Exploiting symmetry for discrete-time reachability computations
For researchers in control theory and robotics, this work provides a general symmetry reduction technique that simplifies dynamic programming for reachability analysis, though the method is domain-specific and incremental.
The paper presents a method for computing backward reachable sets for nonlinear discrete-time control systems with continuous symmetries by exploiting symmetry to reduce the problem to a lower-dimensional space, achieving significant computational speedup. The method is demonstrated on a six-dimensional reach-avoid game of two Dubins vehicles.
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.