Collision Avoidance Strategies for a Three-Player Game

  • Sriram Shankaran
  • Dušan M. Stipanović
  • Claire J. Tomlin
Chapter
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 11)

Abstract

Collision avoidance strategies for a game with three players, two pursuers and one evader, are constructed by determining the semipermeable curves that form the barrier. The vehicles are assumed to have the same capabilities, speed, and turn-rates. The game is assumed to be played on a two-dimensional plane. We consider avoidance strategies for a particular form of the game defined in the following way: the pursuers are assumed to act noncooperatively, the evader upon realizing that one (or both) of the pursuers can cause capture, takes an evasive action. We find states from which the pursuer can cause capture following this evasive action by the evader. The envelope of states that can lead to capture is denoted by the barrier set. Capture is assumed to have occurred when one (or both) pursuers have reached within a circle of radius, l, from the evader. The usable part and its boundary are first determined along with the strategy along the boundary. Semipermeable curves are evolved from the boundary. If two curves intersect (they have a common point), the curves are not extended beyond the intersection point. As in the game of two cars, universal curves and the characteristics that terminate and emanate from the universal curve are used to fill voids on the barrier surface. For the particular game (and associated strategies) considered in this paper, numerical simulations suggest that the enlarged set of initial states that lead to capture is closed. As the game considered here is a subset of the more complete game, when two pursuers try to cause capture of a single evader, the avoidance strategies are most likely to belong to the set of strategies for the complete game.

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References

  1. 1.
    Isaacs, R.: Differential Games. Wiley, New York (1965)MATHGoogle Scholar
  2. 2.
    Merz, A.W.: The Game of Two Identical Cars. Journal of Optimization Theory and Applications 9(5), 324–343 (1972)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Merz, A.W.: The Homicidal Chauffeur. Ph.D. Thesis, Department of Aeronautics and Astronautics, Stanford University (1971)Google Scholar
  4. 4.
    Pachter, M., Getz, W.M.: The Geometry of the Barrier in the “Game of Two Cars”. Optimal Control Applications and Methods 1, 103–118 (1980)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bardi, M., Dolcetta, C.I.: Optimal Control and Viscosity Solutions for Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997)CrossRefGoogle Scholar
  6. 6.
    Falcone, M., Bardi, M.: An approximation scheme for the minimum time function. SIAM Journal of Control and Optimization 28(4), 950–965 (1990)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Patsko, V.S., Turova, V.L.: Families of semipermeable curves in differential games with the homicidal chauffeur dynamics. Automatica 40(12), 2059–2068 (2004)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Mitchell, I., Bayen, A., Tomlin C.: Computing reachable sets for continuous dynamic games using level set methods. IEEE Transactions on Automatic Control 50(7), 947–957 (2005)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Sriram Shankaran
    • 1
  • Dušan M. Stipanović
    • 2
  • Claire J. Tomlin
    • 3
  1. 1.General Electric Research CenterNiskayunaUSA
  2. 2.Department of Industrial and Enterprise Systems Engineering, and Coordinated Science LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA

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