Periodic motion planning for virtually constrained Euler-Lagrange systems

In English The paper suggests an explicit form of a general integral of motion for some classes of dynamical systems including n-degrees of freedom Euler-Lagrange systems subject to (n - 1) virtual holonomic constraints. The knowledge of this integral allows to extend the classical results due to Lyapunov for detecting a presence of periodic solutions for a family of second order systems, and allows to solve the periodic motion planning task for underactuated Euler-Lagrange systems, when there is only one not directly actuated generalized coordinate. As an illustrative example, we have shown how to create a periodic oscillation of the pendulum for a cart-pendulum system and how then to make them orbitally exponentially stable following the machinery developed in A. Shiriaev, J. Perram, C. Canudas-de-Wit, Constructive tool for an orbital stabilization of underactuated nonlinear systems: virtual constraint approach, IEEE Trans. Automat. Control 50 (8) (2005) 1164-1176. The extension here also considers time-varying virtual constraints. (C) 2006 Elsevier B.V. All rights reserved.