From Singleton Obstacles to Clutter: Translation Invariant Compositional Avoid Sets

TL;DR

This paper develops a translation invariant compositional framework for obstacle avoidance, enabling efficient reuse of obstacle avoidance solutions and hierarchical composition to handle cluttered environments.

Contribution

It introduces a novel blockwise composition method that reduces conservatism and provides hierarchical certificates for obstacle avoidance in cluttered spaces.

Findings

Value function equals zero outside obstacles and negative inside

Reusability of obstacle avoidance templates under translation

Hierarchical composition reduces conservatism in clutter avoidance

Abstract

This paper studies obstacle avoidance under translation invariant dynamics using an avoid-side travel cost Hamilton Jacobi formulation. For running costs that are zero outside an obstacle and strictly negative inside it, we prove that the value function is non-positive everywhere, equals zero exactly outside the avoid set, and is strictly negative exactly on it. Under translation invariance, this yields a reuse principle: the value of any translated obstacle is obtained by translating a single template value function. We show that the pointwise minimum of translated template values exactly characterizes the union of the translated single-obstacle avoid sets and provides a conservative inner certificate of unavoidable collision in clutter. To reduce conservatism, we introduce a blockwise composition framework in which subsets of obstacles are merged and solved jointly. This yields a hierarchy of conservative certificates from singleton reuse to the exact clutter value, together with monotonicity under block merging and an exactness criterion based on the existence of a common clutter avoiding control. The framework is illustrated on a Dubins car example in a repeated clutter field.