Trajectory tracking control of parallel manipulator with integral manifold and observer

In view of the problem that flexible displacement will occur for end-effectors of parallel manipulators when operating at a high speed, taking the 3RRR parallel manipulator as the object, the trajectory tracking composite controller based on the integral manifold and high-gain observer is proposed for flexible parallel manipulators. Based on the stiffness matrix, the small variable is introduced to decomposite the rigid-flexible coupling dynamic model of the parallel manipulator into slow subsystem and fast subsystem. For the slow subsystem, the backstepping control is applied for rigid motion tracking of the end-effector. To avoid the influence of the links' flexible displacement comprised of deformation and vibration on the end-effector's motion trajectory, the corrective torque is deduced, and the compensation for the flexible displacement is realized. For the fast subsystem, the sliding mode control is utilized to suppress the vibration. At the same time, the high gain observer is designed to avoid the measurement of curvature rate of the flexible links. Also, the stability of the overall system with the proposed method is proven with the Lyapunov stability theorem and the upper bound of the small variable is obtained. At last, the proposed composite controller together with the singular perturbation control and the rigid model based on backstepping control are simulated, and the vibration suppression and tracking accuracy performances are compared to validate the proposed control scheme.