解决多尺度覆盖优化的数值不稳问题，提升算法适应性。
Search at Scale: Improving Numerical Conditioning of Ergodic Coverage Optimization for Multi-Scale Domains
- 基于MMD提出自适应尺度无关优化方法
- 通过超参数退火与对数空间度量提升数值稳定性
- 适用于复杂约束下的多尺度覆盖任务
近期的遍历覆盖规划方法在应对多种几何覆盖问题及通用约束方面展现出潜力，但对问题空间的数值尺度高度敏感。其根本挑战在于，随着尺度变化，尤其是存在动态非线性约束时，基于核函数的优化公式会变得脆弱且数值不稳定。本文提出一种基于最大均值差异（MMD）的尺度无关、自适应遍历覆盖优化方法。该方法能自动求解微分约束的尺度，并通过退火策略调节超参数，确保物理一致性。此外，我们推导了对数空间下的遍历度量变体，在不损失性能的前提下增强数值条件。我们在多种覆盖问题上对比了现有方法，验证了所提方法的有效性。
原文摘要 · Abstract (English)
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.
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