On the Lagrangian structure of reduced dynamics under virtual holonomic constraints∗∗∗∗∗∗

Author： Mohammadi Alireza Maggiore Manfredi Consolini Luca

E-ISSN： 1262-3377|23|3|913-935

ISSN： 1292-8119

Source： ESAIM: Control, Optimisation and Calculus of Variations, Vol.23, Iss.3, 2017-05, pp. : 913-935

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Abstract

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. In both cases, we show that typical solutions satisfying the virtual constraints lie in a restricted class which we completely characterize.