An equivalence of ADRC and flat filters: an application to a quadrotor UAV with a spherical inverted or suspended pendulum

In this paper, the problem of transporting an inverted or a suspended pendulum on a quadrotor, subject to external crosswind disturbances is addressed. The proposed solution is based on the flatness property of the tangent linearisation model, obtained around a stationary equilibrium point of the system; such an approach allows to define auxiliary outputs called flat outputs associated with the horizontal translational dynamics of the systems, thus simplifying the system model. The simplified dynamics of the two incremental flat outputs are used for the design of a Reduced Order Extended State Observer-based Active Disturbance Rejection Control (ROESO-ADRC) scheme in a trajectory tracking task. The ROESO-ADRC scheme estimates, and properly compensates, the unknown total disturbance; constituted by the crosswind forces, regarded as exogenous disturbances, and neglected nonlinear terms, regarded as endogenous disturbances. An equivalence between the ROESO-ADRC with flat filters is shown. The stability of the trajectory and estimation errors is formally studied and numerical simulations are carried out to verify the performance of the controlled system.