Abstract

In this paper, a position estimation method using a prism-based single-lens stereovision system is proposed. A multifaced prism was considered as a single optical system composed of few refractive planes. A transformation matrix which relates the coordinates of an object point to its coordinates on the image plane through the refraction of the prism was derived based on geometrical optics. A mathematical model which is able to denote the position of an arbitrary faces prism with only seven parameters is introduced. This model further extends the application of the single-lens stereovision system using a prism to other areas. Experimentation results are presented to prove the effectiveness and robustness of our proposed model.

© 2014 Optical Society of America

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References

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  1. E. Trucco and A. Verri, Introductory Techniques for 3D Computer Vision (Prentice Hall, 1998) pp. 139–140.
  2. D. H. Lee, I. Kweon, and R. Cipolla, “Single lens stereo with a biprism,” in Proceedings of IAPR Workshop on Machine Vision Applications (Computer Vision & Robotics, 1998), pp. 17–19.
  3. J. A. Kalomiros and J. Lygouras, “Design and hardware implementation of a stereo-matching system based on dynamic programming,” Microprocess. Microsyst. 35, 496–509 (2011).
    [CrossRef]
  4. L. Cai, L. He, Y. Xu, Y. Zhao, and X. Yang, “Multi-object detection and tracking by stereo vision,” Pattern Recogn. 43, 4028–4041 (2010).
    [CrossRef]
  5. F. A. Hankins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976), pp. 30–31.
  6. E. Hecht, Theory and Problems of Optics, Schaum’s Outline Series (McGraw-Hill, 1975), pp. 142–143.
  7. X. Cao and H. Foroosh, “Camera calibration and light source orientation from solar shadows,” Comput. Vision Image Underst. 105, 60–72 (2007).
  8. Q. Z. Ye, S. H. Ong, and X. Han, “A stereovision system for the inspection of IC bonding wires,” Int. J. Imaging Syst. Technol. 11, 254–262 (2000).
    [CrossRef]
  9. Y. Nishimoto and Y. Shirai, “A feature-based stereo model using disparity histograms of multi-resolution channels,” Adv. Robot. 3, 17–33 (1988).
    [CrossRef]
  10. W. Teoh and X. D. Zhang, “An inexpensive stereoscopic vision system for robots,” in Proceedings of the International Conference on Robotics and Automation (IEEE Computer Society, 1984), Vol. 1, pp. 186–189.
  11. A. Goshtasby and W. A. Gruver, “Design of a single-lens stereo camera system,” Pattern Recogn. 26, 923–937 (1993).
    [CrossRef]
  12. D. H. Lee, I. Kweon, and R. Cipolla, “A biprism-stereo camera system,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 1999), Vol. 1, pp. 82–87.
  13. D. H. Lee and I. Kweon, “A novel stereo camera system by a biprism,” IEEE Trans. Robot. Autom. 16, 528–541 (2000).
    [CrossRef]
  14. K. B. Lim and Y. Xiao, “Virtual stereovision system: new understanding on single-lens stereovision using a biprism,” J. Electron. Imaging 14, 41–52 (2005).
  15. Y. Xiao and K. B. Lim, “A prism-based single-lens stereovision system: from trinocular to multi-ocular,” Image Vis. Comput. 25, 1725–1736 (2007).
  16. X. Li and R. Wang, “Analysis and optimization of the stereo system with a biprism adapter,” Proc. SPIE 7506, 1–8 (2009).
  17. M. Zhao, K. Lim, and W. L. Kee, “Geometrical-analysis-based algorithm for stereo matching of single-lens binocular and multi-Ocular stereovision system,” J. Electron. Sci. Technol. 10, 107–112 (2012).
  18. W. L. Kee, K. B. Lim, and D. Wang, “Virtual epipolar line construction of single-lens bi-prism stereovision system,” J. Electron. Sci. Technol. 10, 97–101 (2012).
  19. X. Cui, K. B. Lim, Q. Guo, and D. Wang, “Accurate geometrical optics model for single-lens stereovision system using a prism,” J. Opt. Soc. Am. A 29, 1828–1837 (2012).
    [CrossRef]
  20. J. J. More, “The Levenberg–Marquardt algorithm, implementation and theory,” in Numerical Analysis, Lecture Notes in Mathematics (Springer, 1978), Vol. 630, pp. 105–116.
  21. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).
    [CrossRef]

2012 (3)

M. Zhao, K. Lim, and W. L. Kee, “Geometrical-analysis-based algorithm for stereo matching of single-lens binocular and multi-Ocular stereovision system,” J. Electron. Sci. Technol. 10, 107–112 (2012).

W. L. Kee, K. B. Lim, and D. Wang, “Virtual epipolar line construction of single-lens bi-prism stereovision system,” J. Electron. Sci. Technol. 10, 97–101 (2012).

X. Cui, K. B. Lim, Q. Guo, and D. Wang, “Accurate geometrical optics model for single-lens stereovision system using a prism,” J. Opt. Soc. Am. A 29, 1828–1837 (2012).
[CrossRef]

2011 (1)

J. A. Kalomiros and J. Lygouras, “Design and hardware implementation of a stereo-matching system based on dynamic programming,” Microprocess. Microsyst. 35, 496–509 (2011).
[CrossRef]

2010 (1)

L. Cai, L. He, Y. Xu, Y. Zhao, and X. Yang, “Multi-object detection and tracking by stereo vision,” Pattern Recogn. 43, 4028–4041 (2010).
[CrossRef]

2009 (1)

X. Li and R. Wang, “Analysis and optimization of the stereo system with a biprism adapter,” Proc. SPIE 7506, 1–8 (2009).

2007 (2)

X. Cao and H. Foroosh, “Camera calibration and light source orientation from solar shadows,” Comput. Vision Image Underst. 105, 60–72 (2007).

Y. Xiao and K. B. Lim, “A prism-based single-lens stereovision system: from trinocular to multi-ocular,” Image Vis. Comput. 25, 1725–1736 (2007).

2005 (1)

K. B. Lim and Y. Xiao, “Virtual stereovision system: new understanding on single-lens stereovision using a biprism,” J. Electron. Imaging 14, 41–52 (2005).

2000 (3)

D. H. Lee and I. Kweon, “A novel stereo camera system by a biprism,” IEEE Trans. Robot. Autom. 16, 528–541 (2000).
[CrossRef]

Q. Z. Ye, S. H. Ong, and X. Han, “A stereovision system for the inspection of IC bonding wires,” Int. J. Imaging Syst. Technol. 11, 254–262 (2000).
[CrossRef]

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).
[CrossRef]

1993 (1)

A. Goshtasby and W. A. Gruver, “Design of a single-lens stereo camera system,” Pattern Recogn. 26, 923–937 (1993).
[CrossRef]

1988 (1)

Y. Nishimoto and Y. Shirai, “A feature-based stereo model using disparity histograms of multi-resolution channels,” Adv. Robot. 3, 17–33 (1988).
[CrossRef]

Cai, L.

L. Cai, L. He, Y. Xu, Y. Zhao, and X. Yang, “Multi-object detection and tracking by stereo vision,” Pattern Recogn. 43, 4028–4041 (2010).
[CrossRef]

Cao, X.

X. Cao and H. Foroosh, “Camera calibration and light source orientation from solar shadows,” Comput. Vision Image Underst. 105, 60–72 (2007).

Cipolla, R.

D. H. Lee, I. Kweon, and R. Cipolla, “Single lens stereo with a biprism,” in Proceedings of IAPR Workshop on Machine Vision Applications (Computer Vision & Robotics, 1998), pp. 17–19.

D. H. Lee, I. Kweon, and R. Cipolla, “A biprism-stereo camera system,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 1999), Vol. 1, pp. 82–87.

Cui, X.

Foroosh, H.

X. Cao and H. Foroosh, “Camera calibration and light source orientation from solar shadows,” Comput. Vision Image Underst. 105, 60–72 (2007).

Goshtasby, A.

A. Goshtasby and W. A. Gruver, “Design of a single-lens stereo camera system,” Pattern Recogn. 26, 923–937 (1993).
[CrossRef]

Gruver, W. A.

A. Goshtasby and W. A. Gruver, “Design of a single-lens stereo camera system,” Pattern Recogn. 26, 923–937 (1993).
[CrossRef]

Guo, Q.

Han, X.

Q. Z. Ye, S. H. Ong, and X. Han, “A stereovision system for the inspection of IC bonding wires,” Int. J. Imaging Syst. Technol. 11, 254–262 (2000).
[CrossRef]

Hankins, F. A.

F. A. Hankins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976), pp. 30–31.

He, L.

L. Cai, L. He, Y. Xu, Y. Zhao, and X. Yang, “Multi-object detection and tracking by stereo vision,” Pattern Recogn. 43, 4028–4041 (2010).
[CrossRef]

Hecht, E.

E. Hecht, Theory and Problems of Optics, Schaum’s Outline Series (McGraw-Hill, 1975), pp. 142–143.

Kalomiros, J. A.

J. A. Kalomiros and J. Lygouras, “Design and hardware implementation of a stereo-matching system based on dynamic programming,” Microprocess. Microsyst. 35, 496–509 (2011).
[CrossRef]

Kee, W. L.

M. Zhao, K. Lim, and W. L. Kee, “Geometrical-analysis-based algorithm for stereo matching of single-lens binocular and multi-Ocular stereovision system,” J. Electron. Sci. Technol. 10, 107–112 (2012).

W. L. Kee, K. B. Lim, and D. Wang, “Virtual epipolar line construction of single-lens bi-prism stereovision system,” J. Electron. Sci. Technol. 10, 97–101 (2012).

Kweon, I.

D. H. Lee and I. Kweon, “A novel stereo camera system by a biprism,” IEEE Trans. Robot. Autom. 16, 528–541 (2000).
[CrossRef]

D. H. Lee, I. Kweon, and R. Cipolla, “A biprism-stereo camera system,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 1999), Vol. 1, pp. 82–87.

D. H. Lee, I. Kweon, and R. Cipolla, “Single lens stereo with a biprism,” in Proceedings of IAPR Workshop on Machine Vision Applications (Computer Vision & Robotics, 1998), pp. 17–19.

Lee, D. H.

D. H. Lee and I. Kweon, “A novel stereo camera system by a biprism,” IEEE Trans. Robot. Autom. 16, 528–541 (2000).
[CrossRef]

D. H. Lee, I. Kweon, and R. Cipolla, “A biprism-stereo camera system,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 1999), Vol. 1, pp. 82–87.

D. H. Lee, I. Kweon, and R. Cipolla, “Single lens stereo with a biprism,” in Proceedings of IAPR Workshop on Machine Vision Applications (Computer Vision & Robotics, 1998), pp. 17–19.

Li, X.

X. Li and R. Wang, “Analysis and optimization of the stereo system with a biprism adapter,” Proc. SPIE 7506, 1–8 (2009).

Lim, K.

M. Zhao, K. Lim, and W. L. Kee, “Geometrical-analysis-based algorithm for stereo matching of single-lens binocular and multi-Ocular stereovision system,” J. Electron. Sci. Technol. 10, 107–112 (2012).

Lim, K. B.

X. Cui, K. B. Lim, Q. Guo, and D. Wang, “Accurate geometrical optics model for single-lens stereovision system using a prism,” J. Opt. Soc. Am. A 29, 1828–1837 (2012).
[CrossRef]

W. L. Kee, K. B. Lim, and D. Wang, “Virtual epipolar line construction of single-lens bi-prism stereovision system,” J. Electron. Sci. Technol. 10, 97–101 (2012).

Y. Xiao and K. B. Lim, “A prism-based single-lens stereovision system: from trinocular to multi-ocular,” Image Vis. Comput. 25, 1725–1736 (2007).

K. B. Lim and Y. Xiao, “Virtual stereovision system: new understanding on single-lens stereovision using a biprism,” J. Electron. Imaging 14, 41–52 (2005).

Lygouras, J.

J. A. Kalomiros and J. Lygouras, “Design and hardware implementation of a stereo-matching system based on dynamic programming,” Microprocess. Microsyst. 35, 496–509 (2011).
[CrossRef]

More, J. J.

J. J. More, “The Levenberg–Marquardt algorithm, implementation and theory,” in Numerical Analysis, Lecture Notes in Mathematics (Springer, 1978), Vol. 630, pp. 105–116.

Nishimoto, Y.

Y. Nishimoto and Y. Shirai, “A feature-based stereo model using disparity histograms of multi-resolution channels,” Adv. Robot. 3, 17–33 (1988).
[CrossRef]

Ong, S. H.

Q. Z. Ye, S. H. Ong, and X. Han, “A stereovision system for the inspection of IC bonding wires,” Int. J. Imaging Syst. Technol. 11, 254–262 (2000).
[CrossRef]

Shirai, Y.

Y. Nishimoto and Y. Shirai, “A feature-based stereo model using disparity histograms of multi-resolution channels,” Adv. Robot. 3, 17–33 (1988).
[CrossRef]

Teoh, W.

W. Teoh and X. D. Zhang, “An inexpensive stereoscopic vision system for robots,” in Proceedings of the International Conference on Robotics and Automation (IEEE Computer Society, 1984), Vol. 1, pp. 186–189.

Trucco, E.

E. Trucco and A. Verri, Introductory Techniques for 3D Computer Vision (Prentice Hall, 1998) pp. 139–140.

Verri, A.

E. Trucco and A. Verri, Introductory Techniques for 3D Computer Vision (Prentice Hall, 1998) pp. 139–140.

Wang, D.

X. Cui, K. B. Lim, Q. Guo, and D. Wang, “Accurate geometrical optics model for single-lens stereovision system using a prism,” J. Opt. Soc. Am. A 29, 1828–1837 (2012).
[CrossRef]

W. L. Kee, K. B. Lim, and D. Wang, “Virtual epipolar line construction of single-lens bi-prism stereovision system,” J. Electron. Sci. Technol. 10, 97–101 (2012).

Wang, R.

X. Li and R. Wang, “Analysis and optimization of the stereo system with a biprism adapter,” Proc. SPIE 7506, 1–8 (2009).

White, H. E.

F. A. Hankins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976), pp. 30–31.

Xiao, Y.

Y. Xiao and K. B. Lim, “A prism-based single-lens stereovision system: from trinocular to multi-ocular,” Image Vis. Comput. 25, 1725–1736 (2007).

K. B. Lim and Y. Xiao, “Virtual stereovision system: new understanding on single-lens stereovision using a biprism,” J. Electron. Imaging 14, 41–52 (2005).

Xu, Y.

L. Cai, L. He, Y. Xu, Y. Zhao, and X. Yang, “Multi-object detection and tracking by stereo vision,” Pattern Recogn. 43, 4028–4041 (2010).
[CrossRef]

Yang, X.

L. Cai, L. He, Y. Xu, Y. Zhao, and X. Yang, “Multi-object detection and tracking by stereo vision,” Pattern Recogn. 43, 4028–4041 (2010).
[CrossRef]

Ye, Q. Z.

Q. Z. Ye, S. H. Ong, and X. Han, “A stereovision system for the inspection of IC bonding wires,” Int. J. Imaging Syst. Technol. 11, 254–262 (2000).
[CrossRef]

Zhang, X. D.

W. Teoh and X. D. Zhang, “An inexpensive stereoscopic vision system for robots,” in Proceedings of the International Conference on Robotics and Automation (IEEE Computer Society, 1984), Vol. 1, pp. 186–189.

Zhang, Z.

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).
[CrossRef]

Zhao, M.

M. Zhao, K. Lim, and W. L. Kee, “Geometrical-analysis-based algorithm for stereo matching of single-lens binocular and multi-Ocular stereovision system,” J. Electron. Sci. Technol. 10, 107–112 (2012).

Zhao, Y.

L. Cai, L. He, Y. Xu, Y. Zhao, and X. Yang, “Multi-object detection and tracking by stereo vision,” Pattern Recogn. 43, 4028–4041 (2010).
[CrossRef]

Adv. Robot. (1)

Y. Nishimoto and Y. Shirai, “A feature-based stereo model using disparity histograms of multi-resolution channels,” Adv. Robot. 3, 17–33 (1988).
[CrossRef]

Comput. Vision Image Underst. (1)

X. Cao and H. Foroosh, “Camera calibration and light source orientation from solar shadows,” Comput. Vision Image Underst. 105, 60–72 (2007).

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330–1334 (2000).
[CrossRef]

IEEE Trans. Robot. Autom. (1)

D. H. Lee and I. Kweon, “A novel stereo camera system by a biprism,” IEEE Trans. Robot. Autom. 16, 528–541 (2000).
[CrossRef]

Image Vis. Comput. (1)

Y. Xiao and K. B. Lim, “A prism-based single-lens stereovision system: from trinocular to multi-ocular,” Image Vis. Comput. 25, 1725–1736 (2007).

Int. J. Imaging Syst. Technol. (1)

Q. Z. Ye, S. H. Ong, and X. Han, “A stereovision system for the inspection of IC bonding wires,” Int. J. Imaging Syst. Technol. 11, 254–262 (2000).
[CrossRef]

J. Electron. Imaging (1)

K. B. Lim and Y. Xiao, “Virtual stereovision system: new understanding on single-lens stereovision using a biprism,” J. Electron. Imaging 14, 41–52 (2005).

J. Electron. Sci. Technol. (2)

M. Zhao, K. Lim, and W. L. Kee, “Geometrical-analysis-based algorithm for stereo matching of single-lens binocular and multi-Ocular stereovision system,” J. Electron. Sci. Technol. 10, 107–112 (2012).

W. L. Kee, K. B. Lim, and D. Wang, “Virtual epipolar line construction of single-lens bi-prism stereovision system,” J. Electron. Sci. Technol. 10, 97–101 (2012).

J. Opt. Soc. Am. A (1)

Microprocess. Microsyst. (1)

J. A. Kalomiros and J. Lygouras, “Design and hardware implementation of a stereo-matching system based on dynamic programming,” Microprocess. Microsyst. 35, 496–509 (2011).
[CrossRef]

Pattern Recogn. (2)

L. Cai, L. He, Y. Xu, Y. Zhao, and X. Yang, “Multi-object detection and tracking by stereo vision,” Pattern Recogn. 43, 4028–4041 (2010).
[CrossRef]

A. Goshtasby and W. A. Gruver, “Design of a single-lens stereo camera system,” Pattern Recogn. 26, 923–937 (1993).
[CrossRef]

Proc. SPIE (1)

X. Li and R. Wang, “Analysis and optimization of the stereo system with a biprism adapter,” Proc. SPIE 7506, 1–8 (2009).

Other (7)

D. H. Lee, I. Kweon, and R. Cipolla, “A biprism-stereo camera system,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 1999), Vol. 1, pp. 82–87.

W. Teoh and X. D. Zhang, “An inexpensive stereoscopic vision system for robots,” in Proceedings of the International Conference on Robotics and Automation (IEEE Computer Society, 1984), Vol. 1, pp. 186–189.

F. A. Hankins and H. E. White, Fundamentals of Optics (McGraw-Hill, 1976), pp. 30–31.

E. Hecht, Theory and Problems of Optics, Schaum’s Outline Series (McGraw-Hill, 1975), pp. 142–143.

E. Trucco and A. Verri, Introductory Techniques for 3D Computer Vision (Prentice Hall, 1998) pp. 139–140.

D. H. Lee, I. Kweon, and R. Cipolla, “Single lens stereo with a biprism,” in Proceedings of IAPR Workshop on Machine Vision Applications (Computer Vision & Robotics, 1998), pp. 17–19.

J. J. More, “The Levenberg–Marquardt algorithm, implementation and theory,” in Numerical Analysis, Lecture Notes in Mathematics (Springer, 1978), Vol. 630, pp. 105–116.

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Figures (9)

Fig. 1.
Fig. 1.

Coordinate system of the virtual camera method.

Fig. 2.
Fig. 2.

Schematic diagram of the ray tracing method. (a) A special assumption of the coordinate system and (b) the practical situation.

Fig. 3.
Fig. 3.

Ray refraction of an arbitrary plane.

Fig. 4.
Fig. 4.

Virtual image points of a five-ocular prism.

Fig. 5.
Fig. 5.

System setup and the multi-ocular prism.

Fig. 6.
Fig. 6.

Segments of the image plane for a two-, three-, and four-ocular prism.

Fig. 7.
Fig. 7.

Images acquired by a two-, three-, and four-ocular prism.

Fig. 8.
Fig. 8.

Positions of the calibration board.

Fig. 9.
Fig. 9.

Comparisons of the length error with and without the position estimation method. (a)–(c) X-axis direction. (d)–(f) Y-axis direction. (g)–(i) Z-axis direction. (a), (d), and (g) Two-ocular prism. (b), (e), and (h) three-ocular prism. (c), (f), and (i) four-ocular prism. PE stands for position estimation method.

Tables (6)

Tables Icon

Table 1. Reprojection Errors of a Two-Ocular Prism

Tables Icon

Table 2. Reprojection Errors of a Three-Ocular Prism

Tables Icon

Table 3. Reprojection Errors of a Four-Ocular Prism

Tables Icon

Table 4. Length Results and Errors of a Two-Ocular Prism

Tables Icon

Table 5. Length Results and Errors of a Three-Ocular Prism

Tables Icon

Table 6. Length Results and Errors of a Four-Ocular Prism

Equations (33)

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

AB=x0nx+y0ny+z0nz+d.
tanα=BC/AB,tanα=BC/AB.
AA=ABAB=ABtanα/tanαAB=AB(tanα/tanα1)=(x0nx+y0ny+z0nz+d)(tanα/tanα1).
AA=[(xx0)2+(yy0)2+(zz0)2]1/2.
(xx0)/nx=(yy0)/ny=(zz0)/nz.
AA=(zz0)(nx2/nz2+ny2/nz2+1)1/2.
z=QnxSx0+QnySy0+Qnz+SSz0+QdS.
x=Qnx2+SnzSnzx0+QnxnySnzy0+QnxSz0+QdnxSnz,y=QnxnySnzx0+Qny2+SnzSnzy0+QnySz0+QdnySnz.
[xyz1]=[Qnx2+SnzSnzQnxnySnzQnxSQdnxSnzQnxnySnzQny2+SnzSnzQnySQdnySnzQnxSQnySQnz+SSQdS0001][x0y0z01].
[xyz1]=[Qnx2+1QnxnyQnxSQdnxQnxnyQny2+1QnySQdnyQnxSQnySQnzS+1QdS0001][x0y0z01],
A=M1A,
tanα/tanα=ncosα/ncosα.
Q=nV·NnV·N1.
AA=ABAB=ABABtanα/tanα=AB(1tanα/tanα)=(x0nx+y0ny+z0nz+d)(tanα/tanα1).
AA=(zz0)(nx2/nz2+ny2/nz2+1)1/2.
A=M2A,
A=M2M1A=MpA,
Vn=MVδVo,
MVδ=[cosδ+vx2ΦvxvyΦ+vzsinδvxvzΦvysinδvxvyΦvzsinδcosδ+vy2ΦvyvzΦ+vxsinδvxvzΦ+vysinδvyvzΦvxsinδcosδ+vz2Φ],Φ=1cosδ,
Vi=MVbαVi1(i=2,3m),
Ni=MViθVb(i=1,2,,m),
Vb·V1=0,
Vb=V1=1.
P=Pb+tVb,
PPb=d,
P={[px,pz,vbx,vby,vbz,d]T(m=2),[px,py,pz,vbx,vby,vbz,d]T(m>2).
D={vbx,vby,vbz,v1x,px,py,pz,n,θ,d}.
Mp=[Rmtm0T1],
sx˜=MintMpMextX˜,
Xc=MextX˜.
sx˜=MintMpXc.
i=1nj=1mxijm^(Mint,Mp,Mext,Xij),
i=1nj=1mxijm^(D,Xijc),

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