Control of Quadrotors Using the Hopf Fibration on SO(3)

Abstract

Geometric, coordinate-free approaches are widely used to control quadrotors on the special Euclidean group (SE(3)). These approaches rely on the construction of an element of the special orthogonal group (SO(3)) from a desired thrust vector direction which lies on a sphere and a desired yaw angle which lies on a circle. The Hairy Ball Theorem can be applied to show that any construction of this type has to be discontinuous or degenerate somewhere. We propose a new geometric control algorithm based on the Hopf fibration, which allows us to place the point of discontinuity as far away from the hover configuration as possible and further than existing approaches. We then use multiple maps from to SO(3) to be able to control the quadrotor through any position and orientation. The proposed Hopf Fibration Control Algorithm (HFCA) is compared to existing geometric control algorithms in experiments and simulation. The HFCA employs multiple charts to allow the quadrotor to execute arbitrary dynamically feasible trajectories on SE(3), including those through configurations in which the vehicle is inverted.