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

Abstract

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.

Controller design under safety specifications for a class of bounded hybrid automata

“…This is however impossible since (w(t), x(t)) ∈ ∂C jt (q(t)) ⊂ C jt (q(t)) c , and establishes (9). Notice that if k = 0 then (9) achieves the desired contradiction.…”

“…Notice that if k = 0 then (9) achieves the desired contradiction. We consider therefore the case when k > 0 and show that (9) implies that…”

“…Ideally, provably safe controllers should also be least restrictive, which means in the context of a driver-assist system that the controller constrains the possible actions of the human driver as little as possible. Due to the computational complexity of the task, the design of provably safe, least restrictive controllers remains a challenge and can in general only be done approximately, see for instance [9,3,11]. However it has been shown that a number of ground transportation systems have the so-called inputoutput order preserving property, in which case exact solutions are possible, see [4,12,5] and the references therein.…”

Incremental Sampling-Based Algorithms for a Class of Pursuit-Evasion Games

“…Some examples include multiagent collision avoidance [3], air combat [4], and path planning in an adversarial environment [5]. The class of pursuit-evasion games that will be considered in this paper is summarized as follows.…”

“…The former reduce the problem to a sequence of finite dimensional optimization problems through discretization [8], whereas the latter solves the Isaacs partial differential equation with boundary conditions using, e.g., multiple shooting [9,10], collocation [11,12], or level-set methods [3,13].…”