This paper presents a coordinate-free formulation of virtual holonomic constraints for underactuated Lagrangian control systems on Riemannian manifolds. It is shown that when a virtual constraint enjoys a regularity property, the constrained dynamics are described by an affine connection dynamical system. The affine connection of the constrained system has an elegant relationship to the Riemannian connection of the original Lagrangian control system. Necessary and sufficient conditions are given for the constrained dynamics to be Lagrangian. A key condition is that the affine connection of the constrained dynamics be metrizable. Basic results on metrizability of affine connections are first reviewed, then employed in three examples in order of increasing complexity. The last example is a double pendulum on a cart with two different actuator configurations. For this control system, a virtual constraint is employed which confines the second pendulum to within the upper half-plane.

A coordinate-free theory of virtual holonomic constraints / Consolini, Luca; Costalunga, Alessandro; Maggiore, Manfredi. - In: JOURNAL OF GEOMETRIC MECHANICS. - ISSN 1941-4889. - 10:4(2018), pp. 467-502. [10.3934/jgm.2018018]

### A coordinate-free theory of virtual holonomic constraints

#### Abstract

This paper presents a coordinate-free formulation of virtual holonomic constraints for underactuated Lagrangian control systems on Riemannian manifolds. It is shown that when a virtual constraint enjoys a regularity property, the constrained dynamics are described by an affine connection dynamical system. The affine connection of the constrained system has an elegant relationship to the Riemannian connection of the original Lagrangian control system. Necessary and sufficient conditions are given for the constrained dynamics to be Lagrangian. A key condition is that the affine connection of the constrained dynamics be metrizable. Basic results on metrizability of affine connections are first reviewed, then employed in three examples in order of increasing complexity. The last example is a double pendulum on a cart with two different actuator configurations. For this control system, a virtual constraint is employed which confines the second pendulum to within the upper half-plane.
##### Scheda breve Scheda completa Scheda completa (DC)
2018
A coordinate-free theory of virtual holonomic constraints / Consolini, Luca; Costalunga, Alessandro; Maggiore, Manfredi. - In: JOURNAL OF GEOMETRIC MECHANICS. - ISSN 1941-4889. - 10:4(2018), pp. 467-502. [10.3934/jgm.2018018]
File in questo prodotto:
File
A COORDINATE-FREE THEORY OF VIRTUAL HOLONOMIC CONSTRAINTS.pdf

accesso aperto

Descrizione: Preprint dell'articolo
Tipologia: Documento in Pre-print
Licenza: Creative commons
Dimensione 1.11 MB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11381/2856283`