This study presents and investigates a six-DOF (degrees of freedom) piezoelectric based stage for positioning error compensation. The relationship between the displacement of the piezoelectric actuators and the stage can be computed according to the geometric relationships of the actuators installed. In this study, a feedforward compensator based on the hysteresis model has been designed for compensation and a PI controller was used for positioning. The combination of a feedforward compensator and PI controller gives the stage good positioning and tracking performance. Stage position information is feedback from a six-DOF optical measurement system comprised of three modular two-dimensional measurement devices. Each module employs a quadrant photodiode (QPD), a laser diode, and a lens. The measurement signal is acquired and processed using an FPGA based processor for real time control. The linear and angular positioning resolution is 0.02 μm and 0.1 arcsec, respectively. When the stage is controlled in a closed loop, the positioning errors are in the range of±0.1 μm and±0.5arcsec. The stage is controlled to track a sinusoidal wave with an amplitude of 2.5 μm and a frequency of 5 Hz; tracking errors were within±0.1 μm and±0.2arcsec.
The aim of this study is to derive the governing equation of an electrostatically actuated micro system by use of the Hamilton principle, and then the natural frequencies of a micro fixed-fixed beam are derived as the solutions to a boundary value problem with prescribed boundary conditions through the differential transformation method (D.T.M.). The differential transformation employed is a transformed function based on the Taylor series that is effective in solving nonlinear problems with fast convergence. The numerical results of the calculated natural frequencies are compared with the analytical data and were found to be in good agreement. Hence, the differential transformation method is one of the most efficient methods of simulating the electrostatic behavior of a micro-structure system, and it has a great potential for use in the analysis of the micro fixed-fixed beam.
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