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Voronoi-based Multi-Robot Formations for 3D Source Seeking via Cooperative Gradient Estimation

Lara Briñón-Arranz, Martin Abou Hamad, Alessandro Renzaglia

TL;DR

A comparative analysis shows how the proposed approach to localizing the source of a three-dimensional signal field with a team of mobile robots able to collect noisy measurements of its strength and share information with each other outperforms an alternative state-of-the-art solution.

Abstract

In this paper, we tackle the problem of localizing the source of a three-dimensional signal field with a team of mobile robots able to collect noisy measurements of its strength and share information with each other. The adopted strategy is to cooperatively compute a closed-form estimation of the gradient of the signal field that is then employed to steer the multi-robot system toward the source location. In order to guarantee an accurate and robust gradient estimation, the robots are placed on the surface of a sphere of fixed radius. More specifically, their positions correspond to the generators of a constrained Centroidal Voronoi partition on the spherical surface. We show that, by keeping these specific formations, both crucial geometric properties and a high level of field coverage are simultaneously achieved and that they allow estimating the gradient via simple analytic expressions. We finally provide simulation results to evaluate the performance of the proposed approach, considering both noise-free and noisy measurements. In particular, a comparative analysis shows how its higher robustness against faulty measurements outperforms an alternative state-of-the-art solution.

Voronoi-based Multi-Robot Formations for 3D Source Seeking via Cooperative Gradient Estimation

TL;DR

A comparative analysis shows how the proposed approach to localizing the source of a three-dimensional signal field with a team of mobile robots able to collect noisy measurements of its strength and share information with each other outperforms an alternative state-of-the-art solution.

Abstract

In this paper, we tackle the problem of localizing the source of a three-dimensional signal field with a team of mobile robots able to collect noisy measurements of its strength and share information with each other. The adopted strategy is to cooperatively compute a closed-form estimation of the gradient of the signal field that is then employed to steer the multi-robot system toward the source location. In order to guarantee an accurate and robust gradient estimation, the robots are placed on the surface of a sphere of fixed radius. More specifically, their positions correspond to the generators of a constrained Centroidal Voronoi partition on the spherical surface. We show that, by keeping these specific formations, both crucial geometric properties and a high level of field coverage are simultaneously achieved and that they allow estimating the gradient via simple analytic expressions. We finally provide simulation results to evaluate the performance of the proposed approach, considering both noise-free and noisy measurements. In particular, a comparative analysis shows how its higher robustness against faulty measurements outperforms an alternative state-of-the-art solution.
Paper Structure (8 sections, 2 theorems, 20 equations, 5 figures, 1 table)

This paper contains 8 sections, 2 theorems, 20 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation & Preliminaries
  3. Voronoi-based Multi-Robot Formation
  4. 3D Cooperative source-seeking
  5. Estimation error analysis
  6. Noise analysis
  7. Simulation Results and Analysis
  8. Conclusion

Key Result

Theorem 1

(from Lara2019TRO) Assume that $\sigma(\textbf{r}):\mathbb{R}^3\rightarrow\mathbb{R}^+$ satisfies Assumption assum:signal3D. Considering a team of $N=2n$ robots with $n\geq 2$ forming a configuration given by eq:Symmetric_formation and defining then it holds

Figures (5)

  • Figure 1: Cooperative source seeking in 3D: a formation of $7$ robots follows the estimated gradient of a signal field to converge toward its source.
  • Figure 2: Constrained CVT on a spherical surface with 30 generator points.
  • Figure 3: Evolution of the three components of the gradient estimate $\widehat{\nabla}\sigma(\textbf{c})$ for a team of $N=7$ robots with CVT-based Spherical formations compared to the real computed gradient of $\sigma_1(\textbf{r})$.
  • Figure 4: Evolution of the norm of the gradient estimation error, i.e., $\|\widehat{\nabla}\sigma(\textbf{c})\-\nabla\sigma(\textbf{c})|$, with a CVT-based Spherical formation for different values of the formation radius $D$.
  • Figure 5: Mean and standard deviation over $100$ trials of the distance between the center and the source location for both Symmetric and CVT-based formations in presence of noisy measurements with $\nu = 0.1$ for all robots (top) and with $\nu = 0.5$ only for one robot and $\nu = 0.1$ for the others (bottom).

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • proof
  • Remark 1