Abstract

This paper presents a computer-vision system to assist reach stackers to automatically align the spreader with the target container. By analyzing infrared lines on the top of the container, the proposed system is able to calculate the relative position between the spreader and the container. The invisible structured lights are equipped in this system to enable all-weather operation, which can avoid environmental factors such as shadows and differences in climate. Additionally, the lateral inclination of the spreader is taken into consideration to offer a more accurate alignment than other competing systems. Estimation errors are reduced through approaches including power series and linear regression. The accuracy can be controlled within 2 cm or 2 deg, which meets the requirements of reach stackers’ operation.

© 2012 Optical Society of America

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References

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  1. P. J. M. Meersmans and R. Dekker, Operations Research Supports Container Handling (Econometric Institute of Erasmus University–Rotterdam, 2001).
  2. K. Uchida, N. Miyata, K. Obata, and H. Yoshikawa, “Container position measuring method and device for cargo crane and container landing/stacking method,” U.S. patent 7,106,883(12 September 2006).
  3. J. Landaluze, V. Del Rio, C. Nicolas, J. Ezkerra, and A. Martinezez, “Mocont location module: a container location system based on DR/DGNSS integration,” in Informatics in Control, Automation and Robotics1, (Springer, 2006), pp. 151–158.
  4. K. Obata, K. Uchida, T. Chikura, H. Yoshikawa, and T. Monzen, “Automated transfer crane,” Tech. Rev. Mitsubishi Heavy Industries 40(0), 1–5 (2003).
  5. K. Obata, N. Miyata, and M. Kobayashi, “Development of automatic container yard crane,” Tech. Rev. Mitsubishi Heavy Industries 38(2), 62–66 (2001).
  6. K. Yasunaga, N. Watanabe, K. Tanouchi, M. Nishioka, and S. Nagashina, “Auto-steering system for transfer crane,” Mitsubishi juko giho 37, 298–301 (2000).
  7. H. Yoshikawa, S. Kunimitsu, H. Hoshina, N. Miyata, and M. Kobayashi, “Development of hoisting load position sensor for container handling cranes,” Tech. Rev. Mitsubishi Heavy Industries 38, 67–72 (2001).
  8. R. Bozzo, A. Derito, R. Nurchi, and N. Ackroyd, “MOCONT: a new system for container terminal monitoring and control,” in Intelligent Transportation Systems Conference Proceedings (IEEE, 2001), pp. 1090–1094.
  9. V. Recagno, A. Derito, and R. Nurchi, “MOCONT: a new system for automatic identification and location of containers,” in Proceedings of Intelligent Transportation Systems (IEEE, 2001).
  10. A. Slutej and F. Kolonić, “Advances in control and sensors technology enabling unmanned operation of containers cranes,” in Proceednigs of the 13th International Conference on Electrical Drives and Power Electronics (2005).
  11. M. B. Duinkerken, J. J. M. Evers, and J. A. Ottjes, “A simulation model for integrating quay transport and stacking policies on automated container terminals,” in Proceedings of the 15th European Simulation Multiconference (2001).
  12. S. Kwasniowski, M. Zajac, and P. Zajac, “Telematic problems of unmanned vehicles positioning at container terminals and warehouses,” in Transport Systems Telematics (2011), pp. 391–399.
    [CrossRef]
  13. O. Wulf and B. Wagner, “Fast 3D scanning methods for laser measurement systems,” in Proceedings of the International Conference on Control Systems and Computer Science (2003), pp. 312–317.
  14. R. Stahlbock and S. Voß, “Operations research at container terminals: a literature update,” OR Spectrum 30, 1–52 (2008).
    [CrossRef]
  15. H. H. Loeffler, W. J. Bakker, S. I. Biba, R. J. Aloisi, D. E. Seals, and C. L. Jones, “Container positioning device,” U.S. patent 10358804 (26 February 2002).
  16. K. Busboom and A. DeBohman, “Container surveillance system and related method,” European patent EP20010107411 (2 October 2002).
  17. T. Takehara, M. W. King, and C. Chung, “Method and apparatus of automated optical container code recognition with positional identification for a transfer container crane,” U.S. patent 7,308,114 (11 December 2007).
  18. P. Thomas, A. Thomas, and L. Scheppmann, “Container tracking and locating systems, methods, and computer program products,” U.S. patent 8,060,263 (27 January 2011).
  19. W. Wichner, “Operating method and device for operating automated container quay cranes,” U.S. patent 6,965,823(15 November 2005).
  20. M. Lindholm and P. Vaha, “Procedure and apparatus for locating a container for lifting,” U.S. patent 5,067,013(19 November 1991).

2008 (1)

R. Stahlbock and S. Voß, “Operations research at container terminals: a literature update,” OR Spectrum 30, 1–52 (2008).
[CrossRef]

2003 (1)

K. Obata, K. Uchida, T. Chikura, H. Yoshikawa, and T. Monzen, “Automated transfer crane,” Tech. Rev. Mitsubishi Heavy Industries 40(0), 1–5 (2003).

2001 (2)

K. Obata, N. Miyata, and M. Kobayashi, “Development of automatic container yard crane,” Tech. Rev. Mitsubishi Heavy Industries 38(2), 62–66 (2001).

H. Yoshikawa, S. Kunimitsu, H. Hoshina, N. Miyata, and M. Kobayashi, “Development of hoisting load position sensor for container handling cranes,” Tech. Rev. Mitsubishi Heavy Industries 38, 67–72 (2001).

2000 (1)

K. Yasunaga, N. Watanabe, K. Tanouchi, M. Nishioka, and S. Nagashina, “Auto-steering system for transfer crane,” Mitsubishi juko giho 37, 298–301 (2000).

Ackroyd, N.

R. Bozzo, A. Derito, R. Nurchi, and N. Ackroyd, “MOCONT: a new system for container terminal monitoring and control,” in Intelligent Transportation Systems Conference Proceedings (IEEE, 2001), pp. 1090–1094.

Aloisi, R. J.

H. H. Loeffler, W. J. Bakker, S. I. Biba, R. J. Aloisi, D. E. Seals, and C. L. Jones, “Container positioning device,” U.S. patent 10358804 (26 February 2002).

Bakker, W. J.

H. H. Loeffler, W. J. Bakker, S. I. Biba, R. J. Aloisi, D. E. Seals, and C. L. Jones, “Container positioning device,” U.S. patent 10358804 (26 February 2002).

Biba, S. I.

H. H. Loeffler, W. J. Bakker, S. I. Biba, R. J. Aloisi, D. E. Seals, and C. L. Jones, “Container positioning device,” U.S. patent 10358804 (26 February 2002).

Bozzo, R.

R. Bozzo, A. Derito, R. Nurchi, and N. Ackroyd, “MOCONT: a new system for container terminal monitoring and control,” in Intelligent Transportation Systems Conference Proceedings (IEEE, 2001), pp. 1090–1094.

Busboom, K.

K. Busboom and A. DeBohman, “Container surveillance system and related method,” European patent EP20010107411 (2 October 2002).

Chikura, T.

K. Obata, K. Uchida, T. Chikura, H. Yoshikawa, and T. Monzen, “Automated transfer crane,” Tech. Rev. Mitsubishi Heavy Industries 40(0), 1–5 (2003).

Chung, C.

T. Takehara, M. W. King, and C. Chung, “Method and apparatus of automated optical container code recognition with positional identification for a transfer container crane,” U.S. patent 7,308,114 (11 December 2007).

DeBohman, A.

K. Busboom and A. DeBohman, “Container surveillance system and related method,” European patent EP20010107411 (2 October 2002).

Dekker, R.

P. J. M. Meersmans and R. Dekker, Operations Research Supports Container Handling (Econometric Institute of Erasmus University–Rotterdam, 2001).

Del Rio, V.

J. Landaluze, V. Del Rio, C. Nicolas, J. Ezkerra, and A. Martinezez, “Mocont location module: a container location system based on DR/DGNSS integration,” in Informatics in Control, Automation and Robotics1, (Springer, 2006), pp. 151–158.

Derito, A.

R. Bozzo, A. Derito, R. Nurchi, and N. Ackroyd, “MOCONT: a new system for container terminal monitoring and control,” in Intelligent Transportation Systems Conference Proceedings (IEEE, 2001), pp. 1090–1094.

V. Recagno, A. Derito, and R. Nurchi, “MOCONT: a new system for automatic identification and location of containers,” in Proceedings of Intelligent Transportation Systems (IEEE, 2001).

Duinkerken, M. B.

M. B. Duinkerken, J. J. M. Evers, and J. A. Ottjes, “A simulation model for integrating quay transport and stacking policies on automated container terminals,” in Proceedings of the 15th European Simulation Multiconference (2001).

Evers, J. J. M.

M. B. Duinkerken, J. J. M. Evers, and J. A. Ottjes, “A simulation model for integrating quay transport and stacking policies on automated container terminals,” in Proceedings of the 15th European Simulation Multiconference (2001).

Ezkerra, J.

J. Landaluze, V. Del Rio, C. Nicolas, J. Ezkerra, and A. Martinezez, “Mocont location module: a container location system based on DR/DGNSS integration,” in Informatics in Control, Automation and Robotics1, (Springer, 2006), pp. 151–158.

Hoshina, H.

H. Yoshikawa, S. Kunimitsu, H. Hoshina, N. Miyata, and M. Kobayashi, “Development of hoisting load position sensor for container handling cranes,” Tech. Rev. Mitsubishi Heavy Industries 38, 67–72 (2001).

Jones, C. L.

H. H. Loeffler, W. J. Bakker, S. I. Biba, R. J. Aloisi, D. E. Seals, and C. L. Jones, “Container positioning device,” U.S. patent 10358804 (26 February 2002).

King, M. W.

T. Takehara, M. W. King, and C. Chung, “Method and apparatus of automated optical container code recognition with positional identification for a transfer container crane,” U.S. patent 7,308,114 (11 December 2007).

Kobayashi, M.

H. Yoshikawa, S. Kunimitsu, H. Hoshina, N. Miyata, and M. Kobayashi, “Development of hoisting load position sensor for container handling cranes,” Tech. Rev. Mitsubishi Heavy Industries 38, 67–72 (2001).

K. Obata, N. Miyata, and M. Kobayashi, “Development of automatic container yard crane,” Tech. Rev. Mitsubishi Heavy Industries 38(2), 62–66 (2001).

Kolonic, F.

A. Slutej and F. Kolonić, “Advances in control and sensors technology enabling unmanned operation of containers cranes,” in Proceednigs of the 13th International Conference on Electrical Drives and Power Electronics (2005).

Kunimitsu, S.

H. Yoshikawa, S. Kunimitsu, H. Hoshina, N. Miyata, and M. Kobayashi, “Development of hoisting load position sensor for container handling cranes,” Tech. Rev. Mitsubishi Heavy Industries 38, 67–72 (2001).

Kwasniowski, S.

S. Kwasniowski, M. Zajac, and P. Zajac, “Telematic problems of unmanned vehicles positioning at container terminals and warehouses,” in Transport Systems Telematics (2011), pp. 391–399.
[CrossRef]

Landaluze, J.

J. Landaluze, V. Del Rio, C. Nicolas, J. Ezkerra, and A. Martinezez, “Mocont location module: a container location system based on DR/DGNSS integration,” in Informatics in Control, Automation and Robotics1, (Springer, 2006), pp. 151–158.

Lindholm, M.

M. Lindholm and P. Vaha, “Procedure and apparatus for locating a container for lifting,” U.S. patent 5,067,013(19 November 1991).

Loeffler, H. H.

H. H. Loeffler, W. J. Bakker, S. I. Biba, R. J. Aloisi, D. E. Seals, and C. L. Jones, “Container positioning device,” U.S. patent 10358804 (26 February 2002).

Martinezez, A.

J. Landaluze, V. Del Rio, C. Nicolas, J. Ezkerra, and A. Martinezez, “Mocont location module: a container location system based on DR/DGNSS integration,” in Informatics in Control, Automation and Robotics1, (Springer, 2006), pp. 151–158.

Meersmans, P. J. M.

P. J. M. Meersmans and R. Dekker, Operations Research Supports Container Handling (Econometric Institute of Erasmus University–Rotterdam, 2001).

Miyata, N.

H. Yoshikawa, S. Kunimitsu, H. Hoshina, N. Miyata, and M. Kobayashi, “Development of hoisting load position sensor for container handling cranes,” Tech. Rev. Mitsubishi Heavy Industries 38, 67–72 (2001).

K. Obata, N. Miyata, and M. Kobayashi, “Development of automatic container yard crane,” Tech. Rev. Mitsubishi Heavy Industries 38(2), 62–66 (2001).

K. Uchida, N. Miyata, K. Obata, and H. Yoshikawa, “Container position measuring method and device for cargo crane and container landing/stacking method,” U.S. patent 7,106,883(12 September 2006).

Monzen, T.

K. Obata, K. Uchida, T. Chikura, H. Yoshikawa, and T. Monzen, “Automated transfer crane,” Tech. Rev. Mitsubishi Heavy Industries 40(0), 1–5 (2003).

Nagashina, S.

K. Yasunaga, N. Watanabe, K. Tanouchi, M. Nishioka, and S. Nagashina, “Auto-steering system for transfer crane,” Mitsubishi juko giho 37, 298–301 (2000).

Nicolas, C.

J. Landaluze, V. Del Rio, C. Nicolas, J. Ezkerra, and A. Martinezez, “Mocont location module: a container location system based on DR/DGNSS integration,” in Informatics in Control, Automation and Robotics1, (Springer, 2006), pp. 151–158.

Nishioka, M.

K. Yasunaga, N. Watanabe, K. Tanouchi, M. Nishioka, and S. Nagashina, “Auto-steering system for transfer crane,” Mitsubishi juko giho 37, 298–301 (2000).

Nurchi, R.

R. Bozzo, A. Derito, R. Nurchi, and N. Ackroyd, “MOCONT: a new system for container terminal monitoring and control,” in Intelligent Transportation Systems Conference Proceedings (IEEE, 2001), pp. 1090–1094.

V. Recagno, A. Derito, and R. Nurchi, “MOCONT: a new system for automatic identification and location of containers,” in Proceedings of Intelligent Transportation Systems (IEEE, 2001).

Obata, K.

K. Obata, K. Uchida, T. Chikura, H. Yoshikawa, and T. Monzen, “Automated transfer crane,” Tech. Rev. Mitsubishi Heavy Industries 40(0), 1–5 (2003).

K. Obata, N. Miyata, and M. Kobayashi, “Development of automatic container yard crane,” Tech. Rev. Mitsubishi Heavy Industries 38(2), 62–66 (2001).

K. Uchida, N. Miyata, K. Obata, and H. Yoshikawa, “Container position measuring method and device for cargo crane and container landing/stacking method,” U.S. patent 7,106,883(12 September 2006).

Ottjes, J. A.

M. B. Duinkerken, J. J. M. Evers, and J. A. Ottjes, “A simulation model for integrating quay transport and stacking policies on automated container terminals,” in Proceedings of the 15th European Simulation Multiconference (2001).

Recagno, V.

V. Recagno, A. Derito, and R. Nurchi, “MOCONT: a new system for automatic identification and location of containers,” in Proceedings of Intelligent Transportation Systems (IEEE, 2001).

Scheppmann, L.

P. Thomas, A. Thomas, and L. Scheppmann, “Container tracking and locating systems, methods, and computer program products,” U.S. patent 8,060,263 (27 January 2011).

Seals, D. E.

H. H. Loeffler, W. J. Bakker, S. I. Biba, R. J. Aloisi, D. E. Seals, and C. L. Jones, “Container positioning device,” U.S. patent 10358804 (26 February 2002).

Slutej, A.

A. Slutej and F. Kolonić, “Advances in control and sensors technology enabling unmanned operation of containers cranes,” in Proceednigs of the 13th International Conference on Electrical Drives and Power Electronics (2005).

Stahlbock, R.

R. Stahlbock and S. Voß, “Operations research at container terminals: a literature update,” OR Spectrum 30, 1–52 (2008).
[CrossRef]

Takehara, T.

T. Takehara, M. W. King, and C. Chung, “Method and apparatus of automated optical container code recognition with positional identification for a transfer container crane,” U.S. patent 7,308,114 (11 December 2007).

Tanouchi, K.

K. Yasunaga, N. Watanabe, K. Tanouchi, M. Nishioka, and S. Nagashina, “Auto-steering system for transfer crane,” Mitsubishi juko giho 37, 298–301 (2000).

Thomas, A.

P. Thomas, A. Thomas, and L. Scheppmann, “Container tracking and locating systems, methods, and computer program products,” U.S. patent 8,060,263 (27 January 2011).

Thomas, P.

P. Thomas, A. Thomas, and L. Scheppmann, “Container tracking and locating systems, methods, and computer program products,” U.S. patent 8,060,263 (27 January 2011).

Uchida, K.

K. Obata, K. Uchida, T. Chikura, H. Yoshikawa, and T. Monzen, “Automated transfer crane,” Tech. Rev. Mitsubishi Heavy Industries 40(0), 1–5 (2003).

K. Uchida, N. Miyata, K. Obata, and H. Yoshikawa, “Container position measuring method and device for cargo crane and container landing/stacking method,” U.S. patent 7,106,883(12 September 2006).

Vaha, P.

M. Lindholm and P. Vaha, “Procedure and apparatus for locating a container for lifting,” U.S. patent 5,067,013(19 November 1991).

Voß, S.

R. Stahlbock and S. Voß, “Operations research at container terminals: a literature update,” OR Spectrum 30, 1–52 (2008).
[CrossRef]

Wagner, B.

O. Wulf and B. Wagner, “Fast 3D scanning methods for laser measurement systems,” in Proceedings of the International Conference on Control Systems and Computer Science (2003), pp. 312–317.

Watanabe, N.

K. Yasunaga, N. Watanabe, K. Tanouchi, M. Nishioka, and S. Nagashina, “Auto-steering system for transfer crane,” Mitsubishi juko giho 37, 298–301 (2000).

Wichner, W.

W. Wichner, “Operating method and device for operating automated container quay cranes,” U.S. patent 6,965,823(15 November 2005).

Wulf, O.

O. Wulf and B. Wagner, “Fast 3D scanning methods for laser measurement systems,” in Proceedings of the International Conference on Control Systems and Computer Science (2003), pp. 312–317.

Yasunaga, K.

K. Yasunaga, N. Watanabe, K. Tanouchi, M. Nishioka, and S. Nagashina, “Auto-steering system for transfer crane,” Mitsubishi juko giho 37, 298–301 (2000).

Yoshikawa, H.

K. Obata, K. Uchida, T. Chikura, H. Yoshikawa, and T. Monzen, “Automated transfer crane,” Tech. Rev. Mitsubishi Heavy Industries 40(0), 1–5 (2003).

H. Yoshikawa, S. Kunimitsu, H. Hoshina, N. Miyata, and M. Kobayashi, “Development of hoisting load position sensor for container handling cranes,” Tech. Rev. Mitsubishi Heavy Industries 38, 67–72 (2001).

K. Uchida, N. Miyata, K. Obata, and H. Yoshikawa, “Container position measuring method and device for cargo crane and container landing/stacking method,” U.S. patent 7,106,883(12 September 2006).

Zajac, M.

S. Kwasniowski, M. Zajac, and P. Zajac, “Telematic problems of unmanned vehicles positioning at container terminals and warehouses,” in Transport Systems Telematics (2011), pp. 391–399.
[CrossRef]

Zajac, P.

S. Kwasniowski, M. Zajac, and P. Zajac, “Telematic problems of unmanned vehicles positioning at container terminals and warehouses,” in Transport Systems Telematics (2011), pp. 391–399.
[CrossRef]

Mitsubishi juko giho (1)

K. Yasunaga, N. Watanabe, K. Tanouchi, M. Nishioka, and S. Nagashina, “Auto-steering system for transfer crane,” Mitsubishi juko giho 37, 298–301 (2000).

OR Spectrum (1)

R. Stahlbock and S. Voß, “Operations research at container terminals: a literature update,” OR Spectrum 30, 1–52 (2008).
[CrossRef]

Tech. Rev. Mitsubishi Heavy Industries (3)

K. Obata, K. Uchida, T. Chikura, H. Yoshikawa, and T. Monzen, “Automated transfer crane,” Tech. Rev. Mitsubishi Heavy Industries 40(0), 1–5 (2003).

K. Obata, N. Miyata, and M. Kobayashi, “Development of automatic container yard crane,” Tech. Rev. Mitsubishi Heavy Industries 38(2), 62–66 (2001).

H. Yoshikawa, S. Kunimitsu, H. Hoshina, N. Miyata, and M. Kobayashi, “Development of hoisting load position sensor for container handling cranes,” Tech. Rev. Mitsubishi Heavy Industries 38, 67–72 (2001).

Other (15)

R. Bozzo, A. Derito, R. Nurchi, and N. Ackroyd, “MOCONT: a new system for container terminal monitoring and control,” in Intelligent Transportation Systems Conference Proceedings (IEEE, 2001), pp. 1090–1094.

V. Recagno, A. Derito, and R. Nurchi, “MOCONT: a new system for automatic identification and location of containers,” in Proceedings of Intelligent Transportation Systems (IEEE, 2001).

A. Slutej and F. Kolonić, “Advances in control and sensors technology enabling unmanned operation of containers cranes,” in Proceednigs of the 13th International Conference on Electrical Drives and Power Electronics (2005).

M. B. Duinkerken, J. J. M. Evers, and J. A. Ottjes, “A simulation model for integrating quay transport and stacking policies on automated container terminals,” in Proceedings of the 15th European Simulation Multiconference (2001).

S. Kwasniowski, M. Zajac, and P. Zajac, “Telematic problems of unmanned vehicles positioning at container terminals and warehouses,” in Transport Systems Telematics (2011), pp. 391–399.
[CrossRef]

O. Wulf and B. Wagner, “Fast 3D scanning methods for laser measurement systems,” in Proceedings of the International Conference on Control Systems and Computer Science (2003), pp. 312–317.

P. J. M. Meersmans and R. Dekker, Operations Research Supports Container Handling (Econometric Institute of Erasmus University–Rotterdam, 2001).

K. Uchida, N. Miyata, K. Obata, and H. Yoshikawa, “Container position measuring method and device for cargo crane and container landing/stacking method,” U.S. patent 7,106,883(12 September 2006).

J. Landaluze, V. Del Rio, C. Nicolas, J. Ezkerra, and A. Martinezez, “Mocont location module: a container location system based on DR/DGNSS integration,” in Informatics in Control, Automation and Robotics1, (Springer, 2006), pp. 151–158.

H. H. Loeffler, W. J. Bakker, S. I. Biba, R. J. Aloisi, D. E. Seals, and C. L. Jones, “Container positioning device,” U.S. patent 10358804 (26 February 2002).

K. Busboom and A. DeBohman, “Container surveillance system and related method,” European patent EP20010107411 (2 October 2002).

T. Takehara, M. W. King, and C. Chung, “Method and apparatus of automated optical container code recognition with positional identification for a transfer container crane,” U.S. patent 7,308,114 (11 December 2007).

P. Thomas, A. Thomas, and L. Scheppmann, “Container tracking and locating systems, methods, and computer program products,” U.S. patent 8,060,263 (27 January 2011).

W. Wichner, “Operating method and device for operating automated container quay cranes,” U.S. patent 6,965,823(15 November 2005).

M. Lindholm and P. Vaha, “Procedure and apparatus for locating a container for lifting,” U.S. patent 5,067,013(19 November 1991).

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Figures (13)

Fig. 1.
Fig. 1.

The alignment between the spreader and the container.

Fig. 2.
Fig. 2.

The function of the proposed system in the overall workflow.

Fig. 3.
Fig. 3.

Bird’s-eye view diagram of structured light installation.

Fig. 4.
Fig. 4.

Three images captured when the spreader does not line up with the container.

Fig. 5.
Fig. 5.

Side-view diagram when the spreader contains lateral inclination.

Fig. 6.
Fig. 6.

Two cases that reach-stacker terminal needs to roughly move the spreader.

Fig. 7.
Fig. 7.

Newly snapped image after adjusting inclination and rotation angle to zero.

Fig. 8.
Fig. 8.

Newly snapped image for estimating horizontal x-shift.

Fig. 9.
Fig. 9.

The principles to align the spreader with the container.

Fig. 10.
Fig. 10.

Laboratory experiment on the lying simulated container.

Fig. 11.
Fig. 11.

Simulator of reach-stacker working condition based on estimation results.

Fig. 12.
Fig. 12.

The shift of laser plane due to the error of emitter’s orientation.

Fig. 13.
Fig. 13.

The top two images show the alignment system mounted on the reach stacker. The red (left) and green (right) circles show the positions where the system locates. The middle image shows the camera and emitter used in our system. The bottom two images show the picture captured by the camera.

Tables (1)

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Table 1. Estimation Error of Deviation in 5 DoF

Equations (17)

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

{MO3kR3C3=fR3MNO2kR2C2=fR2NR3MR2N=R3R2tanβ,
β=arctan(R3C3/MO3R2C2/NO2R3R2fk).
R3M=(h/sinβSR3)tanβ=h/cosβSR3tanβ
h=fR3C3kMO3cosβ+SR3sinβ.
α=arctan(MT3NT4MN).
akMTA=fh0.
bkMTA=fh,
ckMT=fh.
cMT=bMTA.
MTMTA=cbbMTA.
MTMTA=(chah0)k/f,
OUOUA=(nhmh0)k/f.
arctan(x)=xx33+x55++(1)nx2n+12n+1+,1<x<1,
Δβ=βactualβtheoretical(a+Δa/MO3R2C2/NO2R3R2a/MO3R2C2/NO2R3R2)fk=ΔaMO3R3R2fk,
Δβ=βactualβtheoretical[R3C3/(b+Δb)R2C2/NO2R3R2R3C3/bR2C2/NO2R3R2]fk,=Δbb(b+Δb)fR3C3kR3R2.
Δβ=f(h,β).
{Δβ=AX+bX=(h,β),A=(a1,a2).

Metrics