  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. Recent methods either use tools from
machine learning to learn the observable functions or guess a suitable set of
observables that best describes the nonlinear dynamics. 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.
