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Online Learning and Coverage of Unknown Fields Using Random-Feature Gaussian Processes

Ruijie Du, Ruoyu Lin, Yanning Shen, Magnus Egerstedt

TL;DR

This work tackles simultaneous learning and coverage of an unknown spatial density by a multi-robot team when the density may be time-varying. It introduces Online Random Feature GP (O-RFGP) to enable scalable, real-time density estimation and combines it with Voronoi-based coverage and UCB sampling in the ORC framework. A regret-based performance guarantee is established for the time-invariant setting, and extensive simulations plus a hardware-in-the-loop experiment demonstrate that O-RFGP supports accurate tracking of dynamic fields while offering substantial computational savings over full GP methods. The approach enables real-time, adaptive exploration and coverage in dynamic environments, facilitating practical deployment of multi-robot systems for monitoring and sensing tasks.

Abstract

This paper proposes a framework for multi-robot systems to perform simultaneous learning and coverage of a domain of interest characterized by an unknown and potentially time-varying density function. To overcome the limitations of Gaussian Process (GP) regression, we employ Random Feature GP (RFGP) and its online variant (O-RFGP) which enables online and incremental inference. By integrating these with Voronoi-based coverage control and Upper Confidence Bound (UCB) sampling strategy, a team of robots can adaptively focus on important regions while refining the learned spatial field for efficient coverage. The incremental update mechanism of O-RFGP naturally supports time-varying environments, allowing efficient adaptation without retaining historical data. Furthermore, to the best of our knowledge, we provide the first theoretical analysis of online learning and coverage through a regret-based formulation, establishing asymptotic no-regret guarantees in the time-invariant setting. The effectiveness of the proposed framework is demonstrated through simulations with both time-invariant and time-varying density functions, along with a physical experiment with a time-varying density function.

Online Learning and Coverage of Unknown Fields Using Random-Feature Gaussian Processes

TL;DR

This work tackles simultaneous learning and coverage of an unknown spatial density by a multi-robot team when the density may be time-varying. It introduces Online Random Feature GP (O-RFGP) to enable scalable, real-time density estimation and combines it with Voronoi-based coverage and UCB sampling in the ORC framework. A regret-based performance guarantee is established for the time-invariant setting, and extensive simulations plus a hardware-in-the-loop experiment demonstrate that O-RFGP supports accurate tracking of dynamic fields while offering substantial computational savings over full GP methods. The approach enables real-time, adaptive exploration and coverage in dynamic environments, facilitating practical deployment of multi-robot systems for monitoring and sensing tasks.

Abstract

This paper proposes a framework for multi-robot systems to perform simultaneous learning and coverage of a domain of interest characterized by an unknown and potentially time-varying density function. To overcome the limitations of Gaussian Process (GP) regression, we employ Random Feature GP (RFGP) and its online variant (O-RFGP) which enables online and incremental inference. By integrating these with Voronoi-based coverage control and Upper Confidence Bound (UCB) sampling strategy, a team of robots can adaptively focus on important regions while refining the learned spatial field for efficient coverage. The incremental update mechanism of O-RFGP naturally supports time-varying environments, allowing efficient adaptation without retaining historical data. Furthermore, to the best of our knowledge, we provide the first theoretical analysis of online learning and coverage through a regret-based formulation, establishing asymptotic no-regret guarantees in the time-invariant setting. The effectiveness of the proposed framework is demonstrated through simulations with both time-invariant and time-varying density functions, along with a physical experiment with a time-varying density function.

Paper Structure

This paper contains 24 sections, 5 theorems, 69 equations, 15 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Density Learning
  4. Gaussian Process (GP)
  5. Random Feature Gaussian Process (RFGP)
  6. Online Random Feature Gaussian Process (O-RFGP)
  7. Learning-Aided Coverage Algorithm
  8. Experiments
  9. Simulations
  10. Time-invariant Density
  11. Time-varying Density
  12. Hardware-in-the-Loop Experiment
  13. Conclusion
  14. Regret Analysis
  15. Proof of Lemma \ref{['lemma1']}
  16. ...and 9 more sections

Key Result

lemma 1

Pick $\delta \in (0,1]$, and set $\beta_t = 2\log\left( |\mathcal{D}| \frac{\pi_t}{\delta}\right)$, where $\sum_{t \geq 1}\pi_t^{-1}=1, \pi_t >0$. Then, holds for any ${x}\in \mathcal{D}$ and any $t\geq 1$ with probability at least $1-\delta$.

Figures (15)

  • Figure 1: The final configuration (at time step $500$) of ten robots covering the time-invariant density distribution that is: (a) the ground truth, (b) learned using GP, (c) learned using RFGP, and (d) learned using O-RFGP. Areas in red (green) indicate higher (lower) importance.
  • Figure 2: The evolution of the locational cost and the MSE between the learned density and the true density with respect to time steps.
  • Figure 3: The heatmaps of ten robots covering the true time-varying density distribution at different time steps, where the black dots represent the positions of robots and the blue asterisks represent the centers of mass of their Voronoi cells.
  • Figure 4: The heatmaps of $1$ trial of ten robots covering the learned time-varying density distribution using O-RFGP.
  • Figure 5: The heatmaps of $1$ trial of ten robots covering the learned time-varying density distribution using GP based on the measurements collected at the current time step
  • ...and 10 more figures

Theorems & Definitions (9)

  • lemma 1
  • proof
  • theorem 1
  • proof
  • lemma 2
  • lemma 3
  • proof
  • theorem 2
  • proof