Abstract

It is currently difficult to achieve good real-time dynamic angle measurements with high accuracy and large ranges. In this paper, a photoelectric measurement method for dynamic angles based on three laser displacement sensors (LDSs) is proposed. Offline, a dynamic angle vision measurement model is established, and the system is calibrated by using a planar target moved by a 2D moving platform. In the course of measurement, three laser beams emitted from three LDSs are projected onto a rotating plane, and three noncollinear points are acquired synchronously; then the rotation angle is calculated in real time. Simulations verify the feasibility of the method theoretically. Experimental results demonstrate that the method achieves measurement accuracies of 0.008° and 0.046° under quasi-static condition of 80°/s and highly dynamic condition of 1000°/s within the measurement range of about ±40°, respectively.

© 2013 Optical Society of America

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References

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  1. Raja M. Vinyojita, “Vision based landing for unmanned aerial vehicle,” in IEEE Aerospace Conference Proceedings (IEEE, 2011), pp. 3058–3065.
  2. G. J. Zhang, Z. Liu, Z. Z. Wei, H. Lei, and J. Jiang, “Four-channel synchronous dynamic measurement system for rudder angles based on line structured light,” Chin. J. Sci. Instrum. 31, 1910–1915 (2010).
  3. Z. Q. Zhang, G. W. Gao, X. Jiang, and Q. L. Hu, “Research on angle dynamic measurement system for naval gun level platform,” in International Conference on Electronics, Communications and Control (IEEE2011), pp. 4461–4464.
  4. R. B. Zhou and T. B. Xie, “Measurement for non-straightness and muzzle angle of artillery barrel based on laser collimation technology,” in Proceedings of the Third International Symposium on Instrumentation Science and Technology (Academic, 2004), pp. 753–757.
  5. J. J. Lee, H. N. Ho, and J. H. Lee, “A vision-based dynamic rotational angle measurement system for large civil structures,” Sensors 12, 7326–7336 (2012).
    [CrossRef]
  6. Y. Sun, Z. Y. Zhang, S. J. Huang, B. Yu, and S. L. Wang, “Vision measurement technology research for model angle of attack in wind tunnel tests,” Acta Aeronaut. Astronaut. Sin. 34, 1–7 (2013).
  7. A. Just and R. Probst, “Investigation of a ring laser angle measuring instrument for dynamic angle measurements,” in Proceedings of Precision Engineering, Nanotechonology (Academic, 1999), pp. 223–226.
  8. L. R. Qiu, H. R. Guo, Y. Wang, and W. Q. Zhao, “Laser two-dimensional angle dynamic measurement system,” Proc. SPIE 8201, 82010G (1997).
    [CrossRef]
  9. W. Tao, Z. B. Pu, and Z. Zhang, “Dynamic random measurement of angles with HP5528A dual-frequency laser interferometer,” in Proceedings of the Second International Symposium on Instrumentation Science and Technology (2002), vol. 3, pp. 333–336.
  10. J. Y. Zhang, T. Q. Fan, and X. D. Cao, “Dynamic photoelectric autocollimator based on two-dimension position sensitive detector,” Proc. SPIE 6723, 672315 (2011).
    [CrossRef]
  11. H. Y. Shang and G. J. Zhang, “Dynamic angle real-time measuring system with three degrees of freedom,” Opt. Electron. Eng. 33, 98–102 (2006).
  12. H. Y. Wang, J. Jiang, G. J. Zhang, and Y. D. Cui, “Large FOV attitude angle measuring system based on photoelectric collimating,” Infrared Laser Eng. 38, 126–129 (2009).
  13. S. He, Z. B. Wang, X. F. Zhang, Y. Q. Gao, and M. Z. Cao, “Calibration method for rudders angular displacement based on monocular vision,” Opt. Electron. Eng. 36, 73–78 (2009).
  14. G. J. Zhang and F. Q. Zhou, “Position and orientation estimation method for lading of unmanned aerial vehicle with two circle based computer vision,” Acta Aeronaut. Astronaut. Sin. 26, 345–348 (2006).
  15. J. Jiang, Z. Miao, and G. J. Zhang, “Dynamic altitude angle measurement system based on dot-structure light,” Infrared Laser Eng 39, 532–536 (2010).
  16. H. Y. Jin, based on laser trigonometry measuring technology research, Harbin Institute of Technology, Harbin, 2006.
  17. C. S. Dong, “A regression model for analysing the non-linearity of laser triangulation probes,” Int. J. Adv. Manuf. Technol 59, 691–695 (2012).
    [CrossRef]
  18. R. G. Dorsch, G. Häulser, and J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306–1314 (1994).
    [CrossRef]

2013

Y. Sun, Z. Y. Zhang, S. J. Huang, B. Yu, and S. L. Wang, “Vision measurement technology research for model angle of attack in wind tunnel tests,” Acta Aeronaut. Astronaut. Sin. 34, 1–7 (2013).

2012

J. J. Lee, H. N. Ho, and J. H. Lee, “A vision-based dynamic rotational angle measurement system for large civil structures,” Sensors 12, 7326–7336 (2012).
[CrossRef]

C. S. Dong, “A regression model for analysing the non-linearity of laser triangulation probes,” Int. J. Adv. Manuf. Technol 59, 691–695 (2012).
[CrossRef]

2011

J. Y. Zhang, T. Q. Fan, and X. D. Cao, “Dynamic photoelectric autocollimator based on two-dimension position sensitive detector,” Proc. SPIE 6723, 672315 (2011).
[CrossRef]

2010

G. J. Zhang, Z. Liu, Z. Z. Wei, H. Lei, and J. Jiang, “Four-channel synchronous dynamic measurement system for rudder angles based on line structured light,” Chin. J. Sci. Instrum. 31, 1910–1915 (2010).

J. Jiang, Z. Miao, and G. J. Zhang, “Dynamic altitude angle measurement system based on dot-structure light,” Infrared Laser Eng 39, 532–536 (2010).

2009

H. Y. Wang, J. Jiang, G. J. Zhang, and Y. D. Cui, “Large FOV attitude angle measuring system based on photoelectric collimating,” Infrared Laser Eng. 38, 126–129 (2009).

S. He, Z. B. Wang, X. F. Zhang, Y. Q. Gao, and M. Z. Cao, “Calibration method for rudders angular displacement based on monocular vision,” Opt. Electron. Eng. 36, 73–78 (2009).

2006

G. J. Zhang and F. Q. Zhou, “Position and orientation estimation method for lading of unmanned aerial vehicle with two circle based computer vision,” Acta Aeronaut. Astronaut. Sin. 26, 345–348 (2006).

H. Y. Shang and G. J. Zhang, “Dynamic angle real-time measuring system with three degrees of freedom,” Opt. Electron. Eng. 33, 98–102 (2006).

1997

L. R. Qiu, H. R. Guo, Y. Wang, and W. Q. Zhao, “Laser two-dimensional angle dynamic measurement system,” Proc. SPIE 8201, 82010G (1997).
[CrossRef]

1994

Cao, M. Z.

S. He, Z. B. Wang, X. F. Zhang, Y. Q. Gao, and M. Z. Cao, “Calibration method for rudders angular displacement based on monocular vision,” Opt. Electron. Eng. 36, 73–78 (2009).

Cao, X. D.

J. Y. Zhang, T. Q. Fan, and X. D. Cao, “Dynamic photoelectric autocollimator based on two-dimension position sensitive detector,” Proc. SPIE 6723, 672315 (2011).
[CrossRef]

Cui, Y. D.

H. Y. Wang, J. Jiang, G. J. Zhang, and Y. D. Cui, “Large FOV attitude angle measuring system based on photoelectric collimating,” Infrared Laser Eng. 38, 126–129 (2009).

Dong, C. S.

C. S. Dong, “A regression model for analysing the non-linearity of laser triangulation probes,” Int. J. Adv. Manuf. Technol 59, 691–695 (2012).
[CrossRef]

Dorsch, R. G.

Fan, T. Q.

J. Y. Zhang, T. Q. Fan, and X. D. Cao, “Dynamic photoelectric autocollimator based on two-dimension position sensitive detector,” Proc. SPIE 6723, 672315 (2011).
[CrossRef]

Gao, G. W.

Z. Q. Zhang, G. W. Gao, X. Jiang, and Q. L. Hu, “Research on angle dynamic measurement system for naval gun level platform,” in International Conference on Electronics, Communications and Control (IEEE2011), pp. 4461–4464.

Gao, Y. Q.

S. He, Z. B. Wang, X. F. Zhang, Y. Q. Gao, and M. Z. Cao, “Calibration method for rudders angular displacement based on monocular vision,” Opt. Electron. Eng. 36, 73–78 (2009).

Guo, H. R.

L. R. Qiu, H. R. Guo, Y. Wang, and W. Q. Zhao, “Laser two-dimensional angle dynamic measurement system,” Proc. SPIE 8201, 82010G (1997).
[CrossRef]

Häulser, G.

He, S.

S. He, Z. B. Wang, X. F. Zhang, Y. Q. Gao, and M. Z. Cao, “Calibration method for rudders angular displacement based on monocular vision,” Opt. Electron. Eng. 36, 73–78 (2009).

Herrmann, J. M.

Ho, H. N.

J. J. Lee, H. N. Ho, and J. H. Lee, “A vision-based dynamic rotational angle measurement system for large civil structures,” Sensors 12, 7326–7336 (2012).
[CrossRef]

Hu, Q. L.

Z. Q. Zhang, G. W. Gao, X. Jiang, and Q. L. Hu, “Research on angle dynamic measurement system for naval gun level platform,” in International Conference on Electronics, Communications and Control (IEEE2011), pp. 4461–4464.

Huang, S. J.

Y. Sun, Z. Y. Zhang, S. J. Huang, B. Yu, and S. L. Wang, “Vision measurement technology research for model angle of attack in wind tunnel tests,” Acta Aeronaut. Astronaut. Sin. 34, 1–7 (2013).

Jiang, J.

J. Jiang, Z. Miao, and G. J. Zhang, “Dynamic altitude angle measurement system based on dot-structure light,” Infrared Laser Eng 39, 532–536 (2010).

G. J. Zhang, Z. Liu, Z. Z. Wei, H. Lei, and J. Jiang, “Four-channel synchronous dynamic measurement system for rudder angles based on line structured light,” Chin. J. Sci. Instrum. 31, 1910–1915 (2010).

H. Y. Wang, J. Jiang, G. J. Zhang, and Y. D. Cui, “Large FOV attitude angle measuring system based on photoelectric collimating,” Infrared Laser Eng. 38, 126–129 (2009).

Jiang, X.

Z. Q. Zhang, G. W. Gao, X. Jiang, and Q. L. Hu, “Research on angle dynamic measurement system for naval gun level platform,” in International Conference on Electronics, Communications and Control (IEEE2011), pp. 4461–4464.

Jin, H. Y.

H. Y. Jin, based on laser trigonometry measuring technology research, Harbin Institute of Technology, Harbin, 2006.

Just, A.

A. Just and R. Probst, “Investigation of a ring laser angle measuring instrument for dynamic angle measurements,” in Proceedings of Precision Engineering, Nanotechonology (Academic, 1999), pp. 223–226.

Lee, J. H.

J. J. Lee, H. N. Ho, and J. H. Lee, “A vision-based dynamic rotational angle measurement system for large civil structures,” Sensors 12, 7326–7336 (2012).
[CrossRef]

Lee, J. J.

J. J. Lee, H. N. Ho, and J. H. Lee, “A vision-based dynamic rotational angle measurement system for large civil structures,” Sensors 12, 7326–7336 (2012).
[CrossRef]

Lei, H.

G. J. Zhang, Z. Liu, Z. Z. Wei, H. Lei, and J. Jiang, “Four-channel synchronous dynamic measurement system for rudder angles based on line structured light,” Chin. J. Sci. Instrum. 31, 1910–1915 (2010).

Liu, Z.

G. J. Zhang, Z. Liu, Z. Z. Wei, H. Lei, and J. Jiang, “Four-channel synchronous dynamic measurement system for rudder angles based on line structured light,” Chin. J. Sci. Instrum. 31, 1910–1915 (2010).

Miao, Z.

J. Jiang, Z. Miao, and G. J. Zhang, “Dynamic altitude angle measurement system based on dot-structure light,” Infrared Laser Eng 39, 532–536 (2010).

Probst, R.

A. Just and R. Probst, “Investigation of a ring laser angle measuring instrument for dynamic angle measurements,” in Proceedings of Precision Engineering, Nanotechonology (Academic, 1999), pp. 223–226.

Pu, Z. B.

W. Tao, Z. B. Pu, and Z. Zhang, “Dynamic random measurement of angles with HP5528A dual-frequency laser interferometer,” in Proceedings of the Second International Symposium on Instrumentation Science and Technology (2002), vol. 3, pp. 333–336.

Qiu, L. R.

L. R. Qiu, H. R. Guo, Y. Wang, and W. Q. Zhao, “Laser two-dimensional angle dynamic measurement system,” Proc. SPIE 8201, 82010G (1997).
[CrossRef]

Shang, H. Y.

H. Y. Shang and G. J. Zhang, “Dynamic angle real-time measuring system with three degrees of freedom,” Opt. Electron. Eng. 33, 98–102 (2006).

Sun, Y.

Y. Sun, Z. Y. Zhang, S. J. Huang, B. Yu, and S. L. Wang, “Vision measurement technology research for model angle of attack in wind tunnel tests,” Acta Aeronaut. Astronaut. Sin. 34, 1–7 (2013).

Tao, W.

W. Tao, Z. B. Pu, and Z. Zhang, “Dynamic random measurement of angles with HP5528A dual-frequency laser interferometer,” in Proceedings of the Second International Symposium on Instrumentation Science and Technology (2002), vol. 3, pp. 333–336.

Vinyojita, Raja M.

Raja M. Vinyojita, “Vision based landing for unmanned aerial vehicle,” in IEEE Aerospace Conference Proceedings (IEEE, 2011), pp. 3058–3065.

Wang, H. Y.

H. Y. Wang, J. Jiang, G. J. Zhang, and Y. D. Cui, “Large FOV attitude angle measuring system based on photoelectric collimating,” Infrared Laser Eng. 38, 126–129 (2009).

Wang, S. L.

Y. Sun, Z. Y. Zhang, S. J. Huang, B. Yu, and S. L. Wang, “Vision measurement technology research for model angle of attack in wind tunnel tests,” Acta Aeronaut. Astronaut. Sin. 34, 1–7 (2013).

Wang, Y.

L. R. Qiu, H. R. Guo, Y. Wang, and W. Q. Zhao, “Laser two-dimensional angle dynamic measurement system,” Proc. SPIE 8201, 82010G (1997).
[CrossRef]

Wang, Z. B.

S. He, Z. B. Wang, X. F. Zhang, Y. Q. Gao, and M. Z. Cao, “Calibration method for rudders angular displacement based on monocular vision,” Opt. Electron. Eng. 36, 73–78 (2009).

Wei, Z. Z.

G. J. Zhang, Z. Liu, Z. Z. Wei, H. Lei, and J. Jiang, “Four-channel synchronous dynamic measurement system for rudder angles based on line structured light,” Chin. J. Sci. Instrum. 31, 1910–1915 (2010).

Xie, T. B.

R. B. Zhou and T. B. Xie, “Measurement for non-straightness and muzzle angle of artillery barrel based on laser collimation technology,” in Proceedings of the Third International Symposium on Instrumentation Science and Technology (Academic, 2004), pp. 753–757.

Yu, B.

Y. Sun, Z. Y. Zhang, S. J. Huang, B. Yu, and S. L. Wang, “Vision measurement technology research for model angle of attack in wind tunnel tests,” Acta Aeronaut. Astronaut. Sin. 34, 1–7 (2013).

Zhang, G. J.

J. Jiang, Z. Miao, and G. J. Zhang, “Dynamic altitude angle measurement system based on dot-structure light,” Infrared Laser Eng 39, 532–536 (2010).

G. J. Zhang, Z. Liu, Z. Z. Wei, H. Lei, and J. Jiang, “Four-channel synchronous dynamic measurement system for rudder angles based on line structured light,” Chin. J. Sci. Instrum. 31, 1910–1915 (2010).

H. Y. Wang, J. Jiang, G. J. Zhang, and Y. D. Cui, “Large FOV attitude angle measuring system based on photoelectric collimating,” Infrared Laser Eng. 38, 126–129 (2009).

H. Y. Shang and G. J. Zhang, “Dynamic angle real-time measuring system with three degrees of freedom,” Opt. Electron. Eng. 33, 98–102 (2006).

G. J. Zhang and F. Q. Zhou, “Position and orientation estimation method for lading of unmanned aerial vehicle with two circle based computer vision,” Acta Aeronaut. Astronaut. Sin. 26, 345–348 (2006).

Zhang, J. Y.

J. Y. Zhang, T. Q. Fan, and X. D. Cao, “Dynamic photoelectric autocollimator based on two-dimension position sensitive detector,” Proc. SPIE 6723, 672315 (2011).
[CrossRef]

Zhang, X. F.

S. He, Z. B. Wang, X. F. Zhang, Y. Q. Gao, and M. Z. Cao, “Calibration method for rudders angular displacement based on monocular vision,” Opt. Electron. Eng. 36, 73–78 (2009).

Zhang, Z.

W. Tao, Z. B. Pu, and Z. Zhang, “Dynamic random measurement of angles with HP5528A dual-frequency laser interferometer,” in Proceedings of the Second International Symposium on Instrumentation Science and Technology (2002), vol. 3, pp. 333–336.

Zhang, Z. Q.

Z. Q. Zhang, G. W. Gao, X. Jiang, and Q. L. Hu, “Research on angle dynamic measurement system for naval gun level platform,” in International Conference on Electronics, Communications and Control (IEEE2011), pp. 4461–4464.

Zhang, Z. Y.

Y. Sun, Z. Y. Zhang, S. J. Huang, B. Yu, and S. L. Wang, “Vision measurement technology research for model angle of attack in wind tunnel tests,” Acta Aeronaut. Astronaut. Sin. 34, 1–7 (2013).

Zhao, W. Q.

L. R. Qiu, H. R. Guo, Y. Wang, and W. Q. Zhao, “Laser two-dimensional angle dynamic measurement system,” Proc. SPIE 8201, 82010G (1997).
[CrossRef]

Zhou, F. Q.

G. J. Zhang and F. Q. Zhou, “Position and orientation estimation method for lading of unmanned aerial vehicle with two circle based computer vision,” Acta Aeronaut. Astronaut. Sin. 26, 345–348 (2006).

Zhou, R. B.

R. B. Zhou and T. B. Xie, “Measurement for non-straightness and muzzle angle of artillery barrel based on laser collimation technology,” in Proceedings of the Third International Symposium on Instrumentation Science and Technology (Academic, 2004), pp. 753–757.

Acta Aeronaut. Astronaut. Sin.

Y. Sun, Z. Y. Zhang, S. J. Huang, B. Yu, and S. L. Wang, “Vision measurement technology research for model angle of attack in wind tunnel tests,” Acta Aeronaut. Astronaut. Sin. 34, 1–7 (2013).

G. J. Zhang and F. Q. Zhou, “Position and orientation estimation method for lading of unmanned aerial vehicle with two circle based computer vision,” Acta Aeronaut. Astronaut. Sin. 26, 345–348 (2006).

Appl. Opt.

Chin. J. Sci. Instrum.

G. J. Zhang, Z. Liu, Z. Z. Wei, H. Lei, and J. Jiang, “Four-channel synchronous dynamic measurement system for rudder angles based on line structured light,” Chin. J. Sci. Instrum. 31, 1910–1915 (2010).

Infrared Laser Eng

J. Jiang, Z. Miao, and G. J. Zhang, “Dynamic altitude angle measurement system based on dot-structure light,” Infrared Laser Eng 39, 532–536 (2010).

Infrared Laser Eng.

H. Y. Wang, J. Jiang, G. J. Zhang, and Y. D. Cui, “Large FOV attitude angle measuring system based on photoelectric collimating,” Infrared Laser Eng. 38, 126–129 (2009).

Int. J. Adv. Manuf. Technol

C. S. Dong, “A regression model for analysing the non-linearity of laser triangulation probes,” Int. J. Adv. Manuf. Technol 59, 691–695 (2012).
[CrossRef]

Opt. Electron. Eng.

S. He, Z. B. Wang, X. F. Zhang, Y. Q. Gao, and M. Z. Cao, “Calibration method for rudders angular displacement based on monocular vision,” Opt. Electron. Eng. 36, 73–78 (2009).

H. Y. Shang and G. J. Zhang, “Dynamic angle real-time measuring system with three degrees of freedom,” Opt. Electron. Eng. 33, 98–102 (2006).

Proc. SPIE

J. Y. Zhang, T. Q. Fan, and X. D. Cao, “Dynamic photoelectric autocollimator based on two-dimension position sensitive detector,” Proc. SPIE 6723, 672315 (2011).
[CrossRef]

L. R. Qiu, H. R. Guo, Y. Wang, and W. Q. Zhao, “Laser two-dimensional angle dynamic measurement system,” Proc. SPIE 8201, 82010G (1997).
[CrossRef]

Sensors

J. J. Lee, H. N. Ho, and J. H. Lee, “A vision-based dynamic rotational angle measurement system for large civil structures,” Sensors 12, 7326–7336 (2012).
[CrossRef]

Other

Raja M. Vinyojita, “Vision based landing for unmanned aerial vehicle,” in IEEE Aerospace Conference Proceedings (IEEE, 2011), pp. 3058–3065.

Z. Q. Zhang, G. W. Gao, X. Jiang, and Q. L. Hu, “Research on angle dynamic measurement system for naval gun level platform,” in International Conference on Electronics, Communications and Control (IEEE2011), pp. 4461–4464.

R. B. Zhou and T. B. Xie, “Measurement for non-straightness and muzzle angle of artillery barrel based on laser collimation technology,” in Proceedings of the Third International Symposium on Instrumentation Science and Technology (Academic, 2004), pp. 753–757.

W. Tao, Z. B. Pu, and Z. Zhang, “Dynamic random measurement of angles with HP5528A dual-frequency laser interferometer,” in Proceedings of the Second International Symposium on Instrumentation Science and Technology (2002), vol. 3, pp. 333–336.

A. Just and R. Probst, “Investigation of a ring laser angle measuring instrument for dynamic angle measurements,” in Proceedings of Precision Engineering, Nanotechonology (Academic, 1999), pp. 223–226.

H. Y. Jin, based on laser trigonometry measuring technology research, Harbin Institute of Technology, Harbin, 2006.

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Figures (12)

Fig. 1.
Fig. 1.

3D vision measurement model of the LDS.

Fig. 2.
Fig. 2.

Angle calculation model.

Fig. 3.
Fig. 3.

Calibration principle of the dynamic angle measurement system.

Fig. 4.
Fig. 4.

Relative calibration error of the model parameters a0 and r versus noise level.

Fig. 5.
Fig. 5.

Relative calibration error of the model parameter m3 versus noise level.

Fig. 6.
Fig. 6.

Relative calibration error of the model parameter m1 versus noise level.

Fig. 7.
Fig. 7.

Measurement errors of the repetitive calibration testing.

Fig. 8.
Fig. 8.

RMS error versus numerical aperture.

Fig. 9.
Fig. 9.

RMS error versus measured angle.

Fig. 10.
Fig. 10.

Hardware structure of the dynamic angle measurement system: (1) aluminum measured plane, (2) sensor box, (3) high-speed rotating platform, (4)–(6) LDSs, and (7) trigger equipment.

Fig. 11.
Fig. 11.

Repeatability measurement results.

Fig. 12.
Fig. 12.

Measurement error versus rotation speed.

Tables (5)

Tables Icon

Table 1. Ideal Dynamic Angle Vision Measurement Model Parameters

Tables Icon

Table 2. Calibration Results of the Vision Measurement Model of Dynamic Angle Measurement System

Tables Icon

Table 3. Static Angle Measurement Results (deg)

Tables Icon

Table 4. Quasi-Static Angle Measurement Results in 80°/s (deg)

Tables Icon

Table 5. High-Dynamic Angle Measurement Results (deg)

Equations (16)

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

ρ[u1]=[m11m12m13m14m21m22m23m24][xwywzw1],
xw=pyw+qzw+r,
xwa0a3=ywa1a4=zwa2a5,
u=u+(uu0)[k1(x2+ya2)+k2(x2+ya2)2],
{ρ[u1]=[m1m2m3m4m51][ywzw1],xwa0a3=ywa1a4=zwa2a5,u=u+(uu0)[k1(x2+ya2)+k2(x2+ya2)2],
[p1,p2,p3]=f(x1,x2,x3,u1,u2,u3),
l=p1p2×p1p3.
θ1=arccosll0|l||l0|.
l0×l1=l0×l2.
θ=g(x1,x2,x3,u1,u2,u3,u10,u20,u30),
mn1yw+mn2zw+mn3mn4ywumn5zwu=u,
xw=pnyw+qnzw+rn,
[(uun0)(x2+yna2)(uun0)(x2+yna2)2][kn1kn2]=uu,
f(x⃗)=inpiqi(x,ui)2,
g(x⃗)=inθi(x1,x2,x3,u1i,u2i,u3i,u10,u20,u30)φi2,
δx=12πλsinu,

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