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Bearing-Only Solution for Fermat-Weber Location Problem: Generalized Algorithms

Nhat-Minh Le-Phan, Phuoc Doan Nguyen, Hyo-Sung Ahn, Minh Hoang Trinh

TL;DR

Novel algorithms are proposed that let the agent, which can be modeled as either a single-integrator or a double-integrator, follow the Fermat-Weber point of a group of stationary or moving beacons using only bearing measurements.

Abstract

This paper presents novel algorithms for the Fermat-Weber Location Problem, guiding an autonomous agent to the point that minimizes the weighted sum of Euclidean distances to some beacons using only bearing measurements. The existing results address only the simple scenario where the beacons are stationary and the agent is modeled by a single integrator. In this paper, we propose a number of bearing-only algorithms that let the agent, which can be modeled as either a single-integrator or a double-integrator, follow the Fermat-Weber point of a group of stationary or moving beacons. The theoretical results are rigorously proven using Lyapunov theory and supported with simulation examples.

Bearing-Only Solution for Fermat-Weber Location Problem: Generalized Algorithms

TL;DR

Novel algorithms are proposed that let the agent, which can be modeled as either a single-integrator or a double-integrator, follow the Fermat-Weber point of a group of stationary or moving beacons using only bearing measurements.

Abstract

This paper presents novel algorithms for the Fermat-Weber Location Problem, guiding an autonomous agent to the point that minimizes the weighted sum of Euclidean distances to some beacons using only bearing measurements. The existing results address only the simple scenario where the beacons are stationary and the agent is modeled by a single integrator. In this paper, we propose a number of bearing-only algorithms that let the agent, which can be modeled as either a single-integrator or a double-integrator, follow the Fermat-Weber point of a group of stationary or moving beacons. The theoretical results are rigorously proven using Lyapunov theory and supported with simulation examples.

Paper Structure

This paper contains 21 sections, 17 theorems, 54 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notations and Problem Statement
  3. Notations
  4. Problem Statement
  5. Single-Integrator Agent
  6. Stationary Beacons: The Gradient Control Law Revisited and Novel Results
  7. Exponential Stability
  8. Robustness against bearing error
  9. Finite Time Control Law
  10. Moving Beacons
  11. The beacons move with the same constant velocity
  12. The beacons move with the same bounded velocity
  13. Double-Integrator Agent
  14. Stationary Beacons
  15. Moving Beacons
  16. ...and 6 more sections

Key Result

Lemma 1

Plastria2011 There exists a unique Fermat-Weber point $\mathbf{p}^*$ that minimizes $f(\mathbf{p})$ if the following inequalities hold, for all $\mathbf{p} = \mathbf{p}_k$. The Fermat-Weber point $\mathbf{p}^*$ is different from all $\mathbf{p}_i$, $i = 1,\ldots, n$, and satisfies

Figures (7)

  • Figure 1: An example of the sets $\mathcal{C}$ and $\mathcal{B}_R$ of 3 beacons in 2D space, where the Fermat-Weber point is indexed by 0.
  • Figure 2: Simulation 1a: Single-integrator agent model and stationary beacons. Gradient control law (\ref{['eq:control_law_gradient_descent']}) with and without measuring noise (a)-(c)-(e), and comparison with finite time control law (\ref{['eq:finite_time']}) (b)-(d)-(f). (a) and (b) Trajectories of the agent, beacons, and the Fermat-Weber point. The initial and final positions of agents are marked with $'\triangledown'$ and $'\circ'$, respectively; (c) and (d) The function $f(\mathbf{p})$ versus time; (e) and (f) The norm of tracking error $\|\bm{\delta}(t)\|$ versus time.
  • Figure 3: Simulation 1b: Single-integrator agent model and moving beacons with constant velocity: (a) The trajectories of the agent, beacons, and Fermat-Weber point are colored blue, gray, and red, respectively. Their positions at $t = 0$ and $t = 10$ sec are marked with $'\triangledown'$ and $'\circ'$, respectively; (b) The function $f(\mathbf{p})$ versus time; (c) The magnitude of the tracking error $\|\bm{\delta}(t)\|$ versus time.
  • Figure 4: Simulation 1c: Single-integrator agent model and moving beacons with time-varying velocity under the control laws (\ref{['eq:bound moving first order']}) (a)-(c)-(e) and (\ref{['eq:adaptive moving approx first order']}) (b)-(d)-(f). (a) and (b) The trajectories of the agent, beacons, and Fermat-Weber point. Their positions at $t = 0$ and $t = 10$ sec are marked with $'\triangledown'$ and $'\circ'$, respectively; (c) and (d) The function $f(\mathbf{p})$ versus time; (e) and (f) The magnitude of the tracking error $\|\bm{\delta}(t)\|$ versus time.
  • Figure 5: Simulation 2a: Double-integrator agent model and stationary beacons. (a) The trajectories of the agents. The positions of the beacons, the agent, and the Fermat-Weber point are colored in black, blue, and red, respectively. Their positions at $t = 0$ and $t = 15$ sec are marked with $'\triangledown'$ and $'\circ'$, respectively; (b) The function $f(\mathbf{p})$ versus time; (c) The norm of tracking error $\|\bm{\delta}(t)\|$ versus time; (d) The velocities of the agent along the x, y, and z axes.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Corollary 1
  • Theorem 1
  • Remark 1
  • Lemma 4
  • Theorem 2
  • Remark 2
  • Lemma 5
  • ...and 10 more