Skip to main content

Multiplayer Pursuit-Evasion Games in Three-Dimensional Flow Fields

A Correction to this article was published on 22 March 2019

This article has been updated

Abstract

In this paper, we deal with a pursuit-evasion differential game between multiple pursuers and multiple evaders in the three-dimensional space under dynamic environmental disturbances (e.g., winds, underwater currents). We first recast the problem in terms of partitioning the pursuer set and assign each pursuer to an evader. We present two algorithms to partition the pursuer set from either the pursuer’s perspective or the evader’s perspective. Within each partition, the problem is reduced into a multi-pursuer/single-evader game. This problem is then addressed through a reachability-based approach. We give conditions for the game to terminate in terms of reachable set inclusions. The reachable sets of the pursuers and the evader are obtained by solving their corresponding level set equations through the narrow band level set method. We further demonstrate why fast marching or fast sweeping schemes are not applicable to this problem for a general class of disturbances. The time-optimal trajectories and the corresponding optimal strategies can be retrieved afterward by traversing these level sets. The proposed scheme is implemented on problems with both simple and realistic flow fields.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Change history

  • 22 March 2019

    Unfortunately, the acknowledgement text was not submitted during the acceptance of the manuscript.

References

  1. Bakolas E (2014) Optimal guidance of the isotropic rocket in the presence of wind. J Optim Theory Appl 162(3):954–974

    MathSciNet  Article  Google Scholar 

  2. Bakolas E, Tsiotras P (2010) The Zermelo–Voronoi diagram: a dynamic partition problem. Automatica 46(12):2059–2067

    MathSciNet  Article  Google Scholar 

  3. Bakolas E, Tsiotras P (2012) Relay pursuit of a maneuvering target using dynamic Voronoi diagrams. Automatica 48:2213–2220

    MathSciNet  Article  Google Scholar 

  4. Bannikov AS (2009) A non-stationary problem of group pursuit. J Comput Syst Sci Int 48(4):527–532

    MathSciNet  Article  Google Scholar 

  5. Başar T, Olsder GJ (1999) Dynamic noncooperative game theory, vol 23. SIAM, Philadelphia

    MATH  Google Scholar 

  6. Blagodatskikh AI (2009) Simultaneous multiple capture in a simple pursuit problem. J Appl Math Mech 73(1):36–40

    MathSciNet  Article  Google Scholar 

  7. Bopardikar SD, Bullo F, Hespanha JP (2009) A cooperative homicidal chauffeur game. Automatica 45(7):1771–1777

    MathSciNet  Article  Google Scholar 

  8. Chernous’ko FL (1976) A problem of evasion from many pursuers. J Appl Math Mech 40(1):11–20

    MathSciNet  Article  Google Scholar 

  9. Corpetti T, Memin E, Pérez P (2003) Extraction of singular points from dense motion fields: an analytic approach. J Math Imaging Vis 19(3):175–198

    MathSciNet  Article  Google Scholar 

  10. Dong L, Chai S, Zhang B, Nguang SK, Li X (2016) Cooperative relay tracking strategy for multi-agent systems with assistance of voronoi diagrams. J Frankl Inst 353(17):4422–4441

    MathSciNet  Article  Google Scholar 

  11. Earth Observatory (2012) Comparing the winds of Sandy and Katrina. https://earthobservatory.nasa.gov/IOTD/view.php?id=79626&src=ve

  12. Gómez P, Hernandez J, López J (2005) On the reinitialization procedure in a narrow-band locally refined level set method for interfacial flows. Int J Numer Methods Eng 63(10):1478–1512

    MathSciNet  Article  Google Scholar 

  13. Ibragimov GI (2005) Optimal pursuit with countably many pursuers and one evader. Differ Equ 41(5):627–635

    MathSciNet  Article  Google Scholar 

  14. Ibragimov GI, Salimi M, Amini M (2012) Evasion from many pursuers in simple motion differential game with integral constraints. Eur J Oper Res 218(2):505–511

    MathSciNet  Article  Google Scholar 

  15. Isaacs R (1999) Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization. Courier Dover Publications, Mineola

    MATH  Google Scholar 

  16. Khaidarov BK (1984) Positional I-capture in the game of a single evader and several pursuers. J Appl Math Mech 48(4):406–409

    MathSciNet  Article  Google Scholar 

  17. Liberzon D (2011) Calculus of variations and optimal control theory: a concise introduction. Princeton University Press, Princeton, NJ, chap. 4.5

  18. Lolla T (2016) Path planning and adaptive sampling in the coastal ocean. PhD thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology

  19. Lolla T, Ueckermann MP, Yiğit K, Haley PJ Jr, Lermusiaux PFJ (2012) Path planning in time dependent flow fields using level set methods. In: IEEE international conference on robotics and automation. St. Paul, MN, pp 166–173

  20. Lolla T, Lermusiaux PFJ, Ueckermann MP, Haley PJ Jr (2014) Time-optimal path planning in dynamic flows using level set equations: theory and schemes. Ocean Dyn 64(10):1373–1397

    Article  Google Scholar 

  21. Lolla T, Haley PJ Jr, Lermusiaux PFJ (2015) Path planning in multi-scale ocean flows: coordination and dynamic obstacles. Ocean Model 94:46–66. https://doi.org/10.1016/j.ocemod.2015.07.013

    Article  Google Scholar 

  22. Malladi R, Sethian JA, Vemuri BC (1995) Shape modeling with front propagation: a level set approach. IEEE Trans Pattern Anal Mach Intell 17(2):158–175

    Article  Google Scholar 

  23. MATLAB (2015) version 8.6.0 (R2015b). The MathWorks Inc., Natick, Massachusetts

  24. Osher S, Fedkiw R (2006) Level set methods and dynamic implicit surfaces, vol 153. Springer, New York

    MATH  Google Scholar 

  25. Petrov NN, Shuravina IN (2009) On the “soft” capture in one group pursuit problem. J Comput Syst Sci Int 48(4):521–526

    MathSciNet  Article  Google Scholar 

  26. Pittsyk M, Chikrii AA (1982) On a group pursuit problem. J Appl Math Mech 46(5):584–589

    MathSciNet  Article  Google Scholar 

  27. Pshenichnyi BN (1976) Simple pursuit by several objects. Cybern Syst Anal 12(3):484–485

    MathSciNet  Google Scholar 

  28. Sethian JA (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, vol 3. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  29. Sontag ED (2013) Mathematical control theory: deterministic finite dimensional systems, vol 6. Springer, New York

    Google Scholar 

  30. Sun W, Tsiotras P (2017) Sequential pursuit of multiple targets under external disturbances via Zermelo–Voronoi diagrams. Automatica 81:253–260. https://doi.org/10.1016/j.automatica.2017.03.015

    MathSciNet  Article  MATH  Google Scholar 

  31. Sun W, Tsiotras P, Lolla T, Subramani DN, Lermusiaux PFJ (2017) Multiple-pursuer/one-evader pursuit-evasion game in dynamic flowfields. J Guid Control Dyn 40(7):1627–1637

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wei Sun.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work has been supported by NSF award CMMI-1662542.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sun, W., Tsiotras, P. & Yezzi, A.J. Multiplayer Pursuit-Evasion Games in Three-Dimensional Flow Fields. Dyn Games Appl 9, 1188–1207 (2019). https://doi.org/10.1007/s13235-019-00304-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-019-00304-4

Keywords

  • Multiplayer pursuit-evasion
  • Flow field
  • Differential game
  • Reachable set
  • Level set method