  For a fat sub-Riemannian structure, we introduce three canonical Ricci
curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we
prove comparison theorems for conjugate lengths, Bonnet-Myers type results and
Laplacian comparison theorems for the intrinsic sub-Laplacian.
  As an application, we consider the sub-Riemannian structure of $3$-Sasakian
manifolds, for which we provide explicit curvature formulas. We prove that any
complete $3$-Sasakian structure of dimension $4d+3$, with $d>1$, has
sub-Riemannian diameter bounded by $\pi$. When $d=1$, a similar statement holds
under additional Ricci bounds. These results are sharp for the natural
sub-Riemannian structure on $\mathbb{S}^{4d+3}$ of the quaternionic Hopf
fibrations: \begin{equation*} \mathbb{S}^3 \hookrightarrow \mathbb{S}^{4d+3}
\to \mathbb{HP}^d, \end{equation*} whose exact sub-Riemannian diameter is
$\pi$, for all $d \geq 1$.
