On the Lagrangian structure of reduced dynamics under virtual holonomic constraints

Mohammadi, Alireza; Maggiore, Manfredi; Consolini, Luca

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 913-935.

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.

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