Exploiting symmetry for discrete-time reachability computations

TL;DR

This paper introduces a symmetry-exploiting method for efficient backward reachable set computation in nonlinear discrete-time control systems, significantly reducing computational complexity by leveraging Cartan frame-based symmetry reduction.

Contribution

It presents a novel symmetry reduction technique using the Cartan frame for reachability analysis, enabling lower-dimensional computations without algebraic manipulation of system equations.

Findings

Achieved computational speedup in reachability analysis

Successfully computed backward reachable sets for a six-dimensional system

Demonstrated effectiveness on a reach-avoid game with Dubins vehicles

Abstract

We present a method of computing backward reachable sets for nonlinear discrete-time control systems possessing continuous symmetries. The starting point is a dynamic game formulation of reachability analysis where control inputs aim to maintain the state variables within a target tube despite disturbances. Our method exploits symmetry to compute the reachable sets in a lower-dimensional space, enabling a significant computational speedup. To achieve this, we present a general method for symmetry reduction based on the Cartan frame, which simplifies the dynamic programming iteration without algebraic manipulation of the state update equations. We illustrate the results by computing a backward reachable set for a six-dimensional reach-avoid game of two Dubins vehicles.