Locally active globally stable dynamical systems: Theory, learning, and experiments

State-dependent dynamical systems (DSs) offer adaptivity, reactivity, and robustness to perturbations in motion planning and physical human-robot interaction tasks. Learning DS-based motion plans from non-linear reference trajectories is an active research area in robotics. Most approaches focus on learning DSs that can (i) accurately mimic the demonstrated motion, while (ii) ensuring convergence to the target, i.e., they are globally asymptotically (or exponentially) stable. When subject to perturbations, a compliant robot guided with a DS will continue following the next integral curves of the DS towards the target. If the task requires the robot to track a specific reference trajectory, this approach will fail. To alleviate this shortcoming, we propose the locally active globally stable DS (LAGS-DS), a novel DS formulation that provides both global convergence and stiffness-like symmetric attraction behaviors around a reference trajectory in regions of the state space where trajectory tracking is important. This allows for a unified approach towards motion and impedance encoding in a single DS-based motion model, i.e., stiffness is embedded in the DS. To learn LAGS-DS from demonstrations we propose a learning strategy based on Bayesian non-parametric Gaussian mixture models, Gaussian processes, and a sequence of constrained optimization problems that ensure estimation of stable DS parameters via Lyapunov theory. We experimentally validated LAGS-DS on writing tasks with a KUKA LWR 4+ arm and on navigation and co-manipulation tasks with iCub humanoid robots.