On Extremal Controls in the Sub-Riemannian Problem on the Group of Rigid Body Motions

We consider the sub-Riemannian problem on the group of rigid body motions in three–dimensional space. Such a problem is encountered in the analysis of 3D images as well as in describing the motion of a solid body in a fluid. Mathematically, this problem reduces to solving a Hamiltonian system, the vertical part of which is a system of six differential equations with unknown functions — extremal controls. We derive an ordinary differential equation for one of the components of the extremal control vector. The obtained equation admits a solution in elliptic functions. Then we find the expression in the operator form for the remaining components of the extremal control vector.