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Integrable quadratic Hamiltonians on the Euclidean group of motions

Biggs, J.D. and Holderbaum, William (2010) Integrable quadratic Hamiltonians on the Euclidean group of motions. Journal of Dynamical and Control Systems, 16 (3). pp. 301-317. ISSN 1079-2724

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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.