Algebraic Methods for Nonlinear Dynamics and Control

Abstract

Some years ago, experiments with passive dynamic walking convinced me that finding efficient algorithms to reason about the nonlinear dynamics of our machines would be the key to turning a lumbering humanoid into a graceful ballerina. For linear systems (and nearly linear systems), these algorithms already exist—many problems of interest for design and analysis can be solved very efficiently using convex optimization. In this talk, I'll describe a set of relatively recent advances using polynomial optimization that are enabling a similar convex-optimization-based approach to nonlinear systems. I will give an overview of the theory and algorithms, and demonstrate their application to hard control problems in robotics, including dynamic legged locomotion, humanoids and robotic birds. Surprisingly, this polynomial (aka algebraic) view of rigid body dynamics also extends naturally to systems with frictional contact—a problem which intuitively feels very discontinuous.