Dynamic Games and Applications

, Volume 5, Issue 3, pp 297–317 | Cite as

On the Suicidal Pedestrian Differential Game

  • Ioannis Exarchos
  • Panagiotis Tsiotras
  • Meir Pachter
Article

Abstract

We consider the following differential game of pursuit and evasion involving two participating players: an evader, which has limited maneuverability, and an agile pursuer. The agents move on the Euclidean plane with different but constant speeds. Whereas the pursuer can change the orientation of its velocity vector arbitrarily fast, that is, he is a “pedestrian” á la Isaacs, the evader cannot make turns having a radius smaller than a specified minimum turning radius. This problem can be seen as a reversed Homicidal Chauffeur game, hence the name “Suicidal Pedestrian Differential Game.” The aim of this paper is to derive the optimal strategies of the two players and characterize the initial conditions that lead to capture if the pursuer acts optimally, and areas that guarantee evasion regardless of the pursuer’s strategy. Both proximity-capture and point-capture are considered. After applying the optimal strategy for the evader, it is shown that the case of point-capture reduces to a special version of Zermelo’s Navigation Problem (ZNP) for the pursuer. Therefore, the well-known ZNP solution can be used to validate the results obtained through the differential game framework, as well as to characterize the time-optimal trajectories. The results are directly applicable to collision avoidance in maritime and Air Traffic Control applications.

Keywords

Pursuit–evasion Game of two cars Zermelo’s navigation problem 

References

  1. 1.
    Bakolas E, Tsiotras P (2010) Time-optimal synthesis for the Zermelo–Markov–Dubins problem: the constant wind case. In Poceedings of the American control conference, Baltimore, MD, pp 6163–6168Google Scholar
  2. 2.
    Bakolas E, Tsiotras P (2012) Feedback navigation in an uncertain flowfield and connections with pursuit strategies. J Guid Control Dyn 35(4):1268–1279CrossRefGoogle Scholar
  3. 3.
    Basar T, Olsder GJ (1995) Dynamic noncooperative game theory. Academic Press, WalthamGoogle Scholar
  4. 4.
    Bopardikar S, Bullo F, Hespanha J (2007) A cooperative homicidal chauffeur game. In Proceedings of the 46th IEEE conference on decision and control pp 4857–4862Google Scholar
  5. 5.
    Bryson A, Ho Y (1975) Applied optimal control: optimization, estimation and control. Taylor and Francis, New YorkGoogle Scholar
  6. 6.
    Cockayne E (1967) Plane pursuit with curvature constrains. SIAM J Appl Math 15:1511–1516MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Exarchos I, Tsiotras P (2014) An asymmetric version of the two car pursuit-evasion game. In Proceedings of the 53rd IEEE conference on decision and control, Los Angeles, CAGoogle Scholar
  8. 8.
    Ho Y, Bryson A, Baron S (1965) Differential games and optimal pursuit-evasion strategies. IEEE Trans Autom Control 10:385–389MathSciNetCrossRefGoogle Scholar
  9. 9.
    Isaacs R (1965) Differential games. Willey, New YorkMATHGoogle Scholar
  10. 10.
    Lewin J (1994) Differential games—theory and methods for solving game problems with singular surfaces. Springer, LondonGoogle Scholar
  11. 11.
    Li B, Xu C, Kok LT, Chu J (2013) Time optimal Zermelo’s navigation problem with moving and fixed obstacles. J Appl Math Comput 224:866–875CrossRefGoogle Scholar
  12. 12.
    Meier LI (1969) A new technique for solving pursuit-evasion differential games. IEEE Trans Autom Control 14(4):352–359MathSciNetCrossRefGoogle Scholar
  13. 13.
    Merz A (1971) Homicidal chauffeur-a differential game. Ph.D. Thesis, Stanford UniversityGoogle Scholar
  14. 14.
    Merz A (1972) The game of two identical cars. J Opt Theory Appl 9:324–343MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Merz A (1973) Optimal evasive maneuvers in maritime collision avoidance. J Inst Navig 20(2):144–152CrossRefGoogle Scholar
  16. 16.
    Miloh T, Sharma SD (1975) Bericht nr. 319: determination of critical maneuvers for collision avoidance. Tech Rep, Institut fur Schiffbau, Technische Universitat HamburgGoogle Scholar
  17. 17.
    Miloh T, Sharma SD (1976) Bericht nr. 329: maritime collision avoidance as a differential game. Tech Rep, Institut fur Schiffbau, Technische Universitat HamburgGoogle Scholar
  18. 18.
    Mitchell I (2001) Games of two identical vehicles. Technical Report, Department of Aeronautics and Astronautics (SUDAAR), Stanford University, Stanford, CAGoogle Scholar
  19. 19.
    Mitchell I, Bayen A, Tomlin C (2005) A time-dependent Hamilton–Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans Autom Control 50:947–957MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nahin PJ (2007) Chases and escapes: the mathematics of pursuit and evasion. Princeton University Press, PrincetonGoogle Scholar
  21. 21.
    Olsder GJ, Walter JL (1977) Collision avoidance of ships. Tech Rep, Dept of Applied Mathematics, Twente University of TechnologyGoogle Scholar
  22. 22.
    Olsder GJ, Walter JL (1977) A differential game approach to collision avoidance of ships. Optimization techniques. Lecture notes in control and information sciences, vol 6. Springer, Berlin pp 264–271Google Scholar
  23. 23.
    Pachter M (2002) Simple-motion pursuit-evasion differential games. In Proceedings of the 10th Mediterranean conference on control and automationGoogle Scholar
  24. 24.
    Pachter M, Miloh T (1987) The geometric approach to the construction of the barrier surface in differential games. Comput Math Appl 13(1–3):47–67MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Pachter M, Yavin Y (1981) A stochastic homicidal chauffeur pursuit-evasion differential game. J Opt Theory Appl 34(3):405–424MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Patsko V, Turova V (2001) Level sets of the value function in differential games with the homicidal chauffeur dynamics. Int Game Theory Rev 3(1):67–112MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Pshenichnyi B (1976) Simple pursuit by several objects. Kibernetika 3:145–146Google Scholar
  28. 28.
    Rublein G (1972) On pursuit with curvature constrains. SIAM J Control 10:37–39MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Salmon D (1969) Policies and controller design for a pursuing vehicle. IEEE Trans Autom Control 14(5):482–488CrossRefGoogle Scholar
  30. 30.
    Sgall J (2001) Solution of David Gale’s lion and man problem. Theoret Comput Sci 259(1–2):663–670MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Vincent TL, Peng WY (1973) Ship collision avoidance. In Proceedings of the workshop on differential games, Naval Academy, AnnapolisGoogle Scholar
  32. 32.
    Yavin Y (1986) Stochastic pursuit-evasion differential games in the plane. J Opt Theory Appl 50:495–523MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Yavin Y, Villiers RD (1989) Proportional navigation and the game of two cars. J Opt Theory Appl 62:351–369CrossRefMATHGoogle Scholar
  34. 34.
    Zermelo E (1931) Uber das Navigationsproblem bei ruhender oder veranderlicher Windverteilung. Zeitschrift fuer Angewandte Mathematik und Mechanik 11:114–124CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Ioannis Exarchos
    • 1
  • Panagiotis Tsiotras
    • 1
  • Meir Pachter
    • 2
  1. 1.Department of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of Electrical and Computer EngineeringAir Force Institute of TechnologyWright-Patterson A.F.B.USA

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