I am reading a paper which deals with the Geometric Control on $SE(3)$ of a Quadrotor UAV.
At some point it says:
Quaternions do not have singularities but, as the three-sphere double-covers the special orthogonal group, one attitude may be represented by two antipodal points on the three-sphere. This ambiguity should be carefully resolved in quaternion-based attitude control systems and estimator, otherwise they may exhibit unwinding, where a rigid body unnecessarily rotates through a large angle even if the initial attitude error is small.
I did not understand fully what the part in bold means. Someone of you could maybe break it down for me? Why the 3-sphere is mentioned and why it double-covers the SO(3) group?
Thanks in advance.
The $3$ sphere is mentioned because that is what the usual description of quaternions as rotations uses.
The $3$ sphere inside $\mathbb H$ consists of all the elements of length $1$. These elements can be parametrized by a pure quaternion $u$ and a real number $\theta$ in the form $\cos(\theta/2)+u\sin(\theta/2)$. This is the form of a rotation of angle $\theta$ counterclockwise around the vector in the direction $u$.
But the thing is that $q=\cos(\theta/2)+u\sin(\theta/2)$ and $-\cos(\theta/2)-u\sin(\theta/2)$ perform exactly the same rotation! When you apply $q$ as $qxq^{-1}$, it is also obvious that $(-q)x(-q)^{-1}=(-1)^2qxq^{-1}=qxq$. Of course $q$ and $-q$ are antipodes of each other.
This is the source of the $2-1$ correspondence between the quaternions and $SO(3)$. I don't know about the nature of the problem they produce in control systems exactly, but I imagine that it's important to safely choose the version that minimizes motion in the guidance system. Otherwise you might swing "the long way around."