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Energy-Aware Ergodic Search: Continuous Exploration for Multi-Agent Systems with Battery Constraints

Adam Seewald, Cameron J. Lerch, Marvin Chancán, Aaron M. Dollar, Ian Abraham

arXiv:2310.09470v2cs.RO

TL;DR

Battery-powered robots cannot sustain continuous exploration on one charge, while ergodic search does not account for their energy constraints. The paper adds an abstract battery model and battery-aware multi-agent ergodic planning; simulations and physical experiments indicate uninterrupted exploration with spatial coverage, subject to fixed charging stations and centralized information.

  • Problem

    Canonical ergodic search does not account for battery constraints, although continuous exploration may be required and cannot be sustained on a single battery charge.

    Source: §I · p. 1

  • Method

    The approach models battery state of charge and extends canonical ergodic search to battery-constrained multi-agent planning with a finite-horizon framework.

    Source: §I · p. 1, A. Ergodic search · p. 3

  • Results

    Simulations and real-world experiments indicate continuous, uninterrupted multi-agent coverage with at least one agent exploring and the spatial distribution satisfied.

    Source: §V · p. 6

  • Takeaways & Limitations

    The method demonstrates uninterrupted spatially distributed exploration by a multi-agent system under battery constraints.

    Source: §V · p. 6

    The current methods assume fixed-position charging stations and centralized information; energy optimality and online runtime are not addressed.

    Source: §V · p. 6

Abstract

from arXiv · show

Continuous exploration without interruption is important in scenarios such as search and rescue and precision agriculture, where consistent presence is needed to detect events over large areas. Ergodic search already derives continuous trajectories in these scenarios so that a robot spends more time in areas with high information density. However, existing literature on ergodic search does not consider the robot's energy constraints, limiting how long a robot can explore. In fact, if the robots are battery-powered, it is physically not possible to continuously explore on a single battery charge. Our paper tackles this challenge, integrating ergodic search methods with energy-aware coverage. We trade off battery usage and coverage quality, maintaining uninterrupted exploration by at least one agent. Our approach derives an abstract battery model for future state-of-charge estimation and extends canonical ergodic search to ergodic search under battery constraints. Empirical data from simulations and real-world experiments demonstrate the effectiveness of our energy-aware ergodic search, which ensures continuous exploration and guarantees spatial coverage.

I. INTRODUCTION

Ergodic search directs robots toward high-information regions, but battery limits prevent continuous exploration on a single charge. The paper combines battery-aware planning with ergodic search to maintain uninterrupted, spatially distributed coverage.

I. INTRODUCTION

  • Ergodic search allocates more exploration time to regions with high information density and passes quickly through low-information regions.Users can specify areas of interest, including search-and-rescue or precision-agriculture targets.Source: §I · p. 1
  • The paper asks whether battery use can be traded against coverage quality while keeping at least one agent exploring at all times.Continuous presence is often required in coverage-path-planning scenarios.Source: §I · p. 1
  • The approach derives an abstract battery model for future state-of-charge estimation and extends canonical ergodic search to battery-constrained multi-agent exploration.Its finite-horizon framework resembles a model predictive controller.Source: §I · p. 1
  • Simulations and real-world experiments indicate that the method maintains an exploring agent, satisfies the spatial distribution, and achieves uninterrupted exploration.Figure 1 depicts active and recharging agents; sphere colors encode state of charge.Source: §I · p. 1, Figure 1 · p. 1

II. PROBLEM FORMULATION

The paper formulates multi-agent exploration of a bounded space according to a spatial distribution, with at least one agent exploring continuously. It extends canonical ergodic search by imposing battery constraints.

II. PROBLEM FORMULATION

  • Here, “continuously” and “uninterruptedly” mean that at least one agent explores the space at all times, preserving a measurable level of coverage.Canonical ergodic search optimizes an agent’s control to maximize an ergodic metric in the spectral domain.Source: §II · p. 2
  • An ergodic search problem derives a control action that minimizes the ergodic metric for a normalized, nonnegative spatial distribution.The metric compares the robot’s time-average distribution with the spatial distribution using an L2 inner product.Source: §II · p. 2
  • Energy-aware ergodic search requires each agent’s time-average distribution to match the spatial distribution while bounding the multi-agent ergodic metric.The problem is to derive each agent’s control action under these conditions.Source: §II · p. 2
  • The formulation assumes one or more charging stations in the bounded space where agents can land and recharge.Wireless charging is given as an example.Source: §II · p. 2

III. METHODS

The methods target multi-agent exploration that is continuous, uninterrupted, and proportional to a spatial distribution. They address distribution-following and uninterrupted exploration in separate subsections.

III. METHODS

  • The methods section targets continuous, uninterrupted exploration by multiple agents in proportion to a spatial distribution.It divides the approach between achieving the spatial distribution in Sec. III-A and continuous exploration in Sec. III-B.Source: §III · p. 2
  • Sec. III-A addresses how agents achieve exploration proportional to the spatial distribution.The passage identifies the subsection’s aim but does not give its specific method.Source: §III · p. 2
  • Sec. III-B addresses how the multi-agent exploration remains continuous and uninterrupted.The passage identifies the subsection’s aim but does not give its specific method.Source: §III · p. 2

A. Ergodic search

The method represents the target spatial distribution spectrally and optimizes agent trajectories against it. Battery-state constraints and a coverage constraint extend this formulation to uninterrupted multi-agent exploration.

A. Ergodic search

  • The target spatial distribution is modeled as a Gaussian mixture, whose Gaussian components have centers, covariance matrices, and mixing coefficients.The distribution coefficients and trajectory coefficients are evaluated over the frequency-index set.Source: A. Ergodic search · pp. 2–3
  • The ergodic metric compares Fourier coefficients of the spatial distribution and trajectory, with lower frequencies receiving more weight.The bounded search space is represented as a symmetric set, and the coefficients are derived using Fourier series.Source: A. Ergodic search · pp. 2–3
  • Canonical ergodic control is formulated as an optimal control problem using the agent dynamics, trajectory endpoints, and a positive-definite control-penalizing matrix.The trajectory is optimized to minimize the ergodic metric over a finite interval.Source: A. Ergodic search · p. 3
  • The multi-agent extension gives each agent its own control penalty and permits different trajectory durations.The formulation extends the single-agent optimal-control problem and evaluates the ergodic metric across multiple agents.Source: A. Ergodic search · p. 3
  • The battery state is modeled alongside agent trajectories, and the optimization must keep it within its feasible domain while preserving an exploring agent.A desired final state of charge is specified; the optimization horizon is assumed known and may be based on an empirically collected discharge time.Source: A. Ergodic search · p. 3
  • The battery state-of-charge estimation model is depicted as a second-order resistor-capacitor circuit with two RC elements and an additional resistor.The circuit model is cited as [34].Source: Figure 2 · p. 3
  • The energy-aware optimal control problem includes a constraint ensuring that at least one agent explores throughout the optimization horizon.Additional constraints are optional; the ergodic metric is integrated into a constraint as proposed in.Source: A. Ergodic search · p. 3

B. Battery modeling

The battery model uses an abstract equivalent-circuit model to estimate battery metrics over time, supporting control decisions about exploration and recharging.

B. Battery modeling

  • The abstract equivalent-circuit model predicts battery metrics to identify when an agent is exploring and when it should recharge.The model is a second-order resistor-capacitor circuit.Source: B. Battery modeling · p. 3
  • The model uses a battery coefficient, RC-element resistances and capacitances, nominal capacity, and load-dependent internal current.Flight current is assumed constant based on empirical observations of the experimental aerial robot.Source: B. Battery modeling · p. 4
  • The battery state contains two RC-element voltages and normalized state of charge, which evolves from a charged initial value toward discharge.The voltages are measured across the first and second RC elements; state of charge is normalized to (0, 1].Source: B. Battery modeling · p. 4
  • Model parameters are identified to match physical agent behavior, and estimated state of charge informs control while voltage supports physical-system evaluation.Recharging is approximated empirically as bSoC = η bSoC + θ.Source: B. Battery modeling · p. 4

IV. EXPERIMENTAL RESULTS

The experiments evaluate battery-aware exploration with Crazyflie micro aerial vehicles in competing and cooperative Gaussian-coverage scenarios. In the cooperative case, alternating agent pairs sustain continuous exploration across fourteen horizons.

IV. EXPERIMENTAL RESULTS

  • The evaluation uses simulation and physical Crazyflie 2.0 micro aerial vehicles in a three-by-three-meter space with a known environment.The spatial distribution contains either one or two Gaussians in the first scenario and four in the second.Source: §IV · p. 4
  • In the two-Gaussian competing scenario, the blue and red agents swap between the two high-information areas while spending most of their time near the Gaussians.The agents return to charging stations at the end of the first horizon as the other pair explores.Source: Figure 5 · p. 4
  • Four agents sustain continuous, uninterrupted cooperative exploration across fourteen horizons by alternating exploration and charging.Pairs land at charging stations as the other pair proceeds on energy-aware trajectories.Source: Figure 6 · p. 5
  • In the competing one-Gaussian scenario, two selected MAVs explore near the information peak, return to recharge, and hand off exploration to the other pair.The pairs alternate exploration and recharging over successive horizons.Source: Competing exploration · pp. 5–6

Cooperative exploration

Cooperative exploration assigns agents to distinct information-rich regions and rotates them through charging stations. The reported scenario spans fourteen horizons, while the cost balances exploration priorities against battery state of charge.

Cooperative exploration

  • The mixing coefficient δ balances tradeoffs among individual Gaussians, agents, and battery state of charge.In the illustrated setup, α1 explores the area around µ2 in greater detail when δ2 > δ3; α2 shows an analogous pattern.Source: Cooperative exploration · p. 6
  • A prior approach makes the ergodic objective dynamic as information is gathered, whereas this work changes it with battery status.Source: Cooperative exploration · p. 6
  • The agents’ trajectories are selected to end at one another’s charging stations, with the final-point constraint enforcing station proximity.The constraint is evaluated to guarantee that both agents land on the other’s charging station.Source: Cooperative exploration · p. 6
  • Fourteen horizons show two agents landing at each other’s charging stations before the other two are selected to proceed on energy-aware ergodic trajectories.The agents’ past trajectories are shown in gray.Source: Cooperative exploration · p. 6

Ergodicity against battery state of charge

During cooperative exploration, ergodicity changes with agents’ movement and battery-related returns to charging stations. The reported plot separates per-horizon values from their average.

Ergodicity against battery state of charge

  • The figure plots ergodic metric evolution against battery state of charge for cooperative exploration.Source: Ergodicity against battery state of… · p. 6
  • Ergodicity decreases as agents move toward the Gaussian mixture model’s high-information regions, then oscillates as they move between Gaussians.The initial value mostly depends on the distance from charging stations to the Gaussian mixture components.Source: Ergodicity against battery state of… · p. 6
  • Average ergodicity descends during the first half of the horizon and rises in the second, peaking as discharged agents return to charging stations.The return is to areas described as having low information density.Source: Ergodicity against battery state of… · p. 6

V. CONCLUSION AND FUTURE DIRECTIONS

The paper reports empirical evidence of uninterrupted multi-agent coverage while satisfying the spatial distribution. It identifies fixed charging stations and centralized information as current limitations.

V. CONCLUSION AND FUTURE DIRECTIONS

  • Empirical evidence shows multi-agent exploration can maintain at least one exploring agent while satisfying the spatial distribution.Source: §V · p. 6
  • The method combines an abstract battery model with canonical ergodic search in an optimization framework resembling model predictive control.Source: §V · p. 6
  • Current methods assume fixed charging-station locations and centralized information; mobile stations and decentralized systems are planned extensions.Energy optimality and online runtime are also left for future investigation.Source: §V · p. 6

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