New Obstacle Avoidance Motion Planning: A Safe Koopman Planning Method

Robot learning from demonstration (LfD) in obstacle-laden environments presents a fundamental conflict between mimicking expert dynamics and maintaining safety. This article introduces the Safe-Koopman Framework, an operator-theoretic motion planning method that generalizes obstacle-free demonstrations to planar planning tasks with obstacles. Leveraging the Koopman operator with random Fourier features (RFFs), the method lifts nonlinear robot motion into a linear feature space to capture the global kinematic topology of demonstrations. To address safety generalization, two components are proposed. First, a dynamical system modulation mechanism smoothly blends the nominal imitation flow with a local safety-critical flow, reducing discontinuities associated with hard switching. Second, a radial loss is formulated as a data-driven relaxation of control barrier function (CBF) boundary conditions, penalizing inward velocity components near obstacle boundaries. Under the zero-loss idealization and stated regularity assumptions, the radial condition supports forward invariance of the safe set; in implementation, the method empirically maintains positive clearance. The nominal spectral stability of the learned Koopman operator is further analyzed, along with the effect of local nonlinear modulation on this conclusion. Experiments on the LASA handwriting trajectory benchmark show that the proposed method avoids collisions observed in the unconstrained Koopman baseline and reduces jerk compared with artificial potential field (APF)-based obstacle avoidance while maintaining competitive kinematic fidelity.