996 (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 78–83.

Shinohara, S.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. 45, 499–503 (1996).
[CrossRef]

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, M. Sumi, “Compact and high precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeter with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
[CrossRef]

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Automatic measurement of velocity and length of moving plate using self-mixing laser diode,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IMTC’96, IMEKO TC-7), Brussels, 1996 (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 78–83.

T. Takasu, Ş. K. Özdemir, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Length measurement by speckle velocimeter using two self-mixing laser diodes,” in Conference on Lasers and Electro-Optics-Europe, Vol. 6 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 200.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Novel velocity detection of moving object with rough surface vertically illuminated by self-mixing laser diode,” in Proceedings of the Korean Automatic Control Conference (KACC’94), Taejeon, South Korea, 1994 (Korea Association of Automatic Control, Seoul, Korea, 1994), pp. 494–497.

Sumi, M.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. 45, 499–503 (1996).
[CrossRef]

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, M. Sumi, “Compact and high precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeter with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
[CrossRef]

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Automatic measurement of velocity and length of moving plate using self-mixing laser diode,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IMTC’96, IMEKO TC-7), Brussels, 1996 (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 78–83.

T. Takasu, Ş. K. Özdemir, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Length measurement by speckle velocimeter using two self-mixing laser diodes,” in Conference on Lasers and Electro-Optics-Europe, Vol. 6 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 200.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Novel velocity detection of moving object with rough surface vertically illuminated by self-mixing laser diode,” in Proceedings of the Korean Automatic Control Conference (KACC’94), Taejeon, South Korea, 1994 (Korea Association of Automatic Control, Seoul, Korea, 1994), pp. 494–497.

Takai, N.

T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

Takasu, T.

T. Takasu, Ş. K. Özdemir, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Length measurement by speckle velocimeter using two self-mixing laser diodes,” in Conference on Lasers and Electro-Optics-Europe, Vol. 6 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 200.

Wang, W. M.

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single mode laser for optical sensing applications,” IEEE J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

Yoshida, H.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. 45, 499–503 (1996).
[CrossRef]

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, M. Sumi, “Compact and high precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeter with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
[CrossRef]

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Automatic measurement of velocity and length of moving plate using self-mixing laser diode,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IMTC’96, IMEKO TC-7), Brussels, 1996 (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 78–83.

T. Takasu, Ş. K. Özdemir, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Length measurement by speckle velocimeter using two self-mixing laser diodes,” in Conference on Lasers and Electro-Optics-Europe, Vol. 6 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 200.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Novel velocity detection of moving object with rough surface vertically illuminated by self-mixing laser diode,” in Proceedings of the Korean Automatic Control Conference (KACC’94), Taejeon, South Korea, 1994 (Korea Association of Automatic Control, Seoul, Korea, 1994), pp. 494–497.

Appl. Opt. (1)

Appl. Phys. (1)

T. Asakura, N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” Appl. Phys. 25, 179–194 (1981).
[CrossRef]

IEEE J. Lightwave Technol. (1)

W. M. Wang, K. T. V. Grattan, A. W. Palmer, W. J. O. Boyle, “Self-mixing interference inside a single mode laser for optical sensing applications,” IEEE J. Lightwave Technol. 12, 1577–1587 (1994).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. Lang, K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347–355 (1980).
[CrossRef]

S. Donati, G. Giuliani, S. Merlo, “Laser diode feedback interferometer for measurements of displacement without ambiguity,” IEEE J. Quantum Electron. 31, 113–119 (1995).
[CrossRef]

IEEE Trans. Instrum. Meas. (4)

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. 45, 499–503 (1996).
[CrossRef]

N. Servagent, T. Bosch, M. Lescure, “A laser displacement sensor using the self-mixing effect for modal analysis and defect detection,” IEEE Trans. Instrum. Meas. 46, 847–850 (1997).
[CrossRef]

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, M. Sumi, “Compact and high precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

S. Shinohara, H. Naito, H. Yoshida, H. Ikeda, M. Sumi, “Compact and versatile self-mixing type semiconductor laser Doppler velocimeter with direction discrimination circuit,” IEEE Trans. Instrum. Meas. 38, 574–577 (1989).
[CrossRef]

Other (4)

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Novel velocity detection of moving object with rough surface vertically illuminated by self-mixing laser diode,” in Proceedings of the Korean Automatic Control Conference (KACC’94), Taejeon, South Korea, 1994 (Korea Association of Automatic Control, Seoul, Korea, 1994), pp. 494–497.

T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Automatic measurement of velocity and length of moving plate using self-mixing laser diode,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IMTC’96, IMEKO TC-7), Brussels, 1996 (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 78–83.

T. Takasu, Ş. K. Özdemir, S. Shinohara, H. Ikeda, H. Yoshida, M. Sumi, “Length measurement by speckle velocimeter using two self-mixing laser diodes,” in Conference on Lasers and Electro-Optics-Europe, Vol. 6 of 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 200.

K. Petermann, Laser Diode Modulation and Noise (Kluwer Academic, Dordrecht, The Netherlands, 1991).

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Figures (13)

Fig. 1
Fig. 1

Schematic of a semiconductor laser diode with external optical feedback.

Fig. 2
Fig. 2

Schematic configuration of the velocity and the length measurement system with a speckle-signal-processing unit.

Fig. 3
Fig. 3

Speckle signal processing.

Fig. 4
Fig. 4

Typical examples of a speckle signal (lower spectra) and its converted speckle pulse train (upper spectra).

Fig. 5
Fig. 5

Speckle signal frequency versus time plot obtained for a sheet of white paper, 60 mm, pasted on an aluminum plane moving at 200 mm/s.

Fig. 6
Fig. 6

Speckle signal frequency versus time plots of two SM-LD’s vertically illuminating the target surface as shown in Fig. 7.

Fig. 7
Fig. 7

Schematics of the setup used for two SM-LD measurements.

Fig. 8
Fig. 8

Typical example of a speckle signal frequency versus time plot measured by SM-LD1.

Fig. 9
Fig. 9

Rate of change of speckle signal frequency R versus time plots obtained from two SM-LD’s whose speckle signal versus time plots are shown in Fig. 6.

Fig. 10
Fig. 10

Velocity measurement error versus sampling time for various target velocities.

Fig. 11
Fig. 11

Theoretical and experimental error of a velocity measurement at different sampling times for a target moving at 250 mm/s.

Fig. 12
Fig. 12

Standard deviation in length measurements versus sampling time for various target velocities.

Fig. 13
Fig. 13

Theoretical and experimental error values of length measurement at different sampling times for a target 60 mm long moving at 250 mm/s.

Equations (3)

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

v=ΔXTa-Ta=ΔXTb-Tb,
L=vTb-Ta=vTb-Ta.
Ri=Fi+1-FiTsi+1-Tsi+Fi-Fi-1Tsi-Tsi-12, i=1, 2,, n-1,

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