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From gymnastics to virtual nonholonomic constraints: energy injection, dissipation, and regulation for the acrobot

Adan Moran-MacDonald, Manfredi Maggiore, Xingbo Wang

Abstract

In this article we study virtual nonholonomic constraints, which are relations between the generalized coordinates and momenta of a mechanical system that can be enforced via feedback control. We design a constraint which emulates gymnastics giant motion in an acrobot, and prove that this constraint can inject or dissipate energy based on the sign of a design parameter. The proposed constraint is tested both in simulation and experimentally on a real-world acrobot, demonstrating highly effective energy regulation properties and robustness to a variety of disturbances.

From gymnastics to virtual nonholonomic constraints: energy injection, dissipation, and regulation for the acrobot

Abstract

In this article we study virtual nonholonomic constraints, which are relations between the generalized coordinates and momenta of a mechanical system that can be enforced via feedback control. We design a constraint which emulates gymnastics giant motion in an acrobot, and prove that this constraint can inject or dissipate energy based on the sign of a design parameter. The proposed constraint is tested both in simulation and experimentally on a real-world acrobot, demonstrating highly effective energy regulation properties and robustness to a variety of disturbances.

Paper Structure

This paper contains 22 sections, 8 theorems, 51 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Preliminaries on vnhcs
  4. Simply Actuated Hamiltonian Systems
  5. Virtual Nonholonomic Constraints
  6. The Proposed Acrobot vnhc
  7. Energy Injection and Dissipation with the Proposed vnhc
  8. Energy Regulation
  9. Simulation Results
  10. Energy Injection
  11. Energy Dissipation
  12. Oscillation Regulation
  13. Rotation Regulation
  14. Summary of Results
  15. Physical Experiments
  16. ...and 7 more sections

Key Result

Lemma 1

There exists a nonsingular matrix $\hat{T} \in \mathbb{R}^{k \times k}$ so that the regular feedback transformation has a new input matrix $\hat{B}$ for $\hat{\tau}$ which is left semi-orthogonal.

Figures (12)

  • Figure 1: A simplified two-link acrobot as a model for a gymnast. Image modified from xingbo_thesis.
  • Figure 2: The acrobot constraint $q_a = \bar{q}_a \arctan(I p_u)$.
  • Figure 3: Oscillations and rotations gaining energy.
  • Figure 4: The distributed mass acrobot model, represented by two weighted rods differing in both length and mass.
  • Figure 5: A simulation of the acrobot gaining energy.
  • ...and 7 more figures

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Definition 1
  • Definition 2
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 10 more