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Robot Learning on Discrete Surfaces: Theory and Applications

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

arXiv:2610.01910v1cs.RO

TL;DR

Robot learning frameworks often treat surfaces as external constraints, despite the prevalence of discrete polyhedral meshes in practical settings. This paper builds a discrete Riemannian framework for learning directly on meshes and applies it to DMPs, GPs, and RFM. The reported results include more faithful long-horizon DMP behavior than MeshDMP and improved cross-surface generalization.

  • Problem

    Robot learning often treats surface geometry as an external constraint, while polyhedral meshes have singularities that smooth-manifold methods do not address directly.

    Source: §I · p. 1

  • Method

    The paper defines mesh-native Riemannian operators and applies them to Dynamic Movement Primitives, Gaussian Processes, and Riemannian Flow Matching.

    Source: §I · p. 2, §VII · p. 14

  • Results

    The DMP remains faithful to an 8-shaped trajectory’s limit cycle during longer integration, while MeshDMP exhibits undesired rotation around the fixed motion centre.

    Source: A. Comparison against MeshDMP · p. 10

  • Takeaways & Limitations

    Geodesic-based GP kernels support regression at arbitrary surface locations without smoothness assumptions, and are particularly effective on coarse meshes.

    Source: §VII · p. 15

    Geodesic pre-computation has complexity O(N^2 log N) and becomes a significant barrier for surfaces exceeding 10,000 faces.

    Source: §VII · p. 15

Abstract

from arXiv · show

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.

I. INTRODUCTION

Robot learning often treats surfaces as external constraints, while practical CAD and reconstructed surfaces are discrete meshes with singularities that smooth-manifold methods do not address. The paper develops mesh-native geometric operators and extends DMPs, GPs, and RFM to these surfaces.

I. INTRODUCTION

  • Geometry-aware methods for smooth manifolds do not directly fit polyhedral meshes, whose edges and vertices create singularities and ill-defined behavior for naïvely discretized operators.The paper identifies this as a gap between geometry-aware learning theory and practical discrete surfaces.Source: §I · p. 1
  • A discrete Riemannian framework defines geometric operators that remain well-defined across mesh faces, edges, and vertices, using straightest geodesics and tangent-cone projections.The operators support learning across surfaces with different curvature, topology, and discretization resolution without smoothness or spectral assumptions.Source: §I · p. 2

II. RELATED WORK

  • Earlier mesh motion-planning approaches can depend on restrictive topology and sufficient smoothness, avoiding rather than addressing edge and vertex singularities.The paper contrasts its direct treatment of singularities with flattening meshes onto planar domains.Source: §II · p. 2
  • Mesh DMPs encode the forcing term in a fixed tangent space and transport it during execution, addressing instability and degraded generalization associated with moving local coordinates.The formulation also replaces MeshDMP’s bounded-angle approximate exponential map and explicitly handles edge and vertex singularities.Source: §II · p. 2, §I · p. 2
  • Geodesic-distance-based Gaussian Processes support regression at arbitrary mesh locations, including face interiors, without smoothness assumptions or vertex-only evaluation.Practical admissibility criteria address the possibility that geodesic kernels lose positive definiteness beyond a critical lengthscale.Source: §I · p. 2, §II · p. 3
  • Mesh-native operators extend Riemannian Flow Matching and improve vector-field learning relative to spectral premetrics and the spectral baselines of.The introduction reports improved generative quality and reduced training time relative to those baselines.Source: §I · p. 2, §II · p. 3

III. BACKGROUND

The background introduces Riemannian geometry and the operators used to move between a manifold and its tangent spaces. It also illustrates how mesh vertices are classified by their total angle.

A. Riemannian Manifolds

  • A Riemannian manifold pairs a differentiable manifold with a smoothly varying inner product on each tangent space, which determines geometric quantities such as distances, angles, volumes, and curvature.This paper uses the metric induced by the ambient Euclidean space.Source: A. Riemannian Manifolds · p. 3
  • The logarithmic map expresses a manifold point in a base point’s tangent space, the exponential map maps tangent vectors back to the manifold, and parallel transport moves vectors between tangent spaces while preserving geometric properties.The transported vectors preserve properties such as length and angle along the connecting curve.Source: A. Riemannian Manifolds · pp. 3–4
  • Mesh vertices are spherical, Euclidean, or hyperbolic according to whether their total angle is below, equal to, or above 2π.The figure illustrates examples of each vertex type.Source: A. Riemannian Manifolds · p. 4, Figure 1 · p. 4

B. Polyhedral Meshes

The background describes polyhedral meshes as discrete surfaces with singularities at edges and vertices, then recalls geometry-aware DMP and Gaussian Process concepts used later in the paper.

B. Polyhedral Meshes

  • A polyhedral mesh represents a surface as planar polygonal faces joined along shared edges and vertices; its normal field is discontinuous at edges and vertices, where tangent planes are not uniquely defined.The paper focuses on manifold surface meshes that are piecewise-linear two-dimensional surfaces embedded in R3.Source: B. Polyhedral Meshes · p. 4

C. Geometry-Aware Dynamic Movement Primitives

  • A geometry-aware DMP uses a position state, an attractor, a scaled velocity in the position’s tangent space, a phase variable, and a learned forcing function.The forcing function maps the phase variable into the tangent space of the current position.Source: C. Geometry-Aware Dynamic Movement… · p. 4

D. Gaussian Processes on Manifolds

  • A Gaussian Process places a distribution over functions using a kernel that encodes prior assumptions, while Gaussian Process regression predicts function values at test locations from observed input-output pairs.The observations are modeled with noise, and the posterior is also a Gaussian Process.Source: D. Gaussian Processes on Manifolds · p. 4

E. Riemannian Flow Matching

Riemannian Flow Matching transports probability distributions along manifold-respecting flows, while discrete tangent cones provide local geometry at mesh faces, edges, and vertices. The section describes how the flow is learned and how its supervision signal is constructed.

E. Riemannian Flow Matching

  • RFM extends continuous normalizing flows to manifold-supported data by learning a time-dependent vector field that transports a reference distribution toward a target distribution.The learned field approximates the flow’s conditional vector field using a Riemannian-metric norm.Source: E. Riemannian Flow Matching · pp. 4–5
  • A premetric and decreasing scheduler define supervision by shrinking the premetric to a target, steering particles monotonically closer as time progresses.The premetric is positive definite and non-degenerate; the scheduler runs from 1 to 0.Source: E. Riemannian Flow Matching · p. 5

A. Discrete differential operators

  • The paper’s broader goal is to define logarithmic and exponential maps and parallel transport for polyhedral meshes.These operators are needed to extend Riemannian-manifold methods to mesh space.Source: A. Discrete differential operators · p. 5

1) Tangent cone:

  • Mesh tangent cones generalize tangent spaces: they are planar inside faces, unions of half-planes on edges, and generally non-flat piecewise-linear cones at vertices.The construction relies on meshes being Lipschitz manifolds.Source: 1) Tangent cone · p. 5

2) Logarithmic map:

The logarithmic map represents a mesh point by a tangent-cone vector derived from a shortest surface path. The exponential map propagates a vector across mesh singularities using straightest geodesics.

2) Logarithmic map:

  • The logarithmic map sends q to a vector in p’s tangent cone, whose direction follows the shortest path and whose magnitude is the geodesic distance.The shortest path is computed using continuous-Dijkstra methods, with accelerated query structures supporting real-time path generation.Source: 2) Logarithmic map · p. 5

3) Exponential map:

  • On a mesh, ordinary geodesic ODE integration is ill-defined at edges and vertices, while shortest paths alone may not uniquely determine propagation at vertices.Straightest geodesics provide a unified definition for handling edge and vertex singularities.Source: 3) Exponential map · p. 6
  • A straightest geodesic has equal left and right curve angles, each equal to half the total angle at the curve point.The paper uses this condition to define its numerical exponential-map procedure.Source: 3) Exponential map · p. 6
  • The exponential-map algorithm advances within a face when possible; at a boundary, it rotates the remaining vector by half the total angle and continues until reaching q.The construction is described as well-defined even when the vector passes through mesh vertices.Source: 3) Exponential map · pp. 6–7

4) Parallel transport:

Parallel transport moves tangent-cone vectors between mesh points while preserving their geometric properties. Ambient-space projections connect mesh-native quantities with Euclidean representations and return them to the mesh.

4) Parallel transport:

  • Parallel transport is an isometry that moves a vector between tangent cones along a connecting path while preserving its geometric properties.For a mesh embedded in R3, the transformation is encoded as a rotation matrix constructed using the geodesic and surface normals.Source: 4) Parallel transport · p. 7

5) Ambient space and projections:

  • Because manifold vector fields do not readily integrate with Euclidean implementations, the framework defines projections between the mesh and its R3 embedding.Mesh points map trivially to ambient space, while the inverse point projection uses the closest point on the mesh.Source: 5) Ambient space and projections · p. 7
  • Ambient vectors are projected into a point’s tangent cone by selecting the admissible face or edge projection with the largest dot product against the original vector.For points inside a face, the projection reduces to a normal projection onto that face’s plane.Source: 5) Ambient space and projections · p. 7

B. G-DMPs on Meshes

The mesh DMP formulation learns motion forcing terms in a fixed goal tangent cone and parallel-transports them during execution. This avoids limitations of local moving-coordinate encodings and supports transfer across goals and surfaces.

B. G-DMPs on Meshes

  • The standard mesh DMP forcing term can live in ambient R3, with projection either before system integration or after the position state deviates from the manifold.The forcing term is the component that enables learning the demonstrated motion.Source: B. G-DMPs on Meshes · p. 7
  • A locally parameterized forcing term can transfer policies, but may fail to capture global motion behavior; a fixed surface-coordinate encoding can also generalize poorly across differently oriented surfaces.The cited example contrasts a policy learned on the xy plane with execution on the xz plane.Source: B. G-DMPs on Meshes · pp. 7–8
  • The proposed method learns the forcing term at the goal and parallel-transports it to the current tangent cone during execution.The goal-cone representation is encoded in two dimensions, then reconstructed for a new goal or mesh and transported during integration.Source: B. G-DMPs on Meshes · p. 8
  • The rhythmic DMP formulation supports learning and executing periodic motion, with the goal serving as the motion centre and Ω scaling execution frequency.Source: B. G-DMPs on Meshes · p. 8

C. Geodesic-based Gaussian Processes

The section defines a squared-exponential GP kernel by replacing Euclidean distance with geodesic distance on the mesh. Because positive definiteness is not guaranteed for every length scale, it describes a data-dependent procedure for finding an admissible threshold.

Geodesic-based kernel

  • The proposed kernel replaces Euclidean distance with mesh geodesic distance in the squared-exponential GP kernel.The Euclidean squared-exponential kernel uses variance and length scale hyperparameters.Source: C. Geodesic-based Gaussian Processes · p. 8

Admissibility

  • The geodesic kernel may yield negative-definite covariance matrices above a critical length scale κ_th, so admissibility must be checked for the training points.The authors consider κ_th practically large enough in most applications to permit correct regression.Source: C. Geodesic-based Gaussian Processes · p. 8
  • Algorithm 3 uses bisection on the training-point geodesic distance matrix to estimate the largest admissible length scale κ_th.It updates the search interval according to whether the kernel matrix is positive or negative definite, stopping at numerical convergence.Source: C. Geodesic-based Gaussian Processes · pp. 8–9
  • When test points are not known upfront, Monte Carlo sampling estimates κ_th and selects the minimum threshold across sampled points.The procedure repeatedly samples mesh points and computes a critical length scale for each test.Source: C. Geodesic-based Gaussian Processes · p. 9

D. Riemannian Flow Matching on Meshes

The method uses shortest mesh paths as both a premetric and the trajectories that supervise Riemannian Flow Matching. A learned ambient-space vector field is projected onto the mesh tangent cone during generation, then integrated with the mesh exponential map.

Training flows

  • Geodesic distance serves as the premetric, and each reference-to-data sample pair is connected by a shortest path to define its flow and training supervision.Points along the path and their tangential velocities provide the time-indexed training examples.Source: D. Riemannian Flow Matching on Meshes · p. 9
  • The vector field is parameterized in ambient R3, while reference points are sampled uniformly by face area and uniformly within the selected face.This avoids mesh parameterization and restrictions on mesh topology.Source: D. Riemannian Flow Matching on Meshes · p. 9
  • At each training step, the procedure samples time, data and reference points, constructs their geodesic, and uses its point and velocity as supervision.The training algorithm uses the linear scheduler η(t) = 1 − t.Source: D. Riemannian Flow Matching on Meshes · p. 9

Generating samples

  • Generation starts from a reference sample, projects the predicted vector onto the mesh tangent cone, and advances with the exponential map.The procedure repeats these projected mesh-based updates until time reaches 1.Source: D. Riemannian Flow Matching on Meshes · p. 9

V. SIMULATION RESULTS

The simulations compare the proposed mesh methods with established approaches across DMP transfer and GP regression. The results highlight long-run DMP stability and geodesic-kernel performance, alongside a computational trade-off for GP inference.

A. Comparison against MeshDMP

  • After initially converging to the 8-shaped limit cycle, the proposed DMP remains faithful over longer execution, while MeshDMP develops rotational behavior around the motion centre.The paper attributes the difference to MeshDMP's local forcing-term parameterization, which does not capture curvature-induced dynamics changes.Source: Figure 8 · p. 10, A. Comparison against MeshDMP · p. 10
  • The proposed exponential map reconstructs the reference geodesic up to numerical precision, whereas MeshDMP's local approximation is limited to small vectors and face angles below 90°.The comparison applies the two integration schemes to a long vector.Source: Figure 9 · p. 10, A. Comparison against MeshDMP · p. 10

B. Gaussian Processes

  • GP computation trades per-input data-structure construction and roughly linear covariance evaluation in test-point count for the Laplace-Beltrami method's slower eigenfunction setup but faster inference.The geodesic approach is strongest when the mesh is not known upfront and is small to medium sized, up to 5,000 faces.Source: B. Gaussian Processes · pp. 11–12
  • On the Dragon mesh, the geodesic kernel performs slightly better than the Laplace-Beltrami kernel below κ_th = 0.023; on the cube mesh it remains well behaved below its critical length scale.The cube experiment also finds Laplace-Beltrami regression can become unstable at larger length scales, with further mesh upsampling degrading performance.Source: B. Gaussian Processes · p. 11, Figure 10 · p. 11, Figure 11 · p. 12

C. Mesh Flow Matching

Mesh Flow Matching is evaluated on distributions over the Stanford Bunny and Spot the Cow, using mesh-based geodesic supervision and comparison with spectral RFM baselines. The method achieves lower reported test NLL and shorter training time in the described experiments.

C. Mesh Flow Matching

  • Geodesic paths between reference and data samples supply the training flows, with points and tangential velocities sampled along each path.The experiments use 500,000 samples generated from thresholded Laplace-Beltrami eigenfunctions on the Stanford Bunny and Spot the Cow.Source: C. Mesh Flow Matching · p. 12
  • Across the reported experiments, mesh RFM achieves lower test NLL than diffusion- and biharmonic-premetric RFM baselines.Test NLL is reported at the checkpoint with the lowest validation NLL; the table caption identifies the Stanford Bunny results and says lower is better.Source: C. Mesh Flow Matching · pp. 12–13, Table IV · p. 13
  • During integration, exponential-map updates keep paths consistent with the mesh, avoiding the baseline's per-step closest-point projections.The geodesic-path computation during training is amortized by inexpensive temporal upsampling of the vector field.Source: C. Mesh Flow Matching · p. 13
  • Training took 20 hours with the proposed method versus 1 day and 6 hours for the baseline under the reported 100,000-update setup.The authors used batches of 128 and validated every 2,000 steps.Source: C. Mesh Flow Matching · p. 13

VI. EXPERIMENTS

Tablet-drawn trajectories were generalized from a flat plane to target meshes and executed by a robot. The experiments include a surface with a non-differentiable underlying function.

A. Handwritten symbols learning and generalisation

  • Four target surfaces were 3D-printed from function graphs over a 20 × 20 cm domain, using dense print grids and coarser meshes for experiments.The print grid had 500 × 500 points; experimental meshes ranged from 40 × 40 to 80 × 80 points.Source: A. Handwritten symbols learning and… · p. 13
  • A web interface lets users sketch tablet demonstrations and select DMP types; the system typically generates 40,000 trajectory samples in under 5 seconds.The robot used hybrid force-impedance control and surface-normal pencil orientation, smoothed with a symmetric sliding-window filter.Source: A. Handwritten symbols learning and… · pp. 13–14
  • Tablet-drawn paths generalized to target meshes and were robot-executed across four drawing experiments, including on a surface with a non-differentiable underlying function.Figure 13 shows the demonstrated path, generalized trajectory, and corresponding robot execution for each experiment.Source: Figure 13 · p. 13, A. Handwritten symbols learning and… · p. 14

B. Surface polishing

The experiments demonstrate RGB-D-guided polishing on a reconstructed non-flat workpiece. The discussion summarizes the framework’s DMP, GP, and RFM results, alongside computational and kernel limitations and open extensions.

B. Surface polishing

  • An RGB-D-driven pipeline planned and executed polishing on a non-flat workpiece reconstructed online, with before-and-after images documenting the operation.The robot used a soft sponge and hybrid force-impedance control during execution.Source: B. Surface polishing · p. 14, Figure 14 · p. 14
  • A fitted polynomial path guided point-cloud segmentation and mesh reconstruction; a rhythmic DMP shifted its centre along the projected path to generate polishing motion.The centre moved at 5 mm/s; the offline trajectory was generated at 100 Hz, typically in under five seconds.Source: B. Surface polishing · p. 14

VII. DISCUSSION AND CONCLUSION

  • The framework defines discrete-surface logarithmic and exponential maps, parallel transport, and ambient-space projections, applying them to DMPs, GPs, and RFM.Its approach works directly with intrinsic surface geometry rather than treating surfaces as constraints enforced afterward.Source: §VII · p. 14
  • Fixed-tangent-cone forcing terms reconstructed by parallel transport eliminate DMP limit-cycle instability and improve motion-policy generalization across surfaces.The approach was compared with MeshDMP in simulation and tested in a real tablet-drawing pipeline; both demonstrated numerical efficiency.Source: §VII · pp. 14–15
  • Geodesic-based GPs enable regression at arbitrary surface locations without smoothness assumptions, while mesh-specific RFM operators improve generative quality and cut training time by about half.The GP approach is particularly effective on coarse meshes.Source: §VII · p. 15
  • Geodesic pre-computation becomes a significant barrier above 10,000 mesh faces, with complexity O(N^2 log N).Here N denotes the number of mesh faces.Source: §VII · p. 15
  • The geodesic kernel is not guaranteed positive-definite when its lengthscale exceeds κth, risking inconsistent GPs; better threshold characterization is needed.The paper identifies surface curvature, topology, and sample count as factors for future threshold characterization.Source: §VII · p. 15
  • Extending the framework to force-trajectory learning and transfer across different surface topologies, and to time-varying surfaces, remains future work.The paper names online-estimated, evolving meshes for surgical motion planning as one possible time-varying-surface application.Source: §VII · p. 15

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