  We present a new method for motion planning for control systems. The method
aims to provide a natural computational framework in which a broad class of
motion planning problems can be cast; including problems with holonomic and
non-holonomic constraints, drift dynamics, obstacle constraints and constraints
on the magnitudes of the applied controls. The method, which finds its
inspiration in recent work on the so-called geometric heat flows and curve
shortening flows, relies on a hereby introduced partial differential equation,
which we call the affine geometric heat flow, which evolves an arbitrary
differentiable path joining initial to final state in configuration space to a
path that meets the constraints imposed on the problem. From this path,
controls to be applied on the system can be extracted. We provide conditions
guaranteeing that the controls extracted will drive the system arbitrarily
close to the desired final state, while meeting the imposed constraints and
illustrate the method on three canonical examples.
