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Synthesis of Optimal Control for Cooperative Collision Avoidance for Aircraft (Ships) with Unequal Turn Capabilities

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Abstract

This paper presents a synthesis of an optimal control solution for cooperative collision avoidance strategies for aircraft (ships) with unequal turn capabilities in a close proximity coplanar encounter. The analytic expressions for the extremals are presented and their properties are analyzed. Simple algorithms for the synthesis of optimal control are developed. The structure of the synthesis is analyzed and its behavior with a change in the nondimensional turn rate ratio is studied. It is shown that Merz’s solution for identical aircraft (see Merz in Proc. Joint Automatic Control Conf., Paper 15-3, pp. 449–454, 1973; Navigation 20(2):144–152, 1973; Tarnopolskaya and Fulton in J. Optim. Theory Appl. 140(2):355–375, 2009) is a degenerate case of this more general solution. The results of this paper are useful for benchmarking and validating automated proximity management and collision avoidance systems.

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References

  1. Fulton, N.L.: Regional airspace design: a structured systems engineering approach. Doctoral dissertation, University of New South Wales (11 December 2002)

  2. Fulton, N.L.: Airspace design: towards a rigorous specification of conflict complexity based on computational geometry. Aeronaut. J., 75–84 (February 1999)

  3. Merz, A.W.: Optimal aircraft collision avoidance. In: Proc. Joint Automatic Control Conf., Paper 15-3, pp. 449–454 (1973)

  4. Merz, A.W.: Optimal evasive manoeuvres in maritime collision avoidance. Navigation 20(2), 144–152 (1973)

    Google Scholar 

  5. Tarnopolskaya, T., Fulton, N.: Optimal cooperative collision avoidance strategy for coplanar encounter: Merz’s solution revisited. J. Optim. Theory Appl. 140(2), 355–375 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Miele, A., Wang, T., Chao, C.S., Dabney, J.B.: Optimal control of a ship for collision avoidance maneuvers. J. Optim. Theory Appl. 103(3), 495–518 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Miele, A., Wang, T.: Optimal trajectories and guidance schemes for ship collision avoidance. J. Optim. Theory Appl. 129(1), 1–20 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Krozel, J., Peters, M.: Conflict detection and resolution for free flight. Air Traffic Contr. Q. 5(3), 181–211 (1997)

    Google Scholar 

  9. Miele, A., Wang, T., Chao, C.S., Dabney, J.B.: Optimal control of a ship for course change and sidestep maneuvers. J. Optim. Theory Appl. 103(2), 259–282 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Miloh, T., Pachter, M.: Ship collision-avoidance and persuit-evasion differential games with speed-loss in a turn. Comput. Math. Appl. 18(1–3), 77–100 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Krozel, J., Mueller, T., Hunter, G.: Free flight conflict detection and resolution analysis. In: AIAA Guidance, Navigation, and Control Conf., Paper 96-3763, pp. 1–11 (1996)

  12. Clements, J.C.: The optimal control of collision avoidance trajectories in air traffic management. Transp. Res. Part B 33, 265–280 (1999)

    Article  Google Scholar 

  13. Clements, J.C.: Optimal simultaneous pairwise conflict resolution maneuvers in air traffic management. J. Guid. Control Dyn. 25(4), 815–818 (2002)

    Article  Google Scholar 

  14. Menon, P.K., Sweriduk, G.D., Sridhar, B.: Optimal strategies for free-flight air traffic conflict resolution. J. Guid. Control Dyn. 22(2), 202–211 (1999)

    Article  Google Scholar 

  15. Raghunathan, A.U., Gopal, V., Subramanian, D., Biegler, L.T., Samad, T.: Dynamic optimization strategies for three-dimensional conflict resolution of multiple aircraft. J. Guid. Control Dyn. 27(4), 586–594 (2004)

    Article  Google Scholar 

  16. Hu, J., Prandini, M., Sastry, S.: Optimal coordinated maneuvers for three-dimensional aircraft conflict resolution. J. Guid. Control Dyn. 25(5), 888–900 (2002)

    Article  Google Scholar 

  17. Paielli, R.: Modeling maneuver dynamics in air traffic conflict resolution. J. Guid. Control Dyn. 26(3), 407–415 (2003)

    Article  Google Scholar 

  18. Durand, N.: Optimisation de Trajectoires pour la Resolution de Conflits en Route. PhD Dissertation, Institut National Polytechnique de Toulouse (28 May 1996)

  19. Emery, S.: Design aeronautical study for Broome international airport terminal airspace (14 March 2004)

  20. Shukla, U.S., Mahapatra, P.R.: The proportional navigation dilemma—pure or true? IEEE Trans. Aerosp. Electron. Syst. 26(2), 382–392 (1990)

    Article  Google Scholar 

  21. Goodchild, C., Vilaplana, M., Elefante, S.: Co-operative optimal airborne separation assurance in free flight airspace. In: Proc. 3rd USA/Europe Air Traffic Management R&D Seminar, Napoli, 13–16 June 2000

  22. Christodoulou, M.: Automatic commercial aircraft-collision avoidance in free flight: the three-dimensional problem. IEEE Trans. Intell. Transp. Syst. 7(2), 242–249 (2006)

    Article  Google Scholar 

  23. Cesarone, J., Eman, K.F.: Efficient manipulator collision avoidance by dynamic programming. Robot. Comput. Integr. Manuf. 8(1), 35–44 (1991)

    Article  Google Scholar 

  24. Tomlin, C., Pappas, G.J.: Conflict resolution for air traffic management: a study in multiagent hybrid systems. IEEE Trans. Automat. Contr. 43(4), 509–521 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Bicchi, A., Pallotino, L.: On optimal cooperative conflict resolution for air traffic management systems. IEEE Trans. Intell. Transp. Syst. 1(4), 221–232 (2000)

    Article  Google Scholar 

  26. Mitchell, I.M., Bayen, A.M., Tomlin, C.J.: A time-dependent Hamilton-Jocobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Automat. Contr. 50(7), 947–957 (2005)

    Article  MathSciNet  Google Scholar 

  27. Mitchell, I.M., Tomlin, C.J.: Overapproximating reachable sets by Hamilton-Jacobi projections. J. Sci. Comput. 19(1–3), 323–346 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. International Civil Aviation Organization: Annex 2, Rules of the Air. Chapter 3, ninth ed. Amendment 36. Montreal, Canada (November 2001)

  29. Pontryagin, L.S., Boltyanski, W.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1965)

    Google Scholar 

  30. Bryson, A.E.: Dynamic Optimization. Addison-Wesley, Reading (1999)

    Google Scholar 

  31. Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, Berlin (1975)

    MATH  Google Scholar 

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Authors and Affiliations

  1. CSIRO Mathematical & Information Sciences, Locked Bag 17, North Ryde, NSW, 1670, Australia

    T. Tarnopolskaya

  2. CSIRO Mathematical & Information Sciences, GPO Box 664, Canberra, ACT, 2601, Australia

    N. Fulton

Authors
  1. T. Tarnopolskaya
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  2. N. Fulton
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Correspondence to T. Tarnopolskaya.

Additional information

Communicated by H.J. Pesch.

We thank the anonymous referees for helpful comments and suggestions.

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Tarnopolskaya, T., Fulton, N. Synthesis of Optimal Control for Cooperative Collision Avoidance for Aircraft (Ships) with Unequal Turn Capabilities. J Optim Theory Appl 144, 367–390 (2010). https://doi.org/10.1007/s10957-009-9597-1

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