  Dynamical Systems (DS) are an effective and powerful means of shaping
high-level policies for robotics control. They provide robust and reactive
control while ensuring the stability of the driving vector field. The
increasing complexity of real-world scenarios necessitates DS with a higher
degree of non-linearity, along with the ability to adapt to potential changes
in environmental conditions, such as obstacles. Current learning strategies for
DSs often involve a trade-off, sacrificing either stability guarantees or
offline computational efficiency in order to enhance the capabilities of the
learned DS. Online local adaptation to environmental changes is either not
taken into consideration or treated as a separate problem. In this paper, our
objective is to introduce a method that enhances the complexity of the learned
DS without compromising efficiency during training or stability guarantees.
Furthermore, we aim to provide a unified approach for seamlessly integrating
the initially learned DS's non-linearity with any local non-linearities that
may arise due to changes in the environment. We propose a geometrical approach
to learn asymptotically stable non-linear DS for robotics control. Each DS is
modeled as a harmonic damped oscillator on a latent manifold. By learning the
manifold's Euclidean embedded representation, our approach encodes the
non-linearity of the DS within the curvature of the space. Having an explicit
embedded representation of the manifold allows us to showcase obstacle
avoidance by directly inducing local deformations of the space. We demonstrate
the effectiveness of our methodology through two scenarios: first, the 2D
learning of synthetic vector fields, and second, the learning of 3D robotic
end-effector motions in real-world settings.
