Omnidirectional Bats, Point-to-Plane Distances, and the Price of Uniqueness

TL;DR

This paper investigates the challenges of 3D simultaneous localization and mapping using reflections, revealing fundamental non-uniqueness issues in rooms with fewer than nine walls and proposing robust solutions.

Contribution

It extends previous 2D work to 3D, characterizes ambiguities in range-only SLAM, and offers a robust method for inexact measurements in reflection-based localization.

Findings

3D inverse problem differs fundamentally from 2D

Uniqueness is absent in rooms with fewer than nine walls

Proposed a robust solution for inexact measurements

Abstract

We study simultaneous localization and mapping with a device that uses reflections to measure its distance from walls. Such a device can be realized acoustically with a synchronized collocated source and receiver; it behaves like a bat with no capacity for directional hearing or vocalizing. In this paper we generalize our previous work in 2D, and show that the 3D case is not just a simple extension, but rather a fundamentally different inverse problem. While generically the 2D problem has a unique solution, in 3D uniqueness is always absent in rooms with fewer than nine walls. In addition to the complete characterization of ambiguities which arise due to this non-uniqueness, we propose a robust solution for inexact measurements similar to analogous results for Euclidean Distance Matrices. Our theoretical results have important consequences for the design of collocated range-only SLAM systems, and we support them with an array of computer experiments.