🤖 AI Summary
This work addresses the problem of target localization in n-dimensional space under unknown but bounded noise in distance measurements. It proposes a set-membership approach based on direct geometric modeling, constructing a non-convex localization set from range measurements to anchor points and precisely characterizing it via intersections of polyhedra and balls. Avoiding conventional semidefinite relaxation, the method employs convex optimization to efficiently compute tight outer approximations—either bounding boxes or ellipsoids—yielding both inner and outer bounds on the true localization set. The resulting set-valued estimate enjoys rigorous theoretical guarantees, features a compact outer approximation structure, and delivers high-accuracy point estimates, outperforming existing methods that rely on cost-function relaxation.
📝 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.