Recent Trends in Dynamical Systems pp 417-446
Boundary Tracking and Obstacle Avoidance Using Gyroscopic Control
- Cite this paper as:
- 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
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.