ACD-EDMD: Analytical Construction for Dictionaries of Lifting Functions in Koopman Operator-based Nonlinear Robotic Systems

Abstract: Koopman operator theory has been gaining momentum for model extraction, planning, and control of data-driven robotic systems. The Koopman operator's ability to extract dynamics from data depends heavily on the selection of an appropriate dictionary of lifting functions. In this paper we propose ACD-EDMD, a new method for Analytical Construction of Dictionaries of appropriate lifting functions for a range of data-driven Koopman operator based nonlinear robotic systems. The key insight of this work is that information about fundamental topological spaces of the nonlinear system (such as its configuration space and workspace) can be exploited to steer the construction of Hermite polynomial-based lifting functions. We show that the proposed method leads to dictionaries that are simple to implement while enjoying provable completeness and convergence guarantees when observables are weighted bounded. We evaluate ACD-EDMD using a range of diverse nonlinear robotic systems in both simulated and physical hardware experimentation (a wheeled mobile robot, a two-revolute-joint robotic arm, and a soft robotic leg). Results reveal that our method leads to dictionaries that enable high-accuracy prediction and that can generalize to diverse validation sets. The associated GitHub repository of our algorithm can be accessed at \url{https://github.com/UCR-Robotics/ACD-EDMD}.