MODELING OF FLEXIBLE-LINK MANIPULATORS WITH PRISMATIC JOINTS

The axially translating flexible link in flexible manipulators with a prismatic joint can be modeled using the Euler-Bernoulli beam equation together with the convective terms. In general, the method of separat ion of variables cannot be applied to solve this partial differential equation. In this paper, we present a nondimensional form of the Euler -Bernoulli beam equation using the concept of group velocity and prese nt conditions under which separation of variables and assumed modes me thod can be used. The use of clamped-mass boundary conditions lead to a time-dependent frequency equation for the translating flexible beam. We present a novel method to solve this time-dependent frequency equa tion by using a differential form of the frequency equation, We then p resent a systematic modeling procedure for spatial multi-link flexible manipulators having both revolute and prismatic joints. The assumed m odel/Lagrangian formulation of dynamics is employed to derive closed f orm equations of motion, We show, using a model-based control law, tha t the closed-loop dynamic response of modal variables become unstable during retraction of a flexible link, compared to the stable dynamic r esponse during extension of the link, Numerical simulation results are presented for a flexible spatial RRP configuration robot arm. We show that the numerical results compare favorably with those obtained by u sing a finite element-based model.