Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses rigorously computed stability regions to build a sparse tree of LQR-stabilized trajectories. The region of attraction of this nonlinear feedback policy "probabilistically covers" the entire controllable subset of the state space, verifying that all initial conditions that are capable of reaching the goal will reach the goal. We numerically investigate the properties of this systematic nonlinear feedback design algorithm on simple nonlinear systems, prove the property of probabilistic coverage, and discuss extensions and implementation details of the basic algorithm.

We provide an amendment to the first theorem of "Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design" by Manchester & Slotine in the form of an additional technical condition required to show integrability of differential control signals. This technical condition is shown to be satisfied under the original assumptions if the input matrix is constant rank, and also if the strong conditions for a CCM hold. However a simple counterexample shows that if the input matrix drops rank, then the weaker conditions of the original theorem may not imply stabilizability of all trajectories. The remaining claims and illustrative examples of the paper are shown to remain valid with the new condition.

Abstract

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