Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Multi-Player Potential Game Approach for Sensor Network Localization with Noisy Measurements

Gehui Xu, Guanpu Chen, Baris Fidan, Yiguang Hong, Hongsheng Qi, Thomas Parisini, Karl H. Johansson

TL;DR

This work reframes sensor-network localization (SNL) with noisy measurements as a non-convex multi-player potential game, where non-anchor nodes minimize local localization errors. It shows that in the noiseless setting the NE exists, is unique, and coincides with the true network positions, leveraging graph rigidity to guarantee uniqueness. When measurement errors are present but sufficiently small, a unique NE persists and remains close to the true localization, with explicit error bounds that quantify the impact of anchor-position uncertainty and distance-noise. The results are complemented by numerical experiments validating the theoretical bounds and by a detailed graph-theoretic and proof-structure framework (including explicit conditions and an algorithm to compute noise-bounds). Overall, the paper provides a principled, game-theoretic pathway to robust SNL under realistic noisy conditions, with potential extensions to distributed and globally convergent algorithms.

Abstract

Sensor network localization (SNL) is a challenging problem due to its inherent non-convexity and the effects of noise in inter-node ranging measurements and anchor node position. We formulate a non-convex SNL problem as a multi-player non-convex potential game and investigate the existence and uniqueness of a Nash equilibrium (NE) in both the ideal setting without measurement noise and the practical setting with measurement noise. We first show that the NE exists and is unique in the noiseless case, and corresponds to the precise network localization. Then, we study the SNL for the case with errors affecting the anchor node position and the inter-node distance measurements. Specifically, we establish that in case these errors are sufficiently small, the NE exists and is unique. It is shown that the NE is an approximate solution to the SNL problem, and that the position errors can be quantified accordingly. Based on these findings, we apply the results to case studies involving only inter-node distance measurement errors and only anchor position information inaccuracies.

A Multi-Player Potential Game Approach for Sensor Network Localization with Noisy Measurements

TL;DR

This work reframes sensor-network localization (SNL) with noisy measurements as a non-convex multi-player potential game, where non-anchor nodes minimize local localization errors. It shows that in the noiseless setting the NE exists, is unique, and coincides with the true network positions, leveraging graph rigidity to guarantee uniqueness. When measurement errors are present but sufficiently small, a unique NE persists and remains close to the true localization, with explicit error bounds that quantify the impact of anchor-position uncertainty and distance-noise. The results are complemented by numerical experiments validating the theoretical bounds and by a detailed graph-theoretic and proof-structure framework (including explicit conditions and an algorithm to compute noise-bounds). Overall, the paper provides a principled, game-theoretic pathway to robust SNL under realistic noisy conditions, with potential extensions to distributed and globally convergent algorithms.

Abstract

Sensor network localization (SNL) is a challenging problem due to its inherent non-convexity and the effects of noise in inter-node ranging measurements and anchor node position. We formulate a non-convex SNL problem as a multi-player non-convex potential game and investigate the existence and uniqueness of a Nash equilibrium (NE) in both the ideal setting without measurement noise and the practical setting with measurement noise. We first show that the NE exists and is unique in the noiseless case, and corresponds to the precise network localization. Then, we study the SNL for the case with errors affecting the anchor node position and the inter-node distance measurements. Specifically, we establish that in case these errors are sufficiently small, the NE exists and is unique. It is shown that the NE is an approximate solution to the SNL problem, and that the position errors can be quantified accordingly. Based on these findings, we apply the results to case studies involving only inter-node distance measurement errors and only anchor position information inaccuracies.
Paper Structure (13 sections, 6 theorems, 21 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 6 theorems, 21 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Problem formulation
  3. Noiseless SNL problem
  4. Noisy SNL problem
  5. Potential game formulation
  6. Existence, uniqueness, and errors of the solution
  7. Existence and Uniqueness of NE
  8. Numerical Experiments
  9. Concluding Remarks
  10. Graph Theory
  11. Proof of Theorem \ref{['l11']}
  12. Proof of Lemma \ref{['ee1']}
  13. Proof of Theorem \ref{['t11']}

Key Result

Theorem 1

Under Assumption 1, there exists a unique NE $\boldsymbol{x}^{\Diamond}$ of the game $G$, which satisfies $x_i^{\Diamond} =x_i^{\star}$ for $i\in\mathcal{N}_s$ and $x_l =x_l^{\star}$ for $l\in\mathcal{N}_a$ if $\mu_{ij}=0$ for $(i,j) \in\mathcal{E}_{ss}$, ${\mu}_{il}=0$ for $(i,j) \in\mathcal{E}_{as

Figures (3)

  • Figure 1: A sensor network with two non-anchor nodes and three anchor nodes in two configurations.
  • Figure 2: The true localization and noisy localization results.
  • Figure 3: Computed sensor location results with different error levels.

Theorems & Definitions (9)

  • Definition 1: NE
  • Definition 2: local NE
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Remark 1
  • Corollary 1
  • Corollary 2
  • Lemma 2