Synergistic potential functions for hybrid control of rigid-body attitude

Achieving global asymptotic stabilization of rigid body attitude is impossible using smooth feedback; however, this obstruction can be overcome using a hybrid controller that coordinates a "synergistic" family of potential functions and their corresponding feedbacks. In this paper, we show that it is impossible to construct a synergistic family of potential functions from the widely used class of "modified trace functions," despite the fact that one can choose a subset of these functions whose only common critical point is the identity element. With this as motivation, we introduce a parametrized diffeomorphism that is capable of altering the critical behavior of potential functions and generating a synergistic family, paving the way for global asymptotic attitude stabilization of rigid-body attitude by hybrid feedback.