**1. Introduction**

The nanopositioning stage of the piezoelectric ceramic material driver has the advantages of small volume, high displacement resolution, fast response, large bearing capacity, no noise, and high stability [1]. Hence, it is widely used in modern precision machineries as the core device, such as in atomic force microscopy [2,3] and nanolithography processing [4]. However, the inherent hysteresis nonlinearity of the piezoelectric ceramic materials affects the accuracy of this nanopositioning stage [5]. Thus, it is necessary to model effective compensation for the hysteresis [6,7].

In order to improve the positioning accuracy of the nanopositioning platforms, many scholars at home and abroad have conducted extensive research on piezoelectric ceramic hysteresis. There are two popular approaches. One is to study the hysteresis due to the internal mechanism, where the crystal grains constitute the crystal phase of the piezoelectric ceramic and the electric domains appear in the crystal grains. Polarization treatment enables piezoelectric ceramics to exhibit a piezoelectric effect [8]. However, a small number of grains returns to the original direction after the polarization, which is

known as the residual polarization, leading to hysteresis [9]. The second approach is to establish a phenomenological model for the expressed voltage–displacement characteristics, including di fferential equation models such as the Bouc–Wen model [10], operator models such as the Preisach model [11], the Prandtl–Ishlinskii (PI) model [12–14], and others.

Among them, the PI model is weighted and superimposed by the Play operator [15]. The symmetry and hysteresis characteristics of the Play operator can e fficiently describe the appearance characteristics of the hysteresis [16]. Compared with the nonlinear integral of the Preisach model and the nonlinear di fferential equation of the Bouc–Wen model, the PI model has fewer parameters, a simpler structure, and it is easier to find its inverse model. It has been widely used in modeling and compensation of hysteresis features [17,18]. The classical PI model always has symmetry, which makes it hard to accurately approximate the hysteresis characteristics of the boost phase and buck phase at the same time [19].

Many studies have been conducted on how to improve the accuracy of the nanopositioning stage by improving the traditional PI model. There are ways to change the operator by modifying the classical PI model into a generalized PI model, resetting the initial value that can eliminate the influence of the polarization by changing the hysteresis characteristics of the symmetric bias modeling and piecewise fitting method [20–23]. The above researches greatly improve the hysteresis description accuracy of the nanopositioning stage for single-ring linear voltage.

In a previous study [24], a segmented model was carried out by dividing the boost phase hysteresis characteristics by polarization in single-ring linear voltage. However, it is more likely for an improved method to solve the physical problem than a define-based modeling. For the reciprocating linear voltage hysteresis, the hysteresis characteristics of the inner and outer loops are di fferent and complex. Thus, the segmented modeling method is limited in reciprocating linear voltage due to insu fficient theoretical segmentation basis. Therefore, both the classical PI model and the improved model cannot meet the accuracy compensation requirements of the nanopositioning stage under reciprocating linear voltage.

Aiming at the complex hysteresis problem of the piezoelectric stage for the reciprocating linear voltage, this study proposes a mark-segmented PI (MSPI) model that loads the reciprocating linear voltage signal according to the reciprocating linear displacement requirement and compensates using the characteristics of the obtained hysteresis. An approximate hysteresis curve slope can be obtained between the adjacent measurement points of the experimental data, and the segmen<sup>t</sup> identification points corresponding to the slope jump segments are found in a threshold manner; thus, the segments are modeled. The results show that the description accuracy of the MSPI model is high, and it has good performance in both the compensation hysteresis inner loop and hysteresis outer loop.

The structure of this paper is as follows. In the second section, a nanopositioning stage with a double-parallel guiding mechanism is designed, and the mechanical characteristics of the platform are analyzed to optimize the design structure. In the third section, the hysteresis characteristics are analyzed and the MSPI model is designed for fitting the reciprocating nonlinear features. After the inversion, the fourth section gives the control comparison between the MSPI model and the classical PI model. Two other verification experiments are also conducted. Finally, the fifth section summarizes this study.
