On quaternion-based attitude control and the unwinding phenomenon

The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Often, quaternion-based feedbacks require an additional mechanism that lifts a continuous attitude path to the unit quaternion space. When this mechanism is memoryless, it has a limited domain where it remains injective and leads to discontinuities when used globally. To remedy this limitation, we propose a hybrid-dynamic algorithm for lifting a continuous attitude path to the unit quaternion space. We show that this hybrid-dynamic mechanism allows us to directly translate quaternion-based controllers and their asymptotic stability properties (obtained in the unit-quaternion space) to the actual rigid-body-attitude space. We also show that when quaternion-based controllers are not designed to account for the double covering of the rigid-body-attitude space by a unit-quaternion parameterization, they can give rise to the unwinding phenomenon, which we characterize in terms of the projection of asymptotically stable sets.