Efficient Dynamic Simulation of Flexible Link Manipulators with Pid Control

For accurate simulations of the dynamic behavior of flexible manipulators the combination of a perturbation method and modal analysis is proposed. First, the vibrational motion is modeled as a first-order perturbation of a nominal rigid link motion. The vibrational motion is then described by a set of linear timevarying equations. Next, the number of degrees of freedom is reduced by applying a modal reduction technique. The proportional part of the control system is explicitly included in the modal analysis. The applicability of the method is demonstrated by simulating the controlled trajectory motion of a spatial flexible three-degree of freedom manipulator with PID control. INTRODUCTION In this paper a perturbation method is proposed for analyzing the dynamic behavior of flexible manipulators, including the effects of the manipulators control system. It involves a non-linear finite element formulation [1] in which links and joints are considered as specific elements. The governing equations of motion are derived in two sets of coordinates. The first set are the generalized coordinates of the manipulator with rigid links which are used to express control forces and torques. The second set are deformation coordinates that characterize flexible deformations of the links. The non-linear differential equations are solved using a two-step perturbation approach. In this approach the vibrational motion of the manipulator is modeled as a first-order perturbation of the nominal rigid link motion. In the first step the nominal rigid link motion is described by the rigidified manipulator model, that is a non-linear model in which all flexible deformation coordinates are prescribed zero. In a dynamic analysis the equations of motion are symbolically linearized and evaluated numerically in a number of points of the nominal trajectory. During the second step the vibrational motion is described by a set of linear time-varying equations using the linearized description. In order to reduce the dimension of the linearized system, a modal reduction technique is proposed. The mode shape functions are determined by taking into account the proportional feedback gains associated with the rigid link motion. Furthermore, the time-varying nature of the mode shape functions is taken into account. Then modal integration can be applied using only a small number of low-frequency modes. The modeacceleration concept is used to account for the pseudo-static contribution of the high-frequency modes. A spatial flexible three-degree of freedom manipulator (Fig. 1) is analyzed to illustrate the solution method. Results from a non-linear simulation are compared with the perturbation method with and without modal reduction. In earlier papers we discussed the applicability of the non-linear simulations [2, 3] and the perturbation method [4]. In this paper we will focus on the modal analysis. 1 Copyright  2001 by ASME