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Learning Dynamical Systems Encoding Non-Linearity within Space Curvature

Bernardo Fichera, Aude Billard

TL;DR

This work tackles the stability-expressivity trade-off in learning dynamical systems for robot control by encoding non-linearity in the curvature of a latent manifold and modeling the DS as a harmonic damped oscillator on that manifold. By embedding the $d$-D state space into a $(d+1)$-D Euclidean latent space and pulling back the metric, the authors obtain a chart-space DS whose non-linearity arises from curvature rather than a non-convex potential alone, while guaranteeing global asymptotic stability. The framework supports online local deformation to handle obstacles, via kernel-based and velocity-dependent space deformations, and extends naturally to second-order DS with geodesic and harmonic components, including hybrids to manage concave obstacles. Empirically, the method achieves competitive performance with lower training cost, demonstrates clear visualization of the embedded geometry, and translates to real-world 3D robotic end-effector tasks with effective obstacle avoidance and trajectory tracking. Overall, the approach integrates LfD with geometric control to yield expressive, stable policies suitable for reactive robotics in non-ideal environments.

Abstract

Dynamical Systems (DS) are an effective and powerful means of shaping high-level policies for robotics control. They provide robust and reactive control while ensuring the stability of the driving vector field. The increasing complexity of real-world scenarios necessitates DS with a higher degree of non-linearity, along with the ability to adapt to potential changes in environmental conditions, such as obstacles. Current learning strategies for DSs often involve a trade-off, sacrificing either stability guarantees or offline computational efficiency in order to enhance the capabilities of the learned DS. Online local adaptation to environmental changes is either not taken into consideration or treated as a separate problem. In this paper, our objective is to introduce a method that enhances the complexity of the learned DS without compromising efficiency during training or stability guarantees. Furthermore, we aim to provide a unified approach for seamlessly integrating the initially learned DS's non-linearity with any local non-linearities that may arise due to changes in the environment. We propose a geometrical approach to learn asymptotically stable non-linear DS for robotics control. Each DS is modeled as a harmonic damped oscillator on a latent manifold. By learning the manifold's Euclidean embedded representation, our approach encodes the non-linearity of the DS within the curvature of the space. Having an explicit embedded representation of the manifold allows us to showcase obstacle avoidance by directly inducing local deformations of the space. We demonstrate the effectiveness of our methodology through two scenarios: first, the 2D learning of synthetic vector fields, and second, the learning of 3D robotic end-effector motions in real-world settings.

Learning Dynamical Systems Encoding Non-Linearity within Space Curvature

TL;DR

This work tackles the stability-expressivity trade-off in learning dynamical systems for robot control by encoding non-linearity in the curvature of a latent manifold and modeling the DS as a harmonic damped oscillator on that manifold. By embedding the -D state space into a -D Euclidean latent space and pulling back the metric, the authors obtain a chart-space DS whose non-linearity arises from curvature rather than a non-convex potential alone, while guaranteeing global asymptotic stability. The framework supports online local deformation to handle obstacles, via kernel-based and velocity-dependent space deformations, and extends naturally to second-order DS with geodesic and harmonic components, including hybrids to manage concave obstacles. Empirically, the method achieves competitive performance with lower training cost, demonstrates clear visualization of the embedded geometry, and translates to real-world 3D robotic end-effector tasks with effective obstacle avoidance and trajectory tracking. Overall, the approach integrates LfD with geometric control to yield expressive, stable policies suitable for reactive robotics in non-ideal environments.

Abstract

Dynamical Systems (DS) are an effective and powerful means of shaping high-level policies for robotics control. They provide robust and reactive control while ensuring the stability of the driving vector field. The increasing complexity of real-world scenarios necessitates DS with a higher degree of non-linearity, along with the ability to adapt to potential changes in environmental conditions, such as obstacles. Current learning strategies for DSs often involve a trade-off, sacrificing either stability guarantees or offline computational efficiency in order to enhance the capabilities of the learned DS. Online local adaptation to environmental changes is either not taken into consideration or treated as a separate problem. In this paper, our objective is to introduce a method that enhances the complexity of the learned DS without compromising efficiency during training or stability guarantees. Furthermore, we aim to provide a unified approach for seamlessly integrating the initially learned DS's non-linearity with any local non-linearities that may arise due to changes in the environment. We propose a geometrical approach to learn asymptotically stable non-linear DS for robotics control. Each DS is modeled as a harmonic damped oscillator on a latent manifold. By learning the manifold's Euclidean embedded representation, our approach encodes the non-linearity of the DS within the curvature of the space. Having an explicit embedded representation of the manifold allows us to showcase obstacle avoidance by directly inducing local deformations of the space. We demonstrate the effectiveness of our methodology through two scenarios: first, the 2D learning of synthetic vector fields, and second, the learning of 3D robotic end-effector motions in real-world settings.
Paper Structure (24 sections, 2 theorems, 52 equations, 19 figures, 4 tables)

This paper contains 24 sections, 2 theorems, 52 equations, 19 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. Learning the Latent Manifold Embedding
  5. Gradient Systems & Incremental Learning
  6. Online Kernel-Based Local Space Deformation
  7. Obstacle Avoidance via Direct Space Deformation
  8. Local Space Deformation In First Order DS
  9. Local Space Deformation In Second Order DS
  10. Synthetic Example
  11. Locally Active Space Deformation
  12. Directional & Exponential Dissipation
  13. Obstacle Avoidance Online Direct Deformation
  14. Evaluation
  15. Robotics Experiments
  16. ...and 9 more sections

Key Result

proposition 1

$f : \mathcal{M} \rightarrow \mathbb{R}^{d+1}$ is a smooth mapping with local coordinates as in eqn:embedding. $f : \mathcal{M} \hookrightarrow \mathbb{R}^{d+1}$ is an embedding.

Figures (19)

  • Figure 1: End-effector motion of a robotic arm guided by a 2D learned DS. The surface corresponds to the 3D Euclidean space embedded representation of the learnt 2D manifold. The color gradient represents the value of the potential function that drives the linear vector field taking place on the 2D manifold. The manifold's curvature induces "apparent" non-linearity in the 2D chart Euclidean space representation of the vector field taking place on the 2D manifold. During the learning process, the curvature of the manifold adapts so that the streamlines of the 2D chart Euclidean space representation of the vector field follow closely the demonstrated trajectories (red dots), preserving the stability towards a desired equilibrium point (yellow star).
  • Figure 2:
  • Figure 3: Firs-order DS in flat space with localized deformation in the obstacle area: (a) Vector field with one sampled streamline avoiding the obstacle; (b) $3$D embedded representation of the manifold; (c) one sampled trajectory with eigenvalue decomposition ellipses of the inverse of the metric for selected location; (d) metric determinant function with eigenvalue decomposition ellipses of the metric.
  • Figure 4: (a)-(c) Geodesic motion at time instants: $1$s, $5$s and $10$s; (d) 3D embedded representation of case (c). In background the color gradient represents the $d$+1 embedding coordinate.
  • Figure 5:
  • ...and 14 more figures

Theorems & Definitions (4)

  • proposition 1
  • proof
  • theorem 1
  • proof