Motion planning and approximate controllability of a moving cantilever beam with a tip-mass

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 twice continuously differentiable control input which is compatible with the initial state. In this paper, we study the motion planning problem of transferring the beam model from an initial state to a final state over a prescribed time-interval and then employ the results obtained to establish the approximate controllability of this model. We address these problems by extending and applying the generating functions approach to flatness-based control to the beam model. We prove that the transfer described above is feasible if the initial and final states belong to a certain set, which also contains the steady-states of the beam model. We then establish that this set contains all the eigenfunctions of the beam model, which form a Riesz basis for the state-space, and thereby conclude the approximate controllability of the beam model over all time intervals. We illustrate our theoretical results on motion planning using simulations and experiments.