  Autonomous mobile robots must maintain safety, but should not sacrifice
performance, leading to the classical reach-avoid problem: find a trajectory
that is guaranteed to reach a goal and avoid obstacles. This paper addresses
the near danger case, also known as a narrow gap, where the agent starts near
the goal, but must navigate through tight obstacles that block its path. The
proposed method builds off the common approach of using a simplified planning
model to generate plans, which are then tracked using a high-fidelity tracking
model and controller. Existing approaches use reachability analysis to
overapproximate the error between these models and ensure safety, but doing so
introduces numerical approximation error conservativeness that prevents
goal-reaching. The present work instead proposes a Piecewise Affine Reach-avoid
Computation (PARC) method to tightly approximate the reachable set of the
planning model. PARC significantly reduces conservativeness through a careful
choice of the planning model and set representation, along with an effective
approach to handling time-varying tracking errors. The utility of this method
is demonstrated through extensive numerical experiments in which PARC
outperforms state-of-the-art reach avoid methods in near-danger goal reaching.
Furthermore, in a simulated demonstration, PARC enables the generation of
provably-safe extreme vehicle dynamics drift parking maneuvers. A preliminary
hardware demo on a TurtleBot3 also validates the method.
