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Integrating Hamiltonian systems defined on the Lie groups SO(4) and SO(1,3)

Biggs, James D. and Holderbaum, William (2007) Integrating Hamiltonian systems defined on the Lie groups SO(4) and SO(1,3). In: The 45th European Control Conference (ECC '07), 2007-07-02 - 2007-07-05, Kos, Greece.

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    Abstract

    In this paper we study constrained optimal control problems on semi-simple Lie groups. These constrained optimal control problems include Riemannian, sub-Riemannian, elastic and mechanical problems. We begin by lifting these problems, through the Maximum Principle, to their associated Hamiltonian formalism. As the base manifold is a Lie group G the cotangent bundle is realized as the direct product of the dual of the Lie algebra and G. The solutions to these Hamiltonian vector fields are called extremal curves and the projections g(t) in G are the corresponding optimal solutions. The main contribution of this paper is a method for deriving explicit expressions relating the extremal curves to the optimal solutions g(t) in G for the special cases of the Lie groups SO(4) and SO(1;3). This method uses the double cover property of these Lie groups to decouple them into lower dimensional systems. These lower dimensional systems are then solved in terms of the extremals using a coordinate representation and the systems dynamic constraints. This illustrates that the optimal solutions g(t) in G are explicitly dependent on the extremal curves.

    Item type: Conference or Workshop Item (Paper)
    ID code: 7895
    Keywords: geometric methods, optimal control, time-varying systems, Hamiltonian systems, Mechanical engineering and machinery, Motor vehicles. Aeronautics. Astronautics, Aerospace Engineering, Control and Systems Engineering, Space and Planetary Science
    Subjects: Technology > Mechanical engineering and machinery
    Technology > Motor vehicles. Aeronautics. Astronautics
    Department: Faculty of Engineering > Mechanical and Aerospace Engineering
    Related URLs:
    Depositing user: Ms Katrina May
    Date Deposited: 27 Apr 2009 10:25
    Last modified: 30 May 2014 19:58
    URI: http://strathprints.strath.ac.uk/id/eprint/7895

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