Control techniques for impacting flexible systems

Abstract

In this article, a comparative study of the control for the repetitive impacting elastic link with parametrically excited base in rotational motion is considered. First, a sliding mode control strategy based on linearized inverse model is designed and employed to suppress the vibrations of the elastic beam after the impact. The control concept involves the usage of an adaptive plant inverse model as controller in feedforward con®gurations. Next, a linear controller is designed via Lyapunov-Floquet transformation. In this approach, the time-periodic equations of motion are transformed into a time-invariant form, which is suitable for the application of standard time-invariant controller-design techniques. Finally, a fuzzy logic controller is applied for the nonlinear model of the impacting system. The momentum balance method and an empirical coef®cient of restitution is used in the collision.Key words sliding mode, inverse system, fuzzy logic, coef®cient of restitution IntroductionApplication of modern control methodology plays a major role in the development of the manipulators, walking machines and robot arms technology. A special attention is paid to the analysis of elastic links that impact rigid surfaces.An engineering-oriented impact approach for multiple collision industrial applications was developed in [14] using complementary algorithms. The authors replaced the unilateral character of the constraints in the normal direction by an ef®cient bilateral formulation.Planar and nonplanar oscillations of a cantilever beam subjected to a planar periodic excitation were studied in [6,7]. The stability of out-of-plane motion was performed, taking into account damping and geometric nonlinearities in the differential equations of motion.In [5] an elastic arm was considered, modeled as a pinned-free beam attached to a hub. The objective of the work was to carry out experiments designed to determine the necessary control torque applied at the base of the link using only the tip position measurement. A more complex system was analyzed in [3]. The work was related to the problem of controlling plane rotational motions of two rigid bodies connected by an elastic rod.The problem of controlling an elastic arm of two links based on variable structure system theory and pole assignment technique for stabilization was treated in [13]. This design approach was motivated by a simple observation that the nonlinearity in the dynamics of an elastic robotics system is essentially due to rigid modes (joint angles), and, as the time derivatives of the rigid modes vanish, the remaining motion is only due to the elasticity. For the rigid modes, a sliding controller was designed. The controller of the elastic modes was constructed using the pole assignment technique. A similar technique is used in [16] to control ā exible/rigid link robot using sliding-mode and a shaped-input controllers. The ®rst controller was used to control the rigid body motion, while the second one served to control the¯exible motion. Comparative to the two works a...