  A metamorphic robotic system is an aggregate of homogeneous robot units which
can individually and selectively locomote in such a way as to change the global
shape of the system. We introduce a mathematical framework for defining and
analyzing general metamorphic robots. This formal structure, combined with
ideas from geometric group theory, leads to a natural extension of a
configuration space for metamorphic robots -- the state 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 state complexes -- non-positive curvature -- is the key to proving
convergence to the globally time-optimal solution.
