Search at Scale: Improving Numerical Conditioning of Ergodic Coverage Optimization for Multi-Scale Domains

TL;DR

This work tackles the brittleness of kernel-based ergodic coverage caused by scale-dependent numerical conditioning. It introduces a scale-agnostic ergodic optimization built on Maximum Mean Discrepancy (MMD) with four core innovations: domain normalization to a dimensionless space, bandwidth annealing to preserve physical footprints, adaptive time stepping via a log-Delta parameter, and a log-surrogate MMD for stable, size-robust gradients. The method preserves the same optimal coverage as traditional MMD when feasible while dramatically improving conditioning and performance across vastly different spatial scales, demonstrated through simulations and real-world drone experiments. The practical impact spans micro to ocean-scale coverage tasks, enabling reliable, physically consistent ergodic planning for applications in inspection, monitoring, and search tasks.

Abstract

Recent methods in ergodic coverage planning have shown promise as tools that can adapt to a wide range of geometric coverage problems with general constraints, but are highly sensitive to the numerical scaling of the problem space. The underlying challenge is that the optimization formulation becomes brittle and numerically unstable with changing scales, especially under potentially nonlinear constraints that impose dynamic restrictions, due to the kernel-based formulation. This paper proposes to address this problem via the development of a scale-agnostic and adaptive ergodic coverage optimization method based on the maximum mean discrepancy metric (MMD). Our approach allows the optimizer to solve for the scale of differential constraints while annealing the hyperparameters to best suit the problem domain and ensure physical consistency. We also derive a variation of the ergodic metric in the log space, providing additional numerical conditioning without loss of performance. We compare our approach with existing coverage planning methods and demonstrate the utility of our approach on a wide range of coverage problems.

Figures (7)

• Figure 1: Scale-Agnostic Ergodic Search. Our single, scale-invariant optimizer enables proportional coverage guarantees for tasks spanning vastly different spatial scales. A trajectory is generated for a Franka Emika Panda robot inspecting a small rock (right) and for a drone surveying the Atlantic Ocean for global-scale weather monitoring (left).
• Figure 2: Sensitivity of Kernel-Based Methods to Length Scale Selection. Trajectories are generated for identical 2D environments at spatial scale = 1 with varying length scales (h). Length scale selection can lead to underfitting (if too high) or overfitting (if too low) to the utility samples, resulting in degraded performance or complete optimization failure.
• Figure 3: Bandwidth annealing preserves a physical footprint and stabilizes optimization at scale. Left: starting with a large normalized bandwidth $h_{0}=0.05$ yields broad similarity and well-conditioned gradients on a large domain. Right: after continuation to the target physical footprint of $1\,\mathrm{m}$ (which corresponds to $h_{\mathrm{norm}}^{\star}=1.5\times10^{-6}=h_{\mathrm{phys}}^{\star}/e^{2}$), the optimized trajectory (red, $T=500$ knots) achieves fine, proportional coverage of the bunny while constraints are enforced in physical units; domain extent $e=1000\,\mathrm{m}$.
• Figure 4: Adaptive vs. fixed time steps on a sparse, multi-scale target. Philippines silhouette (black) with identical trajectory length, $T=50$. Left (Optimized $\Delta t$): time is concentrated over dense archipelagos; the trajectory refines coverage on many small islands and traverses open water quickly. Right (Fixed $\Delta t$): uniform spacing oversamples empty water and undersamples intricate regions, yielding poorer effective resolution on the islands.
• Figure 5: Comparison of Scale-Invariant EMMD to State-of-the-Art Methods. Quantitative comparison of our scale-agnostic method against state-of-the-art coverage planners on the Stanford bunny mesh (2,503 pts). Trajectories are constrained to an identical maximal length that scales proportionally with environment size. Where existing ergodic exploration methods fail to adapt to vastly different spatial scales, and TSP solvers struggle to generate trajectories that prioritize coverage in length-constrained settings, our method consistently achieves high coverage and maintains numerical stability across scales, while maintaining similar optimization times and coverage quality to ergodic MMD trajectories at scale = 1.

Theorems & Definitions (20)

• Definition 1: Time–averaged (empirical) visitation
• Definition 2: Ergodicity
• Definition 3: Kernel mean embedding gretton2012
• Remark 1: Preview: scale sensitivity
• Definition 4: Maximum Mean Discrepancy
• Proposition 1: Expectation form of MMD hughes2025
• Proposition 2: Ergodic MMD objective
• Proposition 3: Finite-sample MMD estimator
• Proposition 4: Log-surrogate EMMD
• proof