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.
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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
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
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
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
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)
Cheung, W.A.: Constrained pursuit-evasion problems in the plane. Ph.D. thesis, University of British Columbia (2005)
Conway, J.B.: A Course in Functional Analysis. Graduate Texts in Mathematics. Springer, New York (1985)
Dobrin, A.: A review of properties and variations of Voronoi diagrams. Whitman College (2005)
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
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
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
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
Fortune, S.: A sweepline algorithm for Voronoi diagrams. Algorithmica 2(1-4), 153 (1987). https://doi.org/10.1007/BF01840357
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
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
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
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)
Isaacs, R.: Games of Pursuit. Product Page P-257. RAND Corporation, Santa Monica (1951)
Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Optimization, Control and Warfare. Wiley, New York (1965)
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
Oyler, D.: Contributions To Pursuit-Evasion Game Theory. Ph.D. thesis, University of Michigan (2016)
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
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
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)
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
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This paper is based on work performed at the Air Force Research Laboratory (AFRL) Control Science Center of Excellence. Distribution Unlimited. 21 August 2018. Case #88ABW-2018-4153.
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Von Moll, A., Casbeer, D., Garcia, E. et al. The Multi-pursuer Single-Evader Game. J Intell Robot Syst 96, 193–207 (2019). https://doi.org/10.1007/s10846-018-0963-9
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DOI: https://doi.org/10.1007/s10846-018-0963-9