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Abstract

This paper presents trajectory shaping of a surface-to-surface missile attacking a fixed with terminal impact angle constraint. The missile must hit the target from above, subject to the missile dynamics and path constraints. The problem is reinterpreted using optimal control theory resulting in the formulation of minimum integrated altitude. The formulation entails nonlinear, two-dimensional missile flight dynamics, boundary conditions and path constraints. The generic shape of optimal trajectory is: level flight, climbing, diving; this combination of the three flight phases is called the bunt manoeuvre. The numerical solution of optimal control problem is solved by a direct collocation method. The computational results is used to reveal the structure of optimal solution which is composed of several arcs, each of which can be identified by the corresponding manoeuvre executed and constraints active.

References

  1. P. Zarchan. Tactical and Strategic Missile Guidance. Progress in Astronautics and Aeronautics. 5th edition, AIAA, 2007
  2. Chang-Kyung Ryoo; Hangju Cho; Min-Jea Tahk, Optimal Guidance Laws with Terminal Impact Angle Constraint, Journal of Guidance, Control, and Dynamics. 28(4):724-732, 2005
  3. Ben-Asher, J. Z., and Yaesh, I., Advances in Missile Guidance Theory, Progress in Astronautics and Aeronautics, Vol. 180, AIAA, Reston, VA, 1998.
  4. S. Subchan and R. Zbikowski. Computational optimal control of the terminal bunt manoeuvre - Part 1: Minimum altitude case. Optimal Control Applications & Methods, 2007.
  5. S. Subchan and R. Zbikowski. Computational optimal control of the terminal bunt manoeuvre - Part 2: Minimum time case. Optimal Control Applications & Methods, 2007.
  6. H. J Kelley. Methods of gradients. In G. Leitmann, editor, Optimization Techniques with Application to Aerospace Systems, volume 5 of Mathematics in Science and Engineering, pages 206–254. Academic press, New York, 1962.
  7. A. E. Bryson and W. F. Denham. A steepest ascent method for solving optimum programming problems. ASME Journal of Applied Mechanics, Series E:247–257, 1962.
  8. R. Pytlak. Runge-Kutta based procedure for the optimal control of differential algebraic equations. Journal of Optimization Theory and Applications, 97(3):675–705, 1998.
  9. R. Pytlak. Numerical Methods for Optimal Control Problems with State Constraints, volume 1707 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Berlin, 1999.
  10. C. R. Hargraves and S. W. Paris. Direct trajectory optimization using nonlinear programming and collocation. Journal Guidance, Control, and Dynamics, 10(4):338–342, 1987.
  11. E. D. Dickmanns and K. H. Well. Approximate solution of optimal control problems using third order Hermite polynomial functions. In
  12. Proceedings of the IFIP Technical Conference, pages 158–166. Springer-Verlag, 1974.
  13. H. Seywald. Trajectory optimization based on differential inclusion. Journal of Guidance, Control, and Dynamics, 17(3):480–487, 1994.
  14. H. Seywald, E. M. Cliff, and K. H. Well. Range optimal trajectories for an aircraft flying in the vertical plane. Journal of Guidance, Control, and Dynamics, 17(2):389–398, 1994.
  15. J. T. Betts. Survey of numerical methods for trajectory optimization. Journal of Guidance, Control, and Dynamics, 21(2):193–207, 1998.
  16. J. T. Betts. Practical Methods for Optimal Control Using Nonlinear Programming. SIAM, Philadelphia, 2001.
  17. P. J. Enright and B. A. Conway. Discrete approximations to optimal trajectories using direct transcription and nonlinear programming. Journal of Guidance, Control, and Dynamics, 15(4):994–1002, 1991.
  18. A. L. Herman and B. A. Conway. Direct optimization using collocation based on higherorder Gauss-Lobatto quadrature rules. Journal of Guidance, Control,and Dynamics, 19(3):592–599, 1996.
  19. S. Tang and B. A. Conway. Optimization of lowthrust interplanetary trajectories using collocation and nonlinear programming. Journal of Guidance, Control, and Dynamics, 18(3):599–604, 1995.
  20. F. Fahroo and I. M. Ross. Direct trajectory optimization pseudospectral method. Journal of Guidance, Control, and Dynamics, 25(1):160–166, 2002.
  21. G. N. Elnagar. State-control spectral Chebyshev parameterization for linearly constrained quadratic optimal control problems. Journal of Computational and Applied Mathematics, 79(1):19–40, 1997.
  22. G. N. Elnagar and M. A. Kazemi. Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems. Computational Optimization and Applications, 11(2):195–217, 1998.
  23. O. von Stryk. User's guide for DIRCOL - a direct collocation method for the numerical solution of optimal control problems. Technische Universitat Darmstad, November 1999.
  24. O. von Stryk and R. Bulirsch. Direct and indirect methods for trajectory optimization. Annals of Operations Research, 37(1-4):357–373, 1992.
  25. P. E. Gill, W. Murray, and M. H. Wright. SNOPT: An SQP algorithm for large scale constrained optimization. SIAM Journal on Optimization, 12(4):979–1006, 2002.

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