[show abstract][hide abstract] ABSTRACT: The paper presents sufficient and almost necessary conditions for the presence of periodic solutions for zero dynamics of virtually constrained under-actuated Euler-Lagrange system. This result is further extended to detect periodic solutions for a class of hybrid systems in the plane and analyze their orbital stability and instability. Illustrative examples are given.
Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on; 01/2006
[show abstract][hide abstract] ABSTRACT: 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.
[show abstract][hide abstract] ABSTRACT: We present a constructive tool for generation and orbital stabilization of periodic solutions for underactuated nonlinear systems. Our method can be applied to any mechanical system with a number of independent actuators smaller than the number of degrees of freedom by one. The synthesized feedback control law is nonlinear and time-dependent. It is derived from a feedback structure that explicitly uses the general or full integral of the systems zero dynamics. The control law generates a periodic solution and makes it exponentially orbitally stable.
IEEE Transactions on Automatic Control 09/2005; · 2.72 Impact Factor
[show abstract][hide abstract] ABSTRACT: The paper suggests an explicit form of a general integral of motion for some classes of dynamical systems including n-degree-of-freedom mechanical systems subject to (n - 1) virtual holonomic constraints. The computation of this integral opens several possibilities for generating and further exponential orbital stabilization of solutions in nonlinear feedback systems. An illustrative example is given.
Decision and Control, 2004. CDC. 43rd IEEE Conference on; 01/2005