Trajectory Planning for the Bidirectional Quadrotor as a Differentially Flat Hybrid System

The use of bidirectional propellers provides quadrotors with greater maneuverability which is advantageous in constrained environments. This paper addresses the development of a trajectory planning algorithm for quadrotors with bidirectional motors. Previous work has shown that the property of differential flatness can be leveraged for efficient trajectory planning. However, planners that leverage flatness for quadrotors fail at points where the acceleration of the center of mass is equal to gravity, i.e., when the vehicle experiences free fall. The central contribution of this paper is a flatness-based trajectory planning method that allows quadrotors to use bidirectional propellers and pass through the so-called free-fall singularity. We model our system as a differentially flat hybrid system with the aid of coordinate charts derived from the Hopf fibration and develop an algorithm that computes forward and reverse thrusts for each propeller, resulting in smooth trajectories everywhere in SE(3). We demonstrate the planner's versatility by planning knife-edge maneuvers and trajectories passing through the free-fall singularity, while transitioning from forward to reverse thrust.