Skip to main content
SpringerLink
Log in

Differential geometric modeling of guidance problem for interceptors

  • Published:
Science China Technological Sciences Aims and scope Submit manuscript

Abstract

It is a comparatively convenient technique to investigate the motion of a particle with the help of the differential geometry theory, rather than directly decomposing the motion in the Cartesian coordinates. The new model of three-dimensional (3D) guidance problem for interceptors is presented in this paper, based on the classical differential geometry curve theory. Firstly, the kinematical equations of the line of sight (LOS) are gained by carefully investigating the rotation principle of LOS, the kinematic equations of LOS are established, and the concepts of curvature and torsion of LOS are proposed. Simultaneously, the new relative dynamic equations between interceptor and target are constructed. Secondly, it is found that there is an instantaneous rotation plane of LOS (IRPL) in the space, in which two-dimensional (2D) guidance laws could be constructed to solve 3D interception guidance problems. The spatial 3D true proportional navigation (TPN) guidance law could be directly introduced in IRPL without approximation and linearization for dimension-reduced 2D TPN. In addition, the new series of augmented TPN (APN) and LOS angular acceleration guidance laws (AAG) could also be gained in IRPL. After that, the differential geometric guidance commands (DGGC) of guidance laws in IRPL are advanced, and we prove that the guidance commands in arc-length system proposed by Chiou and Kuo are just a special case of DGGC. Moreover, the performance of the original guidance laws will be reduced after the differential geometric transformation. At last, an exoatmospheric interception is taken for simulation to demonstrate the differential geometric modeling proposed in this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Lin C F. Modern Navigation Guidance and Control Processing. NJ: Prentice Hall, 1991

    Google Scholar 

  2. Zarchan P. Tactical and strategic missile guidance. AIAA, 1997. 176

  3. Murtaugh S A, Criel H E. Fundamentals of proportional navigation. IEEE Spectrum, 1966, 3: 75–85

    Article  Google Scholar 

  4. Guelman M. A qualitative study of proportional navigation. IEEE T Aero El Sys, 1971, 7(4): 637–643

    Article  Google Scholar 

  5. Guelman M. Proportional navigation with a maneuvering target. IEEE T Aero El Sys, 1972, 8(4): 364–371

    Article  Google Scholar 

  6. Guelman M. The closed-form solution of true proportional navigation. IEEE T Aero El Sys, 1976, 12(4): 472–482

    Article  MathSciNet  Google Scholar 

  7. Becker K. Closed-form solution of pure proportional navigation. IEEE T Aero El Sys, 1990, 26(3): 526–533

    Article  Google Scholar 

  8. Shukla U S, Mahapatra P R. The proportional navigation dilemmapure or true. IEEE T Aero El Sys, 1990, 26(2): 382–392

    Article  Google Scholar 

  9. Dhar A, Ghose D. Capture region for a realistic TPN guidance law. IEEE T Aero El Sys, 1993, 29(3): 995–1003

    Article  Google Scholar 

  10. Yang C D, Yang C C. Analytical solution of three-dimensional realistic true proportional navigation. J Guid, Contr Dynam, 1996, 19(3): 569–577

    Article  MATH  Google Scholar 

  11. Yang C D, Yang C C. Analytical solution of generalized threedimensional proportional navigation. J Guid, Contr Dynam, 1996, 19(3): 721–724

    Article  MATH  Google Scholar 

  12. Yang C D, Yang C C. Analytical solution of 3D true proportional navigation. IEEE T Aerosp Electron Sys, 1996, 32(4): 1509–1522

    Article  Google Scholar 

  13. Yang C D, Yeh F B. The closed-form solution of generalized proportional navigation. J Guid, Contr Dynam, 1987, 10(2): 216–218

    Article  Google Scholar 

  14. Yang C D, Hsial F B, Yeh F B. Generalized guidance law for homing missiles. IEEE T Aerosp Electron Sys, 1989, 25(2): 197–212

    Article  Google Scholar 

  15. Ghose D. On the generalization of ture proportional navigation. IEEE T Aerosp Electron Sys, 1994, 30(2): 545–555

    Article  Google Scholar 

  16. Yang C D, Yang C C. A unified approach to proportional navigation. IEEE T Aerosp Electron Sys, 1997, 33(2): 557–567

    Article  Google Scholar 

  17. Duflos E. General 3D guidance law modeling. In: Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, Vancouver, SC, 1995. 10: 2013–2018

    Google Scholar 

  18. Duflos E, Penel P, Vanheeghe P. 3D guidance law modeling. IEEE T Aerosp Electron Sys, 1999, 35(1): 72–83

    Article  Google Scholar 

  19. Fend T. The capture region of a general 3D TPN guidance law for missile and target with limited maneuverability. In: Proceedings of the American Control Conference, Arlinton, 2001

  20. Feng T. A unified approach to missile guidance law: A 3D extension. In: Proceedings of the American Control Conference, Anchorage, 2002

  21. Feng T. Unified approach to missile guidance laws: A 3D extension. IEEE T Contr Sys Tech, 2005, 41(4): 1178–1199

    Google Scholar 

  22. Adler F P. Missile guidance by three-dimensional proportional navigation. J Appl Phys, 1956, 27(5): 500–507

    Article  MATH  Google Scholar 

  23. Chiou Y C, Kuo C Y. Geometric approach to three dimensional missile guidance problems. J Guid, Contr, Dynam, 1998, 21(2): 335–341

    Article  Google Scholar 

  24. Kuo C Y, Chiou Y C. Geometric analysis of missile guidance command. IEE P-Contr Theor Ap, 2000, 147(2): 205–211

    Article  Google Scholar 

  25. Kuo C Y, Soetanto D, Chiou Y C. Geometric analysis of flight control command for tactical missile guidance. IEEE T Contr Sys Tech, 2001, 9(2): 234–243

    Article  Google Scholar 

  26. Li C Y, Jing W X. New results on three-dimensional differential geometric guidance and control problem. AIAA Guidance, Navigation, and Control Conference and Exhibit, 2006. 8

  27. Li C Y, Jing W X, Qi Z G, et al. Application of the 3d differential geometric guidance commands (in Chinese). J Astron, 2007, 28(5): 1235–1240

    MATH  Google Scholar 

  28. Li C Y, Qi Z G, Jing W X. Practical study on 2d differential geometric guidance problem (in Chinese). J Harbin Inst Tech, 2007, 39(7): 1031–1035

    Google Scholar 

  29. Omar A, Rafal Z, Antonios T, et al. Differential geometric guidance based on the involute of the target’s trajectory. J Guid, Contr, Dynam, 2005, 28(5): 990–996

    Article  Google Scholar 

  30. White B A, Zbikowski R, Tsourdos A. Direct intercept guidance using differential geometry concepts. AIAA paper 2005-5969, 2005

  31. Hecht C. Homing guidance using angular acceleration of the line of sight. AIAA-91-2701-CP, 1991. 856–869

  32. Struik D J. Lectures on Classical Differential Geometry. New York: Dover, 1988. 10–20

    MATH  Google Scholar 

  33. Chen L, Zhang B. Novel TPN control algorithm for exoatmospheric intercept. J Syst Eng Electr, 2009, 20(6): 1290–1295

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Aerospace and Materials Engineering, National University of Defense Technology, Changsha, 410073, China

    KeBo Li, Lei Chen & XianZong Bai

Authors
  1. KeBo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. XianZong Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Chen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, K., Chen, L. & Bai, X. Differential geometric modeling of guidance problem for interceptors. Sci. China Technol. Sci. 54, 2283–2295 (2011). https://doi.org/10.1007/s11431-011-4451-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11431-011-4451-8

Keywords

Search

Navigation