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Learning Nonholonomic Dynamics with Constraint Discovery

Baiyue Wang, Anthony Bloch

TL;DR

The paper addresses learning nonholonomic dynamics with constraint discovery by leveraging Hamel's formalism and symmetry reduction to a Lie algebra. It introduces a four-step learning procedure that parameterizes the constraint via a neural network, constructs a moving-basis representation, and trains a Neural ODE to recover the dynamics from trajectory data on the tangent bundle $TQ$. The Rolling Disk on $\mathbb{R}\times SE(2)$ serves as a case study, where the approach recovers the constraint map $\tilde{\mathcal{A}}(r,g)$ and matches ground-truth momenta, albeit with some limitations due to numerical integration and data diversity. The work demonstrates constraint discovery from data without explicit constraint specification and discusses practical implications, potential improvements (more diverse trajectories, symmetry-preserving integrators), and broader applicability to real-world nonholonomic systems.

Abstract

We consider learning nonholonomic dynamical systems while discovering the constraints, and describe in detail the case of the rolling disk. A nonholonomic system is a system subject to nonholonomic constraints. Unlike holonomic constraints, nonholonomic constraints do not define a sub-manifold on the configuration space. Therefore, the inverse problem of finding the constraints has to involve the tangent bundle. This paper discusses a general procedure to learn the dynamics of a nonholonomic system through Hamel's formalism, while discovering the system constraint by parameterizing it, given the data set of discrete trajectories on the tangent bundle $TQ$. We prove that there is a local minimum for convergence of the network. We also preserve symmetry of the system by reducing the Lagrangian to the Lie algebra of the selected group.

Learning Nonholonomic Dynamics with Constraint Discovery

TL;DR

The paper addresses learning nonholonomic dynamics with constraint discovery by leveraging Hamel's formalism and symmetry reduction to a Lie algebra. It introduces a four-step learning procedure that parameterizes the constraint via a neural network, constructs a moving-basis representation, and trains a Neural ODE to recover the dynamics from trajectory data on the tangent bundle . The Rolling Disk on serves as a case study, where the approach recovers the constraint map and matches ground-truth momenta, albeit with some limitations due to numerical integration and data diversity. The work demonstrates constraint discovery from data without explicit constraint specification and discusses practical implications, potential improvements (more diverse trajectories, symmetry-preserving integrators), and broader applicability to real-world nonholonomic systems.

Abstract

We consider learning nonholonomic dynamical systems while discovering the constraints, and describe in detail the case of the rolling disk. A nonholonomic system is a system subject to nonholonomic constraints. Unlike holonomic constraints, nonholonomic constraints do not define a sub-manifold on the configuration space. Therefore, the inverse problem of finding the constraints has to involve the tangent bundle. This paper discusses a general procedure to learn the dynamics of a nonholonomic system through Hamel's formalism, while discovering the system constraint by parameterizing it, given the data set of discrete trajectories on the tangent bundle . We prove that there is a local minimum for convergence of the network. We also preserve symmetry of the system by reducing the Lagrangian to the Lie algebra of the selected group.

Paper Structure

This paper contains 21 sections, 3 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Background
  4. Connections and the Horizontal Lift
  5. Connections and the Horizontal Lift
  6. Matrix Representation of Connections and the Horizontal Lift
  7. Reduction to Lie Algebra
  8. The Hamel Equations
  9. Learning Nonholonomic Dynamics
  10. Step One: Data Reduction to Lie Algebra
  11. Step Two: Generate Moving Basis and Quasi-velocities
  12. Step Three: Compute Momentum and Generate Dynamics
  13. Step Four: Training with Neural ODE
  14. Learning the Rolling Disk Dynamics
  15. The $\mathbb{R}\times SE(2)$ version
  16. ...and 6 more sections

Key Result

proposition 1

(1) Both $A$ and $hor$ are projections. (2) They lie in the kernel of each other. (3) Both $V_q$ and $H_q$ are unique given a nonholonomic constraint eq:def-constraint.

Figures (1)

  • Figure 1: The above figure shows the training result after 2000 epochs. The sub plot on the left shows the ground truth momenta (solid lines) and the learned dynamics (dashed lines) generated by a numerical integrator. The sub plot on the right shows the ground truth $\mathcal{A}(r,g)$ (solid lines) and the parameterized $\tilde{\mathcal{A}}(r,g)$ (dashed lines) evaluated after training.

Theorems & Definitions (7)

  • definition 1: Bloch2015
  • definition 2: Lee2000 Page 268
  • definition 3: Bloch2015
  • proposition 1
  • proof
  • theorem 1: Hamel Equations
  • theorem 2: Unique Dynamics Generated by $\mathcal{A}$