  Navigation safety is critical for many autonomous systems such as
self-driving vehicles in an urban environment. It requires an explicit
consideration of boundary constraints that describe the borders of any
infeasible, non-navigable, or unsafe regions. We propose a principled
boundary-aware safe stochastic planning framework with promising results. Our
method generates a value function that can strictly distinguish the state
values between free (safe) and non-navigable (boundary) spaces in the
continuous state, naturally leading to a safe boundary-aware policy. At the
core of our solution lies a seamless integration of finite elements and
kernel-based functions, where the finite elements allow us to characterize
safety-critical states' borders accurately, and the kernel-based function
speeds up computation for the non-safety-critical states. The proposed method
was evaluated through extensive simulations and demonstrated safe navigation
behaviors in mobile navigation tasks. Additionally, we demonstrate that our
approach can maneuver safely and efficiently in cluttered real-world
environments using a ground vehicle with strong external disturbances, such as
navigating on a slippery floor and against external human intervention.
