Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular poin t. This curve evolution is called Euclidean curve shortening and the result is known as the Gage-Hamilton-Gra yson Theorem. Motivated by the rendezvous problem for mobile autonomous robots, we address the proble m of creating a polygon shortening flow. A linear scheme is proposed that exhibits several analogues to Euclidean curve shortening: The polygon shrinks to an elliptical point, convex polygons remain conv ex, and the perimeter of the polygon is monotonically decreasing. This paper studies the rendezvous problem for mobile autonomous robots, in which the goal is to develop a local control strategy that will drive each robots's state (usually its position) to a common value. Research on this problem has been performed in discre te and continuous time.