Koopman Operator Based Modeling for Quadrotor Control on SE(3)

In this letter, we propose a Koopman operator based approach to describe the nonlinear dynamics of a quadrotor on SE(3) in terms of an infinite-dimensional linear system which evolves in the space of observables (lifted space) and which is more appropriate for control design purposes. The major challenge when using the Koopman operator is the characterization of a set of observables that can span the lifted space. Most of the existing methods either start from a set of dictionary functions and then search for a subset that best fits the underlying nonlinear dynamics or they rely on machine learning algorithms to learn these observables. Instead of guessing or learning the observables, in this work we derive them in a systematic way for the quadrotor dynamics on SE(3). In addition, we prove that the proposed sequence of observables converges pointwise to the zero function, which allows us to select only a finite set of observables to form (an approximation of) the lifted space. Our theoretical analysis is also confirmed by numerical simulations which demonstrate that by increasing the dimension of the lifted space, the derived linear state space model can approximate the nonlinear quadrotor dynamics more accurately.