  We present a method for providing statistical guarantees on runtime safety
and goal reachability for integrated planning and control of a class of systems
with unknown nonlinear stochastic underactuated dynamics. Specifically, given a
dynamics dataset, our method jointly learns a mean dynamics model, a
spatially-varying disturbance bound that captures the effect of noise and model
mismatch, and a feedback controller based on contraction theory that stabilizes
the learned dynamics. We propose a sampling-based planner that uses the mean
dynamics model and simultaneously bounds the closed-loop tracking error via a
learned disturbance bound. We employ techniques from Extreme Value Theory (EVT)
to estimate, to a specified level of confidence, several constants which
characterize the learned components and govern the size of the tracking error
bound. This ensures plans are guaranteed to be safely tracked at runtime. We
validate that our guarantees translate to empirical safety in simulation on a
10D quadrotor, and in the real world on a physical CrazyFlie quadrotor and
Clearpath Jackal robot, whereas baselines that ignore the model error and
stochasticity are unsafe.
