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Semi-discrete Optimal Transport for Time-Varying Multi-Agent Coverage Control
Italo Napolitano, Mario di Bernardo
TL;DR
Time-varying coverage has been studied in Voronoi frameworks, but semi-discrete optimal transport coverage has been limited to stationary targets. This paper formulates instantaneous Wasserstein tracking with coupled agent and dual-variable dynamics, and reports exponential tracking along with improved simulation performance over quasi-static OT and Voronoi methods.
Problem
Time-varying coverage has been developed in Voronoi frameworks, while existing semi-discrete OT coverage approaches assume stationary target densities.
Source: §1 · p. 1Method
The paper formulates instantaneous semi-discrete Wasserstein minimization for an evolving target and couples agent-position and dual-variable dynamics with feedback and feedforward terms.
Source: §1 · p. 2, §4.2 · p. 6Results
The proposed dynamics yield exponential tracking of time-varying Laguerre barycenters, and simulations report superior tracking to quasi-static OT and Voronoi-based methods.
Source: §1 · p. 2, §4.2 · p. 7Takeaways & Limitations
The framework tracks time-varying barycenters while maintaining mass balance as the target density evolves.
Source: §6 · p. 9The analysis assumes nonsingularity of M; further analysis of this condition is left for future work.
Source: §4.1 · p. 5
Abstract
from arXiv · showhide
Coverage control algorithms have traditionally focused on static target densities, where agents optimally cover a fixed spatial distribution. However, many applications, such as environmental monitoring, surveillance, and adaptive sensing, involve time-varying densities. While time-varying coverage has been studied in Voronoi-based frameworks, extending recent optimal transport formulations of static coverage control to time-varying target densities remains an open problem. This paper presents a semi-discrete optimal transport framework for time-varying coverage control, in which agents track the first-order optimality conditions associated with minimizing the instantaneous Wasserstein distance from an evolving target density. The proposed approach is based on a coupled system of differential equations governing agent positions and the dual variables defining Laguerre regions. The resulting optimality residuals converge exponentially to zero, with global convergence established for one-dimensional domains. We also derive decentralized approximations and numerical simulations demonstrate improved tracking performance over quasi-static and Voronoi-based methods.
1 Introduction
Coverage control has largely addressed stationary densities, although many applications require tracking changing targets. This paper extends semi-discrete optimal transport coverage to time-varying densities.
1 Introduction
- Prior time-varying coverage extensions used Voronoi-based frameworks, while the cited semi-discrete OT coverage approaches assumed stationary densities [16] [18].Source: §1 · p. 1
- The paper formulates time-varying coverage as instantaneous semi-discrete OT, minimizing Wasserstein distance between agents’ empirical measure and an evolving target measure.The formulation treats the problem as tracking rather than regulation.Source: §1 · p. 2
- Coupled agent and dual-variable dynamics combine proportional feedback with feedforward compensation to track time-varying Laguerre barycenters, with decentralized approximations also derived.The authors report exponential tracking of the barycenters.Source: §1 · p. 2
- Numerical simulations show superior tracking performance over quasi-static OT and Voronoi-based methods.Source: §1 · p. 2
2 Background
The background connects semi-discrete optimal transport to coverage control through weighted spatial partitions and their mass and barycenter properties.
2 Background
- The background defines notation for vectors, matrices, time derivatives, target quantities, probability measures, and Wasserstein distances used in the paper.Source: §2 · p. 2
2.1 Semi-discrete Optimal Transport
- Semi-discrete OT compares a discrete measure with a continuous one, and its transport map partitions the domain into Laguerre cells that generalize Voronoi cells through additive weights.Source: §2.1 · p. 2
2.2 Coverage Control
- Coverage control seeks agent positions minimizing density-weighted coverage cost; static semi-discrete OT coverage uses equal-mass Laguerre regions and converges toward their barycenters.Unlike Voronoi cells, Laguerre partitions use dual variables to enforce mass constraints and adapt to the target density.Source: §2.2 · pp. 2–3
- The paper extends static OT coverage to evolving densities, making the Laguerre partition time-dependent and requiring agents to track a moving saddle point.Source: §2.2 · p. 3
3 Modeling and Problem Statement
The problem statement asks agents to minimize instantaneous Wasserstein mismatch to an evolving target density under stated regularity, actuation, and information assumptions.
3 Modeling and Problem Statement
- At each time, agents minimize the Wasserstein mismatch between their empirical measure and the time-varying target measure, rather than optimizing over an entire time horizon.The stated objective contrasts with the horizon-wide Benamou-Brenier formulation.Source: §3 · p. 3
- The target density is positive on compact convex support and is assumed C1 and Lipschitz in time.Source: §3 · p. 3
- Agents are fully actuated, with their velocities set by control inputs, and are assumed never to overlap.Source: §3 · p. 3
- Each agent needs sufficient sensing and communication to evaluate the target density and its time derivative across its Laguerre cell, exchange states with Laguerre neighbors, and compute cell mass and barycenter.The text notes that required information may come from sensing, model-based estimation, or communication.Source: §3 · p. 3
4 Time-varying Optimal Transport Strategy
The section formulates time-varying coverage as instantaneous semi-discrete optimal transport and tracks a primal-dual stationary point. It develops centralized dynamics and weaker decentralized approximations for the evolving Laguerre regions.
4.1 Proposed solution
- The proposed continuous-time dynamics track a time-varying primal-dual stationary point, whose optimality conditions pair agent barycenters with equal-mass Laguerre regions.Only differences between adjacent dual variables determine the regions, so one dual variable is fixed to zero to remove the additive ambiguity.Source: §4.1 · pp. 3–4
- With M^-1 well-defined, the centralized dynamics use time-derivative feedforward to make position and mass errors converge exponentially at rates Kx and Kϕ almost everywhere.The theorem assumes topological changes occur at a locally finite set of times.Source: §4.1 · p. 5
- The centralized theorem assumes M is nonsingular, while errors in an unavailable target-density time derivative act as bounded disturbances.Further analysis of the nonsingularity assumption is left for future work.Source: §4.1 · p. 5
- In one dimension, the unique ordered optimum is tracked up to agent permutations, and the optimality residuals converge exponentially to zero.The optimal partition is formed by equal-mass quantile intervals.Source: §4.1 · pp. 5–6
4.2 Distributed implementation
- The simpler distributed law numerically estimates barycenter derivatives, tracks the barycentric first-order condition exponentially, and keeps the mass residual bounded.Its feedforward derivative can be estimated from the barycenter history using finite differences or a dynamic differentiator.Source: §4.2 · pp. 6–7
- A truncated Neumann-series approximation makes control inputs depend on data propagated through k-hop Laguerre neighborhoods, assuming the relevant spectral radius is below one.Each multiplication uses exchanges with one-hop Laguerre neighbors.Source: §4.2 · p. 7
5 Numerical validation
The simulations assess how the proposed controllers track changing densities and compare their performance with quasi-static OT and Voronoi-based control. The centralized controller performs best overall, while decentralized and gradient-based variants offer approximate tracking.
5 Numerical validation
- In the 1D example, five agents compress and spread with the changing variance while tracking the oscillating mean without lag.The Wasserstein distance oscillates in response to the varying variance.Source: §5 · p. 8
- TVOT-C's improvement over TVOT-D1 and TVOT-G is 2.34% and 7.41%, respectively; all proposed solutions outperform Voronoi-based approaches in Wasserstein error.TVOT-D1 also achieves barycenter errors 72.7% smaller than OTCC without feedforward.Source: §5 · p. 8
- With feedforward compensation, TVOT-G has significantly smaller errors than OTCC across the tested density-drift velocities and gains, and its mass error tends asymptotically to zero.Without feedforward, acceptable accuracy requires large gains and slow drift; otherwise persistent lag produces substantial steady-state errors.Source: §5 · p. 8
- Under ±20% target-velocity estimation perturbations, error remains bounded for suitable parameters; across N = 50–500 agents, 1D 2-Wasserstein error decays approximately as 1/N.Increasing Kx reduces sensitivity to estimation errors.Source: §5 · pp. 8–9
- TVOT-G agents contract and expand as a unimodal Gaussian shrinks and grows, then divide into two groups after it splits into a bimodal Gaussian and track the separating modes.At the isolated nondifferentiable switching instant t = 3.14 a.u., the strategy continues operating without significant degradation.Source: §5 · pp. 7–8
- In Fig. 2, TVOT-C errors converge exponentially to zero; TVOT-D4 approaches centralized performance, while TVOT-D1 has larger errors and TVOT-G the largest peak errors.Transient spikes occur near the instant when the reference density is nondifferentiable, after which both TVOT-C errors rapidly return to zero.Source: §5 · p. 8, Figure 2 · p. 9
6 Conclusions
The paper formulates time-varying multi-agent coverage within semi-discrete optimal transport and derives coupled dynamics for agent positions and Laguerre-region dual variables. The conclusion identifies practical constraints and decentralized approximation analysis as directions for future work.
6 Conclusions
- The framework uses coupled dynamics for agent positions and Laguerre dual variables to track time-varying densities.The dynamics combine proportional feedback and feedforward compensation to track time-varying barycenters while maintaining mass balance.Source: §6 · p. 9
- Future work includes practical constraints, further characterization of decentralized Neumann-truncation approximation error with hardware experiments, and entropy-regularized formulations.The listed practical constraints include collision avoidance, communication delays, and obstacles.Source: §6 · p. 9
A Simulation parameters
The simulation parameters, quantitative comparison tables, and additional simulations are available at the cited supplementary location.
A Simulation parameters
- Simulation parameters, quantitative comparison tables, and additional simulations are provided in the supplementary materials.The passage points to https://shorturl.at/vgPu8.Source: A Simulation parameters · p. 9