Threshold Strategy for Leaking Corner-Free Hamilton-Jacobi Reachability with Decomposed Computations

TL;DR

This paper addresses the leaking corner issue in high-dimensional Hamilton-Jacobi reachability computations, proposing a threshold strategy and local correction method to improve accuracy while maintaining efficiency.

Contribution

It introduces a necessary condition for the leaking corner issue and a local updating method to correct value functions in decomposed HJ reachability computations.

Findings

The proposed method effectively corrects value functions in numerical simulations.

The approach maintains computational efficiency of dimensionality reduction techniques.

Validated with subsystem decomposition, applicable to other reduction methods.

Abstract

Hamilton-Jacobi (HJ) Reachability is widely used to compute value functions for states satisfying specific control objectives. However, it becomes intractable for high-dimensional problems due to the curse of dimensionality. Dimensionality reduction approaches are essential for mitigating this challenge, whereas they could introduce the ``leaking corner issue", leading to inaccuracies in the results. In this paper, we define the ``leaking corner issue" in terms of value functions, propose and prove a necessary condition for its occurrence. We then use these theoretical contributions to introduce a new local updating method that efficiently corrects inaccurate value functions while maintaining the computational efficiency of the dimensionality reduction approaches. We demonstrate the effectiveness of our method through numerical simulations. Although we validate our method with the self-contained subsystem decomposition (SCSD), our approach is applicable to other dimensionality reduction techniques that introduce the ``leaking corners".