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Towards Optimal Beacon Placement for Range-Aided Localization

Ethan Sequeira, Hussein Saad, Stephen Kelly, Matthew Giamou

TL;DR

This work tackles optimal beacon placement for range-aided localization by formulating a MAP-based design problem using a D-optimal Bayesian objective $f_{\mathrm{D}}$. The authors show that $f_{\mathrm{D}}^+$ is normalized, monotone, and submodular, enabling an efficient greedy algorithm with a $(1-1/e)$ performance guarantee, and also explore a CMA-ES-based continuous-search alternative mapped to a discrete beacon set. They develop a probabilistic framework that integrates Gaussian measurement noise and priors to derive a Fisher information-based objective, enabling offline design over a finite candidate beacon set. Empirical results across benchmarks and large-scale simulations demonstrate that the greedy OBP method outperforms heuristic baselines and approaches brute-force optimality while remaining computationally tractable, with an open-source Python implementation provided for reproducibility and reuse.

Abstract

Range-based localization is ubiquitous: global navigation satellite systems (GNSS) power mobile phone-based navigation, and autonomous mobile robots can use range measurements from a variety of modalities including sonar, radar, and even WiFi signals. Many of these localization systems rely on fixed anchors or beacons with known positions acting as transmitters or receivers. In this work, we answer a fundamental question: given a set of positions we would like to localize, how should beacons be placed so as to minimize localization error? Specifically, we present an information theoretic method for optimally selecting an arrangement consisting of a few beacons from a large set of candidate positions. By formulating localization as maximum a posteriori (MAP) estimation, we can cast beacon arrangement as a submodular set function maximization problem. This approach is probabilistically rigorous, simple to implement, and extremely flexible. Furthermore, we prove that the submodular structure of our problem formulation ensures that a greedy algorithm for beacon arrangement has suboptimality guarantees. We compare our method with a number of benchmarks on simulated data and release an open source Python implementation of our algorithm and experiments.

Towards Optimal Beacon Placement for Range-Aided Localization

TL;DR

This work tackles optimal beacon placement for range-aided localization by formulating a MAP-based design problem using a D-optimal Bayesian objective . The authors show that is normalized, monotone, and submodular, enabling an efficient greedy algorithm with a performance guarantee, and also explore a CMA-ES-based continuous-search alternative mapped to a discrete beacon set. They develop a probabilistic framework that integrates Gaussian measurement noise and priors to derive a Fisher information-based objective, enabling offline design over a finite candidate beacon set. Empirical results across benchmarks and large-scale simulations demonstrate that the greedy OBP method outperforms heuristic baselines and approaches brute-force optimality while remaining computationally tractable, with an open-source Python implementation provided for reproducibility and reuse.

Abstract

Range-based localization is ubiquitous: global navigation satellite systems (GNSS) power mobile phone-based navigation, and autonomous mobile robots can use range measurements from a variety of modalities including sonar, radar, and even WiFi signals. Many of these localization systems rely on fixed anchors or beacons with known positions acting as transmitters or receivers. In this work, we answer a fundamental question: given a set of positions we would like to localize, how should beacons be placed so as to minimize localization error? Specifically, we present an information theoretic method for optimally selecting an arrangement consisting of a few beacons from a large set of candidate positions. By formulating localization as maximum a posteriori (MAP) estimation, we can cast beacon arrangement as a submodular set function maximization problem. This approach is probabilistically rigorous, simple to implement, and extremely flexible. Furthermore, we prove that the submodular structure of our problem formulation ensures that a greedy algorithm for beacon arrangement has suboptimality guarantees. We compare our method with a number of benchmarks on simulated data and release an open source Python implementation of our algorithm and experiments.
Paper Structure (21 sections, 1 theorem, 23 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 1 theorem, 23 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Range-based Localization
  4. Optimal Sensor Placement
  5. Background: Range-Aided Localization
  6. Range-Only Localization
  7. Maximum Likelihood Estimation
  8. Maximum a Posteriori Estimation
  9. Optimal Beacon Placement
  10. Feasible Beacon Positions
  11. An Information-Theoretic Objective
  12. D-Optimal Beacon Placement
  13. Algorithms
  14. Normalized, Monotone, and Submodular Set Functions
  15. Analyzing $f_{\mathrm{D}}$
  16. ...and 6 more sections

Key Result

Theorem 1

Let $S_g \subset \mathcal{S}$ be the greedy solution given by alg:greedy to a set maximization problem like prob:optimal_beacon_placement for some $K = \left\vert S_g\right\vert$ and a set function $f$. If $f$ is NMS, then where $S^\star$ is the optimal solution.

Figures (5)

  • Figure 1: We present an information-theoretic approach to arranging the position $\boldsymbol{\mathbf{a}} _j$ of $K$ sensing beacons in an environment in which an agent relies on range measurements for localization at points $\boldsymbol{\mathbf{x}} _i$ for $i \in [ M ]$. Using a finite set of candidate beacon positions $\mathcal{S}$ and an objective function derived with D-optimal Bayesian experiment design, we cast the optimal beacon placement (OBP) problem as submodular set function maximization with suboptimality guarantees.
  • Figure 2: Range-aided localization of a single unknown position $\boldsymbol{\mathbf{x}} _i$ with prior mean $\check{ \boldsymbol{\mathbf{x}} }_i$ and covariance $\check{ \boldsymbol{\mathbf{\Sigma}} }_i$. For each beacon $\boldsymbol{\mathbf{a}} _j$ in range, a noisy measurement $\tilde{d}_{ij}$ is received.
  • Figure 3: Blueprints of a factory setting retrieved from factoryWebsite overlaid with a simulated robot trajectory in cyan and candidate beacon positions in red. Environments using this trajectory and random samplings of candidate beacon positions are used in the experiments of \ref{['sec:large_scale']}.
  • Figure 4: Information-theoretic objective function $f_{\mathrm{D}}^+$ with respect to the number of selected beacons $K$. The dashed blue line is the suboptimality bound of \ref{['thm:submodularity']}. For all values of $K$ tested, our Greedy method satisfies the suboptimality bound and is very close to the globally optimal solution computed by Brute-Force.
  • Figure 5: Boxplots of RMSE in meters for experiment in \ref{['sec:large_scale']}. Each subplot summarizes the distribution of the RMSE over 50 random trials. Where unspecified, parameters are set to their default values of $K=5$, $C=250$, and $\check{\sigma}_i=8$.

Theorems & Definitions (1)

  • Theorem 1: Greedy Maximization of NMS Functions krause2014submodular