Containment indicator function construction via numerical conformal mapping

Abstract

In optimal-control-based motion planning, it is often desired to obtain a proper (preferably smooth) function, referred to as the containment indicator function in this paper, that describes the shape of the free space. The paper studies the use of numerical conformal mapping for constructing the containment indicator function for an arbitrary two-dimensional geometry. The idea of using conformal mapping is to transform the original shape of interest into a simpler target shape (e.g. disk, rectangle), which can then be characterized by elementary functions. Computational methods for finding the desired conformal maps are also studied. The procedure can be formulated as a convex optimization problem and computed efficiently.