Nonsmooth Analysis, Convex Analysis, and their Applications to Motion Planning

Abstract

Nonsmooth analysis of a broad class of functions taking the form $F(x) = min_i f_i(x)$, where each $f_i$ is a convex function. One element of this class of functions is the distance function, which measures the distance between a point and the nearest point on the nearest obstacle. Many motion planning algorithms are based on the distance function, and thus rigorous analysis of the distance function can provide a better understanding of how to implement traditional motion planning algorithms. Finally, this paper enumerates some useful results in convex analysis.