The operational performance of Flexible Hinge Piezoelectric Stages (FHPSs), essential components in precision engineering, is fundamentally constrained by the inherent hysteresis of the piezoelectric actuator (PEA). A significant deficiency in prevailing characterization methods is their failure to consider the dynamic nature of the mechanical preload exerted by the flexible hinge. This position-dependent preload induces substantial deviations in the PEA’s response characteristics, thereby compromising the predictive accuracy of conventional design frameworks. To address this limitation, this paper proposes a Force-Dependent Prandtl–Ishlinskii (FPI) model that explicitly formulates the PEA’s hysteretic behavior as a function of variable preload conditions. The FPI model is subsequently integrated into a comprehensive FPI-FFEM characterization framework. Within this framework, a Force-analyzed Finite Element Method (FFEM) is utilized to compute the dynamic preload throughout the actuator’s operational stroke. This information, notably neglected in conventional FEM analysis, is essential to the fidelity of the proposed FPI model. Experimental validation demonstrates the superior fidelity of the FPI model in comparison to the traditional PI model for tracking preload-induced nonlinearities. Furthermore, the complete FPI-FFEM framework exhibits substantially enhanced prediction accuracy relative to both conventional PI-FEM and advanced LDPI-FEM methodologies, as demonstrated by a significant reduction in the Mean Absolute Error (MAE).

Piezoelectric-actuated machine tools (PAMTs) exhibit nanoscale motion capabilities, with their S-shaped curve trajectory further enabling smooth path execution and reduced terminal pulse. However, the speed changes inherent in multi-order trajectories introduce an additional non-uniform creep difference (NCD), which differs significantly from conventional hysteresis effects. Traditional models are inadequate for addressing this mixed shape nonlinearity. To overcome this limitation, this paper proposes a hybrid compensation method for the S-shaped curve trajectory of piezoelectric-actuated machine tools. The general deformation law is first established through a comprehensive mechanism analysis. The NCD and hysteresis, induced by speed changes and inherent properties, are decoupled and addressed using a pre-known phenomenon model and a clockwise operator model, respectively. Finally, a hybrid feedforward control strategy is developed to integrate these models for effective compensation. Experimental results demonstrate that the hybrid compensation method achieves a maximum relative error of 5.48% and a maximum mean square error of 0.28%, effectively mitigating the dual nonlinear factors arising from the piezoelectric-actuated machine tool’s trajectory in feedforward control.

Different types of piezoelectric actuators and motors have attracted a great deal of modeling research in applications, with varying types and complexity. This paper specifically summarizes the modelings related to the inchworm piezoelectric motor to clarify the development achievements. It reviews the progress of dynamics models and characteristics models, classifying them by model features and abilities through development history. Finally, conclusions for existing research and possible future fields have been proposed.