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.

Keywords: Underactuated mechanical systems, virtual holonomic constraints, inverse lagrangian problem

^{1}; Maggiore, Manfredi

^{1}; Consolini, Luca

^{2}

@article{COCV_2017__23_3_913_0, author = {Mohammadi, Alireza and Maggiore, Manfredi and Consolini, Luca}, title = {On the {Lagrangian} structure of reduced dynamics under virtual holonomic constraints}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {913--935}, publisher = {EDP-Sciences}, volume = {23}, number = {3}, year = {2017}, doi = {10.1051/cocv/2016020}, zbl = {1407.70036}, mrnumber = {3660454}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016020/} }

TY - JOUR AU - Mohammadi, Alireza AU - Maggiore, Manfredi AU - Consolini, Luca TI - On the Lagrangian structure of reduced dynamics under virtual holonomic constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 913 EP - 935 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016020/ DO - 10.1051/cocv/2016020 LA - en ID - COCV_2017__23_3_913_0 ER -

%0 Journal Article %A Mohammadi, Alireza %A Maggiore, Manfredi %A Consolini, Luca %T On the Lagrangian structure of reduced dynamics under virtual holonomic constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 913-935 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016020/ %R 10.1051/cocv/2016020 %G en %F COCV_2017__23_3_913_0

Mohammadi, Alireza; Maggiore, Manfredi; Consolini, Luca. On the Lagrangian structure of reduced dynamics under virtual holonomic constraints. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 913-935. doi : 10.1051/cocv/2016020. http://archive.numdam.org/articles/10.1051/cocv/2016020/

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