On the Calculation of Differential Parametrizations for the Feedforward Control of an Euler-Bernoulli Beam

This contribution is concerned with the motion planning for underactuated Euler-Bernoulli beams. The design of the feedforward control is based on a differential parametrization of the beam, where all system variables are expressed in terms of a free time function and its infinitely many derivatives. In the paper, we derive an advantageous representation of the set of all formal differential parametrizations of the beam. Based on this representation, we identify a well-known parametrization, for the first time without the use of operational calculus. This parametrization is a flat one, as the corresponding series representations of the system variables converge. Furthermore, we discuss a formal differential parametrization where the free time function allows a physical interpretation as the bending moment at the unactuated boundary. Even though the corresponding series do not converge, a numerical simulation using a least term summation illustrates the usefulness of this formal differential parametrization for the motion planning.