Abstract

The binocular vision galvanometric laser scanning (BGLS) system, which consists of a traditional galvanometric laser scanning (GLS) system and a stereo vision system, is widely applied in various advanced manufacturing fields such as 3-dimensional (3D) laser cutting and 3D laser projection positioning. The BGLS system usually needs to be recalibrated before it is put to a certain use, since the camera parameters, as well as the relative pose between the camera and the GLS, may be changed with different applications. However, a full calibration of the BGLS system requires elaborate modeling and mass calibration data, which considerably affect the efficiency of the BGLS system. A rapid on-site recalibration method is proposed, which can substantially improve the efficiency and flexibility of the BGLS system. With the method, the BGLS system needs to be carefully off-site calibrated for only one time. The on-site precise recalibration can be quickly realized by taking only two images of the laser spots projected by the GLS on a planar board. Moreover, an ingenious linear solving method is proposed to make the whole computation process more stable and timesaving. On-site recalibration and target shooting experiments are respectively conducted to verify the proposed method.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (1)

J. Tu and L. Zhang, “Effective data-driven calibration for galvanometric laser scanning system using binocular stereo vision,” Sensors (Basel) 18(2), 197–214 (2018).
[Crossref] [PubMed]

2017 (1)

2016 (1)

2015 (2)

L. Qi, S. Wang, Y. X. Zhang, Z. Q. Tang, H. Yang, and X. P. Zhang, “Laser cutting of irregular shape object based on stereo vision laser galvanometric scanning system,” Opt. Lasers Eng. 68, 180–187 (2015).
[Crossref]

T. Wissel, B. Wagner, A. Schweikard, P. Stüeber, and F. Ernst, “Data-driven learning for calibrating galvanometric laser scanners,” Sensors (Basel) 15(10), 5709–5717 (2015).
[Crossref]

2013 (2)

G. Cuccolini, L. Orazi, and A. Fortunato, “5 Axes computer aided laser milling,” Opt. Lasers Eng. 51(6), 749–760 (2013).
[Crossref]

F. Ernst, R. Bruder, T. Wissel, P. Stüber, B. Wagner, and A. Schweikard, “Real time contact-free and non-invasive tracking of the human skull: First light and initial validation,” Proc. SPIE 8856, 88561G (2013).
[Crossref]

2012 (1)

2011 (1)

J. Diaci, D. Bračun, A. Gorkič, and J. Možina, “Rapid and flexible laser marking and engraving of tilted and curved surfaces,” Opt. Lasers Eng. 49(2), 195–199 (2011).
[Crossref]

2009 (2)

M. F. Chen, Y. P. Chen, and W. T. Hsiao, “Correction of field distortion of laser marking systems using surface compensation function,” Opt. Lasers Eng. 47(1), 84–89 (2009).
[Crossref]

S. Cui, X. Zhu, W. Wang, and Y. Xie, “Calibration of a laser galvanometric scanning system by adapting a camera model,” Appl. Opt. 48(14), 2632–2637 (2009).
[Crossref] [PubMed]

2007 (1)

M. F. Chen and Y. P. Chen, “Compensating technique of field-distorting error for the CO2 laser galvanometric scanning drilling machines,” Int. J. Mach. Tools Manuf. 47(7), 1114–1124 (2007).
[Crossref]

2006 (1)

G. B. Huang, Q.-Y. Zhu, and C.-K. Siew, “Extreme Learning Machine: Theory and Applications,” Neurocomputing 70(1), 489–501 (2006).
[Crossref]

2005 (1)

J. Xie, S. H. Huang, Z. C. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37(4), 305–311 (2005).
[Crossref]

2004 (1)

A. Kaldos, H. J. Pieper, E. Wolf, and M. Krause, “Laser machining in die making—a modern rapid tooling process,” J. Mater. Process. Technol. 155(1), 1815–1820 (2004).
[Crossref]

2000 (2)

M. A. Stafne, L. D. Mitchell, and R. L. West, “Positional calibration of galvanometric scanners used in laser Doppler vibrometers,” Measurement 28(1), 47–59 (2000).
[Crossref]

J. M. Lee, J. H. Jang, and T. K. Yoo, “Scribing and cutting a blue LED wafer using a Qswitched Nd: YAG laser,” Appl. Phys. A 70(5), 561–564 (2000).
[Crossref]

1997 (1)

R. Setiono, “On the solution of the parity problem by a single hidden layer feedforward neural network,” Neurocomputing 16(3), 225–235 (1997).
[Crossref]

1992 (1)

L. Holmstrom and P. Koistinen, “Using additive noise in back-propagation training,” IEEE Trans. Neural Netw. 3(1), 24–38 (1992).
[Crossref] [PubMed]

1978 (1)

J. J. Moré, “The Levenberg-Marquardt algorithm: Implementation and theory,” Lect. Notes Math. 630, 105–116 (1978).
[Crossref]

Abdel-Aziz, Y. I.

Y. I. Abdel-Aziz and H. M. Karara, “Direct linear transformation from comparator coordinates into object space coordinates in closerange photogrammetry,” in Proc. Symp. Close-Range Photogram., Falls Church, VA, USA, 1971, pp. 1–18.

Anzolin, G.

Bracun, D.

J. Diaci, D. Bračun, A. Gorkič, and J. Možina, “Rapid and flexible laser marking and engraving of tilted and curved surfaces,” Opt. Lasers Eng. 49(2), 195–199 (2011).
[Crossref]

Bruder, R.

F. Ernst, R. Bruder, T. Wissel, P. Stüber, B. Wagner, and A. Schweikard, “Real time contact-free and non-invasive tracking of the human skull: First light and initial validation,” Proc. SPIE 8856, 88561G (2013).
[Crossref]

Chang, Y.

Chang, Y. H.

Chen, M. F.

M. F. Chen, Y. P. Chen, and W. T. Hsiao, “Correction of field distortion of laser marking systems using surface compensation function,” Opt. Lasers Eng. 47(1), 84–89 (2009).
[Crossref]

M. F. Chen and Y. P. Chen, “Compensating technique of field-distorting error for the CO2 laser galvanometric scanning drilling machines,” Int. J. Mach. Tools Manuf. 47(7), 1114–1124 (2007).
[Crossref]

Chen, S.

Chen, Y. P.

M. F. Chen, Y. P. Chen, and W. T. Hsiao, “Correction of field distortion of laser marking systems using surface compensation function,” Opt. Lasers Eng. 47(1), 84–89 (2009).
[Crossref]

M. F. Chen and Y. P. Chen, “Compensating technique of field-distorting error for the CO2 laser galvanometric scanning drilling machines,” Int. J. Mach. Tools Manuf. 47(7), 1114–1124 (2007).
[Crossref]

Cuccolini, G.

G. Cuccolini, L. Orazi, and A. Fortunato, “5 Axes computer aided laser milling,” Opt. Lasers Eng. 51(6), 749–760 (2013).
[Crossref]

Cui, S.

Diaci, J.

J. Diaci, D. Bračun, A. Gorkič, and J. Možina, “Rapid and flexible laser marking and engraving of tilted and curved surfaces,” Opt. Lasers Eng. 49(2), 195–199 (2011).
[Crossref]

Duan, Z. C.

J. Xie, S. H. Huang, Z. C. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37(4), 305–311 (2005).
[Crossref]

Ernst, F.

T. Wissel, B. Wagner, A. Schweikard, P. Stüeber, and F. Ernst, “Data-driven learning for calibrating galvanometric laser scanners,” Sensors (Basel) 15(10), 5709–5717 (2015).
[Crossref]

F. Ernst, R. Bruder, T. Wissel, P. Stüber, B. Wagner, and A. Schweikard, “Real time contact-free and non-invasive tracking of the human skull: First light and initial validation,” Proc. SPIE 8856, 88561G (2013).
[Crossref]

Fortunato, A.

G. Cuccolini, L. Orazi, and A. Fortunato, “5 Axes computer aided laser milling,” Opt. Lasers Eng. 51(6), 749–760 (2013).
[Crossref]

Gorkic, A.

J. Diaci, D. Bračun, A. Gorkič, and J. Možina, “Rapid and flexible laser marking and engraving of tilted and curved surfaces,” Opt. Lasers Eng. 49(2), 195–199 (2011).
[Crossref]

Gu, C.

Holmstrom, L.

L. Holmstrom and P. Koistinen, “Using additive noise in back-propagation training,” IEEE Trans. Neural Netw. 3(1), 24–38 (1992).
[Crossref] [PubMed]

Hsiao, W. T.

M. F. Chen, Y. P. Chen, and W. T. Hsiao, “Correction of field distortion of laser marking systems using surface compensation function,” Opt. Lasers Eng. 47(1), 84–89 (2009).
[Crossref]

Huang, G. B.

G. B. Huang, Q.-Y. Zhu, and C.-K. Siew, “Extreme Learning Machine: Theory and Applications,” Neurocomputing 70(1), 489–501 (2006).
[Crossref]

Huang, S. H.

J. Xie, S. H. Huang, Z. C. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37(4), 305–311 (2005).
[Crossref]

Jang, J. H.

J. M. Lee, J. H. Jang, and T. K. Yoo, “Scribing and cutting a blue LED wafer using a Qswitched Nd: YAG laser,” Appl. Phys. A 70(5), 561–564 (2000).
[Crossref]

Jofre, M.

Kaldos, A.

A. Kaldos, H. J. Pieper, E. Wolf, and M. Krause, “Laser machining in die making—a modern rapid tooling process,” J. Mater. Process. Technol. 155(1), 1815–1820 (2004).
[Crossref]

Karara, H. M.

Y. I. Abdel-Aziz and H. M. Karara, “Direct linear transformation from comparator coordinates into object space coordinates in closerange photogrammetry,” in Proc. Symp. Close-Range Photogram., Falls Church, VA, USA, 1971, pp. 1–18.

Koistinen, P.

L. Holmstrom and P. Koistinen, “Using additive noise in back-propagation training,” IEEE Trans. Neural Netw. 3(1), 24–38 (1992).
[Crossref] [PubMed]

Krause, M.

A. Kaldos, H. J. Pieper, E. Wolf, and M. Krause, “Laser machining in die making—a modern rapid tooling process,” J. Mater. Process. Technol. 155(1), 1815–1820 (2004).
[Crossref]

Lee, J. M.

J. M. Lee, J. H. Jang, and T. K. Yoo, “Scribing and cutting a blue LED wafer using a Qswitched Nd: YAG laser,” Appl. Phys. A 70(5), 561–564 (2000).
[Crossref]

Lo, Y. L.

Mitchell, L. D.

M. A. Stafne, L. D. Mitchell, and R. L. West, “Positional calibration of galvanometric scanners used in laser Doppler vibrometers,” Measurement 28(1), 47–59 (2000).
[Crossref]

Mitchell, M. W.

Moré, J. J.

J. J. Moré, “The Levenberg-Marquardt algorithm: Implementation and theory,” Lect. Notes Math. 630, 105–116 (1978).
[Crossref]

Možina, J.

J. Diaci, D. Bračun, A. Gorkič, and J. Možina, “Rapid and flexible laser marking and engraving of tilted and curved surfaces,” Opt. Lasers Eng. 49(2), 195–199 (2011).
[Crossref]

Oliverio, N.

Orazi, L.

G. Cuccolini, L. Orazi, and A. Fortunato, “5 Axes computer aided laser milling,” Opt. Lasers Eng. 51(6), 749–760 (2013).
[Crossref]

Pieper, H. J.

A. Kaldos, H. J. Pieper, E. Wolf, and M. Krause, “Laser machining in die making—a modern rapid tooling process,” J. Mater. Process. Technol. 155(1), 1815–1820 (2004).
[Crossref]

Pruneri, V.

Qi, L.

L. Qi, S. Wang, Y. X. Zhang, Z. Q. Tang, H. Yang, and X. P. Zhang, “Laser cutting of irregular shape object based on stereo vision laser galvanometric scanning system,” Opt. Lasers Eng. 68, 180–187 (2015).
[Crossref]

Schweikard, A.

T. Wissel, B. Wagner, A. Schweikard, P. Stüeber, and F. Ernst, “Data-driven learning for calibrating galvanometric laser scanners,” Sensors (Basel) 15(10), 5709–5717 (2015).
[Crossref]

F. Ernst, R. Bruder, T. Wissel, P. Stüber, B. Wagner, and A. Schweikard, “Real time contact-free and non-invasive tracking of the human skull: First light and initial validation,” Proc. SPIE 8856, 88561G (2013).
[Crossref]

Setiono, R.

R. Setiono, “On the solution of the parity problem by a single hidden layer feedforward neural network,” Neurocomputing 16(3), 225–235 (1997).
[Crossref]

Shi, Y.

J. Xie, S. H. Huang, Z. C. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37(4), 305–311 (2005).
[Crossref]

Siew, C.-K.

G. B. Huang, Q.-Y. Zhu, and C.-K. Siew, “Extreme Learning Machine: Theory and Applications,” Neurocomputing 70(1), 489–501 (2006).
[Crossref]

Stafne, M. A.

M. A. Stafne, L. D. Mitchell, and R. L. West, “Positional calibration of galvanometric scanners used in laser Doppler vibrometers,” Measurement 28(1), 47–59 (2000).
[Crossref]

Steinlechner, F.

Stüber, P.

F. Ernst, R. Bruder, T. Wissel, P. Stüber, B. Wagner, and A. Schweikard, “Real time contact-free and non-invasive tracking of the human skull: First light and initial validation,” Proc. SPIE 8856, 88561G (2013).
[Crossref]

Stüeber, P.

T. Wissel, B. Wagner, A. Schweikard, P. Stüeber, and F. Ernst, “Data-driven learning for calibrating galvanometric laser scanners,” Sensors (Basel) 15(10), 5709–5717 (2015).
[Crossref]

Tang, Z. Q.

L. Qi, S. Wang, Y. X. Zhang, Z. Q. Tang, H. Yang, and X. P. Zhang, “Laser cutting of irregular shape object based on stereo vision laser galvanometric scanning system,” Opt. Lasers Eng. 68, 180–187 (2015).
[Crossref]

Torres, J. P.

Tu, J.

J. Tu and L. Zhang, “Effective data-driven calibration for galvanometric laser scanning system using binocular stereo vision,” Sensors (Basel) 18(2), 197–214 (2018).
[Crossref] [PubMed]

Wagner, B.

T. Wissel, B. Wagner, A. Schweikard, P. Stüeber, and F. Ernst, “Data-driven learning for calibrating galvanometric laser scanners,” Sensors (Basel) 15(10), 5709–5717 (2015).
[Crossref]

F. Ernst, R. Bruder, T. Wissel, P. Stüber, B. Wagner, and A. Schweikard, “Real time contact-free and non-invasive tracking of the human skull: First light and initial validation,” Proc. SPIE 8856, 88561G (2013).
[Crossref]

Wang, S.

L. Qi, S. Wang, Y. X. Zhang, Z. Q. Tang, H. Yang, and X. P. Zhang, “Laser cutting of irregular shape object based on stereo vision laser galvanometric scanning system,” Opt. Lasers Eng. 68, 180–187 (2015).
[Crossref]

Wang, W.

Wen, C.

Wen, S.

J. Xie, S. H. Huang, Z. C. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37(4), 305–311 (2005).
[Crossref]

West, R. L.

M. A. Stafne, L. D. Mitchell, and R. L. West, “Positional calibration of galvanometric scanners used in laser Doppler vibrometers,” Measurement 28(1), 47–59 (2000).
[Crossref]

Wissel, T.

T. Wissel, B. Wagner, A. Schweikard, P. Stüeber, and F. Ernst, “Data-driven learning for calibrating galvanometric laser scanners,” Sensors (Basel) 15(10), 5709–5717 (2015).
[Crossref]

F. Ernst, R. Bruder, T. Wissel, P. Stüber, B. Wagner, and A. Schweikard, “Real time contact-free and non-invasive tracking of the human skull: First light and initial validation,” Proc. SPIE 8856, 88561G (2013).
[Crossref]

Wolf, E.

A. Kaldos, H. J. Pieper, E. Wolf, and M. Krause, “Laser machining in die making—a modern rapid tooling process,” J. Mater. Process. Technol. 155(1), 1815–1820 (2004).
[Crossref]

Xie, J.

J. Xie, S. H. Huang, Z. C. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37(4), 305–311 (2005).
[Crossref]

Xie, Y.

Yang, H.

L. Qi, S. Wang, Y. X. Zhang, Z. Q. Tang, H. Yang, and X. P. Zhang, “Laser cutting of irregular shape object based on stereo vision laser galvanometric scanning system,” Opt. Lasers Eng. 68, 180–187 (2015).
[Crossref]

Yang, P. M.

Yoo, T. K.

J. M. Lee, J. H. Jang, and T. K. Yoo, “Scribing and cutting a blue LED wafer using a Qswitched Nd: YAG laser,” Appl. Phys. A 70(5), 561–564 (2000).
[Crossref]

Zhang, L.

J. Tu and L. Zhang, “Effective data-driven calibration for galvanometric laser scanning system using binocular stereo vision,” Sensors (Basel) 18(2), 197–214 (2018).
[Crossref] [PubMed]

Zhang, X. P.

L. Qi, S. Wang, Y. X. Zhang, Z. Q. Tang, H. Yang, and X. P. Zhang, “Laser cutting of irregular shape object based on stereo vision laser galvanometric scanning system,” Opt. Lasers Eng. 68, 180–187 (2015).
[Crossref]

Zhang, Y. X.

L. Qi, S. Wang, Y. X. Zhang, Z. Q. Tang, H. Yang, and X. P. Zhang, “Laser cutting of irregular shape object based on stereo vision laser galvanometric scanning system,” Opt. Lasers Eng. 68, 180–187 (2015).
[Crossref]

Zhu, Q.-Y.

G. B. Huang, Q.-Y. Zhu, and C.-K. Siew, “Extreme Learning Machine: Theory and Applications,” Neurocomputing 70(1), 489–501 (2006).
[Crossref]

Zhu, X.

Appl. Opt. (2)

Appl. Phys. A (1)

J. M. Lee, J. H. Jang, and T. K. Yoo, “Scribing and cutting a blue LED wafer using a Qswitched Nd: YAG laser,” Appl. Phys. A 70(5), 561–564 (2000).
[Crossref]

IEEE Trans. Neural Netw. (1)

L. Holmstrom and P. Koistinen, “Using additive noise in back-propagation training,” IEEE Trans. Neural Netw. 3(1), 24–38 (1992).
[Crossref] [PubMed]

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

M. F. Chen and Y. P. Chen, “Compensating technique of field-distorting error for the CO2 laser galvanometric scanning drilling machines,” Int. J. Mach. Tools Manuf. 47(7), 1114–1124 (2007).
[Crossref]

J. Mater. Process. Technol. (1)

A. Kaldos, H. J. Pieper, E. Wolf, and M. Krause, “Laser machining in die making—a modern rapid tooling process,” J. Mater. Process. Technol. 155(1), 1815–1820 (2004).
[Crossref]

Lect. Notes Math. (1)

J. J. Moré, “The Levenberg-Marquardt algorithm: Implementation and theory,” Lect. Notes Math. 630, 105–116 (1978).
[Crossref]

Measurement (1)

M. A. Stafne, L. D. Mitchell, and R. L. West, “Positional calibration of galvanometric scanners used in laser Doppler vibrometers,” Measurement 28(1), 47–59 (2000).
[Crossref]

Neurocomputing (2)

G. B. Huang, Q.-Y. Zhu, and C.-K. Siew, “Extreme Learning Machine: Theory and Applications,” Neurocomputing 70(1), 489–501 (2006).
[Crossref]

R. Setiono, “On the solution of the parity problem by a single hidden layer feedforward neural network,” Neurocomputing 16(3), 225–235 (1997).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

J. Xie, S. H. Huang, Z. C. Duan, Y. Shi, and S. Wen, “Correction of the image distortion for laser galvanometric scanning system,” Opt. Laser Technol. 37(4), 305–311 (2005).
[Crossref]

Opt. Lasers Eng. (4)

M. F. Chen, Y. P. Chen, and W. T. Hsiao, “Correction of field distortion of laser marking systems using surface compensation function,” Opt. Lasers Eng. 47(1), 84–89 (2009).
[Crossref]

G. Cuccolini, L. Orazi, and A. Fortunato, “5 Axes computer aided laser milling,” Opt. Lasers Eng. 51(6), 749–760 (2013).
[Crossref]

L. Qi, S. Wang, Y. X. Zhang, Z. Q. Tang, H. Yang, and X. P. Zhang, “Laser cutting of irregular shape object based on stereo vision laser galvanometric scanning system,” Opt. Lasers Eng. 68, 180–187 (2015).
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Figures (13)

Fig. 1
Fig. 1 BGLS system configuration.
Fig. 2
Fig. 2 The off-site full calibration process.
Fig. 3
Fig. 3 The coordinate transformation between O o X o Y o Z o and O r X r Y r Z r .
Fig. 4
Fig. 4 The two-step method for estimating the initial [ R 0 | T 0 ]. (a) The estimation of rotation matrix R 0 . (b) The estimation of translation vector T 0 .
Fig. 5
Fig. 5 The hardware setup of the BGLS system in the full calibration.
Fig. 6
Fig. 6 Distribution of the 900 control digitals.
Fig. 7
Fig. 7 The situations that the binocular system is changed in the recalibration experiment. (a)-(b) position between the binocular system and the GLS system. (c) Camera lenses with different focal lengths used.
Fig. 8
Fig. 8 Data collection for recalibration. (a) The planar board for acquiring the laser spot grids. (b) An actual laser spot grid.
Fig. 9
Fig. 9 Target shooting experiment. (a) The pattern of target circles. (b) Experiment result of target shooting. (c) The distance between the centers of the laser spot and the target circle.
Fig. 10
Fig. 10 The shooting performances in four different simulated situations. (a) and (b) The shooting performances in the situations shown in Figs. 7(a) and 7(b). (c) and (d) The shooting performances in the situations that the camera lenses of the binocular system are changed from12mm into 17mm and 35mm respectively.
Fig. 11
Fig. 11 The shooting performances in the two additional experiments. (a) The shooting performance wit h M r ' :D V r ' calibrated by two laser spot grids used for recalibration.(b) The shooting performance with M o :D V o obtained in Section 3.1.
Fig. 12
Fig. 12 The projection positioning experiment. (a) The CAD model of the car door. (b) Projection result in linear interpolation scan mode. (c) Projection result in point scan mode.
Fig. 13
Fig. 13 Evaluation of the projection accuracy. (a) The theoretical target point (in red) and the actual laser projection point (in blue). (b) Projection errors of the 137 discrete points.

Tables (2)

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Table 1 Performance of the recalibration method in the representative situations.

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Table 2 Performance measure S d in the six situations.

Equations (12)

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E( V o k )= i=1 N ( d i k ) 2
{ [ v r1 k v r2 k v r3 k ] T =R [ v o1 k v o2 k v o3 k ] T [ v r4 k v r5 k v r6 k ] T =R [ v o4 k v o5 k v o6 k ] T +T ,k=1,2,Q
E( R,T )= k=1 Q ( ( d 1 k ) 2 + ( d 2 k ) 2 )
[ v o1 k v o2 k v o3 k ]= R 0 [ x r2 k x r1 k y r2 k y r1 k z r2 k z r1 k ]
R 0 =( I+S ) ( IS ) 1
[ 0 z 21 k v o3 k y 21 k v o2 k z 21 k v o3 k 0 x 21 k + v o1 k y 21 k + v o2 k x 21 k + v o1 k 0 ][ a b c ]=[ v o2 k x 21 k v o2 k y 21 k v o2 k z 21 k ]
[ b 1 k t 1 k b 2 k t 1 k b 3 k t 1 k ]+[ b 4 k b 5 k b 6 k ]+[ X 0 Y 0 Z 0 ]=[ x r1 k y r1 k z r1 k ]
[ b 1 k t 2 k b 2 k t 2 k b 3 k t 2 k ]+[ b 4 k b 5 k b 6 k ]+[ X 0 Y 0 Z 0 ]=[ x r2 k y r2 k z r2 k ]
[ b 1 1 1 b 2 1 1 b 3 1 1 b 1 1 1 b 2 1 1 b 3 1 1 b 1 Q 1 b 2 Q 1 b 3 Q 1 b 1 Q 1 b 2 Q 1 b 3 Q 1 ] 6Q×2Q+3 [ t 1 1 t 2 1 t 1 Q t 2 Q X 0 Y 0 Z 0 ] 2Q+3 =[ x r1 1 b 4 1 y r1 1 b 5 1 z r1 1 b 6 1 x r2 1 b 4 1 y r2 1 b 5 1 z r2 1 b 6 1 x r1 Q b 4 1 y r1 Q b 5 1 z r1 Q b 6 1 x r2 Q b 4 1 y r2 Q b 5 1 z r2 Q b 6 1 ]
E d = k=1 900 ( ( d 1 k ) 2 + ( d 2 k ) 2 ) / 900
D n dst =arg min D d n ( D )
D j tgt =arg min D d j ( D )

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