Euler-Poincare reduction of externally forced rigid body motion

• Autorzy: Wiśniewski R. , Kulczycki P.
• Języki publikacji: EN

Abstrakty

EN

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-Poincare 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-Poincare reduction to a rigid body motion with forcing. An example of satellite attitude control illustrates usefulness of the Euler-Poincare reduction in control engineering. This work demonstrates how the energy shaping method applies for Euler-Poincare equations.

Słowa kluczowe

EN

modelling; control; rigid body motion; Lagrange mechanics; energy shaping

PL

modelowanie; sterowanie; ruch ciała sztywnego; mechanika Lagrange'a; kształtowanie energetyczne

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2004
• Tom: Vol. 33, no 2
• Strony: 297--310
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Wiśniewski R.

• Aalborg University, Department of Control Engineering, Fredrik Bajers Vej 7, DK-9220 Aalborg 0, Denmark

autor

Kulczycki P.

• Systems Research Institute Polish Academy of Sciences ul. Newelska 6, PL-01-447 Warsaw, Poland

Bibliografia

• Abraham, R. and Marsden, J.E. (1978) Foundations of Mechanics. Perseus.
• Bak, T. (1999) Spacecraft Attitude Determination - a Magnetometer Approach. Ph.D. thesis, Aalborg University.
• Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E. and Murray, T.R.(1996) Nonholonomic Mechanical Systems with Symmetry. Archive for Rational Mechanics and Analysis136, 21-99.
• Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E. and Ratiu, T.S.(1996) The Euler-Poincaŕe Equations and Double Bracket Dissipation. Communications in Mathematical Physics 175, 1-42.
• Bloch, A.M, Leonard, N.E. and Marsden, J.E. (2000) Controlled Lagrangians and the Stabilization of Mechanical Systems; In: The First Matching Theorem. IEEE Transactions on Automatic Control 45.
• Br̈ocker, Th. and Dieck, T. (1985) Representations of Compact Lie Groups. Springer-Verlag.
• Bullo, F. and Murray, R.M. (1999) Tracking for Fully Actuated Mechanical Systems: A Geometric Framework. Automatica 35, 17-34.
• Goldstein, H. (1980) Classical Mechanics. Addison-Wesley.
• Koditschek, D.E. (1989) The Application of Total Energy as a Lyapunov Function for Mechanical Control Systems. Contemporary Mathematics 97, 131-155.
• Koiller, J. (1992) Reduction of Some Classical Non-Holonomic Systems with Symmetry. Archive for Rational Mechanics and Analysis 118, 113-148
• Lee, J.M. (1997) Riemannian Manifolds: An Introduction to Curvature. Springer-Verlag.
• Marsden, J.E. and Ratiu, T.S. (1999) Introduction to Mechanics and Symmetry. Springer-Verlag.
• Marsden, J.E. and Scheurle, J. (1993) The reduced Euler-Lagrange equations. Dynamics and control of mechanical systems, Field Institute Communications 1, 139-164.
• Morton, H. (1994) Hamiltonian and Lagrangian Formulations of Rigid-Body Rotational Dynamics Based on the Euler Parameters. Journal of the Astronautical Sciences 41, 569-592.
• Nijmeijer, H. and van der Schaft, A.J. (1990) Nonlinear Dynamical Control Systems. Springer-Verlag.
• Ting-Yung Wen, J. and Kreutz-Delgado, K. (1991) The Attitude Control Problem. IEEE Transactions on Automatic Control 36, 1148-1162.
• Wertz, J.R. (1990) Spacecraft Attitude Determination and Control. Kluwer Academic Publishers.
• Wiśniewski, R .and Kulczycki, P. (2003) Rotational Motion Control of aSpacecraft . IEEE Trans. on Automatic Control 48.
• Wiśniewski, R. and Kulczycki, P. (2004) Slew Maneuver Control fora Spacecraft Equipped with Star Camera and Reaction Wheels. Control Engineering Practice, in press.