Integrable quadratic Hamiltonians on the Euclidean group of motions
Biggs, J. D. and Holderbaum, W. (2010) Integrable quadratic Hamiltonians on the Euclidean group of motions. Journal of Dynamical and Control Systems, 16 (3). pp. 301-317. ISSN 1573-8698
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To link to this article DOI: 10.1007/s10883-010-9094-8
In this paper, we discuss 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 whose Hamiltonian satisfies a particular condition. In this paper, sub-Riemannian curves are defined in the context of a constrained optimal control problem. The maximum principle is then applied to this problem to yield an 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 extremal curves. These equations are then used to compute sub-Riemannian curves that could potentially be used for motion planning of underwater vehicles.
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