  Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous
Riemannian spaces $(M=G/H,g)$ whose geodesics are orbits of one-parameter
subgroups of $G$. The corresponding metric $g$ is called a geodesic orbit
metric. We study the geodesic orbit spaces of the form $(\Sp(n)/\Sp(n_1)\times
\cdots \times \Sp(n_s), g)$, with $0<n_1+\cdots +n_s\leq n$. Such spaces
include spheres, quaternionic Stiefel manifolds, Grassmann manifolds and
quaternionic flag manifolds. The present work is a contribution to the study of
g.o. spaces $(G/H,g)$ with $H$ semisimple.
