  In this paper, we propose an approach to learn stable dynamical systems
evolving on Riemannian manifolds. The approach leverages a data-efficient
procedure to learn a diffeomorphic transformation that maps simple stable
dynamical systems onto complex robotic skills. By exploiting mathematical tools
from differential geometry, the method ensures that the learned skills fulfill
the geometric constraints imposed by the underlying manifolds, such as unit
quaternion (UQ) for orientation and symmetric positive definite (SPD) matrices
for impedance, while preserving the convergence to a given target. The proposed
approach is firstly tested in simulation on a public benchmark, obtained by
projecting Cartesian data into UQ and SPD manifolds, and compared with existing
approaches. Apart from evaluating the approach on a public benchmark, several
experiments were performed on a real robot performing bottle stacking in
different conditions and a drilling task in cooperation with a human operator.
The evaluation shows promising results in terms of learning accuracy and task
adaptation capabilities.
