  Learning complex robot motions necessarily demands to have models that are
able to encode and retrieve full-pose trajectories when tasks are defined in
operational spaces. Probabilistic movement primitives (ProMPs) stand out as a
principled approach that models trajectory distributions learned from
demonstrations. ProMPs allow for trajectory modulation and blending to achieve
better generalization to novel situations. However, when ProMPs are employed in
operational space, their original formulation does not directly apply to
full-pose movements including rotational trajectories described by quaternions.
This paper proposes a Riemannian formulation of ProMPs that enables encoding
and retrieving of quaternion trajectories. Our method builds on Riemannian
manifold theory, and exploits multilinear geodesic regression for estimating
the ProMPs parameters. This novel approach makes ProMPs a suitable model for
learning complex full-pose robot motion patterns. Riemannian ProMPs are tested
on toy examples to illustrate their workflow, and on real
learning-from-demonstration experiments.
