Boundary tracking control of flexible beams for transferring motions

Although flexible beams for transmitting both translational and rotational large motions are used in practice such as ocean drill pipes, their control has not been considered. This paper develops boundary feedback controllers to stabilise these beams at their reference configurations. Exact nonlinear partial differential equations governing motion of the beams in three-dimensional space are derived and used in the control design. The designed controllers guarantee globally practically asymptotically stability of the beam motions at the reference states (i.e. positions and rotations of a straight beam moving axially with a desired velocity and rotating around its axial axis with a desired velocity). In the control design and analysis of well-posedness and stability, we utilise different transformations between the earth-fixed and body-fixed coordinates, Sobolev embeddings, and a Lyapunov-type theorem developed for a class of evolution systems in Hilbert space. Simulation results are also included to illustrate the effectiveness of the proposed control design.