Adaptive sliding mode control with filtered signals of two manipulators handling a flexible payload

two manipulators rigidly handling a flexible payload to track a predefined trajectory. The control law provides robustness against model parameters uncertainties and external disturbances. To suppress the elastic vibration, an adaptive optimal controller is designed. Implicit proof of the system stability is provided via Lyapunov Stability Theory. Finally, the cooperating control input is shown through two-time scale transform. Simulation results for two three-link manipulators handling a flexible beam are presented to validate the theoretical results.