  This paper presents an active learning strategy for robotic systems that
takes into account task information, enables fast learning, and allows control
to be readily synthesized by taking advantage of the Koopman operator
representation. We first motivate the use of representing nonlinear systems as
linear Koopman operator systems by illustrating the improved model-based
control performance with an actuated Van der Pol system. Information-theoretic
methods are then applied to the Koopman operator formulation of dynamical
systems where we derive a controller for active learning of robot dynamics. The
active learning controller is shown to increase the rate of information about
the Koopman operator. In addition, our active learning controller can readily
incorporate policies built on the Koopman dynamics, enabling the benefits of
fast active learning and improved control. Results using a quadcopter
illustrate single-execution active learning and stabilization capabilities
during free-fall. The results for active learning are extended for automating
Koopman observables and we implement our method on real robotic systems.
