Conjugate points in nilpotent sub-Riemannian problem on the Engel group

The left-invariant sub-Riemannian problem on the Engel group is considered. This problem is very important as nilpotent approximation of nonholonomic systems in four-dimensional space with two-dimensional control, for instance of a system which describes motion of mobile robot with a trailer. We study local optimality of extremal trajectories and estimate conjugate time in this article.