  We consider the problem of robot motion planning in an oriented Riemannian
manifold as a topological motion planning problem in its oriented frame bundle.
For this purpose, we study the topological complexity of oriented frame
bundles, derive an upper bound for this invariant and certain lower bounds from
cup length computations. In particular, we show that for large classes of
oriented manifolds, e.g. for spin manifolds, the topological complexity of the
oriented frame bundle is bounded from below by the dimension of the base
manifold.
