Integrable quadratic Hamiltonians on the Euclidean group of motions

James Biggs, William Holderbaum

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)
77 Downloads (Pure)


This paper tackles the problem of globally computing sub-Riemannian curves on the Euclidean group of motions SE(3). In particular we derive a global result for special sub-Riemannian curves for which their Hamiltonian satisfies a particular condition. The sub-Riemannian curves in this paper are defined in the context of a constrained optimal control problem. The Maximun Principle is then applied to this problem to yield the appropriate left-invariant quadratic Hamiltonian. A number of integrable quadratic Hamiltonians are identified. We then proceed to derive convenient expressions for sub-Riemannian curves in SE(3) that correspond to particular extreme curves. These equations are then used to compute sub-Riemannian curves that could potentially be used for motion planning of underwater vehicles.
Original languageEnglish
Pages (from-to)301-317
Number of pages17
JournalJournal of Dynamical and Control Systems
Issue number3
Publication statusPublished - 23 Jul 2010


  • sub-riemannian curves
  • euclidean group of motions
  • hamiltonian systems
  • motion planning


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