  We address the problem of computing Riemannian normal coordinates on the
real, compact Stiefel manifold of orthogonal frames. The Riemannian normal
coordinates are based on the so-called Riemannian exponential and the
Riemannian logarithm maps and enable to transfer almost any computational
procedure to the realm of the Stiefel manifold. To compute the Riemannian
logarithm is to solve the (local) geodesic endpoint problem. Instead of
restricting the consideration to geodesics with respect to a single selected
metric, we consider a family of Riemannian metrics introduced by H\"uper,
Markina and Silva-Leite that includes the Euclidean and the canonical metric as
prominent examples.
  As main contributions, we provide (1) a unified, structured, reduced formula
for the Stiefel geodesics for the complete family of metrics, (2) a unified
method to tackle the geodesic endpoint problem, (3) an improvement of the
existing Riemannian log map under the canonical metric. The findings are
illustrated by means of numerical examples, where the novel algorithms prove to
be the most efficient methods known to this date.
