When is a Lagrangian Control System With Virtual Holonomic Constraints Lagrangian?

This paper investigates a class of Lagrangian control systems with η degrees-of-freedom (DOF) and η — 1 actuators, assuming that η — 1 virtual holonomic constraints have been enforced via feedback, and a basic regularity condition holds. The reduced dynamics of such systems are described by a second-order unforced differential equation. We present necessary and sufficient conditions under which the reduced dynamics are those of a mechanical system with one DOF and, more generally, under which they have a Lagrangian structure.