Boundary Tracking and Obstacle Avoidance Using Gyroscopic Control

  • Fumin Zhang
  • Eric W. Justh
  • P. S. Krishnaprasad
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 35)


For some time now, control-theoretic studies of collective motion of particles have shown the effectiveness of gyroscopic interactions in establishing stable spatiotemporal patterns in free space. This paper is concerned with strategies (respectively feedback laws) that prescribe (respectively execute) gyroscopic interaction of a single unit-speed particle with a fixed obstacle in space. The purpose of such interaction is to avoid collision with the obstacle and track associated (boundary) curves. Working in a planar setting, using the language of natural frames, we construct a steering law for the particle, based on sensing of curvature of the boundary, to track a Bertrand mate of the same. The curvature data is sensed at the closest point (image particle) on the boundary curve from the current location of the unit-speed particle. This construction extends to the three-dimensional case in which a unit-speed particle tracks a prescribed curve on a spherical obstacle. The tracking results exploit in an essential way, the method of reduction to shape space, and stability analysis of dynamics in shape space.


  1. 1.
    Bishop, R.L.: There is more than one way to frame a curve. Am. Math. Monthly 82(3), 246–251 (1975)CrossRefMATHGoogle Scholar
  2. 2.
    Calini, A.: Recent developments in integrable curve dynamics. In: Geometric Approaches to Differential Equations. Lecture Notes of the Australian Math. Soc., vol. 15, pp. 56–99. Cambridge University Press, Cambridge (2000)Google Scholar
  3. 3.
    Chang, D.E., Shadden, S., Marsden, J.E., Olfati-Saber, R.: Collision avoidance for multiple agent systems. Proc. IEEE Conf. Decis. Control 1, 539–543 (2003)Google Scholar
  4. 4.
    Fajen, B.R., Warren, W.H.: Behavioral dynamics of steering, obstacle avoidance, and route selection. J. Exp. Psychol. Hum. Percept. Perform. 29(2), 343–362 (2003)CrossRefGoogle Scholar
  5. 5.
    Freyman, L., Livingston, S.: Obstacle avoidance and boundary following behavior of the echolocating Bat. Final Report, MERIT BIEN Program, Univ. of Maryland (see (2008)
  6. 6.
    Justh, E.W., Krishnaprasad, P.S.: Equilibria and steering laws for planar formations. Syst. Control Lett. 51, 25–38 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Justh, E.W., Krishnaprasad, P.S.: Natural frames and interacting particles in three dimensions. In: Proc. 44th IEEE Conf. Decision and Control, pp. 2841–2846 (see also arXiv:math.OC/0503390v1) (2005)Google Scholar
  8. 8.
    Justh, E.W., Krishnaprasad, P.S.: Steering laws for motion camouflage. Proc. R. Soc. A 462, 3629–3643 (see also arXiv:math.OC/0508023) (2006)Google Scholar
  9. 9.
    Khalil, H.: Nonlinear Systems. Macmillan Publishing Co., New York (1992)MATHGoogle Scholar
  10. 10.
    Malisoff, M., Mazenc, F., Zhang, F.: Stability and robustness analysis for curve tracking control using input-to-state stability. IEEE Trans. Autom. Control 57(2), 1320–1326 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Micaelli, A., Samson, C.: Trajectory tracking for unicycle-type and two-steering-wheels mobile robots. INRIA-Sophia Antipolis Research Report No. 2097 (1993)Google Scholar
  12. 12.
    Millman, R.S., Parker, G.D.: Elements of Differential Geometry. Prentice Hall, Englewood Cliffs (1977)MATHGoogle Scholar
  13. 13.
    Reddy, P.V.: Steering Laws for Pursuit. M.S. Thesis. University of Maryland (2007)Google Scholar
  14. 14.
    Reddy, P.V., Justh, E.W., Krishnaprasad, P.S.: Motion camouflage in three dimensions. In: Proc. 45th IEEE Conf. Decision and Control, pp. 3327–3332 (see also arXiv:math.OC/0603176) (2006)Google Scholar
  15. 15.
    Samson, C., Ait-Abderrahim, K.: Mobile robot control. Part 1: Feedback control of a nonholonomic wheeled cart in Cartesian space. INRIA-Sophia Antipolis Research Report No. 1288 (1990)Google Scholar
  16. 16.
    Singh, L., Stephanou, H., Wen, J.: Real-time robot motion control with circulatory fields. Proc. IEEE Int. Conf. Robot. Autom. 3, 2737–2742 (1996)CrossRefGoogle Scholar
  17. 17.
    Wei, E., Justh, E.W., Krishnaprasad, P.S.: Pursuit and an evolutionary game. Proc. R. Soc. A 465, 1539–1559 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Wu, W., Zhang, F.: Cooperative exploration of level surfaces of three dimensional scalar fields. Automatica 47(9), 2044–2051 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Zhang, F., O’Connor, A., Luebke, D., Krishnaprasad, P.S.: Experimental study of curvature-based control laws for obstacle avoidance. Proc. IEEE Int. Conf. Robot. Autom. 4, 3849–3854 (2004)Google Scholar
  20. 20.
    Zhang, F., Justh, E.W., Krishnaprasad, P.S.: Boundary following using gyroscopic control. In: Proc. 43rd IEEE Conf. Decision and Control, pp. 5204–5209 (2004)Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Fumin Zhang
    • 1
  • Eric W. Justh
    • 2
  • P. S. Krishnaprasad
    • 3
  1. 1.School of Electrical and Computer EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Naval Research LaboratoryWashington, DCUSA
  3. 3.Institute for Systems Research and Department of Electrical and Computer EngineeringUniversity of MarylandCollege ParkUSA

Personalised recommendations