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