Optimal handling strategies of flexible beams by using a manipulator are discussed. The authors consider the planning of the input torques and the trajectory of the manipulator so as not to excite the vibration of the flexible beam. The equations of the motion of a two-link manipulator with a flexible beam are derived. On the basis of the dynamic equations, the optimal input torques, which accomplish the desired end-position of the manipulator and the diminution of the vibration of the flexible beam, are calculated by using a numerical algorithm for the optimal control of the nonlinear equations. As the equation of the vibration of the flexible beam implies that the acceleration of the end-point of the manipulator excites the vibration, the authors consider an optimal trajectory planning so as to minimize the end-point acceleration of the manipulator. The validity of the optimal input torques and the optimal trajectory are demonstrated by computer simulations and experiments for an air-driven robot arm.
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