There are many equivalent definitions of Riemannian geodesics The definitions can be divided into two classes : geodesics as "shortest curves defined by a variational principle, and geodesics as "straightest curves defined by a connection. All definitions are naturally generalised to sub-Riemannian manifolds, but become non-equivalent. A. Vershik and L. Faddeev showed that for a generic sub-Riemannian manifold (Q, D, g) shortest geodesics ( used in control theory) are different from straightest geodesics (used in non-holonomic mechanics)) on a open dense submanifold. They gave first example (compact Lie group with the bi-invariant metric) when shortest geodesics coincides with straightest geodesics and stated the problem to describe more general class of sub-Riemannian manifolds with this property . We generalised the Vershik-Faddeev example and consider a big class of sub-Riemannian manifolds associated with principal bundle over a Riemannian manifolds, for which shortest geodesics coincides with straightest geodesics. Using the geometry of flag manifolds, we describe some classes of compact homogeneous sub-Riemannian manifolds ( including contact sub-Riemannian manifolds and symmetric sub-Riemannian manifolds ) where straightest geodesics coincides with shortest geodesics. Construction of geodesics in these cases reduces to description of Riemannian geodesics of the Riemannian homogeneous manifold or left-invariant metric on a Lie group.