Distributed parameter models of flexible robot arms

摘要

The modelling of a flexible robot arm is concerned with its rigid body motion as well as its oscillation due to the elasticity of the arm [1-4]. As the elastic deformation is distributed along the arm, it is obvious that this system is a distributed parameter system. Although a set of partial differential equations is presumed to be the best model for this system, it is limited to a relatively low frequency oscillation of a homogeneous, simply shaped arm. In the case of a more complicated flexible arm, the partial differential equation model is not necessarily advantageous, and another method such as the finite element method will be used. First, this paper presents the detailed development of the partial differential equations and boundary conditions for two models of a flexible robot arm. One is a rigorous model obtained through the use of Timoshenko beam theory, which takes into account the effect of transverse shear deformation and rotary inertia; the other is a simpler and more frequently used model obtained by Bernoulli-Euler beam theory, which neglects the effects of shear and rotary inertia. Second, two versions of the model using constrained mode method and an unconstrained mode method are presented. These modal analyses yield sets of ordinary differential equations which are suitable for making reduced-order approximate models for the controller design of the system. Third, by performing Laplace transformation of the partial differential equations of the Bernoulli-Euler beam model, we obtain distributed parameter transfer functions. It is clarified that the zeros and poles of the transfer function coincide with the eigenvalues of the boundary value problems obtained by the constrained mode method and the unconstrained mode method respectively. Finally, the modelling of a multi-link flexible robot arm is described.