Shape Complexes for Metamorhpic Robots

A metamorphic robotic system is an aggregate of identical robot units which can individually detach and reattach in such a way as to change the global shape of the system. We introduce a mathematical framework for defining and analyzing general metamorphic systems. This formal structure combined with ideas from geometric group theory leads to a natural extension of a configuration space for metamorphic systems — the shape complex — which is especially adapted to parallelization. We present an algorithm for optimizing reconfiguration sequences with respect to elapsed time. A universal geometric property of shape complexes — non-positive curvature — is the key to proving convergence to the globally time-optimal solution.