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Breaking Symmetry-Induced Degeneracy in Multi-Agent Ergodic Coverage via Stochastic Spectral Control

Kooktae Lee, Julian Martinez

TL;DR

The paper addresses gradient cancellation in classical SMC that causes directional degeneracy when agents start near symmetry points of a symmetric target density $\rho$, leading to axis-aligned stalling. It introduces a stochastic spectral control framework by smoothing the SMC input with $\varepsilon$ and adding isotropic noise to form the Itô dynamics $d\mathsf{x}_i = u_i dt + \sigma dW_i$. Theoretical results show almost-sure escape from zero-normal-component manifolds (Theorem 1) and infinite-time mean-square boundedness (Theorem 2) under contraction $k>0$. Simulations on symmetric multi-modal references validate that stochastic SMC mitigates transient degeneracy while preserving the ergodic coverage objective, and trajectories remain bounded within $\Omega$.

Abstract

Multi-agent ergodic coverage via Spectral Multiscale Coverage (SMC) provides a principled framework for driving a team of agents so that their collective time-averaged trajectories match a prescribed spatial distribution. While classical SMC has demonstrated empirical success, it can suffer from gradient cancellation, particularly when agents are initialized near symmetry points of the target distribution, leading to undesirable behaviors such as stalling or motion constrained along symmetry axes. In this work, we rigorously characterize the initial conditions and symmetry-induced invariant manifolds that give rise to such directional degeneracy in first-order agent dynamics. To address this, we introduce a stochastic perturbation combined with a contraction term and prove that the resulting dynamics ensure almost-sure escape from zero-gradient manifolds while maintaining mean-square boundedness of agent trajectories. Simulations on symmetric multi-modal reference distributions demonstrate that the proposed stochastic SMC effectively mitigates transient stalling and axis-constrained motion, while ensuring that all agent trajectories remain bounded within the domain.

Breaking Symmetry-Induced Degeneracy in Multi-Agent Ergodic Coverage via Stochastic Spectral Control

TL;DR

The paper addresses gradient cancellation in classical SMC that causes directional degeneracy when agents start near symmetry points of a symmetric target density , leading to axis-aligned stalling. It introduces a stochastic spectral control framework by smoothing the SMC input with and adding isotropic noise to form the Itô dynamics . Theoretical results show almost-sure escape from zero-normal-component manifolds (Theorem 1) and infinite-time mean-square boundedness (Theorem 2) under contraction . Simulations on symmetric multi-modal references validate that stochastic SMC mitigates transient degeneracy while preserving the ergodic coverage objective, and trajectories remain bounded within .

Abstract

Multi-agent ergodic coverage via Spectral Multiscale Coverage (SMC) provides a principled framework for driving a team of agents so that their collective time-averaged trajectories match a prescribed spatial distribution. While classical SMC has demonstrated empirical success, it can suffer from gradient cancellation, particularly when agents are initialized near symmetry points of the target distribution, leading to undesirable behaviors such as stalling or motion constrained along symmetry axes. In this work, we rigorously characterize the initial conditions and symmetry-induced invariant manifolds that give rise to such directional degeneracy in first-order agent dynamics. To address this, we introduce a stochastic perturbation combined with a contraction term and prove that the resulting dynamics ensure almost-sure escape from zero-gradient manifolds while maintaining mean-square boundedness of agent trajectories. Simulations on symmetric multi-modal reference distributions demonstrate that the proposed stochastic SMC effectively mitigates transient stalling and axis-constrained motion, while ensuring that all agent trajectories remain bounded within the domain.
Paper Structure (9 sections, 4 theorems, 40 equations, 2 figures)

This paper contains 9 sections, 4 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Gradient Cancellation in Classical SMC
  3. Example: Quadrimodal Symmetric Gaussian Reference
  4. Symmetry-Induced Invariant Manifolds in Spectral Coverage Control
  5. Stochastic Spectral Multiscale Control
  6. Perturbed SMC Dynamics
  7. Mean-Square Boundedness of Perturbed Multi-Agent SMC Dynamics
  8. Simulation Results
  9. Conclusion and Future Work

Key Result

Proposition 1

Consider the agent dynamics eq:agent_dynamics with control law eq:u_i on a rectangular domain $\Omega = [0,L_x]\times[0,L_y]$ using the Fourier basis eq:f_k, with a target density $\rho(\mathsf{x})$ that is symmetric about the vertical and horizontal midlines. Let an agent be initialized at a locati Specifically, for an agent starting on any of these sets with positions symmetric with respect to $

Figures (2)

  • Figure 1: SMC directional degeneracy example. Crosses: initial positions; circles: final positions. (a) Four agents from corners: axis- or diagonal-constrained motion, with the origin agent stalled. (b) Three agents from non-symmetric positions: prolonged axis-aligned motion, with the left agents eventually increasing $y$ toward the upper-right Gaussian, and the right-most agent moving downward at the boundary before gradually decreasing $x$.
  • Figure 2: Agent trajectories under stochastic spectral control. The undesired early-stage directional bias is eliminated due to perturbation, and all agents remain bounded within the domain.

Theorems & Definitions (11)

  • Remark 1: Axis-Constrained Motion and Stalling in SMC
  • Proposition 1: Symmetry-Induced Invariant Sets in 2D SMC
  • proof
  • Remark 2: Interpretation of Symmetry-Induced Behavior
  • Theorem 1: Almost-Sure Escape from Zero-Normal-Component Manifolds
  • proof
  • Remark 3: Practical Implementation
  • Theorem 2: Infinite-Time Mean-Square Boundedness of Perturbed Multi-Agent SMC with Contraction and Constant Noise
  • proof
  • Corollary 1: Deterministic Contraction Guarantees Uniform Boundedness
  • ...and 1 more