Dynamic modelling and control system of a single non-slender flexible robot’s link rotating in a horizontal plane

Most authors have ignored the effect of rotatory inertia and shear deformation. This practice is justified for slender flexible arms. According to the Timoshenko beam theory, the deflection due to shear force and rotatory inertia should be taken into account in modelling for high speed and high precision requirement when the ratio of the cross-sectional dimensions to length increases. Based on Hamilton's principle and Timoshenko's flexible beam theory, the dynamic model of a single non-slender flexible link is derived, and it is shown that the elastic motion is governed by a pair of coupled partial differential equations with coupled boundary conditions. Then the abstract form of the dynamic equations is studied, and the properties of the spectrum of the elastic operator appearing in the evolution equation are given. Furthermore, the eigenvalue problem of the elastic operator is solved in explicit form. The formulation and well-posedness of the state-space equation, as well as the transfer function of the dynamic control system of the non-slender flexible link, are studied by spectral analysis. Spectral analysis is used to study the well-posedness of the dynamic control system. The tracking control problem is studied and a feedback control scheme that controls the rigid-body motion and elastic behaviors simultaneously is derived based on a n-modal model. Closed-loop configuration of a control system, equivalent circuit of a dc-motor and the overall system block diagram are proposed. The stabilization of the closed-loop system is studied analytically. Finally, the tracking control problem is studied, a stabilizing feedback control law based on a n-modal model to suppress vibrations of the flexible link is derived, and the necessary and sufficient conditions that can guarantee the stability of the closed-loop system, are given. Simulation results are given as well.