In this article, a nonlinear dynamic model of a flexible manipulator is derived through finite element method associated with Lagrange approach. The flexible manipulator is modeled as an Euler-Bernoulli beam driven by a motor at its base and with a point mass tip payload. The generalized coordinates of the system are selected to be the displacements and rotations of the nodes on the considered flexible beam, and such that a state space model is obtained with all the state variables having physical meanings. Based on this model, an effective nonlinear feedback controller is developed to control the tip position. Furthermore, an efficient algorithm is developed to calculate the inverse of the system's inertia matrix for real-time implementation. Numerical simulation results are given to show the effectiveness of the controller and its robustness in handling payload variations.
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Ge et al. (1997) studied this question.