Robot Learning on Discrete Surfaces: Theory and Applications

Matteo Dalle Vedove; Fares J. Abu-Dakka; Luigi Palopoli; Daniele Fontanelli; Matteo Saveriano

cs.ROOct 1, 2026

Robot Learning on Discrete Surfaces: Theory and Applications

Authors: Matteo Dalle Vedove, Fares J. Abu-Dakka, Luigi Palopoli, Daniele Fontanelli, Matteo Saveriano

Organizations: Department of Industrial Engineering, Universit`a di Trento, Trento, Italy. · DRIM, Ph.D. of national interest in Robotics and Intelligent Machines. · Mechanical Engineering Program, Division of Engineering, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates. · Department of Information Engineering and Computer Science, Universit`a di Trento, Trento, Italy.

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.

Figures & tables

$\mathbb{M}$	smooth manifold
$\mathcal{M}$	polyhedral mesh
$M$	manifold (discrete or smooth)
$\mathfrak{g}$	metric on $M$
$T_{{(\cdot)}}\mathbb{M}$	tangent space on $\mathbb{M}$ at ${(\cdot)}$
$T_{{(\cdot)}}\mathcal{M}$	tangent cone on $\mathcal{M}$ at ${(\cdot)}$
$TM$	tangent bundle, set $\left\{(p,v)\ \|\ p\in M,v\in T_{p}M\right\}$

TABLE I: Notation

Panel of Figure 2 — Fig. 7: Relationship between elements defined in the mesh domain and in the ambient space.

Mesh	n. of faces	n. of vertices
3 faces of a cube	$6$	$7$
3 faces of a cube ( $4\times$ upsampling)	$1\,536$	$817$
3 faces of a cube ( $6\times$ upsampling)	$24\,576$	$12\,481$
Spot the Cow [ 51 ]	$5\,856$	$2\,930$
Stanford Dragon [ 52 ]	$9\,983$	$4\,898$
Stanford Bunny [ 52 ]	$5\,000$	$2\,502$

TABLE II: Main meshes used for experiments and corresponding number of faces and vertices. Upsampling achieved by midpoint subdivision.

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.

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.

Panel of Figure 6 — 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.

Panel of Figure 7 — Fig. 11: Experiment on the Stanford Dragon mesh, downsampled to $9\,983$ faces and $4\,898$ vertices. (a) Kernel with length scale $\kappa=0.018$ evaluated at some test points; (b) ground truth computed as the sine of the geodesic distance from a fixed point; (c) noise-free GPR with $\kappa=0.02$ using the geodesic based kernel ( 25 ); (d) median and 10th-90th percentile range MAE across 100 experiments sampling each $48$ points randomly on the mesh.

$m$	setup	$K_{PP}$	$K_{(\cdot)P}$	$K_{PP}$	$K_{(\cdot)P}$
	Geodesic Kernel			Laplace-Beltrami Kernel
Cube ( $240\,$ ms and $87\,$ s)
$20$	$0.343\,$ ms	$0.234\,$ ms	$4.9\,$ ms	$0.739\,$ ms	$4.3\,$ ms
$40$	$0.566\,$ ms	$0.554\,$ ms	$9.08\,$ ms	$1.07\,$ ms	$5.24\,$ ms
$80$	$0.695\,$ ms	$1.13\,$ ms	$9.88\,$ ms	$1.37\,$ ms	$5.59\,$ ms
$160$	$1.37\,$ ms	$4.23\,$ ms	$19.2\,$ ms	$2.82\,$ ms	$7.49\,$ ms

TABLE III: Median computation time of covariance matrices for based on the number of test points

m

on some meshes. For each mesh, we report the eigenfunction computation time;

(\cdot)

here represent the set of all vertices of the mesh.

Panel of Figure 9 — Fig. 12: Target distribution (a) based on the 50th Laplace-Beltrami eigenfunction and $20\,000$ samples (b) generated with our geodesic flow-matching on the Stanford Bunny.

	Geodesic (ours)	Diffusion	Biharmonic
Stanford Bunny
$i=10$	$0.81\pm 0.1$	$1.16\pm 0.02$	$1.06\pm 0.05$
$i=50$	$0.76\pm 0.5$	$1.48\pm 0.01$	$1.55\pm 0.01$
$i=100$	$1.43\pm 0.05$	$1.53\pm 0.01$	$1.49\pm 0.01$
Spot the Cow
$i=10$	$0.54\pm 0.1$	$0.87\pm 0.07$	$1.02\pm 0.06$

TABLE IV: Test NLL on the Stanford Bunny with dataset generated from the

i

-th eigenfunction. Lower is better.

Panel of Figure 11 — Fig. 13: Four experiment of drawing robot; for each experiment we report the demonstrated path (obtained by drawing on a tablet), the trajectory generalised on the target mesh, and the corresponding execution with the robot.

Panel of Figure 12 — Fig. 14: RGB image acquired from the end-effector mounted camera with polynomial path automatically computed by computer-vision algorithms (a). Point cloud with polishing-like trajectory generated on the surface (b). Surface before (c) and after (d) the polishing operation, and snapshot of the robot (e) executing the polishing behaviour.

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Motion Planning Skills

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Robotic surface-interaction tasks, such as spray painting or welding, require both accurate geometric planning and precise motion execution. While modern motion planners generate valid geometric paths, they often lack the expert motor patterns observed in human operators. Conversely, learning from demonstration often tightly couples task execution to the specific training geometry, limiting transferability. We propose a modular framework that decouples geometric motion planning from execution-level expertise. Expert behavior is represented as a vocabulary of interpretable, atomic motor rules, such as velocity scaling and orientation offsets, that systematically modify a geometrically planned reference path. We train a multimodal neural network to infer rule parameters jointly from kinematic trajectory data and CAD model geometry. We evaluate our approach through dynamic simulation on L-shaped and window-shaped objects, demonstrating on simulated data that the model successfully extracts velocity and orientation rules across both topologies.

Abstract

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