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The Multi-pursuer Single-Evader Game

A Geometric Approach
  • Alexander Von MollEmail author
  • David Casbeer
  • Eloy Garcia
  • Dejan Milutinović
  • Meir Pachter
Article
  • 14 Downloads

Abstract

We consider a general pursuit-evasion differential game with three or more pursuers and a single evader, all with simple motion (fixed-speed, infinite turn rate). It is shown that traditional means of differential game analysis is difficult for this scenario. But simple motion and min-max time to capture plus the two-person extension to Pontryagin’s maximum principle imply straight-line motion at maximum speed which forms the basis of the solution using a geometric approach. Safe evader paths and policies are defined which guarantee the evader can reach its destination without getting captured by any of the pursuers, provided its destination satisfies some constraints. A linear program is used to characterize the solution and subsequently the saddle-point is computed numerically. We replace the numerical procedure with a more analytical geometric approach based on Voronoi diagrams after observing a pattern in the numerical results. The solutions derived are open-loop optimal, meaning the strategies are a saddle-point equilibrium in the open-loop sense.

Keywords

Pursuit-evasion Differential game Voronoi diagram Optimization 

Mathematics Subject Classification (2010)

49N70 49N90 49N75 

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References

  1. 1.
    Aurenhammer, F., Edelsbrunner, H.: An optimal algorithm for constructing the weighted voronoi diagram in the plane. Pattern Recogn. 17(2), 251–257 (1984).  https://doi.org/10.1016/0031-3203(84)90064-5 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bakolas, E., Tsiotras, P.: Relay pursuit of a maneuvering target using dynamic Voronoi diagrams. Automatica 48(9), 2213–2220 (2012).  https://doi.org/10.1016/j.automatica.2012.06.003 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Başar, T., Olsder, G.J.: Chapter 8: Pursuit-Evasion Games. In: Mathematics in Science and Engineering, Dynamic Noncooperative Game Theory, vol. 160, pp. 344–398. Elsevier (1982).  https://doi.org/10.1016/S0076-5392(08)62960-4
  4. 4.
    Breakwell, J.V., Hagedorn, P.: Point capture of two evaders in succession. J. Optim. Theory Appl. 27(1), 89–97 (1979).  https://doi.org/10.1007/BF00933327 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brown, K.Q.: Geometric transforms for fast geometric algorithms. Tech. Rep. CMU-CS-80-101. Carnegie-Mellon Univ Pittsburgh PA Dept of Computer Science (1979)Google Scholar
  6. 6.
    Cheung, W.A.: Constrained pursuit-evasion problems in the plane. Ph.D. thesis, University of British Columbia (2005)Google Scholar
  7. 7.
    Conway, J.B.: A Course in Functional Analysis. Graduate Texts in Mathematics. Springer, New York (1985)CrossRefGoogle Scholar
  8. 8.
    Dobrin, A.: A review of properties and variations of Voronoi diagrams. Whitman College (2005)Google Scholar
  9. 9.
    Earl, M.G., D’Andrea, R.: A decomposition approach to multi-vehicle cooperative control. Robot. Auton. Syst. 55(4), 276–291 (2007).  https://doi.org/10.1016/j.robot.2006.11.002 CrossRefGoogle Scholar
  10. 10.
    Festa, A., Vinter, R.B.: A decomposition technique for pursuit evasion games with many pursuers. In: 52nd IEEE Conference on Decision and Control, pp. 5797–5802 (2013),  https://doi.org/10.1109/CDC.2013.6760803
  11. 11.
    Festa, A., Vinter, R.B.: Decomposition of differential games with multiple targets. J. Optim. Theory Appl. 169(3), 848–875 (2016).  https://doi.org/10.1007/s10957-016-0908-z MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fisac, J.F., Sastry, S.S.: The pursuit-evasion-defense differential game in dynamic constrained environments. In: 2015 54th IEEE Conference on Decision and Control (CDC), pp. 4549–4556 (2015),  https://doi.org/10.1109/CDC.2015.7402930
  13. 13.
    Fortune, S.: A sweepline algorithm for Voronoi diagrams. Algorithmica 2(1-4), 153 (1987).  https://doi.org/10.1007/BF01840357 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fuchs, Z.E., Khargonekar, P.P., Evers, J.: Cooperative defense within a single-pursuer, two-evader pursuit evasion differential game. In: 49th IEEE Conference on Decision and Control (CDC), pp. 3091–3097 (2010).  https://doi.org/10.1109/CDC.2010.5717894
  15. 15.
    Ganebny, S.A., Kumkov, S.S., Le Ménec, S., Patsko, V.S.: Model problem in a line with two pursuers and one evader. Dyn. Games Appl. 2(2), 228–257 (2012).  https://doi.org/10.1007/s13235-012-0041-z MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Garcia, E., Fuchs, Z.E., Milutinović, D., Casbeer, D.W., Pachter, M.: A Geometric Approach for the Cooperative Two-Pursuer One-Evader Differential Game. IFAC-PapersOnLine 50(1), 15209–15214 (2017).  https://doi.org/10.1016/j.ifacol.2017.08.2366 CrossRefGoogle Scholar
  17. 17.
    Huang, H., Zhang, W., Ding, J., Stipanović, D.M., Tomlin, C.J.: Guaranteed Decentralized Pursuit-Evasion in the Plane with Multiple Pursuers. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), pp. 4835–4840. IEEE (2011)Google Scholar
  18. 18.
    Isaacs, R.: Games of Pursuit. Product Page P-257. RAND Corporation, Santa Monica (1951)Google Scholar
  19. 19.
    Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Optimization, Control and Warfare. Wiley, New York (1965)zbMATHGoogle Scholar
  20. 20.
    Liu, S.Y., Zhou, Z., Tomlin, C., Hedrick, K.: Evasion as a team against a faster pursuer. In: 2013 American Control Conference, pp. 5368–5373 (2013).  https://doi.org/10.1109/ACC.2013.6580676
  21. 21.
    Oyler, D.: Contributions To Pursuit-Evasion Game Theory. Ph.D. thesis, University of Michigan (2016)Google Scholar
  22. 22.
    Oyler, D.W., Kabamba, P.T., Girard, A.R.: Pursuit-evasion games in the presence of obstacles. Automatica 65(Supplement C), 1–11 (2016).  https://doi.org/10.1016/j.automatica.2015.11.018 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pierson, A., Wang, Z., Schwager, M.: Intercepting Rogue Robots: An Algorithm for Capturing Multiple Evaders With Multiple Pursuers. IEEE Robot. Autom. Lett. 2(2), 530–537 (2017).  https://doi.org/10.1109/LRA.2016.2645516 CrossRefGoogle Scholar
  24. 24.
    Sun, W., Tsiotras, P., Lolla, T., Subramani, D.N., Lermusiaux, P.F.: Multiple-pursuer/one-evader pursuit–evasion game in dynamic flowfields. J. Guid. Control. Dyn. (2017)Google Scholar
  25. 25.
    Von Moll, A., Casbeer, D.W., Garcia, E., Milutinović, D.: Pursuit-evasion of an evader by multiple pursuers. In: 2018 International Conference on Unmanned Aircraft Systems (ICUAS). Dallas (2018),  https://doi.org/10.1109/ICUAS.2018.8453470

Copyright information

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Controls Science Center of ExcellenceAir Force Research LaboratoryWright-Patterson AFBUSA
  2. 2.Department of Computer EngineeringUC Santa CruzSanta CruzUSA
  3. 3.Department of Electrical EngineeringAir Force Institute of TechnologyWright-Patterson AFBUSA

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