A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games

TLDR

The differential game formulation enables treatment of nonlinear systems with inputs and uncertain parameters, addressing limitations of alternative reachable set techniques. The paper describes and implements an algorithm to compute the reachable state set of a continuous dynamic game. The algorithm derives the reachable set as the zero sublevel set of the viscosity solution to a time‑dependent Hamilton‑Jacobi‑Isaacs PDE, leverages continuous solutions to apply level‑set methods for accurate approximations, and is implemented numerically and released online. Correctness is demonstrated on a two‑vehicle, three‑dimensional collision‑avoidance example with an analytic solution.

Abstract

We describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game. The algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation. While alternative techniques for computing the reachable set have been proposed, the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters. Because the time-dependent equation's solution is continuous and defined throughout the state space, methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions. A numerical implementation of our formulation is described and has been released on the web. Its correctness is verified through a two vehicle, three dimensional collision avoidance example for which an analytic solution is available.