A Geometric Approach to Modeling, Simulation and Control

Abstract

In this work, we utilize discrete geometric mechanics to derive a 2nd-order variational integrator so as to simulate rigid body dynamics. The developed integrator is to simulate the motion of a free rigid body and a quad-rotor. We demonstrate the effectiveness of the simulator and its accuracy in long term integration of mechanical systems without energy damping. Furthermore, this work deals with the geometric nonlinear control problem for rigid bodies where backstepping controller is designed for full tracking of position and orientation. The attitude dynamics and control are defined on \textbf{SO(3)} to avoid singularities associated with Euler angles or ambiguities accompanying quaternion representation. The controller is shown to track large rotation attitude signals close to $180^\circ$ achieving almost globally asymptotic stability for rotations. A Quad-rotor is presented as an example of an under-actuated system with nonlinear model on which we apply the backstepping control law. In addition, an aerodynamic model aiming at deriving the aerodynamic forces and torques acting on rotors is added for realistic simulation purposes and to testify the effectiveness of the derived control method.