Robot Learning on Discrete Surfaces: Theory and Applications

Matteo Dalle Vedove, Fares J. Abu-Dakka, Luigi Palopoli, Daniele Fontanelli, Matteo Saveriano · N/A · 2026

Summary

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.

Abstract Summary

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.

Key Points

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.

Methodology

In this work, we aim at extending methods defined for Riemannian manifolds to polyhedral meshes. For this reason, it is necessary to investigate how to properly define the operators described in Sec. III-A , namely logarithmic and exponential maps, and parallel transport, in the space of meshes $\mathcal{M}$.

On smooth manifolds, it is always possible to define a locally linear approximation of $\mathbb{M}$ at a point $p$, the tangent space $T_{p}\mathbb{M}$.

Methodology — Robot Learning on Discrete Surfaces: Theory and Applications — Fig. 8: DMP transferability comparison against [ 9 ] . Both DMP methods have been trained on the same 8-shaped demonstration on the flat plane, and have been generalised onto the Stanford bunny mesh, with equal initial state and centre. The green line represent our method, while the red one is the baseline MeshDMP result.

Simulation Results

In this section, we evaluate and compare the proposed methodology against state-of-the-art approaches. Section V-A compares our DMP with the MeshDMP framework [ 9 ] , Sec V-B compares our geodesic-based squared exponential kernel with the one computed from the Laplace-Beltrami operator introduced in [ 10 ] , while Sec. V-C compares our Mesh Flow Matching against the RFM approach in [ 4 ] .

Our method has been implemented fully in C++ and it relies on the CGAL library for polyhedral mesh operations [ 49 , 50 ] . Experiments in Sec.

Simulation Results — Robot Learning on Discrete Surfaces: Theory and Applications — Fig. 9: Exponential map comparison against [ 9 ] . First we compute the geodesic path (green) between the two orange points, and the logarithmic map $v$ between them. Then we compute the exponential map of $v$ with Algorithm 1 (yellow) and the MeshDMP-proposed algorithm (red). The reconstructed path by our method coincides, up to numerical precision, with the reference geodesic.

Experiments

The key feature of DMPs is their high generalisation capability, which enables the execution of motion policies to new, unseen initial and final condition while preserving the topology of the learned path by exploiting spatial and temporal invariance properties.

As highlighted in ( 22 ) and ( 23 ), exploiting the definition of surface operators on discrete manifolds, it is possible to learn motion policies that can be transferred to different meshes.

Experiments — Robot Learning on Discrete Surfaces: Theory and Applications — Fig. 10: Experiment on a mesh composed of three vertex-adjacent faces of a cube. The ground truth (a) is evaluated at the vertices of a mesh upsampled to $817$ vertices and $1\,536$ faces by computing the sine of the geodesic distance from a fixed point. The noise-free GP predictions are obtained using 70 ground-truth samples with (b) the geodesic based kernel ( 25 ) and (c–d) the Laplace-Beltrami kernel ( 27 ) with $\kappa=0.18,\sigma=1$. Panels (e–f) report the median MAE together with the 10th-90th percentile range as a function of the length scale, computed over 100 experiments, each using 70 randomly selected test vertices. Figures (c) and (e) use the upsampled mesh, while (d) and (f) use a further upsampled mesh with $12\,481$ vertices and $24\,576$ faces. (b–d) show the worst-performing samples in their respective category for the selected length scale.

Sources and Demonstrations

Figures are reproduced from Vedove et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.