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In this article, a novel flexure-based 3-degree-of-freedom elliptical micro/nano-positioning motion stage was developed to obtain translation motions along the x, y, and z directions with high working bandwidth. The stage was configurations with a XYZ vertical motion to ensure the crosstalk between each axis is small enough. The motion stage was driven by three piezoelectric actuators. The corresponding working principles and stiffness modeling were detailed in this article. In addition, experimental testing was conducted to investigate the performances of the motion stage. The obtained strokes in x, y, and z directions are 10.39, 15.43, and 15.55 mm, respectively. The response rise time for each axis is within 10 ms, and the maximum crosstalk all within 1.2% ensured the rapid tracking can be achieved.

In this paper, four-band Toeplitz matrices and four-band Hankel matrices of type I and type II with perturbed rows are introduced. Explicit determinants, inverses and eigenvalues for these matrices are derived by using a nice inverse formula of block bidiagonal Toeplitz matrices.

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