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

Chong He, Mugilan Mariappan, Keval Vora, Mo Chen

TL;DR

This work addresses the prohibitive complexity of Hamilton-Jacobi reachability in high-dimensional systems by analyzing the leaking corner issue that arises when using dimensionality reduction. It formalizes a threshold-based definition and a necessary condition for leakage, and introduces a threshold-driven leak detection and a local updating procedure to recover accurate value functions while preserving the efficiency of low-dimensional decompositions. Numerical experiments on 2D and 6D problems using self-contained subsystem decomposition (SCSD) demonstrate that the approach locates leaking corners and restores the true value functions with substantial speedups compared to full-dimensional computation. The method enables reliable safety and liveness guarantees in higher-dimensional systems and is broadly applicable to other decomposition-based reductions beyond SCSD, with potential extensions to parallelism and learning-assisted updates.

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".

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

TL;DR

This work addresses the prohibitive complexity of Hamilton-Jacobi reachability in high-dimensional systems by analyzing the leaking corner issue that arises when using dimensionality reduction. It formalizes a threshold-based definition and a necessary condition for leakage, and introduces a threshold-driven leak detection and a local updating procedure to recover accurate value functions while preserving the efficiency of low-dimensional decompositions. Numerical experiments on 2D and 6D problems using self-contained subsystem decomposition (SCSD) demonstrate that the approach locates leaking corners and restores the true value functions with substantial speedups compared to full-dimensional computation. The method enables reliable safety and liveness guarantees in higher-dimensional systems and is broadly applicable to other decomposition-based reductions beyond SCSD, with potential extensions to parallelism and learning-assisted updates.

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".
Paper Structure (17 sections, 3 theorems, 41 equations, 3 figures, 6 tables, 1 algorithm)

This paper contains 17 sections, 3 theorems, 41 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. System Dynamics
  4. Value Functions in Hamilton-Jacobi Reachability
  5. Computational Dimensionality Reduction
  6. Value Function Reconstruction
  7. leaking corners
  8. Allowable Control and the "Leaking Corner"
  9. Correcting of Leaking Corners
  10. Detecting the Leaking Corners
  11. Local Updating Procedure
  12. Numerical Examples
  13. Running Example: 2D Single Integrator
  14. Single time step case
  15. 10 time steps case
  16. ...and 2 more sections

Key Result

Lemma 1

Suppose $\tilde{w}_1(t) = w_1^*(t):=(u_1(t), u_c(t))$ for all $t$ and $(\tilde{w}_1(\cdot), \tilde{w}_2(\cdot))$ is a pair of allowable control functions. Then, $\tilde{w}_2(t)=\tilde{w}_2^*(t)=(u_2(t), u_c(t))$, where $u_2(t)=\mathbf{0}$ for all $t$.

Figures (3)

  • Figure 1: The results from the low-dimensional computation are in the first row. The third row shows the local updated results in the full-dimensional space. The combined results, which equal the true results, are in the second row.
  • Figure 2: The 2 figures illustrate the value functions. The left figure displays the approximated value function, with the leaking corners" $\mathcal{L}(t)$—where the values deviate from the ground truth—highlighted in gray. The right figure shows the value function after applying our correction method, where no "leaking corners" remain, demonstrating the effectiveness of our method in aligning with the ground truth.
  • Figure 3: The figure demonstrates the local updating results for the backward computation of $0.02$ seconds with $\Delta=0.04$, $0.06$ seconds with $\Delta=0.1212$ and $0.1$ seconds with $\Delta=0.204$. The dimensions shown are $x$ and $y$. We take a slice of $v_x=-1$, $v_y=-1$, $\theta=0$, and $\omega=0.4$.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 3 more