An efficient algorithm for the generalized distance measure

Abstract

This paper presents an efficient algorithm for computing a distance measure between two compact convex sets Q and A, defined as the minimum scale factor such that the scaled Q is not disjoint from A. An important application of this algorithm in robotics is the computation of the minimum distance between two objects, which can be performed by taking A as the Minkowski difference of the objects and Q as a set containing the origin in its interior. In this generalized definition, the traditional Euclidean distance is a special case where Q is the unit ball. While this distance measure was proposed almost a decade ago, there has been no efficient algorithm to compute it in general cases. Our algorithm fills this void and we demonstrate its superior efficiency compared to approaches based on general-purpose optimization.

Generalized Distance Between Compact Convex Sets: Algorithms and Applications

“…While the generalized distance was proposed almost a decade ago, the only known algorithm was direct application of numerical optimization. The high accuracy and efficiency of our algorithms have been verified with a number of numerical examples in spaces of different dimensions in this paper and our preliminary work [19], [20]. Here, we have also presented various practical applications of the distance algorithms, including two central problems in robotics, i.e., collision detection and optimal grasp planning, and a fundamental problem in manufacturing, i.e., flatness error evaluation.…”

“…Preliminary versions of the algorithms have been proposed in [19] and [20], and their superior efficiency over methods based on numerical optimization has been verified by several numerical examples. In this paper, besides more numerical examples, we further verify the performance in practical applications by applying the algorithms to two problems in robotics, namely collision detection and grasp planning, as well as the flatness error evaluation problem in manufacturing.…”

“…Therefore, σ k must converge to 1 and η k to d + Q (A, B). The condition σ k − 1 < can be used to terminate the iteration [19], where > 0 is the termination tolerance. Here, we propose a new stopping criterion that better controls the accuracy of results.…”

“…As a comparison, we write d + Q (S, 0) and d − Q (S, 0) as the following optimization problems as in [17] and [19] and compute them using the active-set algorithm provided by the Optimization Toolbox of MATLAB: where bd(Q) denotes the boundary of Q. To obtain results on the same level of accuracy as our algorithms, the objective function value tolerance for stopping the active-set algorithm is set to the same value as .…”

“…Noticing this, Zhu et al [3], [17], [18] replaced the unit ball with a general compact convex set, called the gauge set, and presented a unified distance definition including Euclidean, one-norm, and infinity-norm distances. However, computation of the generalized distance has been limited to polytope pairs based on general-purpose optimization until recently Zheng and Yamane [19], [20] proposed algorithms that can handle general convex shapes.…”