Geodesic
On the Lagrangian structure of reduced dynamics under virtual holonomic constraints
Mohammadi, Alireza1 ; Maggiore, Manfredi1 ; Consolini, Luca2
1 Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, ON, M5S 3G4, Canada.
2 Dipartimento di Ingegneria dell’Informazione, via Usberti 181/a, 43124 Parma, Italy.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 913-935

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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.

DOI : 10.1051/cocv/2016020
Classification : 70Q05, 93C10, 93C15, 49Q99
Keywords: Underactuated mechanical systems, virtual holonomic constraints, inverse lagrangian problem

Mohammadi, Alireza 1 ; Maggiore, Manfredi 1 ; Consolini, Luca 2

1 Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, ON, M5S 3G4, Canada.
2 Dipartimento di Ingegneria dell’Informazione, via Usberti 181/a, 43124 Parma, Italy. 
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     title = {On the {Lagrangian} structure of reduced dynamics under virtual holonomic constraints},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {913--935},
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Mohammadi, Alireza; Maggiore, Manfredi; Consolini, Luca. On the Lagrangian structure of reduced dynamics under virtual holonomic constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 913-935. doi: 10.1051/cocv/2016020

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