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Cooperative Localisation of a GPS-Denied UAV using Direction of Arrival Measurements

James S. Russell, Mengbin Ye, Brian D. O. Anderson, Hatem Hmam, Peter Sarunic

TL;DR

Semidefinite programming and the orthogonal Procrustes algorithm are employed, and accuracy is improved through maximum likelihood estimation, and a three-agent extension is explored.

Abstract

A GPS-denied UAV (Agent B) is localised through INS alignment with the aid of a nearby GPS-equipped UAV (Agent A), which broadcasts its position at several time instants. Agent B measures the signals' direction of arrival with respect to Agent B's inertial navigation frame. Semidefinite programming and the Orthogonal Procrustes algorithm are employed, and accuracy is improved through maximum likelihood estimation. The method is validated using flight data and simulations. A three-agent extension is explored.

Cooperative Localisation of a GPS-Denied UAV using Direction of Arrival Measurements

TL;DR

Semidefinite programming and the orthogonal Procrustes algorithm are employed, and accuracy is improved through maximum likelihood estimation, and a three-agent extension is explored.

Abstract

A GPS-denied UAV (Agent B) is localised through INS alignment with the aid of a nearby GPS-equipped UAV (Agent A), which broadcasts its position at several time instants. Agent B measures the signals' direction of arrival with respect to Agent B's inertial navigation frame. Semidefinite programming and the Orthogonal Procrustes algorithm are employed, and accuracy is improved through maximum likelihood estimation. The method is validated using flight data and simulations. A three-agent extension is explored.

Paper Structure

This paper contains 26 sections, 29 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Linear System Method
  4. Forming a system of linear equations
  5. Example of LS method in noiseless case using real flight trajectories
  6. Semidefinite programming method
  7. Quadratic constraints on entries of $\bm{\Psi}$
  8. Formulation of the Semidefinite Program
  9. Rank Relaxation of Semidefinite Program
  10. Orthogonal Procrustes Algorithm
  11. Example of SDP+O method with noisy DOA measurements
  12. Maximum Likelihood Estimation
  13. Likelihood Function Derivation
  14. Gradient descent and adaptive step size
  15. Example of ML refinement of SDP+O solution
  16. ...and 11 more sections

Figures (11)

  • Figure 1: Illustration of coordinate frames in a two-dimensional space
  • Figure 2: Illustration of azimuth and elevation components of a DOA measurement
  • Figure 3: Recovery of global coordinates of Agent B for recorded trajectories. Errors are $\sigma_\theta = 0.5^\circ$ and $\sigma_\phi = 2^\circ$ with respect to body fixed frame for the DOA measurements
  • Figure 4: Convergence of negative log-likelihood function using ML for real trajectory pair
  • Figure 5: Improvement in rotation error in degrees using ML for real trajectory pair
  • ...and 6 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2