Consensus tracking and vibration control for multiple 3D flexible rotating manipulator systems with input amplitude and rate constraints over directed graph

This paper addresses the angle consensus tracking and vibration control problems of a class of leader-following three-dimensional (3D) flexible rotating manipulator (FRM) systems subject to input amplitude and rate constraints (IARCs) over a directed communication graph. Each follower FRM consists of a rotating base and a flexible manipulator, modeled as a distributed parameter system governed by coupled partial differential equations (PDEs). The desired angular positions of the bases and manipulators are regarded as the virtual leader’s signals. To overcome the difficulty in constructing a suitable Lyapunov function caused by the asymmetry of the Laplacian matrix in the directed topology, a set of distributed observers is proposed to estimate the leader’s angular positions. Based on these observers, a distributed boundary control strategy is developed using the backstepping method combined with a smooth hyperbolic tangent function, which effectively handles the IARCs and achieves both consensus tracking and vibration suppression. The stability of the closed-loop systems is rigorously analyzed using Lyapunov’s direct method and LaSalle’s invariance principle. Finally, several simulation examples are presented to verify the effectiveness of the proposed control scheme.