  The left-invariant sub-Riemannian problem on the Engel group is considered.
The problem gives the nilpotent approximation to generic nonholonomic systems
in four-dimensional space with two-dimensional control, for instance to a
system which describes motion of mobile robot with a trailer. The global
optimality of extremal trajectories is studied via geometric control theory.
The global diffeomorphic structure of the exponential mapping is described. As
a consequence, the cut time is proved to be equal to the first Maxwell time
corresponding to discrete symmetries of the exponential mapping.
