A Modal Expansion Approach Of The Dynamic Model For Flexible Manipulators

Abstract

This paper presents a methodology for mathematical modelling of n-link flexible robotic manipulators or n-link manipulators containing both rigid and flexible links. The kinetic and potential energies of the links are expressed in terms of generalized coordinates. The Lagrangian approach is then employed to derive the joint motion equations of the links, including the effects of their elastic motion. Also, the equations for transverse vibration of the links are developed, including the effects of their joint motion. The overall dynamics of the links is given in terms of a coupled system of nonlinear ordinary and partial differential equations. The modal expansion approach is utilized to express the solutions of the transverse vibration as an infinite sum in terms of the modes and generalized coordinates, and then to approximate the partial differential equations by ordinary differential equations. The method will lead to rigorous analytical investigations using infinite-dimensional system theory. The finite dimensional model can be obtained from the results using truncation of the modes. This will reduce the original set of infinite equations to a set of finite, coupled nonlinear ordinary differential equations. An example is given to illustrate an application of the proposed approach, to derive the complete equations of motion for a two-link planar flexible manipulator.