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Cooperative Localisation of a GPS-Denied UAV in 3-Dimensional Space Using Direction of Arrival Measurements

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

TL;DR

A novel approach for localising a GPS (Global Positioning System)-denied Unmanned Aerial Vehicle (UAV) with the aid of a GPS-equipped UAV in three-dimensional space and the Orthogonal Procrustes algorithm is employed.

Abstract

This paper presents a novel approach for localising a GPS (Global Positioning System)-denied Unmanned Aerial Vehicle (UAV) with the aid of a GPS-equipped UAV in three-dimensional space. The GPS-equipped UAV makes discrete-time broadcasts of its global coordinates. The GPS-denied UAV simultaneously receives the broadcast and takes direction of arrival (DOA) measurements towards the origin of the broadcast in its local coordinate frame (obtained via an inertial navigation system (INS)). The aim is to determine the difference between the local and global frames, described by a rotation and a translation. In the noiseless case, global coordinates were recovered exactly by solving a system of linear equations. When DOA measurements are contaminated with noise, rank relaxed semidefinite programming (SDP) and the Orthogonal Procrustes algorithm are employed. Simulations are provided and factors affecting accuracy, such as noise levels and number of measurements, are explored.

Cooperative Localisation of a GPS-Denied UAV in 3-Dimensional Space Using Direction of Arrival Measurements

TL;DR

A novel approach for localising a GPS (Global Positioning System)-denied Unmanned Aerial Vehicle (UAV) with the aid of a GPS-equipped UAV in three-dimensional space and the Orthogonal Procrustes algorithm is employed.

Abstract

This paper presents a novel approach for localising a GPS (Global Positioning System)-denied Unmanned Aerial Vehicle (UAV) with the aid of a GPS-equipped UAV in three-dimensional space. The GPS-equipped UAV makes discrete-time broadcasts of its global coordinates. The GPS-denied UAV simultaneously receives the broadcast and takes direction of arrival (DOA) measurements towards the origin of the broadcast in its local coordinate frame (obtained via an inertial navigation system (INS)). The aim is to determine the difference between the local and global frames, described by a rotation and a translation. In the noiseless case, global coordinates were recovered exactly by solving a system of linear equations. When DOA measurements are contaminated with noise, rank relaxed semidefinite programming (SDP) and the Orthogonal Procrustes algorithm are employed. Simulations are provided and factors affecting accuracy, such as noise levels and number of measurements, are explored.

Paper Structure

This paper contains 17 sections, 26 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Localisation with Noiseless Measurements
  4. Forming a system of linear equations
  5. Example with noiseless DOA measurements
  6. Nongeneric trajectories
  7. Localisation with Noisy DOA Measurements
  8. Quadratic constraints on entries of $\Psi$
  9. Formulation of the Semidefinite Program
  10. Rank Relaxation of Semidefinite Program
  11. Orthogonal Procrustes Problem
  12. Metrics for error in $\overline{R}$ and $\overline{T}$
  13. Simulation Results
  14. Example with noisy DOA measurements
  15. Monte Carlo Simulations on Random Trajectories
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Recovery of global coordinates of Agent B for example trajectories (error of $\sigma = 3^{\circ}$ in noisy case)
  • Figure 2: Median $d(\overline{R},{\bm{R^B_A}})$ vs. number of DOA measurements used to solve SDP+O.
  • Figure 3: Median $error(\overline{P_B^A})$ vs. number of DOA measurements used to solve SDP+O (average distance of 1.38km between agents).
  • Figure 4: Comparison of median $d(\overline{R},\bm{R_A^B})$ using SDP+O and LS+O methods for example trajectory ($K$ = 6).