Localization with Single or Antipodal Distance Measurements

TL;DR

The paper introduces a preprocessing framework for localizing a depth sensor inside a known polygonal workspace by exploiting a 3D configuration space $W\times\mathbb{S}^1$ and a Rotational Trapezoidal Decomposition (RTD). It presents both simple and advanced output-sensitive data structures for single and antipodal depth measurements, enabling efficient retrieval of pose preimages $h^{-1}(d)$ and related planar projections, with preprocessing in $O((E+n)\log n)$ time and space $O(E+n)$ where $E$ is the number of visibility-graph edges. Key ideas include analytic opening functions within RTD cells, combinatorial-change handling via interval trees, and antipodal-measurement optimization that yields $O(\log n+k)$ query times, where $k$ is the number of relevant RTD cells. The authors also provide an implementation (open source) and experimental results, highlighting practical performance and the potential to localize with few depth measurements using inexpensive sensors. Overall, the work advances exact, geometry-based localization by reducing reliance on full visibility polygons and enabling scalable, output-sensitive queries in the robot localization setting.

Abstract

Given a polygonal workspace $W$, a depth sensor placed at point $p=(x,y)$ inside $W$ and oriented in direction $θ$ measures the distance $d=h(x,y,θ)$ between $p$ and the closest point on the boundary of $W$ along a ray emanating from $p$ in direction $θ$. We study the following problem: For a polygon $W$ with $n$ vertices, possibly with holes, preprocess it such that given a query real value $d> 0$, one can efficiently compute the preimage $h^{-1}(d) \subset W\times \mathbb{S}^1$, namely determine all the possible poses (positions and orientations) of a depth sensor placed in $W$ that would yield the reading $d$, in an output-sensitive fashion. We describe such an output-sensitive data structure, which answers queries in $O(k \log n)$ time, where $k$ is the number of vertices and maximal arcs of low degree algebraic curves constituting the answer. We also obtain analogous results for the more useful case (narrowing down the set of possible poses), where the sensor performs two antipodal depth measurements from the same point in $W$. We then describe simpler data structures for the same two problems, where we employ a decomposition of $W\times \mathbb{S}^1$, and where the query time is output-sensitive relative to this decomposition. Our software implementation for these latter structures is open source and publicly available. Although robot localization is often carried out by exploring the full visibility polygon of a sensor placed at a point of the environment, the approach that we propose here opens the door to sufficing with only few depth measurements, which is advantageous as it allows for usage of inexpensive sensors and could also lead to savings in storage and communication costs.

Figures (20)

• Figure 1: Given a sensor drawn as a gray disc in a unit square workspace (a), and a single distance measurement $d=0.15$ drawn as a dotted line sourced at the sensor and hitting the wall, the possible locations and configurations of the sensor are shown in the two-dimensional workspace (b), and in the three-dimensional C-space (c,d), respectively. The gray scale corresponds to the measurement angle: configurations with greater measurement angle are drawn in darker gray. The gray shades are projected onto the two-dimensional workspace at (b); note that, as there are multiple configurations with different measurement angles that are projected onto the same position $p$, the shade at $p$ is the darkest shade among all of them.
• Figure 2: Type I events: Two vertices of the same edge align along the rotating ray. Rays are rotating counterclockwise. The event vertices are drawn as small black discs. Each column depicts one example, before (top) and after (bottom) the event.
• Figure 3: Type II events: Two vertices of two distinct edges align along the rotating ray. Rays are rotating counterclockwise. The event vertices are drawn as small black discs. Each column depicts one example, before (top), during (middle), and after (bottom) the event.
• Figure 6: Example of a vertical decomposition of a polygonal workspace with a single hole (a) and the line segments (bold, dash-dotted) along which the sensor may be when it reads the distance $d$ in the upward vertical direction (b).
• Figure 7: (a) Type A curve: when the limiting vertex lies on the cell's ceiling, the curve is a circular arc. (b)(c) Type B curve: when the limiting vertex does not lie on the cell's ceiling, the curve is a conchoid of Nicomedes.

Theorems & Definitions (6)

• Lemma 2.1
• Definition 2.2
• Theorem 3.1
• Theorem 4.1
• Theorem 5.1
• Theorem 5.2