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A Unified Formulation of Geometry-aware Dynamic Movement Primitives

Fares J. Abu-Dakka, Matteo Saveriano, Ville Kyrki

TL;DR

This work extends dynamic movement primitives to operate on arbitrary Riemannian manifolds by embedding geometry-aware dynamics directly in the manifold's structure. The proposed gadmp unifies handling of diverse data types (e.g., orientations, SPD matrices, and composite manifolds) and preserves key DMP properties, including convergence to a goal and on line goal switching, without requiring data reparametrization. A formal stability analysis underpins the approach, and extensive validation on synthetic LASA-like datasets and real robot tasks demonstrates accurate reproduction, smooth goal switching, and the ability to learn manipulability ellipsoids within composite manifolds. The framework enables robust learning and execution of complex skills in unstructured settings, with potential for integration with iterative learning control and manifold-aware exploration strategies.

Abstract

Learning from demonstration (LfD) is considered as an efficient way to transfer skills from humans to robots. Traditionally, LfD has been used to transfer Cartesian and joint positions and forces from human demonstrations. The traditional approach works well for some robotic tasks, but for many tasks of interest, it is necessary to learn skills such as orientation, impedance, and/or manipulability that have specific geometric characteristics. An effective encoding of such skills can be only achieved if the underlying geometric structure of the skill manifold is considered and the constrains arising from this structure are fulfilled during both learning and execution. However, typical learned skill models such as dynamic movement primitives (DMPs) are limited to Euclidean data and fail in correctly embedding quantities with geometric constraints. In this paper, we propose a novel and mathematically principled framework that uses concepts from Riemannian geometry to allow DMPs to properly embed geometric constrains. The resulting DMP formulation can deal with data sampled from any Riemannian manifold including, but not limited to, unit quaternions and symmetric and positive definite matrices. The proposed approach has been extensively evaluated both on simulated data and real robot experiments. The performed evaluation demonstrates that beneficial properties of DMPs, such as convergence to a given goal and the possibility to change the goal during operation, apply also to the proposed formulation.

A Unified Formulation of Geometry-aware Dynamic Movement Primitives

TL;DR

This work extends dynamic movement primitives to operate on arbitrary Riemannian manifolds by embedding geometry-aware dynamics directly in the manifold's structure. The proposed gadmp unifies handling of diverse data types (e.g., orientations, SPD matrices, and composite manifolds) and preserves key DMP properties, including convergence to a goal and on line goal switching, without requiring data reparametrization. A formal stability analysis underpins the approach, and extensive validation on synthetic LASA-like datasets and real robot tasks demonstrates accurate reproduction, smooth goal switching, and the ability to learn manipulability ellipsoids within composite manifolds. The framework enables robust learning and execution of complex skills in unstructured settings, with potential for integration with iterative learning control and manifold-aware exploration strategies.

Abstract

Learning from demonstration (LfD) is considered as an efficient way to transfer skills from humans to robots. Traditionally, LfD has been used to transfer Cartesian and joint positions and forces from human demonstrations. The traditional approach works well for some robotic tasks, but for many tasks of interest, it is necessary to learn skills such as orientation, impedance, and/or manipulability that have specific geometric characteristics. An effective encoding of such skills can be only achieved if the underlying geometric structure of the skill manifold is considered and the constrains arising from this structure are fulfilled during both learning and execution. However, typical learned skill models such as dynamic movement primitives (DMPs) are limited to Euclidean data and fail in correctly embedding quantities with geometric constraints. In this paper, we propose a novel and mathematically principled framework that uses concepts from Riemannian geometry to allow DMPs to properly embed geometric constrains. The resulting DMP formulation can deal with data sampled from any Riemannian manifold including, but not limited to, unit quaternions and symmetric and positive definite matrices. The proposed approach has been extensively evaluated both on simulated data and real robot experiments. The performed evaluation demonstrates that beneficial properties of DMPs, such as convergence to a given goal and the possibility to change the goal during operation, apply also to the proposed formulation.
Paper Structure (30 sections, 39 equations, 12 figures, 3 tables, 1 algorithm)

This paper contains 30 sections, 39 equations, 12 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Dynamic Movement Primitives
  5. Riemannian manifolds
  6. Mapping operators
  7. Cartesian products in Riemannian geometry
  8. Computing in Riemannian manifolds
  9. Riemannian geometric mean
  10. Proposed approach
  11. Geometry-aware formulation
  12. Goal switching
  13. Stability analysis
  14. on Riemannian manifold products
  15. Validation
  16. ...and 15 more sections

Figures (12)

  • Figure 1: A Riemannian manifold ${\mathcal{M}}$ and its tangent space ${\mathcal{T}_{\bm{P}}\mathcal{M}}$ defined at point $\bm{P}$.
  • Figure 2: Results of gadmp while learning and producing trajectories that cover both south and north hemispheres. Black dashed curves denote demonstrations, while brown curves represent reproduction. Green point ${\bm{Y}}_1$ denotes the starting point of the trajectory, while the blue one indicates the goal ${\bm{G}}$. The red point illustrates the antipodal point of the goal. The figure shows gadmp while executing a trajectory that (a) does not contain an antipodal of the goal ${\bm{G}}$, and (b) contains an antipodal of the goal.
  • Figure 3: Illustrates the performance of gadmp when executing Riemannian LASA dataset. 1$^{\text{st}}$row: Euclidean 2D trajectory. 2$^{\text{nd}}$row: Unit quaternion trajectory. 3$^{\text{rd}}$ row: Rotation matrix trajectory. 4$^{\text{th}}$row: spd trajectory. 1$^{\text{st}}$column: Trajectories from different manifolds. 2$^{\text{nd}}$column: first-derivative in different manifolds. 3$^{\text{rd}}$ column: The distance in each manifold between the demonstration and the gadmp reproduction. 4$^{\text{th}}$column: The Cartesian representation of the gadmp reproduction. In 1$^{\text{st}}$ and 2$^{\text{nd}}$columns, dashed lines represent demonstration data while colored solid lines represent the gadmp results.
  • Figure 4: gadmp execution of the same unit quaternion trajectory tested in koutras2020correct. The first three rows show the error between the current unit quaternion and the goal (left) and new goal (right). The bottom four rows show the evolution of each unit quaternion element, over time, toward the goal and new goal. Dashed black lines represent information related to the demonstration trajectory.
  • Figure 5: Comparison between the proposed gadmp and koutras2020correct. The first three rows show more stable starting using gadmp. Bottom: Compares the mean error of gadmp (in red) and koutras2020correct (dashed black lines).
  • ...and 7 more figures

Theorems & Definitions (1)

  • proof