Geometric Modeling and Attitude Stabilization of Quadcopters

The dynamics of an underactuated mechanical system, e.g. quadcopters, naturally evolve on a nonlinear manifold. Hence, differential geometric modeling concepts are better suited to arrive at the set of equations describing the dynamic behaviour of such systems. Resorting to a differential geometric approach, a generalized dynamical model for a quadcopter is proposed in this paper. The rotation matrices in forward kinematic map are represented using Euler angles. The modeling has been presented on a single chart. The Riemannian metric being a crucial ingredient for model development on a chart is derived from the total kinetic energy. A complete set of dynamic equations of motion neglecting external disturbances is derived for the quadcopter by representing the Levi-Civita affine connection in local coordinates. Further, based on the derived model, a proportional-derivative (PD) controller is designed for attitude stabilization. Simulation studies validate the proposed model and also demonstrate its robustness to wide variations in initial conditions.