Learning Dynamical System-Based Robot Motions from Demonstrations Via ODE-Driven Diffeomorphic Mappings

Abstract

Learning from Demonstrations (LfD) has emerged as a prominent paradigm for imparting motion skills to robotic systems. Dynamical systems (DS) offer a potent mathematical framework for representing point-to-point motions, a critical requirement for numerous practical applications in robotics. While existing approaches typically construct DS models by employing diffeomorphic mappings to morph stable reference systems toward observed demonstrations, the requirement to preserve strict diffeomorphic properties introduces architectural constraints on neural network design, thereby constraining their expressiveness. To address this limitation, we present a DS-based LfD formulation that relaxes traditional diffeomorphism constraints. Our framework employs bidirectional temporal integration of ordinary differential equations (ODEs) to simultaneously satisfy stability guarantees and trajectory alignment objectives. A key innovation lies in a variational calculus framework for Jacobian estimation, enabling efficient computation of DS vector fields while maintaining numerical stability. Comprehensive evaluations demonstrate that our method achieves 33.7% improvement in trajectory reproduction accuracy compared to state-of-the-art baselines while preserving Lyapunov stability. The proposed methodology significantly expands the representational capacity of DS-based learning systems, enabling robust reproduction of complex motion patterns.