A Note on the Consensus Protocol with Some Applications to Agent Orbit Pattern Generation

Part of the Springer Tracts in Advanced Robotics book series (STAR, volume 83)

Abstract

We propose an extension to the standard feedback control for consensus problems for multi-agent systems in the plane. The proposed extension allows for a richer class of trajectories including periodic and quasi-periodic solutions, as well as agreement to consensus states outside the convex hull of the initial positions of the agents. We investigate in great detail the special case of three agents, which results in non-trivial geometric patterns described by ellipsoidal, epitrochoidal and hypotrochoidal curves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arcak, M.: Passivity as a design tool for group coordination. IEEE Transactions on Automatic Control 52(8), 1380–1390 (2007), doi:10.1109/TAC.2007.902733MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bernstein, D.S.: Matrix Mathematics: Theory, Facts, and Formulas, 2nd edn. Princeton University Press, Princeton (2009)MATHGoogle Scholar
  3. 3.
    Fax, J.A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Transactions on Automatic Control 49(9), 1465–1476 (2004), doi:10.1109/TAC.2004.834433MathSciNetCrossRefGoogle Scholar
  4. 4.
    Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (2001)MATHCrossRefGoogle Scholar
  5. 5.
    Hall, L.: Trochoids, Roses, and Thorns–Beyond the Spirograph. College Mathematics Journal 23(1), 20–35 (1992)CrossRefGoogle Scholar
  6. 6.
    Justh, E., Krishnaprasad, P.: Equilibria and Steering Laws for Planar Formations. Systems & Control Letters 52(1), 25–38 (2004), doi:10.1016/j.sysconle.2003.10.004MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Justh, E.W., Krishnaprasad, P.S.: Natural frames and interacting particles in three dimensions. In: Proc. 44th IEEE Conf. and 2005 European Control Conf. Decision and Control CDC-ECC 2005, pp. 2841–2846 (2005), doi:10.1109/CDC.2005.1582594Google Scholar
  8. 8.
    Lawrence, J.: A Catalog of Special Plance Curves, 1st edn. Dover Publications (1972)Google Scholar
  9. 9.
    Leonard, N.E., Fiorelli, E.: Virtual leaders, artificial potentials, and coordinated control of groups. In: Proc. of the 40th IEEE Conference on Decision and Control, pp. 2968–2973 (2001), doi:10.1109/.2001.980728Google Scholar
  10. 10.
    Marshall, J.A., Broucke, M.E., Francis, B.A.: Formations of vehicles in cyclic pursuit. IEEE Trans. on Automatic Control 49(11), 1963–1974 (2004), doi:10.1109/TAC.2004.837589MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mesbahi, M., Egerstedt, M.: Graph Theoretic Methods in Multiagent Networks. Princeton University Press, Princeton (2010)MATHGoogle Scholar
  12. 12.
    Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in multi-agent networked systems. Proceedings of the IEEE 97, 215–233 (2006), doi:10.1109/JPROC.2006.887293Google Scholar
  13. 13.
    Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. on Automatic Control 49(9), 1520–1533 (2004), doi:10.1109/TAC.2004.834113MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pavone, M., Frazzoli, E.: Decentralized policies for geometric pattern formation and path coverage. Journal of Dynamic Systems, Measurement, and Control 129, 633–643 (2007), doi:10.1115/1.2767658CrossRefGoogle Scholar
  15. 15.
    Ren, W.: Collective motion from consensus with cartesian coordinate coupling - part i: Single-integrator kinematics. In: Proc. 47th IEEE Conf. Decision and Control CDC 2008, pp. 1006–1011 (2008), doi:10.1109/CDC.2008.4738708Google Scholar
  16. 16.
    Ren, W., Beard, R.W.: Consensus seeking in multi-agent systems using dynamically changing interaction topologies. IEEE Trans. on Automatic Control 50(5), 655–661 (2005), doi:10.1109/TAC.2005.846556MathSciNetCrossRefGoogle Scholar
  17. 17.
    Scardovi, L., Leonard, N.E., Sepulchre, R.: Stabilization of collective motion in three dimensions: A consensus approach. In: Proc. 46th IEEE Conf. Decision and Control, pp. 2931–2936 (2007), doi:10.1109/CDC.2007.4434721Google Scholar
  18. 18.
    Sepulchre, R., Paley, D.A., Leonard, N.E.: Stabilization of planar collective motion: All-to-all communication 52(5), 811–824 (2007), doi:10.1109/TAC.2007.898077MathSciNetGoogle Scholar
  19. 19.
    Singh, L., Stephanou, H., Wen, J.: Real-time robot motion control with circulatory fields. In: Proceedings of 1996 IEEE International Conference on Robotics and Automation, pp. 2737–2742 (1996), doi:10.1109/ROBOT.1996.506576Google Scholar
  20. 20.
    Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots: Formation of geometric patterns. SIAM J. on Computing 28(4), 1347–1363 (1999), doi:10.1137/S009753979628292XMathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Wang, L.S., Krishnaprasad, P.S.: Gyroscopic control and stabilization. Journal of Nonlinear Science 2, 367–415 (1992), doi:10.1007/BF01209527MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Panagiotis Tsiotras
    • 1
  • Luis Ignacio Reyes Castro
    • 1
  1. 1.School of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations