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Exploiting Sparsity for Localization of Large-Scale Wireless Sensor Networks

Shiraz Khan, Inseok Hwang, James Goppert

TL;DR

It is shown that typical WSN configurations (which can be modelled as random geometric graphs) can be localised in a scalable manner using the proposed LB‐EKF approach.

Abstract

Wireless Sensor Network (WSN) localization refers to the problem of determining the position of each of the agents in a WSN using noisy measurement information. In many cases, such as in distance and bearing-based localization, the measurement model is a nonlinear function of the agents' positions, leading to pairwise interconnections between the agents. As the optimal solution for the WSN localization problem is known to be computationally expensive in these cases, an efficient approximation is desired. In this paper, we show that the inherent sparsity in this problem can be exploited to greatly reduce the computational effort of using an Extended Kalman Filter (EKF) for large-scale WSN localization. In the proposed method, which we call the Low-Bandwidth Extended Kalman Filter (LB-EKF), the measurement information matrix is converted into a banded matrix by relabeling (permuting the order of) the vertices of the graph. Using a combination of theoretical analysis and numerical simulations, it is shown that typical WSN configurations (which can be modeled as random geometric graphs) can be localized in a scalable manner using the proposed LB-EKF approach.

Exploiting Sparsity for Localization of Large-Scale Wireless Sensor Networks

TL;DR

It is shown that typical WSN configurations (which can be modelled as random geometric graphs) can be localised in a scalable manner using the proposed LB‐EKF approach.

Abstract

Wireless Sensor Network (WSN) localization refers to the problem of determining the position of each of the agents in a WSN using noisy measurement information. In many cases, such as in distance and bearing-based localization, the measurement model is a nonlinear function of the agents' positions, leading to pairwise interconnections between the agents. As the optimal solution for the WSN localization problem is known to be computationally expensive in these cases, an efficient approximation is desired. In this paper, we show that the inherent sparsity in this problem can be exploited to greatly reduce the computational effort of using an Extended Kalman Filter (EKF) for large-scale WSN localization. In the proposed method, which we call the Low-Bandwidth Extended Kalman Filter (LB-EKF), the measurement information matrix is converted into a banded matrix by relabeling (permuting the order of) the vertices of the graph. Using a combination of theoretical analysis and numerical simulations, it is shown that typical WSN configurations (which can be modeled as random geometric graphs) can be localized in a scalable manner using the proposed LB-EKF approach.
Paper Structure (15 sections, 4 theorems, 34 equations, 7 figures, 4 algorithms)

This paper contains 15 sections, 4 theorems, 34 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Topology of the WSN
  4. Motion and Measurement Models
  5. Extended Kalman Filter (EKF)
  6. Exploiting Sparsity through Vertex Relabeling
  7. Approximation of $\lbrace M_k \rbrace$ as Banded Matrices
  8. Minimal Bandwidth of $\mathcal{G}$
  9. Vertex Relabeling
  10. Random Geometric Graphs
  11. Numerical Simulations
  12. L-Banded Matrix Inversion
  13. LB-EKF with Vertex Relabeling
  14. Vertex Relabeling of Large-Scale WSNs
  15. Conclusions

Key Result

Lemma 1

The matrix $S^{(2)}_{k}$ in (eq:Iblock) has the same sparsity pattern as $\mathcal{L}\otimes \mathbbm{1}_d$, where $\mathbbm 1_d$ is the $d \times d$ matrix where each entry is $1$.

Figures (7)

  • Figure 1: Labeling a cycle graph and a grid graph using Algorithm \ref{['alg:vr']} achieves the minimal bandwidth (2 and 4, respectively) on either graph.
  • Figure 2: The error (in Frobenius norm) in the approximation of the inverse of a $500\times 500$ covariance matrix $A^{-1}$ by another covariance matrix $\breve A^{-1}$ of a given bandwidth.
  • Figure 3: Initial positions of the WSN; the red squares indicate the beacons (agents which can measure their own position).
  • Figure 4: Covariance ellipses of the estimates computed using LB-EKF+VR (blue shade) plotted over those of EKF (orange shade). The resulting green hue indicates that the ellipses of the two algorithms are overlapping.
  • Figure 5: MSE of the position estimates of the agents, computed using EKF, LB-EKF with vertex relabeling and LB-EKF without vertex relabeling, averaged over 5000 Monte Carlo trials.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition : Bandwidth of a matrix
  • Definition : Minimal bandwidth of a graph
  • Lemma 1
  • proof
  • Theorem 1: Upper bound of $\varphi(\mathcal{G}, \mathcal{X})$
  • proof
  • Corollary 1.1: Upper bound of $\varphi_{\textrm{min}}(\mathcal{G})$
  • proof
  • Definition : Random Geometric Graph gilbert1961random
  • Lemma 2: Properties of random geometric graphs daley2008poissonTheory