Motion Planning Using Interval Analysis

This paper deals with characterizing the connected component AS(~a) of a set S that contains a given point ~a. The set S is assumed to be de…ned by nonlinear inequalities, so that interval analysis can be used to prove that a given box is inside or outside S. Further, the characterization of AS(~a) is used to …nd a feasible path (i.e. a path that lies within S) that links ~a to any other point ~b 2 AS(~a). Keywords: Connexity, Interval analysis, Motion Planning, Set inversion, Subpavings. 1. Introduction In this paper, we present a new interval-based approach to …nd a collision-free path for a robot in a given space with obstacles. The issue of path planning in a known environment has been addressed by many researchers (see e.g. [7], [8] and [4]). Most of the current approaches to path planning are based on the concept of con…guration space (C-space) introduced by LozanoPérez and Wesley [5]. C-space is the set S of all possible con…gurations of a robot. The start con…guration and the goal con…guration become two points ~a and ~b of S. The number of independent parameters needed to fully specify a robot con…guration is the dimension of the C-space. The problem formulated in the C-space amounts to …nd a path included in S that links ~a to ~b. Many approaches to solve this problem are based on the use of potential functions, …rst introduced by Khatib [3]. In the potential …eld approach, the obstacles to be avoided are represented by a repulsive potential, and the goal is represented by an attractive potential. According to the force generated by the sum of these potential …elds robot attains (if the method does not stop at any local minimum) the goal without collding with obstacles. Except for some special classes of problems, these methods are not guaranteed and interval analysis could be helpful to deal with motion planning problems in a global and guaranteed way. Interval analysis has already been used for parametric paths in [2], but the former method requires a model for the path and is limited to small dimensional path models. In this paper a new nonparametric approach based on interval analysis is considered. This approach …rst characterizes the set S of all feasible con…gurations for the robot as presented on Section 3. Then, a color propagation algorithm, presented in Section 4, is performed to characterize the connected component AS(~a) of S that contains the start con…guration point ~a. At last, a feasible path that links ~a to ~b is computed using the procedure presented on Section 5. 2. Inclusion test Interval arithmetic provides an extension to real intervals and vector intervals of the classical arithmetical operations on real numbers and vectors. (see e.g. [6]). It makes it possible to build inclusion tests to prove that a given box ~ X is inside or outside the con…guration space S, when S is given by nonlinear inequalities. Box: A box or vector interval ~ X of Rn is the Cartesian product of n intervals: ~ X = x¡1 ; x+1 ¤£ ¢ ¢ ¢ £ x¡n ; x+n ¤ : (2.1) The set of all boxes of Rn is denoted by IRn. To bisect a box ~ X means to cut it along the symmetry plane normal to the side of maximum length. The length of this side is the width of ~ X and denoted by w( ~ X): A bisection of ~ X generates two non-overlapping boxes ~ XL and ~ XR such that ~ X = ~ XL [ ~ XR. A box ~ Y is a left neighbor of ~ X with respect to the ith dimension if and only if y+ i = x¡i and 8j 2 [1; n]; Xj \ Yj 6= ; (2.2) We shall also say that ~ Y is a i-left neighbor of ~ X. ~ Y is a i-right neighbor of ~ X if and only if ~ X is a i-left neighbor of ~ Y : Boolean Interval: A Boolean number is an element of B = f0; 1g where 0 stands for false and 1 for true. By extension, a Boolean interval is an element of IB = f0; 1; [0; 1]g: If a is a Boolean interval, 0:a = 0; 1:a = a; 0 + a = a; 1 + a = 1; a:a = a+ a = a: (2.3) Inclusion test: An inclusion test for the Boolean function (or test) t : Rn ! f0; 1g is a function T : IRn ! IB such that for all boxes ~ X 2 IRn, T ( ~ X) = 1 ) 8~x 2 ~ X; t(~x) = 1; T ( ~ X) = 0 ) 8~x 2 ~ X; t(~x) = 0: (2.4) Moreover, for all vectors ~a, T (~a) = t(~a). Interval analysis provides inclusion tests for a large class of Boolean tests. 3. Characterization of the con…guration space 3.1. Paving A guaranteed characterization of compact sets of Rn can be done by using pavings. A paving of a box ~ X0 is partition of ~ X0 with boxes. On Figure 1, a paving of the box ~ X0 = [0; 8]£ [0; 6] is presented. This paving contains 3 boxes ~ A; ~ B and ~ X. In the algorithm to be presented, topological informations on boxes are required. Each box has to memorize the address of its neighbors. Before de…ning how is implemented the box class, let us have a look to the Figure 2. It illustrates how these topological informations take place in the memory of the computer. Boxes ~ X; ~ A and ~ B are rounded by four grey trapezoid, each of them representing a list of neighbors. For example, since ~ A has two right neighbors in the direction 1, two arrows have their origins located on the right trapezoid of ~ A.