Extremal paths in the nilpotent sub-Riemannian problem on the Engel group (subcritical case of pendulum oscillations)

We consider a left-invariant sub-Riemannian problem on an Engel group. This problem arises as a nilpotent approximation of nonholonomic systems in the four-dimensional space with two-dimensional control (e.g., the system describing the motion of a mobile robot with a trailer). For the sub-Riemannian problem on the Engel group, abnormal extremal paths are calculated. The subsystem for conjugate variables of normal Hamiltonian system of Pontryagin's maximum principle is reduced to the pendulum equation. Normal extremal paths corresponding to subcritical pendulum oscillations were calculated.