Gerardo De La Torre

Ian Abraham, Gerardo De La Torre, Todd D. Murphey

This paper explores the application of Koopman operator theory to the control of robotic systems. The operator is introduced as a method to generate data-driven models that have utility for model-based control methods. We then motivate the use of the Koopman operator towards augmenting model-based control. Specifically, we illustrate how the operator can be used to obtain a linearizable data-driven model for an unknown dynamical process that is useful for model-based control synthesis. Simulated results show that with increasing complexity in the choice of the basis functions, a closed-loop controller is able to invert and stabilize a cart- and VTOL-pendulum systems. Furthermore, the specification of the basis function are shown to be of importance when generating a Koopman operator for specific robotic systems. Experimental results with the Sphero SPRK robot explore the utility of the Koopman operator in a reduced state representation setting where increased complexity in the basis function improve open- and closed-loop controller performance in various terrains, including sand.

Gerardo De La Torre, Todd Murphey

Variational integrators are momentum-preserving and symplectic numerical methods used to propagate the evolution of Hamiltonian systems. In this paper, we introduce a new class of variational integrators that achieve fourth-order convergence despite having the same integration scheme as traditional second-order variational integrators. The new class of integrators are created by replacing a dynamical system's Lagrangian in the variational integration algorithm with its surrogate Lagrangian. By incorporating the surrogate Lagrangian the propagation errors induced by variational integrators, up to a given order, are eliminated. Furthermore, no assumption on the Lagrangian's structure is made and, therefore, the proposed approach is applicable to a large range of dynamical systems. In addition, surrogate variational integrators are also constructed for Hamiltonian systems subjected to holonomic constraints and external forces. Finally, the methodology is extended to derive higher-order surrogate variational integrators that achieve an arbitrary order of accuracy but retain second-order complexity in the integration scheme. Several numerical experiments are presented to demonstrate the efficacy of our approach.