A Robust Controlled Backward Reach Tube with (Almost) Analytic Solution for Two Dubins Cars

Benchmark Proposal: We describe how a well-known backward reachability problem with nonlinear dynamics and adversarial inputs—based on a pursuit evasion game with two identical vehicles that have Dubins car dynamics—can be viewed as a robust controlled backward reach tube. The resulting set is nonconvex with a surface that is nondifferentiable in places, yet (mostly explicit) closed form solutions for points on the surface of this set have been derived based on a classical differential game analysis, and so these points can be sampled with high accuracy at arbitrary density. We propose this problem as a benchmark because few existing reachability algorithms can tackle robust controlled backward reach tubes despite their potential for proving the robust safety of systems, and this (almost) analytic solution exists against which to compare prospective solutions. We then describe some extensions to the problem to provide additional future challenges. Code is provided.