Global Stabilization of Antipodal Points on $n$-Sphere With Application to Attitude Tracking

Existing approaches to robust global asymptotic stabilization of a pair of antipodal points on unit $n$-sphere $\mathbb {S}^{n}$ typically involve the noncentrally synergistic hybrid controllers for attitude tracking on unit quaternion space. However, when switching faults occur due to parameter errors, the noncentrally synergistic property can lead to the unwinding problem or, in some cases, destabilize the desired set. In this work, a hybrid controller is first proposed based on a novel centrally synergistic family of potential functions on $\mathbb {S}^{n}$, which is generated from a basic potential function through angular warping. The synergistic parameter can be explicitly expressed if the warping angle has a positive lower bound at the undesired critical points of the family. Next, the proposed approach induces a new quaternion-based controller for global attitude tracking. It has three advantageous features over the existing synergistic designs: it is consistent, i.e., free from the ambiguity of unit quaternion representation; it is switching fault tolerant, i.e., the desired closed-loop equilibria remain asymptotically stable even when the switching mechanism does not work; and it relaxes the assumption on the parameter of the basic potential function in the literature. Comprehensive simulation confirms the high robustness of the proposed centrally synergistic approach compared with the existing noncentrally synergistic approaches.