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MMP++: Motion Manifold Primitives with Parametric Curve Models

Yonghyeon Lee

TL;DR

The paper addresses the limited temporal modulation and via-point handling of discrete-time Motion Manifold Primitives (MMP) by introducing Motion Manifold Primitives++ (MMP++) that uses parametric curve models to map latent coordinates to curve parameters and then to trajectories. It further introduces Isometric Motion Manifold Primitives++ (IMMP++) by applying CurveGeom-based isometric regularization to preserve the manifold geometry in the latent space, reducing geometric distortion. The approach demonstrates superior trajectory generation across 2-DoF planar, 7-DoF robot-arm, and SE(3) trajectory tasks, with effective latent-coordinate modulation and online re-planning capabilities under dynamic constraints. These methods reduce data dimensionality, enable efficient density estimation in latent space (e.g., GMM or KDE), and enable rapid online adaptation, with potential extensions to vision-conditioned conditioning and matrix Lie group data.

Abstract

Motion Manifold Primitives (MMP), a manifold-based approach for encoding basic motion skills, can produce diverse trajectories, enabling the system to adapt to unseen constraints. Nonetheless, we argue that current MMP models lack crucial functionalities of movement primitives, such as temporal and via-points modulation, found in traditional approaches. This shortfall primarily stems from MMP's reliance on discrete-time trajectories. To overcome these limitations, we introduce Motion Manifold Primitives++ (MMP++), a new model that integrates the strengths of both MMP and traditional methods by incorporating parametric curve representations into the MMP framework. Furthermore, we identify a significant challenge with MMP++: performance degradation due to geometric distortions in the latent space, meaning that similar motions are not closely positioned. To address this, Isometric Motion Manifold Primitives++ (IMMP++) is proposed to ensure the latent space accurately preserves the manifold's geometry. Our experimental results across various applications, including 2-DoF planar motions, 7-DoF robot arm motions, and SE(3) trajectory planning, show that MMP++ and IMMP++ outperform existing methods in trajectory generation tasks, achieving substantial improvements in some cases. Moreover, they enable the modulation of latent coordinates and via-points, thereby allowing efficient online adaptation to dynamic environments.

MMP++: Motion Manifold Primitives with Parametric Curve Models

TL;DR

The paper addresses the limited temporal modulation and via-point handling of discrete-time Motion Manifold Primitives (MMP) by introducing Motion Manifold Primitives++ (MMP++) that uses parametric curve models to map latent coordinates to curve parameters and then to trajectories. It further introduces Isometric Motion Manifold Primitives++ (IMMP++) by applying CurveGeom-based isometric regularization to preserve the manifold geometry in the latent space, reducing geometric distortion. The approach demonstrates superior trajectory generation across 2-DoF planar, 7-DoF robot-arm, and SE(3) trajectory tasks, with effective latent-coordinate modulation and online re-planning capabilities under dynamic constraints. These methods reduce data dimensionality, enable efficient density estimation in latent space (e.g., GMM or KDE), and enable rapid online adaptation, with potential extensions to vision-conditioned conditioning and matrix Lie group data.

Abstract

Motion Manifold Primitives (MMP), a manifold-based approach for encoding basic motion skills, can produce diverse trajectories, enabling the system to adapt to unseen constraints. Nonetheless, we argue that current MMP models lack crucial functionalities of movement primitives, such as temporal and via-points modulation, found in traditional approaches. This shortfall primarily stems from MMP's reliance on discrete-time trajectories. To overcome these limitations, we introduce Motion Manifold Primitives++ (MMP++), a new model that integrates the strengths of both MMP and traditional methods by incorporating parametric curve representations into the MMP framework. Furthermore, we identify a significant challenge with MMP++: performance degradation due to geometric distortions in the latent space, meaning that similar motions are not closely positioned. To address this, Isometric Motion Manifold Primitives++ (IMMP++) is proposed to ensure the latent space accurately preserves the manifold's geometry. Our experimental results across various applications, including 2-DoF planar motions, 7-DoF robot arm motions, and SE(3) trajectory planning, show that MMP++ and IMMP++ outperform existing methods in trajectory generation tasks, achieving substantial improvements in some cases. Moreover, they enable the modulation of latent coordinates and via-points, thereby allowing efficient online adaptation to dynamic environments.
Paper Structure (19 sections, 4 theorems, 33 equations, 16 figures, 3 tables, 1 algorithm)

This paper contains 19 sections, 4 theorems, 33 equations, 16 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Movement Primitives
  4. Manifold-based Movement Primitives
  5. Manifold Learning and Latent Space Distortion
  6. Geometric Preliminaries
  7. Riemannian Manifolds
  8. Immersion and Embedding
  9. Riemannian Geometry of Parametric Curves
  10. Isometry and Coordinate-Invariant Distortion Measure
  11. Isometric Motion Manifold Primitives++
  12. Discrete-Time MMPs and Their Limitations
  13. Motion Manifold Primitives++
  14. Isometric Regularization
  15. Experiments
  16. ...and 4 more sections

Key Result

Proposition 1

Suppose a curve $x(t;w)$ is smooth in both $t$ and $w$ and $x(t;\cdot): {\cal W} \to {\cal X}$ is injective, i.e., if $x(t;w_1)=x(t;w_2)$ for all $t \in [0, T]$, then $w_1 = w_2$. Let $w=(w^1,\ldots, w^n)$ and $v=(v^1,\ldots, v^n)\in \mathbb{R}^{n}$, if for all $w \in {\cal W}$ and ${\cal W}$ is compact, then ${\cal X}_{{\cal W}}$ is an $n$-dimensional smooth manifold.

Figures (16)

  • Figure 1: MMP++: A latent coordinate space ${\cal Z}$ is mapped to a subspace of the curve parameter space ${\cal W}$; the parameter space ${\cal W}$ is mapped to a subspace of the infinite-dimensional trajectory space. The motion manifold and parametric curve space are visualized as a curve and surface, not because their actual dimensions are one and two, but only to indicate the relative size relationships of their dimensions.
  • Figure 2: Left: There are 15 demonstration trajectories (red, green, and blue trajectories) that travel from the start to the goal, avoiding the obstacle. Middle and Right: MMP++ and IMMP++ learn two-dimensional manifolds in the curve parameter space and produce two-dimensional latent coordinate spaces. Latent values of the demonstration trajectories are visualized in the latent coordinate spaces, marked as $\times$. GMMs of three components are fitted in the latent spaces, and the sampled points are visualized as stars $*$. The corresponding generated trajectories are also visualized.
  • Figure 3: A local coordinate system for an $m$-dimensional Riemannian manifold ${\cal M}$. The Riemannian metric at coordinates $x$, $G(x)$, is visualized as a red equidistant ellipse that is $\{y \in \mathbb{R}^{m} \: | \: (y-x)^T G(x) (y-x) = {\rm constant}\}$.
  • Figure 4: An illustration of immersion and embedding between two manifolds ${\cal M}$ and ${\cal N}$.
  • Figure 5: An illustration of Riemannian geometry of the parametric curve manifold ${\cal X}_{\cal W}$.
  • ...and 11 more figures

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Definition 1
  • Proposition 4
  • proof