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Optimal Cooperative Collision Avoidance Strategy for Coplanar Encounter: Merz’s Solution Revisited

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Abstract

Analytic solutions for optimal collision avoidance strategies are of great importance when setting and validating air traffic rules and as a benchmark when validating automated proximity management and collision avoidance systems. Such a solution for optimal air collision avoidance strategies for a coplanar cooperative encounter between two identical aircraft (or ships) was first presented by Merz (Proc. Joint Automatic Control Conf., Pap. 15-3:449–454, 1973; Navigation 20(2):144–152, 1973). Unfortunately, Merz provided only a very brief indicative justification for his solution. This paper presents a rigorous analysis of the problem. New results include a characterization of a complete set of extremals, justification for optimal strategies and an analysis of the properties of the regions of different optimal strategies. A simple, practical and sufficiently accurate closed form approximation for dispersal curves that partition the plane of initial positions into the regions of different optimal strategies is also presented.

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References

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

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. 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  MathSciNet  Google Scholar 

  5. 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  MathSciNet  Google Scholar 

  6. 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  MathSciNet  Google Scholar 

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

    Google Scholar 

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

  9. 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 

  10. 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 

  11. 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 

  12. 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 

  13. 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 

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

    Article  Google Scholar 

  15. Durand, N.: Optimisation de trajectoires pour la resolution de conflits en route. Ph.D. Dissertation, Institut National Polytechnique de Toulouse, 28 May 1996

  16. Emery, S.: Design Aeronautical Study for Broome International Airport Terminal Airspace. 14 March 2004

  17. 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 

  18. 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

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  21. Merz, A.W., Hague, D.S.: Coplanar tail-chase aerial combat as a differential game. AIAA J. 15(10), 1419–1423 (1977)

    Article  MathSciNet  Google Scholar 

  22. 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  MathSciNet  Google Scholar 

  23. 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 

  24. Fulton, N.L.: Regional airspace design: a structured systems engineering approach. Doctoral Dissertation, University of New South Wales, 11 December 2002

  25. Fulton, N.L.: Airspace design: towards a rigorous specification of conflict complexity based on computational geometry. Aeronaut. J., February 1999

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

    Google Scholar 

  27. Bryson, A.E., Ho, Y.-C.: Optimal Control. Wiley, New York (1975)

    Google Scholar 

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

    Google Scholar 

  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 

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Authors and Affiliations

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

    T. Tarnopolskaya

  2. Senior Principal Research Engineer, 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 Q.C. Zhao.

We thank the anonymous referees and our colleagues Drs. R.S. Anderssen and R. Jarrett for helpful comments and suggestions.

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Tarnopolskaya, T., Fulton, N. Optimal Cooperative Collision Avoidance Strategy for Coplanar Encounter: Merz’s Solution Revisited. J Optim Theory Appl 140, 355–375 (2009). https://doi.org/10.1007/s10957-008-9452-9

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