Planning with Learned Dynamics: Probabilistic Guarantees on Safety and Reachability via Lipschitz Constants

We present a method for feedback motion planning of systems with unknown\ndynamics which provides probabilistic guarantees on safety, reachability, and\ngoal stability. To find a domain in which a learned control-affine\napproximation of the true dynamics can be trusted, we estimate the Lipschitz\nconstant of the difference between the true and learned dynamics, and ensure\nthe estimate is valid with a given probability. Provided the system has at\nleast as many controls as states, we also derive existence conditions for a\none-step feedback law which can keep the real system within a small bound of a\nnominal trajectory planned with the learned dynamics. Our method imposes the\nfeedback law existence as a constraint in a sampling-based planner, which\nreturns a feedback policy around a nominal plan ensuring that, if the Lipschitz\nconstant estimate is valid, the true system is safe during plan execution,\nreaches the goal, and is ultimately invariant in a small set about the goal. We\ndemonstrate our approach by planning using learned models of a 6D quadrotor and\na 7DOF Kuka arm. We show that a baseline which plans using the same learned\ndynamics without considering the error bound or the existence of the feedback\nlaw can fail to stabilize around the plan and become unsafe.\n