Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Here and in the rest of the paper, \(\vert \mathbf{w}\vert = {(\mathbf{w} \cdot \mathbf{w})}^{1/2}\) for any \(\mathbf{w} \in {\mathbb{R}}^{3}\).
References
Bishop, R.L.: There is more than one way to frame a curve. Am. Math. Monthly 82(3), 246–251 (1975)
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)
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)
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)
Freyman, L., Livingston, S.: Obstacle avoidance and boundary following behavior of the echolocating Bat. Final Report, MERIT BIEN Program, Univ. of Maryland (see http://scottman.net/pub/BIEN2008-Freyman-Livingston.pdf) (2008)
Justh, E.W., Krishnaprasad, P.S.: Equilibria and steering laws for planar formations. Syst. Control Lett. 51, 25–38 (2004)
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)
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)
Khalil, H.: Nonlinear Systems. Macmillan Publishing Co., New York (1992)
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)
Micaelli, A., Samson, C.: Trajectory tracking for unicycle-type and two-steering-wheels mobile robots. INRIA-Sophia Antipolis Research Report No. 2097 (1993)
Millman, R.S., Parker, G.D.: Elements of Differential Geometry. Prentice Hall, Englewood Cliffs (1977)
Reddy, P.V.: Steering Laws for Pursuit. M.S. Thesis. University of Maryland (2007)
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)
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)
Singh, L., Stephanou, H., Wen, J.: Real-time robot motion control with circulatory fields. Proc. IEEE Int. Conf. Robot. Autom. 3, 2737–2742 (1996)
Wei, E., Justh, E.W., Krishnaprasad, P.S.: Pursuit and an evolutionary game. Proc. R. Soc. A 465, 1539–1559 (2009)
Wu, W., Zhang, F.: Cooperative exploration of level surfaces of three dimensional scalar fields. Automatica 47(9), 2044–2051 (2011)
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)
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)
Acknowledgements
This research was supported in part by the Naval Research Laboratory under Grant Nos. N00173-02-1G002, N00173-03-1G001, N00173-03-1G019, and N00173-04-1G014; by the Air Force Office of Scientific Research under AFOSR Grant Nos. F49620-01-0415, FA95500410130, and FA95501010250; by the Army Research Office under ODDR&E MURI01 Program Grant No. DAAD19-01-1-0465 to the Center for Communicating Networked Control Systems (through Boston University); and by ONR MURI Grant No. N000140710734. P.S. Krishnaprasad also gratefully acknowledges the support of the Bernoulli Center at EPFL, Lausanne during the early stages of this body of work. E.W. Justh was supported in part by the Office of Naval Research.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this paper
Cite this paper
Zhang, F., Justh, E.W., Krishnaprasad, P.S. (2013). Boundary Tracking and Obstacle Avoidance Using Gyroscopic Control. In: Johann, A., Kruse, HP., Rupp, F., Schmitz, S. (eds) Recent Trends in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 35. Springer, Basel. https://doi.org/10.1007/978-3-0348-0451-6_16
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0451-6_16
Published:
Publisher Name: Springer, Basel
Print ISBN: 978-3-0348-0450-9
Online ISBN: 978-3-0348-0451-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)