Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 362437, 23 pages
Research Article

Simultaneous Piezoelectric Actuator and Sensor Placement Optimization and Control Design of Manipulators with Flexible Links Using SDRE Method

1Department of Mathematics and Statistics, Federal University of Pelotas, s/n°, 354, 96010-900 Pelotas, RS, Brazil
2Department of Mechanical Engineering, Federal University of Rio Grande do Sul, R. Sarmento Leite, 425, 90050-170 Porto Alegre, RS, Brazil

Received 17 June 2010; Accepted 22 November 2010

Academic Editor: Sergio Preidikman

Copyright © 2010 Alexandre Molter 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 paper presents a control design for flexible manipulators using piezoelectric actuators bonded on nonprismatic links. The dynamic model of the manipulator is obtained in a closed form through the Lagrange equations. Each link is discretized using finite element modal formulation based on Euler-Bernoulli beam theory. The control uses the motor torques and piezoelectric actuators for controlling vibrations. An optimization problem with genetic algorithm (GA) is formulated for the location and size of the piezoelectric actuator and sensor on the links. The natural frequencies and mode shapes are computed by the finite element method, and the irregular beam geometry is approximated by piecewise prismatic elements. The State-Dependent Riccati Equation (SDRE) technique is used to derive a suboptimal controller for a robot control problem. A state-dependent equation is solved at each new point obtained for the variables from the problem, along the trajectory to obtain a nonlinear feedback controller. Numerical tests verify the efficiency of the proposed optimization and control design.