A coordinate-free theory of virtual holonomic constraints

TL;DR

This paper develops a coordinate-free framework for virtual holonomic constraints in underactuated Lagrangian systems on Riemannian manifolds, linking affine connections to the original system's geometry and providing conditions for Lagrangian dynamics.

Contribution

It introduces a novel coordinate-free formulation of virtual holonomic constraints, relating affine and Riemannian connections, and establishes criteria for the constrained system to be Lagrangian.

Findings

Affine connection of constrained system relates to original Riemannian connection.

Conditions for constrained dynamics to be Lagrangian are established.

Demonstrated with examples including a double pendulum on a cart.

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.