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Cooperative Relative Localization in MAV Swarms with Ultra-wideband Ranging

Changrui Liu, Sven U. Pfeiffer, Guido C. H. E. de Croon

TL;DR

The paper tackles 3-D relative localization for MAV swarms in GPS-denied environments by introducing cooperative relative localization (CRL) that fuses pairwise relative motion dynamics with inter-neighbor distances. It develops three CRL models (fCRL,hCRL,nCRL) and performs differential-geometry–based observability analysis to show that indirect neighbor measurements expand observable subspaces, enabling improved localization. To address heavy-tailed UWB noise, the authors propose a kernel-induced extended Kalman filter using a novel Logarithmic-Versoria (LV) kernel and derive convergence conditions for the fixed-point update, demonstrating robustness to outliers and measurement covariance initialization. Through Monte Carlo simulations, CRL with LV-EKF consistently outperforms baseline methods in accuracy and reliability, validating the practical viability of cooperative, distributed RL in swarming MAVs with limited computation.

Abstract

Relative localization (RL) is essential for the successful operation of micro air vehicle (MAV) swarms. Achieving accurate 3-D RL in infrastructure-free and GPS-denied environments with only distance information is a challenging problem that has not been satisfactorily solved. In this work, based on the range-based peer-to-peer RL using the ultra-wideband (UWB) ranging technique, we develop a novel UWB-based cooperative relative localization (CRL) solution that integrates the relative motion dynamics of each host-neighbor pair to build a unified dynamic model and takes the distances between the neighbors as \textit{bonus information}. Observability analysis using differential geometry shows that the proposed CRL scheme can expand the observable subspace compared to other alternatives using only direct distances between the host agent and its neighbors. In addition, we apply the kernel-induced extended Kalman filter (EKF) to the CRL state estimation problem with the novel-designed Logarithmic-Versoria (LV) kernel to tackle heavy-tailed UWB noise. Sufficient conditions for the convergence of the fixed-point iteration involved in the estimation algorithm are also derived. Comparative Monte Carlo simulations demonstrate that the proposed CRL scheme combined with the LV-kernel EKF significantly improves the estimation accuracy owing to its robustness against both measurement outliers and incorrect measurement covariance matrix initialization. Moreover, with the LV kernel, the estimation is still satisfactory when performing the fixed-point iteration only once for reduced computational complexity.

Cooperative Relative Localization in MAV Swarms with Ultra-wideband Ranging

TL;DR

The paper tackles 3-D relative localization for MAV swarms in GPS-denied environments by introducing cooperative relative localization (CRL) that fuses pairwise relative motion dynamics with inter-neighbor distances. It develops three CRL models (fCRL,hCRL,nCRL) and performs differential-geometry–based observability analysis to show that indirect neighbor measurements expand observable subspaces, enabling improved localization. To address heavy-tailed UWB noise, the authors propose a kernel-induced extended Kalman filter using a novel Logarithmic-Versoria (LV) kernel and derive convergence conditions for the fixed-point update, demonstrating robustness to outliers and measurement covariance initialization. Through Monte Carlo simulations, CRL with LV-EKF consistently outperforms baseline methods in accuracy and reliability, validating the practical viability of cooperative, distributed RL in swarming MAVs with limited computation.

Abstract

Relative localization (RL) is essential for the successful operation of micro air vehicle (MAV) swarms. Achieving accurate 3-D RL in infrastructure-free and GPS-denied environments with only distance information is a challenging problem that has not been satisfactorily solved. In this work, based on the range-based peer-to-peer RL using the ultra-wideband (UWB) ranging technique, we develop a novel UWB-based cooperative relative localization (CRL) solution that integrates the relative motion dynamics of each host-neighbor pair to build a unified dynamic model and takes the distances between the neighbors as \textit{bonus information}. Observability analysis using differential geometry shows that the proposed CRL scheme can expand the observable subspace compared to other alternatives using only direct distances between the host agent and its neighbors. In addition, we apply the kernel-induced extended Kalman filter (EKF) to the CRL state estimation problem with the novel-designed Logarithmic-Versoria (LV) kernel to tackle heavy-tailed UWB noise. Sufficient conditions for the convergence of the fixed-point iteration involved in the estimation algorithm are also derived. Comparative Monte Carlo simulations demonstrate that the proposed CRL scheme combined with the LV-kernel EKF significantly improves the estimation accuracy owing to its robustness against both measurement outliers and incorrect measurement covariance matrix initialization. Moreover, with the LV kernel, the estimation is still satisfactory when performing the fixed-point iteration only once for reduced computational complexity.
Paper Structure (19 sections, 1 theorem, 66 equations, 21 figures, 8 tables, 3 algorithms)

This paper contains 19 sections, 1 theorem, 66 equations, 21 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries & Problem Formulation
  3. Notation and Definitions
  4. Peer-to-peer Relative Motion Dynamics
  5. Heavy-tailed UWB Measurement Noise & Delay Effects
  6. Cooperative Relative Localization
  7. Modeling of CRL Schemes
  8. Knowledge from Differential Geometry
  9. Observability Analysis
  10. Filtering Based State Estimation
  11. Extended Kalman Filter
  12. Kernel-induced Extended Kalman Filter
  13. Convergence of The Fixed-point Iteration
  14. Computational Complexity
  15. Simulation Results & Analysis
  16. ...and 4 more sections

Key Result

Theorem 1

Given $\rho > \xi$ (cf. eq:xi_definition) and $\tau \geq \max\{\tau^\ast, \tau^\dagger\}$, where $\tau^\ast$ satisfies $\Upsilon(\tau^\ast) = \rho$ (cf. eq:tau_limits1) and $\tau^\dagger$ satisfies $\Phi(\tau^\dagger; \rho) = \zeta\; (0 < \zeta < 1)$ (cf. eq:tau_limits2). Then for $\hat{\mathbf{x}} According to the Banach fixed-point theorem agarwal2001fixed, with a large enough kernel bandwidth

Figures (21)

  • Figure 1: Cooperative relative localization problem, where agent $i$ aims to localize its neighbors $j_1$ and $j_2$ in its body centered horizontal frame, i.e., agent $i$ needs to compute $\mathbf{p}_{ij_1}$ and $\mathbf{p}_{ij_2}$.
  • Figure 2: 2-D visualization of the spherical cap which is marked in red.
  • Figure 3: PDFs of noise $\nu_d$
  • Figure 4: Unobservable motions: case (1) (top left); case (2) (top right); case (3) (bottom left); case (4) (bottom right).
  • Figure 5: Kernel Function Comparison
  • ...and 16 more figures

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 1: Lie Derivative
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Definition 2: Information Package
  • ...and 6 more