Kernel-VEC: Autonomous Scanning via Kernelized Ergodic Search

Autonomous scanning robots search for structures they cannot directly see: sensing is confined to a small footprint, motion is constrained to a surface, and the region of interest lies in the volume beneath it. Ergodic control addresses this by planning a trajectory whose time-averaged sensing coverage matches a target distribution over that volume. Existing volumetric formulations evaluate their objective through a basis-function expansion whose cost grows sharply with planning dimension, making full $SE(3)$ pose planning impractical. We devise a new closed-form kernel objective over Gaussian-mixture sensor footprints and targets, removing the basis expansion entirely and yielding exact gradients by automatic differentiation. On a representative $SE(3)$ coverage task modeled on robotic cardiac ultrasound, our method achieves $1.58\times$ better coverage than all baselines tested while converging up to $13.5\times$ faster.