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Mathematical Problems in Engineering
Volume 2009, Article ID 727385, 16 pages
Research Article

Vibration Control of Manipulators with Flexible Nonprismatic Links Using Piezoelectric Actuators and Sensors

1Department of Mathematics and Statistics, University of Caxias do Sul (UCS), R. Francisco GetĂșlio Vargas, 1130, 95570-560 Caxias do Sul, RS, Brazil
2Department of Mechanical Engineering, Federal University of Rio Grande do Sul (UFRGS), Sarmento Leite, 425, 90050-170 Porto Alegre, RS, Brazil

Received 17 February 2009; Accepted 29 April 2009

Academic Editor: Mohammad Younis

Copyright © 2009 Valdecir Bottega et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This work presents a tracking control model for a flexible nonprismatic link robotic manipulator using simultaneously motor torques and piezoelectric actuators. The dynamic model of the flexible manipulator is obtained in a closed form through the Lagrange equations. The control uses the motor torques for the joints tracking control and also to reduce the low-frequency vibration induced in the manipulator links. The stability of this control is guaranteed by the Lyapunov stability theory. Piezoelectric actuators and sensors are added for controlling vibrations with frequencies beyond the reach of motor torque control. The naturals frequencies are calculated by the finite element method, and the approximated eigenfunctions are interpolated by polynomials. Three eigenfunctions are used for the dynamics of the arm, while only two are used for the control. Numerical experiments on Matlab/Simulink are used to verify the efficiency of the control model.