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

TL;DR

Addresses continuous exploration under battery constraints by extending ergodic search to energy-aware multi-agent planning with a battery SoC model and finite-horizon optimization. The method combines Gaussian Mixture spatial density $\phi$ with an ergodic metric $\mathcal{E}$ across agents and enforces at least one agent to satisfy $b_{SoC} \in (0,b_f]$ at horizon end. A second-order RC circuit ECM provides SOC dynamics $\dot{\mathbf{b}}(t) = A \mathbf{b}(t) + B I(t)$ and recharging $\mathbf{b}_{SoC} = \eta \mathbf{b}_{SoC} + \theta$, integrated into a finite-horizon optimization. Experiments in simulation and real Crazyflie MAVs demonstrate sustained coverage consistent with $\phi$ while ensuring uninterrupted exploration by at least one agent.

Abstract

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.

Figures (7)

• Figure 1: Example of energy-aware ergodic search. A set of agents explores $\mathcal{Q}$, focusing on areas with high information density $\mu_1$, $\mu_2$, $\mu_3$, and $\mu_4$, employing ergodic search. The exploration is continuous and uninterrupted so that there is always one agent exploring -- $\alpha_1$, whereas $\alpha_2$, $\alpha_3$, and $\alpha_4$ are recharging. The colors of the spheres indicate the state of charge.
• Figure 2: Abstract equivalent circuit model for state-of-charge estimation seewaldphdthesis. The model consists of a second-order resistor-capacitor circuit with two resistors $R_1$ and $R_2$ and two capacitors $C_1$ and $C_2$ in two separate circuit elements. An additional resistor $R$ is also employed.
• Figure 3: Information spatial distribution and search space in our experimental evaluation. The distribution consists of four Gaussians in a Gaussian mixture model $\phi$. The Gaussians are centered in $\mu_1$, $\mu_2$, $\mu_3$, and $\mu_4$, as depicted by the cyan empty squares. The search space $\mathcal{Q}$ is a three-by-three area. The resulting ergodic trajectories are expected to be such that the robot spends more time close to the Gaussians.
• Figure 4: Experimental evaluation of competing exploration with one Gaussian. Four agents $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ explore the space two-by-two first, and they compete for one area with high information density. The agents $\alpha_1$ blue and $\alpha_2$ red explore the space in the first horizon $t_0$ (left of the figure), spending most of the time close to the Gaussian. The agents then return to the charging station to recharge the battery. The other two agents $\alpha_3$ dark green and $\alpha_4$ magenta proceed in the next time horizon.
• Figure 5: Experimental evaluation of competing exploration with two Gaussians. Four agents explore the space and compete for two areas with high information density (instead of one in Fig. \ref{['fig:res2']}). The agents blue and red are selected first. One can note how both agents swap between the areas but spend most time near the Gaussians. At the end of the first horizon, they return to the charging stations with the other two agents dark green and magenta exploring the space.

Theorems & Definitions (2)

• Definition 2.1: Ergodicity
• Definition 2.2: Ergodic metric