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Geometric stabilization of virtual linear nonholonomic constraints

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

Abstract

In this paper, we give sufficient conditions for and deduce a control law under which a mechanical control system converges exponentially fast to a virtual linear nonholonomic constraint that is control invariant via the same feedback control. Virtual constraints are relations imposed on a control system that become invariant via feedback control, as opposed to physical constraints acting on the system. Virtual nonholonomic constraints, similarly to mechanical nonholonomic constraints, are a class of virtual constraints that depend on velocities rather than only on the configurations of the system.

Geometric stabilization of virtual linear nonholonomic constraints

Abstract

In this paper, we give sufficient conditions for and deduce a control law under which a mechanical control system converges exponentially fast to a virtual linear nonholonomic constraint that is control invariant via the same feedback control. Virtual constraints are relations imposed on a control system that become invariant via feedback control, as opposed to physical constraints acting on the system. Virtual nonholonomic constraints, similarly to mechanical nonholonomic constraints, are a class of virtual constraints that depend on velocities rather than only on the configurations of the system.

Paper Structure

This paper contains 5 sections, 3 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Nonholonomic mechanical systems
  3. Virtual nonholonomic constraints
  4. Stabilization of virtual nonholonomic constraints
  5. Conclusions and Future Work

Key Result

Theorem 1

If the distribution $\mathcal{D}$ and the control input distribution $\mathcal{F}$ are transversal, then there exists a unique control function $u:\mathcal{D}\to\mathbb{R}^{m}$ making the distribution a virtual nonholonomic constraint associated with the mechanical control system mechanical:control:

Figures (4)

  • Figure 1: The projection of a trajectory of the closed-loop system into the plane $xy$ in Example \ref{['nh:robot']}.
  • Figure 2: The constraint function $\hat{\mu}(t)$ along the same trajectory in Example \ref{['nh:robot']}.
  • Figure 3: Projection of a trajectory of the closed-loop system into the plane $xy$. Example \ref{['disk:example']}.
  • Figure 4: Constraint functions in Example \ref{['disk:example']}.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Remark 2
  • ...and 4 more