Ergodic exploration of dynamic distribution

TL;DR

This work tackles dynamic search in oceanic environments by coupling a time-varying target probability density $m(x,t)$ with a gradient-based ergodic controller (HEDAC) to guide multi-agent search. The framework integrates an advection–diffusion PDE for $m$, incorporating flow drift and sensing as a sink, with a potential-field PDE that yields collision-free, ergodic trajectories for UAVs; diffusion accounts for drift and sensing uncertainties. Two scenarios—synthetic cavity flow and a realistic Unije Channel SAR mission—demonstrate substantial gains over static-probability baselines, including improved target detection rates and accurate survey completion metrics, even under delayed starts and multi-phase operations. The approach remains computationally feasible in real time on standard hardware, enabling practical deployment for ocean SAR and resource-aware mission planning.

Abstract

This research addresses the challenge of performing search missions in dynamic environments, particularly for drifting targets whose movement is dictated by a flow field. This is accomplished through a dynamical system that integrates two partial differential equations: one governing the dynamics and uncertainty of the probability distribution, and the other regulating the potential field for ergodic multi-agent search. The target probability field evolves in response to the target dynamics imposed by the environment and accomplished sensing efforts, while being explored by multiple robot agents guided by the potential field gradient. The proposed methodology was tested on two simulated search scenarios, one of which features a synthetically generated domain and showcases better performance when compared to the baseline method with static target probability over a range of agent to flow field velocity ratios. The second search scenario represents a realistic sea search and rescue mission where the search start is delayed, the search is performed in multiple robot flight missions, and the procedure for target drift uncertainty compensation is demonstrated. Furthermore, the proposed method provides an accurate survey completion metric, based on the known detection/sensing parameters, that correlates with the actual number of targets found independently.

Figures (6)

• Figure 1: Cavity lid-driven flow field represented with a vector field, streamlines, and velocity intensity contour plot. Positions of simulated search targets at $t=0$ are displayed as red dots.
• Figure 2: Comparison of the static probability search and the proposed method. The figure features the search domain at $t=500$ s, with the underlay of the potential field displayed with the purple-white shaded heat-map for both cases. It displays agents' trajectories and the simulated search targets. Under the domain plots, the figure includes a search performance analysis graphs containing survey accomplishment $\eta$ and target detection rate $\kappa$ metrics in relation to search time. The difference in the agent behavior recognized by observing the agents' trajectories, where the agents in the static probability scenario focus their search only on the area covered by the initial target probability distribution $m_0$, while in the agents' motion in the dynamic case. The search performance improvement can be seen by comparing the target detection rate between the cases, furthermore, the survey accomplishment in the proposed method stays true to the target detection rate.
• Figure 3: Survey accomplishment rate and target detection rate in relation to $\lambda$ which represents the ratio of UAV velocity and average flow field velocity. The dashed line at $\lambda=50$ represents the average estimated value for maritime search mission utilizing multirotor UAVs. The graph was generated using the synthetic cavity test case for different $\lambda$ values, by scaling the flow field. It showcases increased performance of the proposed method, compared to the baseline static probability approach, in the range of realistic values at different times of the search $T$.
• Figure 4: Visual representation of the flow field and simulated targets' positions at the start of advection $t=0$, and at the end of the search $t=T$ for the Unije Channel test case. The executed search mission is started at $t=3$ h.
• Figure 5: The Unije Channel test case search analysis including the comparison for both cases with and without uncertainty compensation. The thick black lines on the domain plots represent the coastline and the breaks in those lines represent the open water. The blue arrows in the background represent the flow field vectors, while the purple-white filled contour plot is a representation of the potential field that governs the motion of UAV search agents. The situation represents the search state at $t=21150$ s, and the UAVs' trajectories during the final search phase are represented with the thin black lines inside the search domain. The graphs below the domain plot showcase the target detection rate and survey accomplishment metrics throughout the whole duration of the search.