Learning to Draw on the Curved Surface: A Riemannian Adaptive Moment Iterative Learning Control Approach

Precise trajectory tracking on curved surfaces is essential in robotic operations such as drawing, polishing, and manufacturing. While classical Euclidean iterative learning control (ILC) improves repeatable tracking accuracy through trial-to-trial error correction, it ignores the underlying surface geometry, often producing command updates that violate geometric feasibility. This paper proposes R-ILC, a Riemannian adaptive moment ILC framework that embeds surface constraints directly into the command update rule via Riemannian optimization. Theoretical analysis establishes almost-sure convergence of the Riemannian gradient norm to zero, with avoidance of strict saddles. Experiments on curved-surface robotic drawing validate the effectiveness of our approach while demonstrating its advantages over classical Euclidean ILC in tracking accuracy and geometric feasibility.