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Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modelling

Daniel Ordoñez-Apraez, Vladimir Kostic, Giulio Turrisi, Pietro Novelli, Carlos Mastalli, Claudio Semini, Massimiliano Pontil

TL;DR

This paper tackles the challenge of learning globally linear models for robotic dynamics by exploiting discrete morphological symmetries. It introduces Dynamics Harmonic Analysis (DHA), which decomposes the state space into isotypic subspaces corresponding to irreducible representations of a symmetry group, enabling independent linear dynamics on each subspace. Building on this, the authors propose the equivariant Dynamics Autoencoder (eDAE), a neural architecture that learns a Koopman-operator-based model while enforcing $\mathbb{G}$-equivariance, leading to improved generalization, sample efficiency, and interpretability. Experiments on synthetic symmetric systems and a quadruped robot (mini-cheetah) demonstrate the method’s ability to capture both transient and steady locomotion dynamics with fewer parameters and lower computational cost, aided by an open-access library for reproducibility. The approach offers a principled, symmetry-aware framework for data-driven robotics with broad applicability to systems possessing discrete symmetry groups.

Abstract

We introduce the use of harmonic analysis to decompose the state space of symmetric robotic systems into orthogonal isotypic subspaces. These are lower-dimensional spaces that capture distinct, symmetric, and synergistic motions. For linear dynamics, we characterize how this decomposition leads to a subdivision of the dynamics into independent linear systems on each subspace, a property we term dynamics harmonic analysis (DHA). To exploit this property, we use Koopman operator theory to propose an equivariant deep-learning architecture that leverages the properties of DHA to learn a global linear model of the system dynamics. Our architecture, validated on synthetic systems and the dynamics of locomotion of a quadrupedal robot, exhibits enhanced generalization, sample efficiency, and interpretability, with fewer trainable parameters and computational costs.

Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modelling

TL;DR

This paper tackles the challenge of learning globally linear models for robotic dynamics by exploiting discrete morphological symmetries. It introduces Dynamics Harmonic Analysis (DHA), which decomposes the state space into isotypic subspaces corresponding to irreducible representations of a symmetry group, enabling independent linear dynamics on each subspace. Building on this, the authors propose the equivariant Dynamics Autoencoder (eDAE), a neural architecture that learns a Koopman-operator-based model while enforcing -equivariance, leading to improved generalization, sample efficiency, and interpretability. Experiments on synthetic symmetric systems and a quadruped robot (mini-cheetah) demonstrate the method’s ability to capture both transient and steady locomotion dynamics with fewer parameters and lower computational cost, aided by an open-access library for reproducibility. The approach offers a principled, symmetry-aware framework for data-driven robotics with broad applicability to systems possessing discrete symmetry groups.

Abstract

We introduce the use of harmonic analysis to decompose the state space of symmetric robotic systems into orthogonal isotypic subspaces. These are lower-dimensional spaces that capture distinct, symmetric, and synergistic motions. For linear dynamics, we characterize how this decomposition leads to a subdivision of the dynamics into independent linear systems on each subspace, a property we term dynamics harmonic analysis (DHA). To exploit this property, we use Koopman operator theory to propose an equivariant deep-learning architecture that leverages the properties of DHA to learn a global linear model of the system dynamics. Our architecture, validated on synthetic systems and the dynamics of locomotion of a quadrupedal robot, exhibits enhanced generalization, sample efficiency, and interpretability, with fewer trainable parameters and computational costs.
Paper Structure (9 sections, 1 theorem, 3 equations, 1 figure)

This paper contains 9 sections, 1 theorem, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related work
  4. Preliminaries
  5. Modelling of dynamical systems
  6. Linear models and the Koopman operator
  7. Symmetry groups and their representations
  8. Isotypic decomposition and its basis
  9. Symmetries of dynamical systems

Key Result

Theorem 1

Let $\mathbb{G}_{\hbox{$$}}$ be a compact symmetry group and ${\mathcal{H}}$ a symmetric separable Hilbert space with a unitary group representation ${\rho_{ {\vcenter{\hbox{${\mathcal{H}}$}}} } } : \mathbb{G}_{\hbox{$$}}\rightarrow \mathbb{U}({\mathcal{H}})$. Then we can identify ${k} \leq |\mathbb

Figures (1)

  • Figure 1: (a) Diagram of the discrete symmetry group $\mathbb{G}_{\hbox{$$}}:=\mathbb{K}_{4}$ of the mini-cheetah robot (see https://github.com/Danfoa/MorphoSymm/blob/devel/docs/static/animations/mini_cheetah-Klein4-symmetries_anim_static.gif?utm_source=l4dc&utm_medium=l4dc&utm_campaign=l4dc). Each symmetry $g\in \mathbb{G}_{\hbox{$$}}$ relates states that evolve identically under physics laws; see https://github.com/Danfoa/MorphoSymm/blob/devel/docs/static/dynamic_animations/mini-cheetah-dynamic_symmetries_forces.gif?utm_source=l4dc&utm_medium=l4dc&utm_campaign=l4dc of states related by $g_{s}$. This results in the decomposition of the set of states ${\Omega}$ into $g$-transformed copies of the quotient set ${\Omega}/\mathbb{G}_{\hbox{$$}}$, encompassing all unique system states. (b) Isotypic decomposition of the robot's space of generalized position coordinates $\mathcal{Q}$ into isotypic subspaces: $\mathcal{Q}:= \vcenter{\hbox{$\bm{\oplus}$}}_{{i}=1}^{4} \mathcal{Q}_{i}$. Each subspace, $\mathcal{Q}_{i}$ describes a space of symmetry-constrained synergistic motions. Consequently, any position configuration ${\bm{q}}_{}(\omega) \in \mathcal{Q}$, can be decomposed into projections within these subspaces: ${\bm{q}}_{}(\omega) := \vcenter{\hbox{$\bm{\oplus}$}}_{{i}=1}^{4} {\bm{q}}_{}^{(i)}(\omega)$ (see https://github.com/Danfoa/DynamicsHarmonicsAnalysis/blob/main/media/DynamicsHarmonicAnalysis_mini_cheetah_K4.md?utm_source=l4dc&utm_medium=l4dc&utm_campaign=l4dc). (c) Joint-space Kinetic energy distribution across isotypic subspaces for two gait/motion trajectories in the real world: jumping and trotting (see https://github.com/Danfoa/DynamicsHarmonicsAnalysis/blob/main/media/DynamicsHarmonicAnalysis_mini_cheetah_K4.md?utm_source=l4dc&utm_medium=l4dc&utm_campaign=l4dc). Both gaits primarily evolve within one or two lower-dimensional isotypic subspaces, with less significant subspaces engaged temporarily during disturbances.

Theorems & Definitions (2)

  • Theorem 1: Isotypic Decomposition
  • Definition 1: Symmetric dynamical systems