Search at Scale: Improving Numerical Conditioning of Ergodic Coverage Optimization for Multi-Scale Domains
Yanis Lahrach, Christian Hughes, Ian Abraham
AI summary
Problem
Kernel-based ergodic coverage planning methods are highly sensitive to domain scale and hyperparameter selection, causing numerical instability, brittle optimization, and loss of physical meaning when applied to multi-scale robotic tasks.
Approach
The authors reformulate the ergodic Maximum Mean Discrepancy objective using domain normalization, automatic bandwidth annealing, adaptive time-step optimization, and a log-space surrogate metric to decouple numerical conditioning from physical scale.
Key results
- Decouples kernel hyperparameters from physical domain scale via automatic normalization and geometric annealing
- Introduces an adaptive time-step formulation that optimizes differential constraints for scale-robust trajectory planning
- Derives a log-surrogate MMD metric that provides numerically stable, bounded gradients across orders of magnitude
- Demonstrates reliable coverage planning across micro-scale inspection and macro-scale weather monitoring tasks
Why it matters
Enables robust, general-purpose ergodic coverage planning for robotic exploration and inspection tasks spanning vastly different spatial scales without manual retuning.
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 hyperpa- rameters 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.