Abstract

This study proposes a pose-relay videometric method that uses a parallel camera series and is applicable to measuring deformation between nonintervisible and intervisible objects with a very wide angle of view. The measuring system is constructed by adding symmetrical cameras to the pose-relay stations of a single camera measuring system to improve its robustness and precision. An adjustment data fusion method is suggested to take full advantage of the data redundancy among neighboring relay stations in the proposed system. Simulated results show that the adjusted method enhances the measuring precision achieved with the classic weighted average data fusion method owing to its use of the restraint condition inherent in the system.

© 2010 Optical Society of America

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  1. Q. Yu, G. Jiang, S. Fu, Z. Chao, Y. Shang, and X. Sun, “Fold-ray videometrics method for the deformation measurement of nonintervisible large structures,” Appl. Opt. 48, 4683–4687(2009).
    [CrossRef] [PubMed]
  2. Q. Yu, G. Jiang, S. Fu, S. Yang, Y. Liang, and L. Li, “Broken-ray videometric method and system for measuring the three-dimensional position and pose of the non-intervisible object,” in the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Science (Beijing, 2008), 37(B5), pp. 145–148.
  3. Q. Yu, G. Jiang, S. Fu, and Y. Shang, “Measuring deformation of large vessels with innovative broken-ray videometrics,” presented at the 22nd International Congress of Theoretical and Applied Mechanics (ICTAM2008), Adelaide, Australia, 24–30 August 2008.
  4. Q. Yu, G. Jiang, Z. Chao, S. Fu, Y. Shang, and X. Yang, “Deformation monitoring system of tunnel rocks with innovative broken-ray videometrics,” Proc. SPIE 7375, 73752C (2008).
    [CrossRef]
  5. G. Jiang, “Study on pose relay videometrics method by camera-series and ship deformations measurement,” Ph.D. dissertation (National University of Defence Technology, 2010).
  6. Y. Shiu and H. Ahmad, “Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX=XB,” IEEE Trans. Rob. Autom. 5, 16–29 (1989).
    [CrossRef]
  7. F. Park and B. Martin, “Robot sensor calibration: solving AX=XB on the Euclidean group,” IEEE Trans. Rob. Autom. 10, 717–721 (1994).
    [CrossRef]
  8. C. Lu, G. Hager, and E. Mjolsness, “Fast and globally convergent pose estimation from video images,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 610–622 (2000).
    [CrossRef]
  9. A. Ansar and K. Daniilidis, “Linear pose estimation from points or lines,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 578–589 (2003).
    [CrossRef]
  10. F. Moreno-Noguer, V. Lepetit, and P. Fua, “Accurate non-iterative o(n) solution to the pnp problem,” in Proceeding of the IEEE International Conference on Computer Vision (IEEE, 2007), pp. 1–8.
  11. J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
    [CrossRef]
  12. S. Hu, Z. Jing, and H. Leung, “A robust fusion algorithm for multi-sensor tracking,” in Proceedings of IEEE Intelligent Transportation Systems, 2003 (IEEE, 2003), Vol. 2, pp. 919–923.
    [CrossRef]
  13. T. Luhmann, S. Roboson, S. Kyle, and L. Harley, Close Range Photogrammetry: Principles, Techniques and Applications (Whittles, 2006).

2009 (1)

2008 (1)

Q. Yu, G. Jiang, Z. Chao, S. Fu, Y. Shang, and X. Yang, “Deformation monitoring system of tunnel rocks with innovative broken-ray videometrics,” Proc. SPIE 7375, 73752C (2008).
[CrossRef]

2003 (1)

A. Ansar and K. Daniilidis, “Linear pose estimation from points or lines,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 578–589 (2003).
[CrossRef]

2000 (1)

C. Lu, G. Hager, and E. Mjolsness, “Fast and globally convergent pose estimation from video images,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 610–622 (2000).
[CrossRef]

1994 (1)

F. Park and B. Martin, “Robot sensor calibration: solving AX=XB on the Euclidean group,” IEEE Trans. Rob. Autom. 10, 717–721 (1994).
[CrossRef]

1992 (1)

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
[CrossRef]

1989 (1)

Y. Shiu and H. Ahmad, “Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX=XB,” IEEE Trans. Rob. Autom. 5, 16–29 (1989).
[CrossRef]

Ahmad, H.

Y. Shiu and H. Ahmad, “Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX=XB,” IEEE Trans. Rob. Autom. 5, 16–29 (1989).
[CrossRef]

Ansar, A.

A. Ansar and K. Daniilidis, “Linear pose estimation from points or lines,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 578–589 (2003).
[CrossRef]

Chao, Z.

Q. Yu, G. Jiang, S. Fu, Z. Chao, Y. Shang, and X. Sun, “Fold-ray videometrics method for the deformation measurement of nonintervisible large structures,” Appl. Opt. 48, 4683–4687(2009).
[CrossRef] [PubMed]

Q. Yu, G. Jiang, Z. Chao, S. Fu, Y. Shang, and X. Yang, “Deformation monitoring system of tunnel rocks with innovative broken-ray videometrics,” Proc. SPIE 7375, 73752C (2008).
[CrossRef]

Cohen, P.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
[CrossRef]

Daniilidis, K.

A. Ansar and K. Daniilidis, “Linear pose estimation from points or lines,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 578–589 (2003).
[CrossRef]

Fu, S.

Q. Yu, G. Jiang, S. Fu, Z. Chao, Y. Shang, and X. Sun, “Fold-ray videometrics method for the deformation measurement of nonintervisible large structures,” Appl. Opt. 48, 4683–4687(2009).
[CrossRef] [PubMed]

Q. Yu, G. Jiang, Z. Chao, S. Fu, Y. Shang, and X. Yang, “Deformation monitoring system of tunnel rocks with innovative broken-ray videometrics,” Proc. SPIE 7375, 73752C (2008).
[CrossRef]

Q. Yu, G. Jiang, S. Fu, S. Yang, Y. Liang, and L. Li, “Broken-ray videometric method and system for measuring the three-dimensional position and pose of the non-intervisible object,” in the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Science (Beijing, 2008), 37(B5), pp. 145–148.

Q. Yu, G. Jiang, S. Fu, and Y. Shang, “Measuring deformation of large vessels with innovative broken-ray videometrics,” presented at the 22nd International Congress of Theoretical and Applied Mechanics (ICTAM2008), Adelaide, Australia, 24–30 August 2008.

Fua, P.

F. Moreno-Noguer, V. Lepetit, and P. Fua, “Accurate non-iterative o(n) solution to the pnp problem,” in Proceeding of the IEEE International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Hager, G.

C. Lu, G. Hager, and E. Mjolsness, “Fast and globally convergent pose estimation from video images,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 610–622 (2000).
[CrossRef]

Harley, L.

T. Luhmann, S. Roboson, S. Kyle, and L. Harley, Close Range Photogrammetry: Principles, Techniques and Applications (Whittles, 2006).

Herniou, M.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
[CrossRef]

Hu, S.

S. Hu, Z. Jing, and H. Leung, “A robust fusion algorithm for multi-sensor tracking,” in Proceedings of IEEE Intelligent Transportation Systems, 2003 (IEEE, 2003), Vol. 2, pp. 919–923.
[CrossRef]

Jiang, G.

Q. Yu, G. Jiang, S. Fu, Z. Chao, Y. Shang, and X. Sun, “Fold-ray videometrics method for the deformation measurement of nonintervisible large structures,” Appl. Opt. 48, 4683–4687(2009).
[CrossRef] [PubMed]

Q. Yu, G. Jiang, Z. Chao, S. Fu, Y. Shang, and X. Yang, “Deformation monitoring system of tunnel rocks with innovative broken-ray videometrics,” Proc. SPIE 7375, 73752C (2008).
[CrossRef]

G. Jiang, “Study on pose relay videometrics method by camera-series and ship deformations measurement,” Ph.D. dissertation (National University of Defence Technology, 2010).

Q. Yu, G. Jiang, S. Fu, S. Yang, Y. Liang, and L. Li, “Broken-ray videometric method and system for measuring the three-dimensional position and pose of the non-intervisible object,” in the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Science (Beijing, 2008), 37(B5), pp. 145–148.

Q. Yu, G. Jiang, S. Fu, and Y. Shang, “Measuring deformation of large vessels with innovative broken-ray videometrics,” presented at the 22nd International Congress of Theoretical and Applied Mechanics (ICTAM2008), Adelaide, Australia, 24–30 August 2008.

Jing, Z.

S. Hu, Z. Jing, and H. Leung, “A robust fusion algorithm for multi-sensor tracking,” in Proceedings of IEEE Intelligent Transportation Systems, 2003 (IEEE, 2003), Vol. 2, pp. 919–923.
[CrossRef]

Kyle, S.

T. Luhmann, S. Roboson, S. Kyle, and L. Harley, Close Range Photogrammetry: Principles, Techniques and Applications (Whittles, 2006).

Lepetit, V.

F. Moreno-Noguer, V. Lepetit, and P. Fua, “Accurate non-iterative o(n) solution to the pnp problem,” in Proceeding of the IEEE International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Leung, H.

S. Hu, Z. Jing, and H. Leung, “A robust fusion algorithm for multi-sensor tracking,” in Proceedings of IEEE Intelligent Transportation Systems, 2003 (IEEE, 2003), Vol. 2, pp. 919–923.
[CrossRef]

Li, L.

Q. Yu, G. Jiang, S. Fu, S. Yang, Y. Liang, and L. Li, “Broken-ray videometric method and system for measuring the three-dimensional position and pose of the non-intervisible object,” in the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Science (Beijing, 2008), 37(B5), pp. 145–148.

Liang, Y.

Q. Yu, G. Jiang, S. Fu, S. Yang, Y. Liang, and L. Li, “Broken-ray videometric method and system for measuring the three-dimensional position and pose of the non-intervisible object,” in the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Science (Beijing, 2008), 37(B5), pp. 145–148.

Lu, C.

C. Lu, G. Hager, and E. Mjolsness, “Fast and globally convergent pose estimation from video images,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 610–622 (2000).
[CrossRef]

Luhmann, T.

T. Luhmann, S. Roboson, S. Kyle, and L. Harley, Close Range Photogrammetry: Principles, Techniques and Applications (Whittles, 2006).

Martin, B.

F. Park and B. Martin, “Robot sensor calibration: solving AX=XB on the Euclidean group,” IEEE Trans. Rob. Autom. 10, 717–721 (1994).
[CrossRef]

Mjolsness, E.

C. Lu, G. Hager, and E. Mjolsness, “Fast and globally convergent pose estimation from video images,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 610–622 (2000).
[CrossRef]

Moreno-Noguer, F.

F. Moreno-Noguer, V. Lepetit, and P. Fua, “Accurate non-iterative o(n) solution to the pnp problem,” in Proceeding of the IEEE International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Park, F.

F. Park and B. Martin, “Robot sensor calibration: solving AX=XB on the Euclidean group,” IEEE Trans. Rob. Autom. 10, 717–721 (1994).
[CrossRef]

Roboson, S.

T. Luhmann, S. Roboson, S. Kyle, and L. Harley, Close Range Photogrammetry: Principles, Techniques and Applications (Whittles, 2006).

Shang, Y.

Q. Yu, G. Jiang, S. Fu, Z. Chao, Y. Shang, and X. Sun, “Fold-ray videometrics method for the deformation measurement of nonintervisible large structures,” Appl. Opt. 48, 4683–4687(2009).
[CrossRef] [PubMed]

Q. Yu, G. Jiang, Z. Chao, S. Fu, Y. Shang, and X. Yang, “Deformation monitoring system of tunnel rocks with innovative broken-ray videometrics,” Proc. SPIE 7375, 73752C (2008).
[CrossRef]

Q. Yu, G. Jiang, S. Fu, and Y. Shang, “Measuring deformation of large vessels with innovative broken-ray videometrics,” presented at the 22nd International Congress of Theoretical and Applied Mechanics (ICTAM2008), Adelaide, Australia, 24–30 August 2008.

Shiu, Y.

Y. Shiu and H. Ahmad, “Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX=XB,” IEEE Trans. Rob. Autom. 5, 16–29 (1989).
[CrossRef]

Sun, X.

Weng, J.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
[CrossRef]

Yang, S.

Q. Yu, G. Jiang, S. Fu, S. Yang, Y. Liang, and L. Li, “Broken-ray videometric method and system for measuring the three-dimensional position and pose of the non-intervisible object,” in the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Science (Beijing, 2008), 37(B5), pp. 145–148.

Yang, X.

Q. Yu, G. Jiang, Z. Chao, S. Fu, Y. Shang, and X. Yang, “Deformation monitoring system of tunnel rocks with innovative broken-ray videometrics,” Proc. SPIE 7375, 73752C (2008).
[CrossRef]

Yu, Q.

Q. Yu, G. Jiang, S. Fu, Z. Chao, Y. Shang, and X. Sun, “Fold-ray videometrics method for the deformation measurement of nonintervisible large structures,” Appl. Opt. 48, 4683–4687(2009).
[CrossRef] [PubMed]

Q. Yu, G. Jiang, Z. Chao, S. Fu, Y. Shang, and X. Yang, “Deformation monitoring system of tunnel rocks with innovative broken-ray videometrics,” Proc. SPIE 7375, 73752C (2008).
[CrossRef]

Q. Yu, G. Jiang, S. Fu, and Y. Shang, “Measuring deformation of large vessels with innovative broken-ray videometrics,” presented at the 22nd International Congress of Theoretical and Applied Mechanics (ICTAM2008), Adelaide, Australia, 24–30 August 2008.

Q. Yu, G. Jiang, S. Fu, S. Yang, Y. Liang, and L. Li, “Broken-ray videometric method and system for measuring the three-dimensional position and pose of the non-intervisible object,” in the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Science (Beijing, 2008), 37(B5), pp. 145–148.

Appl. Opt. (1)

IEEE Trans. Pattern Anal. Mach. Intell. (3)

C. Lu, G. Hager, and E. Mjolsness, “Fast and globally convergent pose estimation from video images,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 610–622 (2000).
[CrossRef]

A. Ansar and K. Daniilidis, “Linear pose estimation from points or lines,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 578–589 (2003).
[CrossRef]

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 965–980 (1992).
[CrossRef]

IEEE Trans. Rob. Autom. (2)

Y. Shiu and H. Ahmad, “Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX=XB,” IEEE Trans. Rob. Autom. 5, 16–29 (1989).
[CrossRef]

F. Park and B. Martin, “Robot sensor calibration: solving AX=XB on the Euclidean group,” IEEE Trans. Rob. Autom. 10, 717–721 (1994).
[CrossRef]

Proc. SPIE (1)

Q. Yu, G. Jiang, Z. Chao, S. Fu, Y. Shang, and X. Yang, “Deformation monitoring system of tunnel rocks with innovative broken-ray videometrics,” Proc. SPIE 7375, 73752C (2008).
[CrossRef]

Other (6)

G. Jiang, “Study on pose relay videometrics method by camera-series and ship deformations measurement,” Ph.D. dissertation (National University of Defence Technology, 2010).

Q. Yu, G. Jiang, S. Fu, S. Yang, Y. Liang, and L. Li, “Broken-ray videometric method and system for measuring the three-dimensional position and pose of the non-intervisible object,” in the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Science (Beijing, 2008), 37(B5), pp. 145–148.

Q. Yu, G. Jiang, S. Fu, and Y. Shang, “Measuring deformation of large vessels with innovative broken-ray videometrics,” presented at the 22nd International Congress of Theoretical and Applied Mechanics (ICTAM2008), Adelaide, Australia, 24–30 August 2008.

F. Moreno-Noguer, V. Lepetit, and P. Fua, “Accurate non-iterative o(n) solution to the pnp problem,” in Proceeding of the IEEE International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

S. Hu, Z. Jing, and H. Leung, “A robust fusion algorithm for multi-sensor tracking,” in Proceedings of IEEE Intelligent Transportation Systems, 2003 (IEEE, 2003), Vol. 2, pp. 919–923.
[CrossRef]

T. Luhmann, S. Roboson, S. Kyle, and L. Harley, Close Range Photogrammetry: Principles, Techniques and Applications (Whittles, 2006).

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Figures (4)

Fig. 1
Fig. 1

Pose-relay videometrics by single camera series (B, reference; G, target; T i , relay stations; M i , markers; C i , cameras).

Fig. 2
Fig. 2

Artificial cooperative marker (left) constructed by infrared LED crosses and its corresponding image (right) on a camera through an infrared filter.

Fig. 3
Fig. 3

Pose-relay videometrics by parallel camera series (B, reference; G, target; T i , relay stations; M i , markers; C i , cameras; C i , symmetrical cameras).

Fig. 4
Fig. 4

Survey adjustment data fusion ( T i , relay stations; M i , markers; C i , cameras).

Tables (2)

Tables Icon

Table 1 RMS of Errors Made by a Single-Level Pose-Relay System

Tables Icon

Table 2 RMS of Errors Made by a Five-Level Pose-Relay System

Equations (31)

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

P N = R M , N P M + t M , N .
P L = R N , L P N + t N , L .
P L = R N , L R M , N P M + R N , L t M , N + t N , L ,
{ R M , L = R N , L R M , N t M , L = R N , L t M , N + t N , L ,
{ R 1 , n = i = 1 n 1 R n i , n i + 1 = R n 1 , n R n 2 , n 1 R 1 , 2 t 1 , n = i = 1 n 2 [ ( j = 0 n 2 i R n j 1 , n j ) t i , i + 1 ] + t n 1 , n ,
R G , B = R M n , M 1 = R C 1 , M 1 R M 2 , C 1 R C 2 , M 2 R M 3 , C 2 R C 3 , M 3 R M 4 , C 3 R C 4 , M 4 R M n , C 4 ,
t G , B = t M n , M 1 = R C 1 , M 1 R M 2 , C 1 R C 2 , M 2 R M 3 , C 2 R C 3 , M 3 R M 4 , C 3 R C 4 , M 4 t M n , C 4 + R C 1 , M 1 R M 2 , C 1 R C 2 , M 2 R M 3 , C 2 R C 3 , M 3 R M 4 , C 3 t C 4 , M 4 + R C 1 , M 1 R M 2 , C 1 R C 2 , M 2 R M 3 , C 2 R C 3 , M 3 t M 4 , C 3 + R C 1 , M 1 R M 2 , C 1 R C 2 , M 2 R M 3 , C 2 t C 3 , M 3 + R C 1 , M 1 R M 2 , C 1 R C 2 , M 2 t M 3 , C 2 + R C 1 , M 1 R M 2 , C 1 t C 2 , M 2 + R C 1 , M 1 t M 2 , C 1 + t C 1 , M 1 .
R G , B = R M n , M 1 = R C 1 , M 1 R M 2 , C 1 R C 2 , M 2 R M 3 , C 2 R C 3 , M 3 R M 4 , C 3 R C 4 , M 4 R M n , C 4 .
{ R G , B = R B , G 1 t G , B = R B , G 1 t B , G .
A q G , B = 1 σ 1 2 A q ( G , B ) 1 + 1 σ 2 2 A q ( G , B ) 2 1 σ 1 2 + 1 σ 2 2 ( q = x , y , z ) .
A q G , B = A q ( G , B ) 1 + A q ( G , B ) 2 2 ( q = x , y , z ) .
R C 2 , M 1 ( A x C 2 , M 1 , A y C 2 , M 1 , A z C 2 , M 1 ) · R M 2 , C 2 · R C 1 , M 2 ( A x C 1 , M 2 , A y C 1 , M 2 , A z C 1 , M 2 ) · R M 1 , C 1 = I .
R C 2 M 1 · R M 2 C 2 · δ R C 1 M 2 δ A x C 1 M 2 · R M 1 C 1 · δ A x C 1 M 2 + R C 2 M 1 · R M 2 C 2 · δ R C 1 M 2 δ A y C 1 M 2 · R M 1 C 1 · δ A y C 1 M 2 + R C 2 M 1 · R M 2 C 2 · δ R C 1 M 2 δ A z C 1 M 2 · R M 1 C 1 · δ A z C 1 M 2 + δ R C 2 M 1 δ A x C 2 M 1 · R M 2 C 2 · R C 1 M 2 · R M 1 C 1 · δ A x C 2 M 1 + δ R C 2 M 1 δ A y C 2 M 1 · R M 2 C 2 · R C 1 M 2 · R M 1 C 1 · δ A y C 2 M 1 + δ R C 2 M 1 δ A z C 2 M 1 · R M 2 C 2 · R C 1 M 2 · R M 1 C 1 · δ A z C 2 M 1 = I R C 2 M 1 · R M 2 C 2 · R C 1 M 2 · R M 1 C 1 .
P x = R C 2 M 1 · R M 2 C 2 · δ R C 1 M 2 δ A x C 1 M 2 · R M 1 C 1 ,
P y = R C 2 M 1 · R M 2 C 2 · δ R C 1 M 2 δ A y C 1 M 2 · R M 1 C 1 ,
P z = R C 2 M 1 · R M 2 C 2 · δ R C 1 M 2 δ A z C 1 M 2 · R M 1 C 1 ,
Q x = δ R C 2 M 1 δ A x C 2 M 1 · R M 2 C 2 · R C 1 M 2 · R M 1 C 1 ,
Q y = δ R C 2 M 1 δ A y C 2 M 1 · R M 2 C 2 · R C 1 M 2 · R M 1 C 1 ,
Q z = δ R C 2 M 1 δ A z C 2 M 1 · R M 2 C 2 · R C 1 M 2 · R M 1 C 1
L = R C 2 M 1 · R M 2 C 2 · R C 1 M 2 · R M 1 C 1 ,
( P 11 x P 11 y P 11 z Q 11 x Q 11 y Q 11 z P 12 x P 12 y P 12 z Q 12 x Q 12 y Q 12 z P 13 x P 13 y P 13 z Q 13 x Q 13 y Q 13 z P 21 x P 21 y P 21 z Q 21 x Q 21 y Q 21 z P 22 x P 22 y P 22 z Q 22 x Q 22 y Q 22 z P 23 x P 23 y P 23 z Q 23 x Q 23 y Q 23 z P 31 x P 31 y P 31 z Q 31 x Q 31 y Q 31 z P 32 x P 32 y P 32 z Q 32 x Q 32 y Q 32 z P 33 x P 33 y P 33 z Q 33 x Q 33 y Q 33 z ) ( δ A x C 1 M 2 δ A y C 1 M 2 δ A z C 1 M 2 δ A x C 2 M 1 δ A y C 2 M 1 δ A z C 2 M 1 ) = ( 1 L 11 L 12 L 13 L 21 1 L 22 L 23 L 31 L 32 1 L 33 ) ,
A 9 × 6 X 6 × 1 = B 9 × 1 ,
X 6 × 1 = ( A x C 1 M 2 A y C 1 M 2 A z C 1 M 2 A x C 2 M 1 A y C 2 M 1 A z C 2 M 1 ) T
X 6 × 1 = ( δ A x C 1 M 2 δ A y C 1 M 2 δ A z C 1 M 2 δ A x C 2 M 1 δ A y C 2 M 1 δ A z C 2 M 1 ) T
X 6 × 1 = A 9 × 6 + B 9 × 1 .
( X 6 × 1 ) k + 1 = ( X 6 × 1 ) k + X 6 × 1 ,
R C 2 , M 1 ( A x C 2 , M 1 , A y C 2 , M 1 , A z C 2 , M 1 )
[ R M 1 , M 2 ( A x M 1 , M 2 , A y M 1 , M 2 , A z M 1 , M 2 ) ]
[ R M 1 , M 2 SCS ( A x M 1 , M 2 SCS , A y M 1 , M 2 SCS , A z M 1 , M 2 SCS ) ]
R M 1 , M 2 WA ( A x M 1 , M 2 WA , A y M 1 , M 2 WA , A z M 1 , M 2 WA ) ,
R M 1 , M 2 SA ( A x M 1 , M 2 SA , A y M 1 , M 2 SA , A z M 1 , M 2 SA ) ,

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