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Stabilité des systèmes dynamiques non-réguliers, application aux robots marcheurs

Abstract : Originating in the analysis of non permanent contact between perfectly
rigid bodies, the mathematical analysis of nonsmooth Lagrangian
dynamical systems concerns Lagrangian dynamical systems with
coordinates constrained to stay inside some closed sets, what leads to
introduce mathematical tools which are unusual in control theory,
velocities with locally bounded variations, measure accelerations,
measure differential inclusions to name a few. The control theory for
such dynamical systems is just beginning to appear and even the basic
Lyapunov stability theory still needs to be stated.

In this work we thus propose to establish some first bases for the
Lyapunov stability analysis of the nonsmooth dynamical systems. We
will see that it is possible, provided some additional assumptions, to
extend some classical results. We will propose for example a Lyapunov
stability theorem and an extension of the LaSalle theorem for
dynamical systems described by flows which can undergo
discontinuities.

Building on these theorems, we then propose a Lagrange-Dirichlet
theorem for nonsmooth Lagrangian dynamical systems by showing that
their energy can be naturally taken as Lyapunov functions. Based on
these results we are then able to prove the stability of simple
control laws for the position and force regulation of a robotic
manipulator and a walking robot without any assumptions on the state
of the contacts. We will also underline the interest of a passivity
based control law for nonsmooth lagrangian dynamical systems.
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https://tel.archives-ouvertes.fr/tel-00011984
Contributor : Isabelle Rey <>
Submitted on : Monday, March 20, 2006 - 9:05:38 AM
Last modification on : Friday, June 26, 2020 - 4:56:01 PM
Long-term archiving on: : Saturday, April 3, 2010 - 10:57:12 PM

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  • HAL Id : tel-00011984, version 1

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Citation

Sophie Chareyron. Stabilité des systèmes dynamiques non-réguliers, application aux robots marcheurs. Automatique / Robotique. Institut National Polytechnique de Grenoble - INPG, 2005. Français. ⟨tel-00011984⟩

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