Symplectic reduction of holonomic open-chain multi-body systems with constant momentum

JOURNAL OF GEOMETRY AND PHYSICS

卷:89

Article

Chhabra, Robin¹  Emami, M. Reza^2,3 

[1] MacDonald Dettwiler & Associates Ltd, Brampton, ON L6S 4J3, Canada

[2] Univ Toronto, Inst Aerosp Studies, Toronto, ON M3H 5T6, Canada

[3] Lulea Univ Technol, Space Technol Div, Dept Comp Sci Elect & Space Engn, Kiruna 98128, Sweden

关键词: Open-chain multi-body system;    Differential forms;    Hamiltonian dynamics;    Momentum map;    Symplectic reduction;   

DOI  :  10.1016/j.geomphys.2014.12.011

来源: Elsevier

【 摘 要 】

This paper presents a two-step symplectic geometric approach to the reduction of Hamilton's equation for open-chain, multi-body systems with multi-degree-of-freedom holonomic joints and constant momentum. First, symplectic reduction theorem is revisited for Hamiltonian systems on cotangent bundles. Then, we recall the notion of displacement subgroups, which is the class of multi-degree-of-freedom joints considered in this paper. We briefly study the kinematics of open-chain multi-body systems consisting of such joints. And, we show that the relative configuration manifold corresponding to the first joint is indeed a symmetry group for an open-chain multi-body system with multi-degree-of-freedom holonomic joints. Subsequently using symplectic reduction theorem at a non-zero momentum, we express Hamilton's equation of such a system in the symplectic reduced manifold, which is identified by the cotangent bundle of a quotient manifold. The kinetic energy metric of multi-body systems is further studied, and some sufficient conditions are introduced, under which the kinetic energy metric is invariant under the action of a subgroup of the configuration manifold. As a result, the symplectic reduction procedure for open-chain, multi-body systems is extended to a two-step reduction process for the dynamical equations of such systems. Finally, we explicitly derive the reduced dynamical equations in the local coordinates for an example of a six-degree-of-freedom manipulator mounted on a spacecraft, to demonstrate the results of this paper. (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

Free