A Finite Element Dynamic Analysis of Flexible Manipulators

A finite element - based method is presented for analyzing the dynamic behavior of flexible manipulators, including the effects of the manipulator's control system. The method involves a nonlinear finite-element formulation in which both links and joints are considered as specific finite elements. The governing equations of motion are formulated in terms of two sets of coordinates, namely generalized coordinates of the manipulator with rigid links and deformation mode coordinates that characterize deformation of the links. The method also permits generation of locally linearized models about a nominal trajectory. The numerical performance of the presented finite element method is demonstrated by analyzing the closed-loop behav ior of a flexible three-degree-of-freedom manipulator in which the position of the end effector is to be sensed. On the basis of the linearized system description, controllability is discussed in terms of the modal representations of the system. Uncontrollable manipulator configurations have been indi cated for bending vibrations of the out-of-plane motion of the manipulator. Fast and precise closed-loop tip position control has been obtained for the in-plane motion with a bandwidth that is three times the bandwidth achieved with actuator joint feedback. However, no significant increase in the bandwidth has been obtained for the out-of-plane motion. The improve ment for the latter motion is strongly limited by the nonmini mum phase behavior.