Comparison of the Assumed Modes and Finite Element Models for Flexible Multilink Manipulators

Abstract

The objective of this paper is to compare two discretization models—namely, the assumed modes and finite element mod els—to efficiently represent the link flexibility of robot manip ulators. We present a systematic modeling procedure based on homogeneous transformation matrices for spatial multilink flexible manipulators with both revolute and prismatic joints. The Lagrangian formulation of dynamics and computer algebra are employed to derive closed-form equations of motion. We show that fewer mathematical operations are required for in ertia matrix computation in the finite element model compared with the assumed modes formulation; however, because the number of state space equations are more, the numerical sim ulation time may be greater for finite element models. Use of the finite element model to approximate flexibility usually gives rise to overestimated stiffness matrix. We analytically show that overestimation of structure stiffness may lead to unstable closed-loop response of the original manipulator system, using a model-based control law. We illustrate the complexity owing to the time-dependent frequency equation of the assumed modes model arising in a prismatic jointed flexible link with payload and in manipulators with more than one link with revolute joints. We describe a novel method based on the differential form of the frequency equation to simulate such systems. A model-based decoupling control law is used to compare the dynamic responses of the manipulator system. The results are illustrated by numerical simulation of a flexible spatial RRP configuration robot.