Incremental Reference Generation for Nonsingular Control on $SE (3)$

Rotational singularities inherent to three different geometric controllers on $SE (3)$ are studied. A new singularity-free method of incrementally generating a reference rotation is proposed, removing a class of singularities without affecting proven stability properties of the closed-loop systems. If satisfying a bound on attitude rate and maximum loop rate in the reference generation, the proposed synthesis of a controlled rotation enables complex manoeuvres on the entirety of $SE (3)$, demonstrated with high-velocity loops on a toroid surface.