Sub-Riemannian Geodesics on the Group of Motions of Euclidean Space

We consider the sub-Riemannian problem on the group of motions of three-dimensional Euclidean space SE(3).  We prove Liouville integrability of the Hamiltonian system of PMP, and present explicit formulas for the extremal controls u1, ... , u5 in the particular case u6=0. This case is important in applications: tracking of neural fibers and blood vessels in MRI and CT images of human brain; and in motion planning problem for an aircraft, that can move forward/backward.

Next, we show a relationship between the sub-Riemann problem in SE(3) and problem \textbf{Pcurve} of minimizing the compromise between length and geodesic curvature for a curve in three-dimensional space with fixed boundary points and directions. We give explicit formulas for extremals in problem \textbf{Pcurve} and investigate their geometric properties.

The talk is based on joint works with R. Duits, A. Ghosh, T. Dela Haije and A. Popov.

Автор: Alexey Mashtakov
Дата: 7th June, 2018

The Seventh International Conference
“Geometry, Dynamics, Integrable Systems – GDIS 2018”
5–9 June, 2018, Moscow

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