  This research presents a novel, analytical, Koopman Operator based
formulation for position and attitude dynamics which can be used to derive
control strategies for underactuated systems. Compared to data driven Koopman
based techniques, the analytical approach presented in this work is model based
and allows for an exact linear representation of the original nonlinear
position and attitude dynamics. In fact, the resulting infinite dimensional
model, defined in the lifted state space, is linear in the autonomous component
and state dependent in the control. A boundary study is carried on to define
the range of validity of the finite truncation of the Koopman based model
followed by a controllability and stabilizability analysis to show the
feasibility of employing the derived model for control system design. Compared
to existing literature formulation, the presented model results in a better
approximation of the original dyanmics using a more compact truncation of the
lifted state space. Moreover, the model is derived using the Koopman approach
on the entirety of the dynamics and does not require the need of angular
velocity dynamic compensation. A case study involving an underactuated
quadrotor unmanned aerial vehicle (UAV) is provided to show that, for practical
use, a truncated subset of the infinite dimensional model, embeds most of the
original nonlinear dynamics and can be used to design linear control strategies
in the lifted space which results in nonlinear controllers in the original
state space. The main advantages of the presented approach reside in the
effective use of linear control strategies for nonlinear plats and the solution
of the underactuation problem employing a single control loop.
