  Dexterous and autonomous robots should be capable of executing elaborated
dynamical motions skillfully. Learning techniques may be leveraged to build
models of such dynamic skills. To accomplish this, the learning model needs to
encode a stable vector field that resembles the desired motion dynamics. This
is challenging as the robot state does not evolve on a Euclidean space, and
therefore the stability guarantees and vector field encoding need to account
for the geometry arising from, for example, the orientation representation. To
tackle this problem, we propose learning Riemannian stable dynamical systems
(RSDS) from demonstrations, allowing us to account for different geometric
constraints resulting from the dynamical system state representation. Our
approach provides Lyapunov-stability guarantees on Riemannian manifolds that
are enforced on the desired motion dynamics via diffeomorphisms built on neural
manifold ODEs. We show that our Riemannian approach makes it possible to learn
stable dynamical systems displaying complicated vector fields on both
illustrative examples and real-world manipulation tasks, where Euclidean
approximations fail.
