Distance Between a Point and a Convex Cone in $n$ -Dimensional Space: Computation and Applications

Abstract

This paper presents an algorithm to compute the minimum distance from a point to a convex cone in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional space. The convex cone is represented as the set of all nonnegative combinations of a given set. The algorithm generates a sequence of simplicial cones in the convex cone, such that their distances to the single point converge to the desired distance. In many cases, the generated sequence is finite, and therefore, the algorithm has finite-convergence property. Recursive formulas are derived to speed up the computation of distances between the single point and the simplicial cones. The superior efficiency and effectiveness of this algorithm are demonstrated by applications to force-closure test, system equilibrium test, and contact force distribution, which are fundamental problems in the research of multicontact robotic systems. Theoretical and numerical comparisons with previous work are provided.