Ergodic Trajectory Design by Learned Pushforward Maps: Provable Coverage via Conditional Flow Matching

2026-05-13

AI Summary

This work addresses the continuous trajectory ergodic coverage problem under energy limitations, no-fly zones, and acceleration constraints by proposing a novel decoupling approach that separates density matching from ergodicity enforcement. The method analytically constructs uniformly distributed trajectories in a latent space to guarantee ergodicity and employs an offline-learned pushforward map to transform the latent occupancy distribution into the target spatial density. Built upon conditional flow matching (CFM) and Lipschitz neural vector fields, the framework incorporates multiple differentiable constraints via soft penalties, enabling a single training phase to yield asymptotically ergodic trajectories that satisfy all constraints and support unlimited tasks and multi-agent deployment. Theoretical analysis establishes bounds on acceleration-energy trade-offs, an ergodic convergence rate of $O(1/\sqrt{K})$, and approximation error guarantees, with overall performance end-to-end estimable and certifiable through CFM training metrics.

Abstract

Designing continuous trajectories whose time-averaged occupancy provably matches a prescribed spatial density (the \emph{ergodic coverage} problem) is central to UAV-assisted data collection and sensing, robotic exploration, and mobile monitoring. For flying agents in particular, this challenge is acute: trajectories must balance coverage fidelity against tight energy budgets, no-fly zones, and acceleration limits. Existing methods either re-optimize each trajectory online (with cost growing in the horizon and re-running for every target, agent, and realization) or rely on bespoke analytical constructions that must be re-derived for each new constraint. We propose a \emph{epushforward} framework that decouples ergodicity from density matching: an analytic latent trajectory provides exact uniform ergodicity on a simple annular domain, and a single map, learned offline by optimal-transport conditional flow matching, transports this latent occupancy onto the prescribed target density. The composed trajectory is then asymptotically ergodic with respect to the learned pushforward distribution, with deviation from the target controlled by the flow-matching training loss. Once trained for a given target density and constraint set, the map serves an unbounded number of trajectories and a multi-agent fleet without per-agent retraining, and many differentiable operational constraints (no-fly zones, acceleration ceilings, or fairness penalties) enter as additive soft penalties in the training loss without re-deriving the design. We prove three results (an acceleration-energy bound, an $O(1/\sqrt{K})$ ergodic convergence rate in the number of trajectory cycles $K$, and an approximation-error bound) that combine into an end-to-end coverage bound estimable from CFM training diagnostics (certified given an architectural Lipschitz bound on $v_θ$).

Problem

• ergodic coverage
• trajectory design
• spatial density
• UAV
• motion constraints

Innovation

• pushforward map
• conditional flow matching
• ergodic trajectory
• optimal transport
• coverage guarantee