Next generation large-scale spectroscopic survey projects may require smaller and high-precision fiber-positioning units. In this paper， a design scheme of the precise positioning part of the R mechanism of an R-θ fiber-positioning unit is proposed. Stacked piezoelectric ceramics combined with a flexible hinge lever displacement amplifying mechanism are used to achieve positional accuracy of the precise positioning part of the R mechanism. To minimize the influence of hysteresis nonlinearity on the precision of precise positioning， the classical Preisach model is used to establish the driving model of the amplified output displacement， and the corresponding optimization is conducted on this basis. The average error of the final established precise positioning displacement model is within 4 μm， and this error is significantly reduced compared with the maximum nonlinear error of 40 μm. The experimental results reveal that the displacement-driven model established by the Preisach model further improves the positioning accuracy and meets the error design requirements of precision positioning and also allows for a larger design space for later coarse positioning.