  This paper is about the construction of a family of centrally synergistic
potential functions from a single modified trace function on SO(3). First, we
demonstrate that it is possible to complete the construction through angular
warping with multiple directions, particularly effective in the unresolved
cases in the literature. Second, it can be shown that for each potential
function in the family, there exists a subset of the family such that the
synergistic gap is positive at the unwanted critical points. This allows the
switching condition to be checked within the selected subsets while
implementing synergistic hybrid control. Furthermore, the positive lower bound
of synergistic gap is explicitly expressed by selecting a traditional warping
angle function. Finally, we apply the proposed synergistic potential functions
to obtain robust global attitude tracking.
