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From Singleton Obstacles to Clutter: Translation Invariant Compositional Avoid Sets

Prashant Solanki, Jasper Van Beers, Coen De Visser

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.

From Singleton Obstacles to Clutter: Translation Invariant Compositional Avoid Sets

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.
Paper Structure (24 sections, 9 theorems, 97 equations, 4 figures)

This paper contains 24 sections, 9 theorems, 97 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Controlled dynamics
  4. Template obstacle and translated copies
  5. Running cost and translation structure
  6. Template, clutter, and composed value functions
  7. Translation and Singleton Composition of Avoid Value Functions
  8. Sign characterization of the template value
  9. Translation of values and avoid sets
  10. Singleton obstacle min-composition
  11. Comparison with the true clutter value
  12. Hierarchical Blockwise Composition for Clutter Avoidance
  13. Partitions of the obstacle set and induced block values
  14. Exact sign characterization of blockwise composition
  15. Conservatism with respect to the true clutter value
  16. ...and 9 more sections

Key Result

Proposition 1

For every $t\in[0,T)$, and In particular, $V(t,x)\le 0$ for all $(t,x)\in[0,T]\times\mathbb R^n$.

Figures (4)

  • Figure 1: Translated template value on the $1\times 2$ tiled configuration at the slice $\theta=\pi$.
  • Figure 2: Value learned directly on the corresponding $1\times 2$ two-obstacle configuration at the slice $\theta=\pi$.
  • Figure 3: Absolute pointwise error between the translated template value and the directly trained $1\times 2$ value at the slice $\theta=\pi$.
  • Figure 4: Hybrid closed-loop rollout for the Dubins-car example in a tiled clutter field. Circular obstacles are centered at the lattice points \ref{['eq:dubins_example_centers']}. The blue marker denotes the initial state and the orange marker denotes the terminal state at which the goal region is reached.

Theorems & Definitions (19)

  • Proposition 1: Negative sublevel and zero level of $V$
  • Proof 1
  • Theorem 1: Translation of single-obstacle value and avoid set
  • Proof 2
  • Theorem 2: Exact sign characterization of the composed value
  • Proof 3
  • Corollary 1: Sign characterization of the true clutter value
  • Proof 4
  • Proposition 2: Conservatism of the composed value
  • Proof 5
  • ...and 9 more