  Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which
span the tangent space at every point, we consider the question of when there
is locally a coordinate system in which these vector fields have a higher level
of smoothness. For example, when is there a coordinate system in which the
vector fields are smooth, or real analytic, or have Zygmund regularity of some
finite order? We address this question in a quantitative way, which strengthens
and generalizes previous works on the quantitative theory of sub-Riemannian
(aka Carnot-Carath\'eodory) geometry due to Nagel, Stein, and Wainger, Tao and
Wright, the second author, and others. Furthermore, we provide a diffeomorphism
invariant version of these theories. This is the first part in a three part
series of papers. In this paper, we study a particular coordinate system
adapted to a collection of vector fields (sometimes called canonical
coordinates) and present results related to the above questions which are not
quite sharp; these results from the backbone of the series. The methods of this
paper are based on techniques from ODEs. In the second paper, we use additional
methods from PDEs to obtain the sharp results. In the third paper, we prove
results concerning real analyticity and use methods from ODEs.

Differential Geometry;Classical Analysis and ODEs;Metric Geometry