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This paper studies the problem of control strategy synthesis for dynamicalsystems with differential constraints to fulfill a given reachability goalwhile satisfying a set of safety rules. Particular attention is devoted togoals that become feasible only if a subset of the safety rules are violated.The proposed algorithm computes a control law, that minimizes the level ofunsafety while the desired goal is guaranteed to be reached. This problem ismotivated by an autonomous car navigating an urban environment while followingrules of the road such as "always travel in right lane'' and "do not changelanes frequently''. Ideas behind sampling based motion-planning algorithms,such as Probabilistic Road Maps (PRMs) and Rapidly-exploring Random Trees(RRTs), are employed to incrementally construct a finite concretization of thedynamics as a durational Kripke structure. In conjunction with this, a weightedfinite automaton that captures the safety rules is used in order to find anoptimal trajectory that minimizes the violation of safety rules. We prove thatthe proposed algorithm guarantees asymptotic optimality, i.e., almost-sureconvergence to optimal solutions. We present results of simulation experimentsand an implementation on an autonomous urban mobility-on-demand system.

Incremental Sampling-based Algorithm for Minimum-violation Motion Planning