Fast Reachable Set Approximations via State Decoupling Disturbances

TL;DR

This paper introduces a novel method for approximating reachable sets in high-dimensional systems by simplifying dynamics through state decoupling, balancing accuracy and computational efficiency.

Contribution

The paper proposes a flexible approach that simplifies system dynamics by treating state variables as disturbances, enabling faster reachable set approximations.

Findings

The method is conservative in the approximation direction.

Demonstrated on a four-dimensional plane model.

Reduces computational complexity for high-dimensional systems.

Abstract

With the recent surge of interest in using robotics and automation for civil purposes, providing safety and performance guarantees has become extremely important. In the past, differential games have been successfully used for the analysis of safety-critical systems. In particular, the Hamilton-Jacobi (HJ) formulation of differential games provides a flexible way to compute the reachable set, which can characterize the set of states which lead to either desirable or undesirable configurations, depending on the application. While HJ reachability is applicable to many small practical systems, the curse of dimensionality prevents the direct application of HJ reachability to many larger systems. To address computation complexity issues, various efficient computation methods in the literature have been developed for approximating or exactly computing the solution to HJ partial differential equations, but only when the system dynamics are of specific forms. In this paper, we propose a flexible method to trade off optimality with computation complexity in HJ reachability analysis. To achieve this, we propose to simplify system dynamics by treating state variables as disturbances. We prove that the resulting approximation is conservative in the desired direction, and demonstrate our method using a four-dimensional plane model.