MeshDMP: Motion Planning on Discrete Manifolds using Dynamic Movement Primitives

TL;DR

MeshDMP addresses the challenge of reliable in-contact motions on complex workpiece geometries by learning surface-aware policies directly on triangulated meshes. It extends Dynamic Movement Primitives to discrete manifolds by introducing geometry-aware operators, including $\log$ and $\exp$ maps and parallel transport, and by using a covariant derivative-based transformation with an isometric projection $\mathbf{T}(\boldsymbol{y},\boldsymbol{z})$ to adapt the forcing term across surfaces. The approach learns the forcing term from demonstrations projected onto the mesh via least-squares fitting of Gaussian basis weights, and validates the method through synthetic experiments and real car fender polishing with a KUKA manipulator, showing robust generalisation to different topologies. Real-time considerations are discussed, highlighting microsecond-scale geodesic queries but potential runtime costs for end-point updates, suggesting precomputation and caching for online deployment. Overall, MeshDMP provides a geometry-aware framework for efficient, adaptable motion on arbitrary mesh surfaces with practical industrial relevance.

Abstract

An open problem in industrial automation is to reliably perform tasks requiring in-contact movements with complex workpieces, as current solutions lack the ability to seamlessly adapt to the workpiece geometry. In this paper, we propose a Learning from Demonstration approach that allows a robot manipulator to learn and generalise motions across complex surfaces by leveraging differential mathematical operators on discrete manifolds to embed information on the geometry of the workpiece extracted from triangular meshes, and extend the Dynamic Movement Primitives (DMPs) framework to generate motions on the mesh surfaces. We also propose an effective strategy to adapt the motion to different surfaces, by introducing an isometric transformation of the learned forcing term. The resulting approach, namely MeshDMP, is evaluated both in simulation and real experiments, showing promising results in typical industrial automation tasks like car surface polishing.

Figures (5)

• Figure 1: (a) Demonstrated trajectory (orange) and path obtained by MeshDMP (blue) after learning the forcing term. (b) Position trajectory of the demonstration (dashed lines) and the one obtained through MeshDMP integration (solid lines).
• Figure 2: Execution of a MeshDMP learned on an 8-shaped trajectory on the flat surfaces, then generalised to a low (a) and high (b) polygonal density meshed torus, as well as on a simplified Stanford bunny (c).
• Figure 3: Results obtained in the car fender polishing with fixed centre experiment. (a) MeshDMP (blue) executed on the front fender of a car considering a fixed centre position. The purple line marks the cusp-shaped crest on the fender. (b) Position and orientation trajectories generated by the MeshDMP (dashed lines) and followed by the robot (solid line).
• Figure 4: End-effector configurations while traversing the crest present in the surface.
• Figure 5: Results obtained in the car fender polishing with shifted centre experiment. (a) MeshDMP (blue) executed on the fender of a car while shifting the centre from $\boldsymbol{g}_{\textrm{start}}$ to $\boldsymbol{g}_{\textrm{final}}$ (orange). (b) Pose trajectories generated by MeshDMP (dashed) and followed by the robot (solid lines).