Mutational equations for shapes and vision-based control

Basic idea of vision-based control in robotics is to include the vision system directly in the control servo loop of the robot. When images are binary, this problem corresponds to the control of the evolution of a geometric domain. The present paper proposes mathematical tools derived from shape analysis and optimization to study this problem in a quite general way, i.e., without any regularity assumptions or modelsa priori on the domains that we deal with. Indeed, despite the lackness of a vectorial structure, one can develop a differential calculus in the metric space of all non-empty compact subsets of a given domain ofR ⁿ , and adapt ideas and results of classical differential systems to study and control the evolution of geometric domains. For instance, a shape Lyapunov characterization allows to investigate the asymptotic behavior of these geometric domains using the notion of directional shape derivative. We apply this inR ² to the visual servoing problem using the optical flow equations and some experimental simulations illustrate this approach.