From Singleton Obstacles to Clutter: Translation Invariant Compositional Avoid Sets

2026-03-23

AI Summary

This work addresses reachability analysis for collision avoidance in translation-invariant dynamical systems operating in complex obstacle-rich environments. The authors propose a scalable verification framework based on Hamilton–Jacobi equations augmented with an obstacle-avoidance side cost function. By exploiting the system’s translational invariance, a single-obstacle template value function is reused and combined via pointwise minima to construct multi-obstacle avoidance sets. A hierarchical block-composition scheme is further introduced, yielding a sequence of increasingly less conservative certificates that balance computational efficiency against solution accuracy. Theoretical analysis and simulations on Dubins vehicles demonstrate that the approach significantly reduces the conservatism inherent in traditional superposition strategies in cluttered scenes, with the computed inner approximation of the inevitable collision set monotonically converging toward the exact solution as obstacle blocks are merged.

Technology Category

• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Intelligent Robots: Motion and Path Planning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• obstacle avoidance
• translation invariance
• Hamilton-Jacobi
• avoid sets
• clutter

Innovation

• translation invariance
• Hamilton-Jacobi
• avoid sets
• compositional framework
• value function reuse