Learning stable robotic skills on Riemannian manifolds

In this paper, we propose an approach to learn stable dynamical systems that evolve on Riemannian manifolds. Our approach leverages a data-efficient procedure to learn a diffeomorphic transformation, enabling the mapping of simple stable dynamical systems onto complex robotic skills. By harnessing mathematical techniques derived from differential geometry, our method guarantees that the learned skills fulfill the geometric constraints imposed by the underlying manifolds, such as unit quaternions (UQ) for orientation and symmetric positive definite (SPD) matrices for impedance. Additionally, the method preserves convergence towards a given target. Initially, the proposed methodology is evaluated through simulation on a widely recognized benchmark, which involves projecting Cartesian data onto UQ and SPD manifolds. The performance of our proposed approach is then compared with existing methodologies. Apart from that, a series of experiments were performed to evaluate the proposed approach in real-world scenarios. These experiments involved a physical robot tasked with bottle stacking under various conditions and a drilling task performed in collaboration with a human operator. The evaluation results demonstrate encouraging outcomes in terms of learning accuracy and the ability to adapt to different situations.