Rafal Wisniewski, Piotr Kulczycki Euler-Poincare reduction of externally forced rigid body motion Control and Cybernetics, vol. 33, nr 2, ss. 297-310, 2004
Streszczenie:
If a mechanical system experiences symmetry, the Lagrangian becomes invariant under a certain group action. This property leads to substantial simplification of the description of movement. The standpoint in this article is a mechanical system
affected by an external force of a control action. Assuming that the system possesses symmetry and the configuration manifold corresponds to a Lie group, the Euler-Poincaré reduction breaks up the motion into separate equations of dynamics and kinematics. This becomes of particular interest for modeling, estimation and
control of mechanical systems. A control system generates an external force, which may break the symmetry in the dynamics. This paper shows how to model and to control a mechanical system on the reduced phase space, such that complete state space asymptotic stabilization can be achieved. The paper comprises a specialization of the well-known Euler-Poincaré reduction to a rigid body motion with forcing.
An example of satellite attitude control illustrates usefulness of the Euler-Poincaré reduction in control engineering. This work demonstrates how the energy shaping method applies for Euler-Poincaré equations.