  In this paper, 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 observable
functions (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 observable functions 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 observable
functions. 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 observable functions converges
pointwise to the zero function, which allows us to select only a finite set of
observable functions 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.
