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.

翻译：本文研究平移不变动力学下基于回避侧行驶代价的Hamilton-Jacobi公式障碍物规避问题。对于障碍物外部为零、内部严格负的运行时成本，我们证明值函数处处非正，在回避集外严格为零，在回避集上严格为负。在平移不变性下，这产生重用原则：任何平移障碍物的值可通过平移单个模板值函数获得。我们证明平移模板值函数的逐点最小值精确刻画平移单障碍物回避集的并集，并提供杂乱环境中不可避碰撞的保守内证书。为降低保守性，我们引入分块组合框架，将障碍物子集合并求解。该框架建立从单障碍物重用到精确杂乱值的保守证书层级结构，同时具备块合并下的单调性，以及基于共同杂乱规避控制存在性的精确性判据。通过重复杂乱场中的Dubins车辆实例验证该框架。