  Consider a non-uniform Euler-Bernoulli beam with a tip-mass at one end and a
cantilever joint at the other end. The cantilever joint is not fixed and can
itself be moved along an axis perpendicular to the beam. The position of the
cantilever joint is the control input to the beam. The dynamics of the beam is
governed by a coupled PDE-ODE model with boundary input. On a natural
state-space, there exists a unique state trajectory for this beam model for
every initial state and each smooth control input which is compatible with the
initial state. In this paper, we study the motion planning problem of
transferring the beam from an initial state to a final state over a prescribed
time interval. We address this problem by extending the generating functions
approach to flatness-based control, originally proposed in the literature for
motion planning of parabolic PDEs, to the beam model. We prove that such a
transfer is possible if the initial and final states belong to a certain set,
which also contains steady-states of the beam. We illustrate our theoretical
results using simulations and experiments.
