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Nonholonomic mechanics and virtual constraints on Riemannian homogeneous spaces

Efstratios Stratoglou, Alexandre Anahory Simoes, Anthony Bloch, Leonardo J. Colombo

Abstract

Nonholonomic systems are, so to speak, mechanical systems with a prescribed restriction on the velocities. A virtual nonholonomic constraint is a controlled invariant distribution associated with an affine connection mechanical control system. A Riemannian homogeneous space is, a Riemannian manifold that looks the same everywhere, as you move through it by the action of a Lie group. These Riemannian manifolds are not necessarily Lie groups themselves, but nonetheless possess certain symmetries and invariances that allow for similar results to be obtained. In this work, we introduce the notion of virtual constraint on Riemannian homogeneous spaces in a geometric framework which is a generalization of the classical controlled invariant distribution setting and we show the existence and uniqueness of a control law preserving the invariant distribution. Moreover we characterize the closed-loop dynamics obtained using the unique control law in terms of an affine connection. We illustrate the theory with new examples of nonholonomic control systems inspired by robotics applications.

Nonholonomic mechanics and virtual constraints on Riemannian homogeneous spaces

Abstract

Nonholonomic systems are, so to speak, mechanical systems with a prescribed restriction on the velocities. A virtual nonholonomic constraint is a controlled invariant distribution associated with an affine connection mechanical control system. A Riemannian homogeneous space is, a Riemannian manifold that looks the same everywhere, as you move through it by the action of a Lie group. These Riemannian manifolds are not necessarily Lie groups themselves, but nonetheless possess certain symmetries and invariances that allow for similar results to be obtained. In this work, we introduce the notion of virtual constraint on Riemannian homogeneous spaces in a geometric framework which is a generalization of the classical controlled invariant distribution setting and we show the existence and uniqueness of a control law preserving the invariant distribution. Moreover we characterize the closed-loop dynamics obtained using the unique control law in terms of an affine connection. We illustrate the theory with new examples of nonholonomic control systems inspired by robotics applications.

Paper Structure

This paper contains 17 sections, 14 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Nonholonomic systems in a coordinate-free setting
  3. Background on Riemannian manifolds
  4. Riemannian geometry and Lie groups
  5. Nonholonomic systems on Lie Groups
  6. Nonholonomic systems on Riemannian homogeneous spaces
  7. Homogeneous spaces
  8. Riemannian Homogeneous Spaces
  9. Mechanical systems on Riemannian homogeneous spaces
  10. Nonholonomic systems on Riemannian homogeneous spaces
  11. Virtual nonholomic constraints on Riemannian homogeneous spaces
  12. Virtual constraints
  13. Virtual nonholonomic constraints on Riemannian homogeneous spaces
  14. The induced constrained connection
  15. Example: A sphere rolling over another sphere
  16. ...and 2 more sections

Key Result

Lemma 1

The Riemannian $\mathfrak{g}-$connection satisfies: for all $\xi,\eta\in\mathfrak{g}$, where $\sharp_{\mathfrak{g}}:\mathfrak{g}^{*}\to\mathfrak{g}$ is the inverse map of $\flat_{\mathfrak{g}}$.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2: see goodman2024reduction
  • Theorem 1
  • Example 1
  • Example 2
  • Theorem 2
  • Proof 1
  • Example 3
  • ...and 23 more