Applied Math Colloquium
Path planning in control systems: The Combinatorics of high order iterated integrals
Fri, 02/08/2019 - 3:00pm
We consider path planning problems for highly nonlinear control systems that are governed by finite dimensional differential equations. Think of parallel parking as a prototypical, simple example that every driver understands. Common strategies involve nilpotent approximating systems and steering with sinusoids. For low order terms, both iterated Lie brackets and iterated integral functionals, for most practical purposes, are characterized by their finely homogeneous degrees. But starting with length five, the internal combinatorial structures present new challenges to deciding controllability, and in steering algorithms, e.g. the labeled map to independent Lie brackets. We review some combinatorial and algebraic structures, including Zinbiel and combinatorial Hopf algebras. For systems with two control inputs we present novel choices for steering with sinusoids, that are orders of magnitude more efficient than the most general known algorithm for generating motions in most directions.