Guaranteed Ergodic Exploration Under Dynamic Disturbances Using Reachability Analysis

Core Concepts

This paper proposes a method called RAnGE (Reachability Analysis for Guaranteed Ergodicity) that can generate robust ergodic trajectories for autonomous exploration under dynamic disturbances by formulating the ergodic exploration problem as a differential game and solving it using Hamilton-Jacobi-Isaacs reachability analysis.

Abstract

The paper investigates performance guarantees on coverage-based ergodic exploration methods in environments containing disturbances. Ergodic exploration methods generate trajectories for autonomous robots such that time spent in an area is proportional to the utility of exploring in the area. However, providing formal performance guarantees for ergodic exploration methods is still an open challenge due to the complexities in the problem formulation.

The key highlights and insights are:

1. The authors propose to formulate ergodic search as a differential game, in which a controller and external disturbance force seek to minimize and maximize the ergodic metric, respectively.
2. Through an extended-state Bolza-form transform of the ergodic problem, the authors demonstrate it is possible to use techniques from reachability analysis to solve for optimal controllers that guarantee coverage and are robust against disturbances.
3. The authors leverage neural-network based methods to obtain approximate value function solutions for reachability problems that mitigate the increased computational scaling due to the extended state.
4. The authors are able to compute continuous value functions for the ergodic exploration problem and provide performance guarantees for coverage under disturbances.
5. Simulated and experimental results demonstrate the efficacy of the authors' approach to generate robust ergodic trajectories for search and exploration with external disturbance force.

Stats

The time-averaged spatial distribution spends time in each part of the search area proportional to the information to be gained in the area.
The ergodic metric is defined as the sum of squared differences between the time-averaged trajectory statistics and the information density in the Fourier domain.

Quotes

"Ergodic exploration methods generate trajectories for autonomous robots such that time spent in an area is proportional to the utility of exploring in the area."
"Through an extended-state Bolza-form transform of the ergodic problem, we demonstrate it is possible to use techniques from reachability analysis to solve for optimal controllers that guarantee coverage and are robust against disturbances."
"We leverage neural-network based methods to obtain approximate value function solutions for reachability problems that mitigate the increased computational scaling due to the extended state."

Key Insights Distilled From

RAnGE

by Henry Berger... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03186.pdf

Deeper Inquiries

How can the proposed method be extended to higher-dimensional exploration spaces and more complex disturbance models?

To extend the proposed method to higher-dimensional exploration spaces, one could increase the number of dimensions in the state space and augment the state variables accordingly. This would involve adding more Fourier modes for encoding the time-averaged trajectory statistics and information density. Additionally, the dynamics of the system and disturbance would need to be adjusted to accommodate the higher-dimensional space. Complex disturbance models could be incorporated by introducing more variability and dynamics into the disturbance forces, requiring a more sophisticated approach to calculating the worst-case disturbance. By adapting the augmented state formulation and the reachability analysis techniques to handle higher dimensions and more complex disturbances, the RAnGE framework could be effectively applied to these scenarios.

What are the limitations of the current value function approximation approach, and how could it be improved to handle longer time horizons?

One limitation of the current value function approximation approach is the constraint on the time horizon for training the DNN. Due to the complexity of the value function, training was only successful for a one-second time horizon. To handle longer time horizons, improvements could be made in the training process. One approach could involve breaking down the longer time horizon into smaller segments and training the DNN incrementally on each segment. This would allow for the gradual learning of the value function over the entire time horizon. Additionally, more sophisticated neural network architectures or training algorithms could be explored to improve the efficiency and accuracy of the value function approximation, enabling it to handle longer time horizons more effectively.

Could the RAnGE framework be applied to other trajectory-based optimization problems beyond ergodic exploration?

Yes, the RAnGE framework could be applied to a wide range of trajectory-based optimization problems beyond ergodic exploration. The key strength of the RAnGE framework lies in its ability to provide performance guarantees for coverage-based problems in the presence of disturbances. This capability is valuable in various applications such as path planning, motion control, and autonomous navigation. By formulating the optimization problem as a differential game and leveraging reachability analysis techniques, the RAnGE framework can be adapted to different scenarios where robust trajectory planning is essential. Whether in search and rescue missions, environmental monitoring, or robotic exploration tasks, the RAnGE framework can offer a systematic and robust approach to trajectory optimization under dynamic disturbances.