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

TL;DR

This paper introduces a probabilistic motion planning method that uses Lipschitz constants to provide safety and reachability guarantees for systems with learned dynamics, demonstrated on a quadrotor and robotic arm.

Contribution

It develops a novel approach combining Lipschitz constant estimation with sampling-based planning to ensure safety and goal reachability with learned models.

Findings

The method guarantees safety and goal reachability under certain probabilistic conditions.

It outperforms baseline planners that ignore error bounds and feedback law existence.

Demonstrated successful planning on complex robotic systems.

Abstract

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.