Learning Provably Robust Motion Planners Using Funnel Libraries

Abstract

This paper presents an approach for learning motion planners that are accompanied with probabilistic guarantees of success on new environments that hold uniformly for any disturbance to the robot's dynamics within an admissible set. We achieve this by bringing together tools from generalization theory and robust control. First, we curate a library of motion primitives where the robustness of each primitive is characterized by an over-approximation of the forward reachable set, i.e., a "funnel". Then, we optimize probably approximately correct (PAC)-Bayes generalization bounds for training our planner to compose these primitives such that the entire funnels respect the problem specification. We demonstrate the ability of our approach to provide strong guarantees on two simulated examples: (i) navigation of an autonomous vehicle under external disturbances on a five-lane highway with multiple vehicles, and (ii) navigation of a drone across an obstacle field in the presence of wind disturbances.

Disturbance-Parametrized Robust Lattice-Based Motion Planning

“…The constraint (18c) ensures the feasibility of the state transitions, which guarantees that the uncertain state x(τ ), associated with the solution x(t) of COCP (18), satisfies x(τ ) ∈ R(A(x k )), ∀τ ∈ [0, T ] (necessary to satisfy Corollary 1). The solution of the COCP (18) constitutes a motion primitive m p (defined in (16)), in which the motion is governed by the following…”

“…Although these approaches are efficient robust motion-planning solutions, they are not suitable to handle systems with nonlinear models, which is mostly the case in practise. The approaches in [12]- [16] address the robust motion-planning problem for nonlinear systems with uncertainties. A learning-based robust motion planner is proposed in [12], which utilizes contraction theory to generate a safety certificate for trajectories of a nonlinear system affected by additive disturbances.…”

“…A control barrier function based robust RRT strategy is proposed for uncertain nonlinear systems in [14] and the performance is validated through numerical simulations. The strategies in [15], [16] generate funnel libraries to handle uncertainties in robust motion planning. These strategies can efficiently handle nonlinear systems affected by external disturbances during runtime.…”

“…Furthermore, all the solutions guarantee robustness based on worst-case disturbance knowledge and is apparently incapable of utilizing additional disturbance information (for example local estimates of disturbance affecting a certain region of the navigable state space) during planning. In the robust motion planning strategies in [12]- [16], the planning performance can be improved apparently only if the worst-case bound of the disturbance is decreased.…”

Sim-to-Lab-to-Real: Safe Reinforcement Learning with Shielding and Generalization Guarantees

“…Majumdar et al [14] apply the PAC-Bayes framework in policy learning settings and provide generalization guarantees for control policies in unseen environments. Follow-up work has provided strong guarantees in different robotics settings including for learning neural network policies for vision-based control [16,31,32]. However, previous work has not adopted safety-related policy architectures nor considered safety during training.…”