Set-Membership Localization via Range Measurements

TL;DR

This paper introduces a set-membership approach for localizing a point in space using range measurements with bounded errors, providing guaranteed convex outer bounds for the feasible location set.

Contribution

It develops convex programming methods to compute tight outer bounds for the localization set, improving over existing relaxation-based approaches.

Findings

The localization set is contained within an intersection of balls and a polytope.

Convex programming can efficiently compute outer bounds like boxes or ellipsoids.

The method offers guaranteed set estimates without relying on semidefinite relaxations.

Abstract

In this paper we discuss a classical geometrical problem of estimating an unknown point's location in $\Real{n}$ from several noisy measurements of the Euclidean distances from this point to a set of known reference points (anchors). We approach the problem via a set-mem\-ber\-ship methodology, in which we assume the distance measurements to be affected by unknown-but-bounded errors, and we characterize the set of all points that are consistent with the measurements and their assumed error model. This set is nonconvex, but we show in the paper that it is contained in a region given by the intersection of certain closed balls and a polytope, which we call the {\em localization set}. Then, we develop efficient methods, based on convex programming, for computing a tight outer-bounding set of simple structure (a box, or an ellipsoid) for the localization set, which then acts as a guaranteed set-valued location estimate. % The center of the bounding set also serves as a point location estimate. Related problems of inner approximation of the localization set via balls and ellipsoids are also posed as convex programming problems. Different from existing methods based on semidefinite programming relaxations of a nonconvex cost minimization problem, our approach is direct, geometric and based on a polyhedral set of points that satisfy pairwise differences of the measurement equations.