An Euclidean norm based criterion to assess robots’ 2D path-following performance

Abstract

Abstract. A current need in the robotics field is the definition of methodologies for quantitatively evaluating the results of experiments. This paper contributes to this by defining a new criterion for assessing path-following tasks in the planar case, that is, evaluating the performance of robots that are required to follow a desired reference path. Such criterion comes from the study of the local differential geometry of the problem. New conditions for deciding whether or not the zero locus of a given polynomial intersects the neighbourhood of a point are defined. Based on this, new algorithms are presented and tested on both simulated data and experiments conducted at sea employing an Unmanned Surface Vehicle.2000 Mathematics Subject Classifications: 26C10, 15A60, 93C85, 62P99.

r-norm bounds and metric properties for zero loci of real analytic functions

“…We are interested in finding conditions on a positive number ε to ensure that the closed r-ball (or a subset of it) centered at p and of radius ε either intersects X or does not intersect X. Indeed, this turns out to be useful in practical applications, for instance, in the setting of the Hough transform, a standard technique to detect curves in images (e.g., see [2], [16], [17], [18]), or for evaluating the performance of robots that are required to follow a desired reference path (see [13]), or even in the field of robot motion planning (see [3]).…”

“…In order to approach the problem, we take X as the zero locus of a finite collection of real functions. In [13] and [16] the problem is solved for the case where X is an algebraic hypersurface and the balls are taken in 2-and ∞-norms, respectively. In this paper, inspired by the papers quoted above, we deal with a more general situation, namely, X is the zero locus of a finite collection of real analytic functions and the balls are taken in r-norms, with r ∈ [1, ∞].…”