We consider a sub-Riemannian problem on the Lie group SE(3) of rigid
body motions in R3. By given two orthonormal frames N0 and N1б attached respectively at two given points q0 = (x0; y0; z0) and q1 = (x1; y1; z1) in space R3, we aim to find an optimal motion that transfers q0 to q1 such that the frame N0 is transferred to the frame N1. The
frame can move forward or backward along one of the vector chosen in the
frame and rotate simultaneously via the remaining two (of three) prescribed
axes. The required motion should be optimal in the sense of minimal length
in the 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 Pcurve of minimizing the compromise between length and ab-
solute curvature for a curve in R3 with fixed boundary points and directions.
We give explicit formulas for extremals in problem Pcurve and investigate
their geometric properties.
The talk is based on joint works with R. Duits,A. Ghosh, T. Dela Haije and A.Popov.
International Conference "Optimal Control and Differential Games" dedicated to the 110th anniversary of L. S. Pontryagin (December 12–14, 2018, Steklov Mathematical Institute of RAS, ul. Gubkina, 8, Moscow)