A Unified Framework for Normative-Imitative Trajectory Learning from Demonstration

Abstract

Robot skill generation is often approached from two distinct perspectives: normative trajectory optimization, which emphasizes smoothness-based criteria such as minimum jerk, and imitation-based learning, which prioritizes fidelity to demonstrated behaviors. While both paradigms aim to produce feasible and meaningful motions, they are typically formulated separately. In practice, however, many robotic skills require trajectories that are both dynamically smooth and faithful to demonstrations. We propose a unified framework for normative–imitative trajectory optimization that makes this trade-off explicit and tunable. Our framework formulates trajectory generation as a constrained quadratic program combining weighted linear-operator smoothness penalties, a quadratic imitation anchoring term, and affine equality constraints for feasibility. For affine-constrained instances, the resulting problem is strictly convex, admits a unique global minimizer, and can be solved efficiently using standard quadratic programming techniques. Our proposed formulation unifies a broad class of smoothness objectives, including minimum velocity, acceleration, jerk, snap, and elastic energy models, within a single operator-based representation, while incorporating demonstration fidelity in a principled manner. Simulation and real-world experiments on a UR5e robotic arm demonstrate that our framework provides predictable interpolation between purely normative and purely imitative behaviors, offering a compact and extensible foundation for trajectory learning from demonstration.