Robot learning improves motion control on curved and flat surfaces
Robot Learning on Discrete Surfaces: Theory and Applications
Robotics
Summary
Most robot learning systems do not consider the true shape and structure of surfaces, especially those made of many flat faces like 3D models. The authors created a new mathematical framework that lets robots learn movements directly on these complex surfaces, better respecting their shapes. They showed this improves the robot’s ability to replicate and plan motions smoothly across edges and curves, tested in simulations and real robots. This helps robots perform tasks like following drawn paths or polishing uneven objects more accurately.
What this means in practice
- •For robotic system designers: Improve robot motion planning on CAD and 3D scanned surfaces by directly learning from discrete mesh geometry.
- •For industrial automation engineers: Plan polishing and surface treatment robot motions precisely on reconstructed RGB-D surfaces to better follow complex object shapes.
Authors
Matteo Dalle Vedove, Fares J. Abu-Dakka, Luigi Palopoli, Daniele Fontanelli, Matteo Saveriano
Abstract
All the objects composing our world are enclosed within surfaces. Yet, most robot learning and motion generation frameworks treat surfaces as constraints ignoring their intrinsic geometry. This gap is acute for polyhedral meshes--the standard output of CAD and 3D reconstruction--whose discrete geometric structure remains unexploited. In this paper, we propose a unified discrete Riemannian framework that enables robot learning directly on polyhedral surface meshes. Using discrete differential geometry, we define logarithmic and exponential maps, parallel transport, and ambient-space projections that remain well-defined across faces, edges, and vertices. We instantiate the framework in three learning paradigms: (i) Dynamic Movement Primitives (DMPs), an improved exponential-map computation and a fixed-tangent-cone forcing-term encoding with parallel transport yield better cross-surface generalisation and stability over prior mesh-based approaches. (ii) Gaussian Process (GP), a geodesic-based kernel with practical admissibility control, enables regression at arbitrary mesh locations without smoothness assumptions. (iii) Riemannian Flow Matching (RFM), mesh-native operators improve generative quality over spectral baselines while reducing training time. The framework is validated in simulation against state-of-the-art methods and demonstrated on two real-robot scenarios: generalising user-drawn trajectories across different surfaces and planning polishing motions on RGB-D-reconstructed surfaces.