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

James Biggs, William Holderbaum

Research output: Contribution to conferencePaper

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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.
Original languageEnglish
Pages1427-1432
Number of pages5
Publication statusPublished - 5 Jul 2007
EventThe 45th European Control Conference (ECC '07) - Kos, Greece
Duration: 2 Jul 20075 Jul 2007

Conference

ConferenceThe 45th European Control Conference (ECC '07)
CityKos, Greece
Period2/07/075/07/07

Keywords

  • geometric methods
  • optimal control
  • time-varying systems
  • Hamiltonian systems

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