This paper investigates a class of Lagrangian control systems with n degrees-of-freedom (DOF) and n- 1 actuators, assuming that n - 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.
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