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.

翻译：本文研究了利用声音在由平面结构构成的未知环境中重建车辆路径的问题。现有文献中的许多系统通过车载扬声器和麦克风实现这一目标，但环境中的对称性会导致此类系统的解存在歧义。我们提出将扬声器固定于环境中的某个位置而非车辆上以解决该问题。这一方法能否在不考虑环境几何形状的情况下消除歧义，引出了可用纯数学语言描述的对称性破缺问题。我们证明：在三维或更高维几何中，该问题有肯定解；在二维几何中给出反例。排除反例出现的罕见情形后，二维几何同样有肯定解。我们的研究成果可推导出一种适用于携带四个麦克风的车辆在固定位置扬声器发射短脉冲声波环境中导航的简易路径重建算法。该算法可与文献中的其他方法结合，构建适用于潜在对称环境中导航车辆的路径追踪系统。