  Let $X_1,\cdots,X_m$ be vector fields satisfying H\"ormander's Lie bracket
generating condition on a smooth manifold $M$. We generalise Connes's tangent
groupoid, by constructing a completion of the space $M\times M\times
\mathbb{R}_+^\times$ using the sub-Riemannian metric. We use our space to
calculate all the tangent cones of the sub-Riemannian metric in the sense of
the Gromov-Hausdorff distance. This generalises a result of Bella\"iche.
