Geometric Control and Trajectory Optimization for Bidirectional Thrust Quadrotors

Geometric control of quadrotors provides a way to control the 3D position and a yaw angle of the robot with a larger stability region than linearized controllers, but is still singular for some orientations. This singularity exists even if SO(3) is parameterized in a non-singular way such as with unit quaternions. Recent advances using the Hopf Fibration have eliminated these singularities using multiple coordinate charts and switching between them to avoid singularities. We further extend the envelope of geometric control for quadrotors to a system where the propellers can reverse direction and produce negative forces using additional charts. We show that our proposed controller is stable in hovering a robot upright and upside down. We then show how to extend trajectory generation using differentially flat coordinates to optimize trajectories using multiple charts to produce trajectories which can transition between upright and upside down.