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Measure Preserving Flows for Ergodic Search in Convoluted Environments

Albert Xu, Bhaskar Vundurthy, Geordan Gutow, Ian Abraham, Jeff Schneider, Howie Choset

TL;DR

This work presents a modified ergodic metric using the Laplace-Beltrami eigenfunctions to capture map geometry and obstacle locations within the ergodic metric and introduces an approach to generate trajectories that minimize the ergodic metric while guaranteeing obstacle avoidance using measure-preserving vector fields.

Abstract

Autonomous robotic search has important applications in robotics, such as the search for signs of life after a disaster. When \emph{a priori} information is available, for example in the form of a distribution, a planner can use that distribution to guide the search. Ergodic search is one method that uses the information distribution to generate a trajectory that minimizes the ergodic metric, in that it encourages the robot to spend more time in regions with high information and proportionally less time in the remaining regions. Unfortunately, prior works in ergodic search do not perform well in complex environments with obstacles such as a building's interior or a maze. To address this, our work presents a modified ergodic metric using the Laplace-Beltrami eigenfunctions to capture map geometry and obstacle locations within the ergodic metric. Further, we introduce an approach to generate trajectories that minimize the ergodic metric while guaranteeing obstacle avoidance using measure-preserving vector fields. Finally, we leverage the divergence-free nature of these vector fields to generate collision-free trajectories for multiple agents. We demonstrate our approach via simulations with single and multi-agent systems on maps representing interior hallways and long corridors with non-uniform information distribution. In particular, we illustrate the generation of feasible trajectories in complex environments where prior methods fail.

Measure Preserving Flows for Ergodic Search in Convoluted Environments

TL;DR

This work presents a modified ergodic metric using the Laplace-Beltrami eigenfunctions to capture map geometry and obstacle locations within the ergodic metric and introduces an approach to generate trajectories that minimize the ergodic metric while guaranteeing obstacle avoidance using measure-preserving vector fields.

Abstract

Autonomous robotic search has important applications in robotics, such as the search for signs of life after a disaster. When \emph{a priori} information is available, for example in the form of a distribution, a planner can use that distribution to guide the search. Ergodic search is one method that uses the information distribution to generate a trajectory that minimizes the ergodic metric, in that it encourages the robot to spend more time in regions with high information and proportionally less time in the remaining regions. Unfortunately, prior works in ergodic search do not perform well in complex environments with obstacles such as a building's interior or a maze. To address this, our work presents a modified ergodic metric using the Laplace-Beltrami eigenfunctions to capture map geometry and obstacle locations within the ergodic metric. Further, we introduce an approach to generate trajectories that minimize the ergodic metric while guaranteeing obstacle avoidance using measure-preserving vector fields. Finally, we leverage the divergence-free nature of these vector fields to generate collision-free trajectories for multiple agents. We demonstrate our approach via simulations with single and multi-agent systems on maps representing interior hallways and long corridors with non-uniform information distribution. In particular, we illustrate the generation of feasible trajectories in complex environments where prior methods fail.
Paper Structure (15 sections, 3 theorems, 17 equations, 7 figures, 3 tables)

This paper contains 15 sections, 3 theorems, 17 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Ergodic search amidst obstacles
  4. Laplace-Beltrami basis functions for non-Euclidean spaces
  5. Theory
  6. Measure-preserving flows
  7. Infinite-Limit Ergodicity from Random Flows
  8. Ergodic metric on arbitrary 2D metric space
  9. Ergodic metric minimization for finite time horizon
  10. Results and Discussion
  11. Obstacle-free Map Experiment Results
  12. Maze Map Experiment Results
  13. Rooms Map Experiment Results
  14. Conclusions and Future Work
  15. Proof of \ref{['thm:measure-preserving-divergence']}

Key Result

theorem thmcountertheorem

Let $(X,\mathscr B,\mu)$ be a measure space (we assume $X\subseteq \mathbb R^n$). Let $p(x)$ be a continuous and differentiable probability density function associated with $\mu$ such that $\mu(A)=\int_A p(x)dx$. The set of all measure-preserving flows on this measure space can be described through

Figures (7)

  • Figure 1: An interior environment with two rooms connected by a narrow hallway. An ergodicity-minimizing trajectory generated using measure-preserving vector fields translates the three robots, starting from the red X's, to the other room through the narrow hallway for efficient information gathering.
  • Figure 2: Example measure-preserving flows for a uniform information distribution on a maze-like space, drawn in blue. (A) Starting from an initial condition marked by the brown X, the brown line represents a trajectory traced by following the flow field lines. Note that the flow lines form stationary limit cycles, as shown by the faded brown extension, so the robot will simply trace the same cycle repeatedly. (B) By cleverly switching the underlying flow field when the robot reaches the magenta diamond, we can escape the limit cycle and cover more space. (C) We can repeat this process when the robot reaches the purple plus, extending the trajectory to cover more of the map. This process can be repeated again when the robot reaches the white star ad infinitum.
  • Figure 3: Effect of modifying the ergodic metric to use the Laplace-Beltrami eigenfunctions. (A) A C-shaped map with an information peak (yellow) and test curve (red) are separated by a gap. Both basis function choices depend only on the geometry of the obstacle map and can be compared in figures B and D. The Laplace-Beltrami eigenfunctions (B) have the correct sense of continuity influenced by the boundaries of the free space, while the ordinary Fourier basis functions (D) try to maintain continuity within the gap. As a result, the Laplace-Beltrami loss landscape (C) smoothly pushes the test curve around the central obstacle, while the Fourier loss landscape (E) gets trapped in a local minima.
  • Figure 4: Obstacle maps used for testing. From left to right, we have the obstacle-free map, the Maze map, and the Rooms map.
  • Figure 5: Results for obstacle-free square map comparing our and the baseline methods. From left to right, we have the trajectories generated for the uniform, nonuniform, and multi-agent nonuniform test cases. The ergodic metric value for each trajectory is shown in the figure legend.
  • ...and 2 more figures

Theorems & Definitions (10)

  • definition thmcounterdefinition: Flow
  • definition thmcounterdefinition: Measure-preserving
  • theorem thmcountertheorem: Measure-preserving flows
  • proof
  • definition thmcounterdefinition: Random flow
  • theorem thmcountertheorem: Unique Ergodicity
  • proof
  • corollary thmcountercorollary: Constructing a Uniquely Ergodic Random Flow
  • proof
  • proof