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Bearing-Based Network Localization Under Randomized Gossip Protocol

Nhat-Minh Le-Phan, Minh Hoang Trinh, Phuoc Doan Nguyen

TL;DR

The paper tackles bearing-based network localization using a randomized gossip protocol to estimate node positions from bearing measurements and beacon information. It introduces an expected matrix-weighted graph formulation and leverages infinitesimal bearing rigidity to guarantee identifiability. The main contributions are a two-agent gossip update with a carefully chosen step size, proofs of convergence in both expectation and second moment, and an explicit bound on convergence rate via an epsilon-consensus time. Simulations on a 1089-node 3D network corroborate the theory, showing rapid, exponential decay of bearing error. Overall, the approach yields a scalable, distributed localization method with low communication cost suitable for large sensor networks.

Abstract

In this paper, we consider a randomized gossip algorithm for the bearing-based network localization problem. Let each sensor node be able to obtain the bearing vectors and communicate its position estimates with several neighboring agents. Each update involves two agents, and the update sequence follows a stochastic process. Under the assumption that the network is infinitesimally bearing rigid and contains at least two beacon nodes, we show that when the updating step-size is properly selected, the proposed algorithm can successfully estimate the actual sensor nodes' positions with probability one. The randomized update provides a simple, distributed, and cost-effective method for localizing the network. The theoretical result is supported with a simulation of a 1089-node sensor network.

Bearing-Based Network Localization Under Randomized Gossip Protocol

TL;DR

The paper tackles bearing-based network localization using a randomized gossip protocol to estimate node positions from bearing measurements and beacon information. It introduces an expected matrix-weighted graph formulation and leverages infinitesimal bearing rigidity to guarantee identifiability. The main contributions are a two-agent gossip update with a carefully chosen step size, proofs of convergence in both expectation and second moment, and an explicit bound on convergence rate via an epsilon-consensus time. Simulations on a 1089-node 3D network corroborate the theory, showing rapid, exponential decay of bearing error. Overall, the approach yields a scalable, distributed localization method with low communication cost suitable for large sensor networks.

Abstract

In this paper, we consider a randomized gossip algorithm for the bearing-based network localization problem. Let each sensor node be able to obtain the bearing vectors and communicate its position estimates with several neighboring agents. Each update involves two agents, and the update sequence follows a stochastic process. Under the assumption that the network is infinitesimally bearing rigid and contains at least two beacon nodes, we show that when the updating step-size is properly selected, the proposed algorithm can successfully estimate the actual sensor nodes' positions with probability one. The randomized update provides a simple, distributed, and cost-effective method for localizing the network. The theoretical result is supported with a simulation of a 1089-node sensor network.
Paper Structure (11 sections, 10 theorems, 27 equations, 2 figures)

This paper contains 11 sections, 10 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. Expected Matrix-weighted Graph
  4. Bearing Rigidity Theory
  5. MAIN RESULTS
  6. Bearing-Based Network Localization Algorithm
  7. Convergence in Expectation
  8. Convergence of Second Moment
  9. Convergence Rate
  10. SIMULATION EXAMPLE
  11. Conclusion

Key Result

Lemma 1

HMT2018 The expected Laplacian matrix $\mathbf{L}^{\rm M}$ is symmetric and positive semi-definite, and its null space is given as: ${\rm null}(\mathbf{L}^{\rm M}) = {\rm span}\{ {\rm range}(\mathbf{1}_n \otimes \mathbf{I}_d), \{ \mathbf{v}=[v_1^\top,\dots, v_n^\top]^\top \in \mathbb{R}^{nd}|~(v_j-v

Figures (2)

  • Figure 1: Examples of infinitesimally/non-infinitesimally bearing rigid frameworks in two-dimensional space.
  • Figure 2: Simulation of a sensor network consisting of 1089 nodes under the gossip-based network localization protocol \ref{['algorithm0']}, \ref{['algorithm1']}, \ref{['algorithm2']}: (a) - the graph $\mathcal{G}$; (b) - the actual configuration $p$; (c) - the bearing error vs time; From (d) to (i) - the estimate configurations at different time instances.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Definition 3
  • Theorem 4
  • Remark 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • Lemma 11
  • Theorem 12
  • ...and 3 more