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Euler-Poincaré reduction of externally forced rigid body motion. (English) Zbl 1167.70331
Summary: 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 wellknown 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.
MSC:
70Q05 Control of mechanical systems
93B28 Operator-theoretic methods
93B11 System structure simplification
37N05 Dynamical systems in classical and celestial mechanics
70E50 Stability problems in rigid body dynamics
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
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