Spherical Inverted Pendulum on a Quadrotor UAV: A Flatness and Discontinuous Extended State Observer Approach

This article addresses the problem of balancing an inverted spherical pendulum on a quadrotor. The full dynamic model is obtained via the Euler-Lagrange formalism, where the dynamics of the pendulum is coupled to the dynamics of the quadrotor, taking as control inputs the torques associated with the yaw, roll, and pitch dynamics, and a control input for the vertical displacement in height. A trajectory tracking control scheme is proposed by means of an active disturbance rejection control based on a discontinuous extended state observer (ADRC-DESO) that allows controlling the system in the translational dynamics of the quadrotor including the rotational dynamics and the inverted pendulum dynamics. To address this problem, the dynamic model is linearized around an equilibrium point, taking into consideration that the system operates in close vicinity of the equilibrium points, thus considerably simplifying the dynamic model. Proving that the linear model is controllable and therefore differentiable flat, flat outputs are proposed around the displacements associated with the three cartesian axes of the Euclidean space, including a dynamic associated with the yaw dynamics of the quadrotor allowing to parameterize the full linear system. Simulation results as well as a convergence analysis validate the performance of the strategy.