Learning orbitally stable dynamics via transverse contraction criteria for modeling periodic tasks

This paper presents a novel framework for learning orbitally stable nonlinear dynamical systems from demonstrations for rhythmic tasks in robotics. The core innovation is a reproducing kernel Hilbert space (RKHS) parametrization method for rhythmic dynamics modeling, ensuring the existence of stable closed-loop orbits within the generated trajectories. By leveraging transverse contraction theory, we provide theoretical guarantees for the orbital stability of the learned dynamics. To address computational inefficiencies associated with linear matrix inequalities (LMI) constraints, we relax the semi-infinite constraints and simplify the parametrization, transforming the problem into iterative solutions of convex quadratic optimization problems, which can be efficiently solved. We validate the proposed algorithm through simulations and a series of real-world rhythmic tasks. The simulation results indicate that our method significantly outperforms existing approaches in accurately replicating demonstrated behaviors. Additionally, real-world experiments consistently show high performance in completing rhythmic tasks, demonstrating the method’s potential to address challenges in reproducing periodic movements and advancing rhythmic motion replication.