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Multi-modal Swarm Coordination via Hopf Bifurcations

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Abstract

This paper outlines a methodology for the construction of vector fields that can enable a multi-robot system moving on the plane to generate multiple dynamical behaviors by adjusting a single scalar parameter. This parameter essentially triggers a Hopf bifurcation in an underlying time-varying dynamical system that steers a robotic swarm. This way, the swarm can exhibit a variety of behaviors that arise from the same set of continuous differential equations. Other approaches to bifurcation-based swarm coordination rely on agent interaction which cannot be realized if the swarm members cannot sense or communicate with one another. The contribution of this paper is to offer an alternative method for steering minimally instrumented multi-robot collectives with a control strategy that can realize a multitude of dynamical behaviors without switching their constituent equations. Through this approach, analytical solutions for the bifurcation parameter are provided, even for more complex cases that are described in the literature, along with the process to apply this theory in a multi-agent setup. The theoretical predictions are confirmed via simulation and experimental results with the latter also demonstrating real-world applicability.

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References

  1. Das, S., Hunter, E.E., DeLateur, N.A., Steager, E.B., Weiss, R., Kumar, V.: Cellular expression through morphogen delivery by light activated magnetic microrobots. J. Micro-Bio. Robot. 15(2), 79–90 (2019)

    Article  Google Scholar 

  2. Kokkoni, E., Mavroudi, E., Zehfroosh, A., Galloway, J.C., Vidal, R., Heinz, J., Tanner, H.G.: GEARing smart environments for pediatric motor rehabilitation. J. Neuroeng. Rehabil. 17(1), 16 (2020)

    Article  Google Scholar 

  3. Kokkoni, E., Zehfroosh, A., Kannappan, P., Mavroudi, E., Galloway, J.C., Heinz, J., Vidaly, R., Tanner, H.G.: Principles for building “smart” learning environments in pediatric early rehabilitation. Robot.: Sci. Sys.; Workshop Percept. Interact. Dyn. Child-Robot Interact. (2017)

  4. Zehfroosh, A., Tanner, H.G.: Reactive Motion Planning for Temporal Logic Tasks Without Workspace Discretization. In: Proceedings of the IEEE American Control Conference, pp. 4872–4877 (2019)

  5. Tanner, H., Christodoulakis, D.: Cooperation Between Aerial and Ground Vehicle Groups for Reconnaissance Missions. In: Proceedings of the IEEE Conference on Decision and Control, pp. 5918–5923 (2006)

  6. Valbuena Reyes, L.A., Tanner, H.G.: Flocking, formation control, and path following for a group of mobile robots. IEEE Trans. Control. Syst. Technol. 23(4), 1268–1282 (2015)

    Article  Google Scholar 

  7. Yadav, I., Tanner, H.G.: Mobile Radiation Source Interception by Aerial Robot Swarms. In: Proceedings of the 2nd IEEE International Symposium on Multi-Robot and Multi-Agent Systems, pp. 63–69 (2019)

  8. Marshall, J.A., Broucke, M.E., Francis, B.A.: Formations of vehicles in cyclic pursuit. IEEE Trans. Autom. Control 49(11), 1963–1974 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Halder, U., Schlotfeldt, B., Krishnaprasad, P.: Steering for beacon pursuit under limited sensing. In: Proceedings of the IEEE 55th Conference on Decision and Control, pp. 3848–3855 (2016)

  10. Galloway, K.S., Dey, B.: Collective motion under beacon-referenced cyclic pursuit. Automatica 91, 17–26 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gazi, V.: Swarm aggregations using artificial potentials and sliding-mode control. IEEE Trans. Robot. 21(6), 1208–1214 (2005)

    Article  Google Scholar 

  12. Topaz, C.M., Bertozzi, A.L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65(1), 152–174 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Virágh, C., Vásárhelyi, G., Tarcai, N., Szörényi, T., Somorjai, G., Nepusz, T., Vicsek, T.: Flocking algorithm for autonomous flying robots. Bioins. Biomim. 9(2), 025012 (2014)

    Article  Google Scholar 

  14. Edwards, V., deZonia, P., Hsieh, M.A., Hindes, J., Triandaf, I., Schwartz, I.B.: Delay induced swarm pattern bifurcations in mixed reality experiments. Chaos Interdiscip. J. Nonlinear Sci. 30(7), 073126 (2020)

    Article  MathSciNet  Google Scholar 

  15. Szwaykowska, K., Schwartz, I.B., Romero, L.M.-y-T., Heckman, C.R., Mox, D., Hsieh, M.A.: Collective motion patterns of swarms with delay coupling: Theory and experiment. Phys. Rev. E 93(3), 032307 (2016)

  16. Schwartz, I.B., Edwards, V., Kamimoto, S., Kasraie, K., Ani Hsieh, M., Triandaf, I., Hindes, J.: Torus bifurcations of large-scale swarms having range dependent communication delay. Chaos Interdiscip. J. Nonlinear Sci. 30(5), 051106 (2020)

    Article  MathSciNet  Google Scholar 

  17. Tanner, H.G., Loizou, S., Kyriakopoulos, K.J.: Nonholonomic navigation and control of cooperating mobile manipulators. IEEE Trans. Robot. Autom. 19(1), 53–64 (2003)

    Article  Google Scholar 

  18. Tanner, H.G., Jadbabaie, A., Pappas, G.J.: Stable Flocking of Mobile Agents, Part I: Fixed Topology. In: Proceedings of the IEEE Conference on Decision and Control, pp. 2010–2015 (2003)

  19. Tanner, H.G., Jadbabaie, A., Pappas, G.J.: Stable Flocking of Mobile Agents, Part II: Dynamic Topology. In: Proceedings of the IEEE Conference on Decision and Control, pp. 2016–2021 (2003)

  20. Saber, R.O.: Flocking for multi-agent dynamic systems: algorithms and theory. IEEE Trans Autom Control 51(3), 401–420 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Stager, A., Tanner, H.: Composition of Local Potential Functions with Reflection. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 5558–5564 (2019)

  22. Panagou, D., Tanner, H.G., Kyriakopoulos, K.J.: Nonholonomic control design via reference vector fields and output regulation. ASME J. Dyn. Sys. Meas. Control 137(8), 2831–2836 (2015)

    Article  Google Scholar 

  23. Kingston, P., Egerstedt, M.: Index-free multi-agent systems: An eulerian approach. IFAC Proc. 43(19), 215–220 (2010)

    Google Scholar 

  24. Panagou, D., Tanner, H.G., Kyriakopoulos, K.J.: Control of nonholonomic systems using reference vector fields. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp. 2831–2836 (2011)

  25. Stager, A., Tanner, H.G.: Stochastic Behavior of Robots that Navigate by Interacting with Their Environment. In: Proceedings of the IEEE Conference on Decision and Control, pp. 6871–6876 (2016)

  26. Karydis, K., Zarouk, D., Poulakakis, I., Fearing, R.S., Tanner, H.G.: Planning with the Star(s). In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3033–3038 (2014)

  27. Goebel, R., Sanfelice, R.G., Teel, A.R.: Hybrid Dynamical Systems. In: Princeton University Press (2012)

  28. Reverdy, P.B.: A route to limit cycles via unfolding the pitchfork with feedback. In: Proceedings of the American Control Conference, pp. 3057–3062 (2019)

  29. Reverdy, P.B., Koditschek, D.E.: A dynamical system for prioritizing and coordinating motivations. SIAM J. Appl. Dyn. Sys. 17, 1683–1715 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Thompson, C., Reverdy, P.B.: Drive-based motivation for coordination of limit cycle behaviors. In: Proceedings of the IEEE Conference on Decision and Control (CDC), pp. 244–249 (2019)

  31. Baxevani, K., Tanner, H.G.: Multi-behavioral multi-robot systems driven by motivation dynamics. In: Proceedings of the American Control Conference, (in Print) (2023)

  32. Baxevani, K., Tanner, H.G.: Bifurcating vector fields driven by time-scale separated motivational dynamics. In: Proceedings of the IFAC World Congress, (in Print) (2023)

  33. Guckenheimer, J., Holmes, P.: Nonlinear Oscilations, Dynamical Systems, and Bifurcation of Vector Fields. In: Springer-Verlag (1983)

  34. Verduzco, F.: The First Lyapunov Coefficient for a Class of Systems. In: Triennial IFAC World Congress, pp. 1205–1209 (2005)

  35. Valbuena, L.R., Tanner, H.G.: Flocking, formation control and path following for a group of mobile robots. IEEE Trans. Control. Syst. Technol. 23(4), 1268–1282 (2015)

    Article  Google Scholar 

  36. d’Andréa-Novel, B., Bastin, G., Campion, G.: Modelling and Control of Non Holonomic Wheeled Mobile Robots. In: Proceedings of the 1991 IEEE International Conference on Robotics and Automation, pp. 1130–1135 (1991)

  37. Baxevani, K., Tanner, H.G.: Constructing continuous multi-behavioral planar systems through motivation dynamics and bifurcations. In: Proceedings of the IEEE Conference on Decision and Control, pp. 1095–1100 (2021)

  38. Baxevani, K., Otto, G.E., Tanner, H.G., , Trembanis, A.C.: Development and Field Testing of an Optimal Path Following ASV Controller for Marine Surveys. In: Proceedings of the 2022 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 6861–6866 (2022)

  39. OpenCV Developers Team: Open Source Computer Vision (OpenCV) Library. Available: http://opencv.org. Accessed 2000

  40. Loizou, S.G.: The navigation transformation. IEEE Trans. Robot. 33(6), 1516–1523 (2017)

    Article  Google Scholar 

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Acknowledgements

This work is supported by the National Science foundation’s Smart and Connected Health program via award # 2014264.

Funding

This work is supported by the National Science foundation’s Smart and Connected Health program via award # 2014264.

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Authors and Affiliations

  1. Center for Autonomous and Robotics Systems, University of Delaware, Newark, DE, USA

    Kleio Baxevani & Herbert G. Tanner

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  1. Kleio Baxevani
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  2. Herbert G. Tanner
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Kleio Baxevani and Herbert Tanner. The first draft of the manuscript was written by Kleio Baxevani and all authors commented and edited the manuscript. All authors read and approved the final manuscript.

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Correspondence to Kleio Baxevani.

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Baxevani, K., Tanner, H.G. Multi-modal Swarm Coordination via Hopf Bifurcations. J Intell Robot Syst 109, 34 (2023). https://doi.org/10.1007/s10846-023-01966-4

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