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Distributed Finite-Time Cooperative Localization for Three-Dimensional Sensor Networks

Jinze Wu, Lorenzo Zino, Zhiyun Lin, Alessandro Rizzo

TL;DR

This work tackles distributed localization in three-dimensional sensor networks using range measurements by first establishing a necessary-sufficient node localizability condition in barycentric coordinates. It then delivers a fully distributed verification procedure that operates in finite time via a novel sum-consensus algorithm, followed by a distributed finite-time localization method based on conjugate gradient on a reduced, localizable subgraph. The approach yields rigorous convergence guarantees and shows strong scalability in simulations, outperforming existing methods in convergence speed while gracefully handling unlocalizable nodes. The results enable scalable, robust localization in large 3D networks and introduce a general finite-time consensus tool with potential applications beyond localization.

Abstract

This paper addresses the distributed localization problem for a network of sensors placed in a three-dimensional space, in which sensors are able to perform range measurements, i.e., measure the relative distance between them, and exchange information on a network structure. First, we derive a necessary and sufficient condition for node localizability using barycentric coordinates. Then, building on this theoretical result, we design a distributed localizability verification algorithm, in which we propose and employ a novel distributed finite-time algorithm for sum consensus. Finally, we develop a distributed localization algorithm based on conjugate gradient method, and we derive a theoretical guarantee on its performance, which ensures finite-time convergence to the exact position for all localizable nodes. The efficiency of our algorithm compared to the existing ones from the state-of-the-art literature is further demonstrated through numerical simulations.

Distributed Finite-Time Cooperative Localization for Three-Dimensional Sensor Networks

TL;DR

This work tackles distributed localization in three-dimensional sensor networks using range measurements by first establishing a necessary-sufficient node localizability condition in barycentric coordinates. It then delivers a fully distributed verification procedure that operates in finite time via a novel sum-consensus algorithm, followed by a distributed finite-time localization method based on conjugate gradient on a reduced, localizable subgraph. The approach yields rigorous convergence guarantees and shows strong scalability in simulations, outperforming existing methods in convergence speed while gracefully handling unlocalizable nodes. The results enable scalable, robust localization in large 3D networks and introduce a general finite-time consensus tool with potential applications beyond localization.

Abstract

This paper addresses the distributed localization problem for a network of sensors placed in a three-dimensional space, in which sensors are able to perform range measurements, i.e., measure the relative distance between them, and exchange information on a network structure. First, we derive a necessary and sufficient condition for node localizability using barycentric coordinates. Then, building on this theoretical result, we design a distributed localizability verification algorithm, in which we propose and employ a novel distributed finite-time algorithm for sum consensus. Finally, we develop a distributed localization algorithm based on conjugate gradient method, and we derive a theoretical guarantee on its performance, which ensures finite-time convergence to the exact position for all localizable nodes. The efficiency of our algorithm compared to the existing ones from the state-of-the-art literature is further demonstrated through numerical simulations.
Paper Structure (23 sections, 6 theorems, 35 equations, 7 figures, 6 algorithms)

This paper contains 23 sections, 6 theorems, 35 equations, 7 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Notation
  4. Graph Theory
  5. Barycentric Coordinates
  6. Problem Setup and Statement
  7. Problem Setup
  8. Problem Statement
  9. Localizability Verification Algorithm
  10. Necessary and Sufficient Node Localizability Condition
  11. Distributed Verification Algorithm for Node Localizability
  12. Local sum
  13. Finite-Time K-Max-Consensus Sum
  14. Distributed Localizablity Verification
  15. Finite-Time Localization Algorithm
  16. ...and 8 more sections

Key Result

Theorem 1

Supposing that node $i$ belongs to at least one clique of order $5$, then node $i$ is localizable iff $\mathbf{e_i} \bot \textrm{ker}\left( M^\top M \right)$, where $\textrm{ker}(M^\top M)$ denotes the kernel of the matrix $M^\top M$.

Figures (7)

  • Figure 1: Illustration of the barycentric coordinates of node $i$ with respect to nodes $j, k, l$, and $h$ in a three-dimensional space.
  • Figure 2: Flow chart of the Distributed Localizablity Verification Algorithm.
  • Figure 3: Configuration and topology of the sensor network for Case Study I. Anchor nodes are denoted in green, unlocalizable nodes (detected by Algorithm \ref{['Alg: Distributed Localizablity Verification Algorithm']}) are in red, and localizable nodes in blue.
  • Figure 4: Output of our localization algorithm compared to the absolute positions for Case Study I. The blue circles denote the real absolute position and the red asterisks denote the final estimated position. The plot also depicts a sample trajectory: the initial estimate is represented by a grey square, the estimates at each 5 iterations by orange triangles.
  • Figure 5: Evolution of the estimation error ratio using Algorithm \ref{['Alg: Distributed Localization Algorithm']} (red), compared to JU xia2022exploratory (blue) and RI algorithms han2017barycentric (green) for Case Study I.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 8 more