Planning with Learned Dynamics: Guaranteed Safety and Reachability via Lipschitz Constants
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We present an approach for feedback motion planning of systems with unknown dynamics which provides guarantees on safety, reachability, and stability about the goal. Given a learned control-affine approximation of the true dynamics, we estimate the Lipschitz constant of the difference between the true and learned dynamics to determine a trusted domain for our learned model. Provided the system has at least as many controls as states, we further derive the conditions under which a one-step feedback law exists. This allows fora small bound on the tracking error when the trajectory is executed on the real system. Our method imposes a check for the existence of the feedback law as constraints in a sampling-based planner, which returns a feedback policy ensuring that under the true dynamics, the goal is reachable, the path is safe in execution, and the closed-loop system is 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.
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