If the question is still relevant to you, have a look at the paper:
R. Zimmermann: A Matrix-Algebraic Algorithm for the Riemannian Logarithm on the Stiefel Manifold under the Canonical Metric, SIAM. J. Matrix Anal. & Appl. 38-2 (2017), pp. 322-342, arXiv:1604.05054, doi: 10.1137/16M1074485
