Robot Learning on Discrete Surfaces: Theory and Applications

2026-10-01

AI Summary

This work addresses the limitation of existing robot learning approaches that treat polyhedral mesh surfaces merely as constraints, neglecting their intrinsic geometry and hindering efficient motion generation and generalization directly on discrete surfaces. We propose the first unified discrete Riemannian framework for polyhedral meshes, defining discrete logarithmic and exponential maps alongside parallel transport to exploit surface geometric priors for native learning. Building upon this foundation, we instantiate three paradigms: an enhanced Dynamic Movement Primitive (DMP), a geodesic Gaussian process kernel, and Riemannian flow matching. Experiments demonstrate that our approach surpasses state-of-the-art methods in simulation and validates cross-surface trajectory generalization on real-robot polishing tasks. The proposed framework significantly improves generation quality and stability while substantially reducing training costs.

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.

Problem

• Robot Learning
• Discrete Surfaces
• Polyhedral Meshes
• Motion Generation
• Riemannian Geometry

Innovation

• Discrete Riemannian Framework
• Polyhedral Meshes
• Dynamic Movement Primitives
• Gaussian Process
• Riemannian Flow Matching