Distance computing in the multivalue number space

Abstract

Abs$ruct: Distancc computing in robotics appears in the trcatmcnt of most of problems. In this article we propose a distance computing algorithm between any f " n e n t point and any shape obstacle boundary. The algorithm prcscntcd is fast and not obstacle shape dependent. elements of Ki are disjoint. Let us consider C a set of classes allowing to differentiate each object according to particular characteristics. We note Ki(a,) the object K, belonging to theclassoj, Let K the set of obstacles such as m K = U k i i=l I INTRODUCTIONWe note 0 the set of the objects K. W* is the Distance computing in robotics is frequently presented complement of K in W such as as the minimal Euclidean distance computing between two convex polygonal objects. This an essential problem in robotics that covers most real cases because each polygonal object is always decomposable in more convex polygonal objccts. This problem occurs in other fields such as CAD and computer graphics. In robotics, distance computing is uscd in various cases, for example in artificial field computing, in minimal distance before collision, in simulation ... Other applications are open. There are a number of works concerning distance computing between polytopes, a few of the latest examples of which are : [GIL 871 [ZEG92] [LIN91]. In these different works, the authors consider the problem of distance computing between two convex polytopes, each is modeled by edges and vertices. The results are interesting, the complexity is in O(log(M)) with M the total vertices number. In [SRI931 the authors extcnd distance computing to a polytope penetrating another. In this paper we propose a method of distance computing between a point belonging to an object of any shape and the nearest point belonging to its boundary. The algorithm uses properties of the Multivalue Numbers which we use for modeling the environment. W' = Complement(W nK)Let us consider the set Q={{k,,k,, ..., k,},W') each element of S2 which belongs to a class Q. The same class may appear more than once.Problem 1: Let z = (zi,zi, ..., z;) belonging to 3" and to a compact element of W such as it belongs to the class oj. We note dis(z,aj) with o1 f o , the minimal distance between the point z and an element of class Q. E R.The distance used may be the Eucliiean distance dis( z, a,) = / -or another distance defined for a particular application. The algorithm we present here is based on the decomposition of the element of W according to a rectangloid structure. A rectanglold is an object Ri such as:Each element of R is composed of a set of rectangloids with its class Q. The previous problem leads to problem 2. I1 PROBLEM STATEMENTLet us consider a workspace W in 33" and a set of objects each represented by a compact set Ki on 31". All Problem 2: Let R the set of rectangloids and let a point z of %" belonging to a rectangloid Ri of class o,, the problem consists in computing the minimal distance dis(z,a ) 9 belonging to a rectangloid of class os 0, . 0-7803-2129-4/94 $3.00 0 1994 IEEE