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Path Tracking using Echoes in an Unknown Environment: the Issue of Symmetries and How to Break Them

Mireille Boutin, Gregor Kemper

Abstract

This paper deals with the problem of reconstructing the path of a vehicle in an unknown environment consisting of planar structures using sound. Many systems in the literature do this by using a loudspeaker and microphones mounted on a vehicle. Symmetries in the environment lead to solution ambiguities for such systems. We propose to resolve this issue by placing the loudspeaker at a fixed location in the environment rather than on the vehicle. The question of whether this will remove ambiguities regardless of the environment geometry leads to a question about breaking symmetries that can be phrased in purely mathematical terms. We solve this question in the affirmative if the geometry is in dimension three or bigger, and give counterexamples in dimension two. Excluding the rare situations where the counterexamples arise, we also give an affirmative answer in dimension two. Our results lead to a simple path reconstruction algorithm for a vehicle carrying four microphones navigating within an environment in which a loudspeaker at a fixed position emits short bursts of sounds. This algorithm could be combined with other methods from the literature to construct a path tracking system for vehicles navigating within a potentially symmetric environment.

Path Tracking using Echoes in an Unknown Environment: the Issue of Symmetries and How to Break Them

Abstract

This paper deals with the problem of reconstructing the path of a vehicle in an unknown environment consisting of planar structures using sound. Many systems in the literature do this by using a loudspeaker and microphones mounted on a vehicle. Symmetries in the environment lead to solution ambiguities for such systems. We propose to resolve this issue by placing the loudspeaker at a fixed location in the environment rather than on the vehicle. The question of whether this will remove ambiguities regardless of the environment geometry leads to a question about breaking symmetries that can be phrased in purely mathematical terms. We solve this question in the affirmative if the geometry is in dimension three or bigger, and give counterexamples in dimension two. Excluding the rare situations where the counterexamples arise, we also give an affirmative answer in dimension two. Our results lead to a simple path reconstruction algorithm for a vehicle carrying four microphones navigating within an environment in which a loudspeaker at a fixed position emits short bursts of sounds. This algorithm could be combined with other methods from the literature to construct a path tracking system for vehicles navigating within a potentially symmetric environment.
Paper Structure (4 sections, 7 theorems, 25 equations, 2 figures)

This paper contains 4 sections, 7 theorems, 25 equations, 2 figures.

Table of Contents

  1. Problem Setup
  2. Breaking symmetries
  3. The Cayley-Menger matrix and affine subspaces
  4. A Reconstruction Algorithm for the Vehicle Path and Source Position

Key Result

Theorem 1

Let $\mathcal{H}$ be a finite set of affine hyperplanes in ${\mathbb R}^n$. Then there is a nonzero polynomial $f \in {\mathbb R}[x_1 ,\ldots, x_n]$ such that for all $\mathbf{v} \in {\mathbb R}^n$ with $f(\mathbf{v}) \ne 0$ and for $H_1 ,\ldots, H_m,H'_1 ,\ldots, H'_m \in \mathcal{H}$ such that the then $H_i = H'_i$ and therefore $\mathbf{w}_i = \mathbf{w}'_i$ for all $i$. Moreover, for $H_1,H_2,

Figures (2)

  • Figure 1: Loudspeaker on the vehicle. The microphones (red) and the loudspeaker (blue) are positioned on the vehicle. In both vehicle positions indicated, the echoes from a sound heard by the microphones will be exactly the same. So the positions are indistinguishable.
  • Figure 2: Loudspeaker at a fixed position. The sound travels along the dashed lines. Virtually, it comes from the mirror points (violet). The fixed position of the loudspeaker (blue) is such that there is no symmetry among the mirror points. Vehicle positions are distinguishable.

Theorems & Definitions (16)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • proof : Proof of \ref{['tSymmetries3D', 'tSymmetries2D']}
  • Lemma 4
  • proof
  • Proposition 5
  • Remark 6
  • proof : Proof of \ref{['pCM', 'rQuadratic']}
  • Proposition 7
  • ...and 6 more