  We present a method for feedback motion planning of systems with unknown
dynamics which provides probabilistic guarantees on safety, reachability, and
goal stability. To find a domain in which a learned control-affine
approximation of the true dynamics can be trusted, we estimate the Lipschitz
constant of the difference between the true and learned dynamics, and ensure
the estimate is valid with a given probability. Provided the system has at
least as many controls as states, we also derive existence conditions for a
one-step feedback law which can keep the real system within a small bound of a
nominal trajectory planned with the learned dynamics. Our method imposes the
feedback law existence as a constraint in a sampling-based planner, which
returns a feedback policy around a nominal plan ensuring that, if the Lipschitz
constant estimate is valid, the true system is safe during plan execution,
reaches the goal, and is ultimately invariant in a small set about the goal. We
demonstrate our approach by planning using learned models of a 6D quadrotor and
a 7DOF Kuka arm. We show that a baseline which plans using the same learned
dynamics without considering the error bound or the existence of the feedback
law can fail to stabilize around the plan and become unsafe.
