Omnidirectional bats, point-to-plane distances, and the price of uniqueness

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.

The hybrid Cramér-Rao lower bound for simultaneous self-localization and room geometry estimation

“…In general, if nothing is known about the trajectory, this problem is not identifiable for some important classes of room geometries, i.e., parallelogram room-geometry cannot be uniquely determined from the distances between walls and trajectory points (cf. [21,22], and for a comprehensive analysis, see [23]). Nevertheless, the robot movement commands provide partial information about the robot trajectory.…”

“…Consider first the deterministic parameters; we note that in the finite sample regime, there are useful estimators that do not satisfy (22), e.g., the maximum likelihood (ML) in the pure classic case and the joint ML/MAP in the hybrid case (see [25]). Nevertheless, in the large-sample regime, an estimator that does not satisfy (22) concerning the deterministic static parameters is not consistent. Since inconsistent estimators are useless in the large-sample regime for these parameters, the assumption in (22) is not restricting.…”

Shapes From Echoes: Uniqueness From Point-to-Plane Distance Matrices

“…in 3D; N is written out in (5). The wall normals N 0 and N of the two room-trajectory configurations R 0 and R are in 3D, where ϕ 0 k , ϕ k ∈ [0, 2π) and θ 0 k , θ k ∈ [0, π).…”

“…Prior work on localization from point-to-plane distances has so far been mostly computational [2], [3]. Although several papers point out problems with uniqueness [4], [5], a complete study was up to now absent. The most notable result is presented in [6], which shows that one can reconstruct a room from the first-order echoes from one omnidirectional loudspeaker to four non-planar microphones, placed together on a drone with generic position and orientation.…”

Joint Estimation of the Room Geometry and Modes with Compressed Sensing

“…On the other hand, the solutions for estimating the shape of the room usually rely on knowing the location of early reflections [11], [12], but finding the true reflections within an echogram is not a trivial problem and is still an open research question.…”