  In the present paper we focus our attention on the design of the
feedback-based feed-forward controller asymptotically stabilizing the
double-pendulum-type crane system with the time-varying rope length in the
desired end position of payload (the origin of the coordinate system). In
principle, we will consider two cases, in the first case, the sway angle of
payload is uncontrolled and the second case, when the sway angle of payload is
controlled by an external force. Mathematical modelling in the framework of
Lagrange formalism and numerical simulation in the Matlab environment indicate
the substantial reduction of the transportation time to the desired end
position. Another principal novelty of this paper lies in deriving and analysis
of a complete mathematical model without approximating the nonlinear terms and
without neglecting some structural parameters of systems for the reasons
described in the Remark 4.2 and Remark 5.1.
