Set-Valued Shadow Matching Using Zonotopes for 3-D Map-Aided GNSS Localization

TL;DR

This paper introduces zonotope shadow matching (ZSM), a set-valued GNSS localization method using 3-D maps and convex polytope representations to improve robustness and accuracy over traditional point-based approaches.

Contribution

ZSM is the first to use constrained zonotopes for set-valued GNSS shadow matching, enabling efficient propagation of localization estimates in 3-D environments.

Findings

ZSM effectively refines position estimates using set operations.

ZSM outperforms grid-based methods in simulated environments.

ZSM successfully localizes in complex 3-D urban scenarios.

Abstract

Unlike many urban localization methods that return point-valued estimates, a set-valued representation enables robustness by ensuring that a continuum of possible positions obeys safety constraints. One strategy with the potential for set-valued estimation is GNSS-based shadow matching~(SM), where one uses a three-dimensional (3-D) map to compute GNSS shadows (where line-of-sight is blocked). However, SM requires a point-valued grid for computational tractability, with accuracy limited by grid resolution. We propose zonotope shadow matching (ZSM) for set-valued 3-D map-aided GNSS localization. ZSM represents buildings and GNSS shadows using constrained zonotopes, a convex polytope representation that enables propagating set-valued estimates using fast vector concatenation operations. Starting from a coarse set-valued position, ZSM refines the estimate depending on the receiver being inside or outside each shadow as judged by received carrier-to-noise density. We demonstrated our algorithm's performance using simulated experiments on a simple 3-D example map and on a dense 3-D map of San Francisco.