## 1 Introduction  
Piezoelectric-actuated machine tools (PAMTs) use piezoelectric actuators as their core components. Their high resolution, fast response, high stiffness, and micro deformation characteristics enable precise nanoscale displacement control [1, 2, 3], making them a key technological carrier in the field of ultra-precision machining. In practical machining [4, 5], the tool must balance the requirements of \"flexible feed\" and \"flexible retreat\" during the reciprocating high-speed movement [6, 7]. It is necessary to avoid rigid impact damage to the workpiece surface at the moment of contact and maintain machining efficiency through high-frequency micro motion. S-shaped curretrajectory planning effectively solves the inertia impact problem caused by sudden acceleration changes in traditional trapezoidal trajectories through continuous and smooth velocity and acceleration curve design. It not only reduces the vibration of the machining, but also shortens the idle travel time through dynamic optimization of the velocity curves [8, 9]. This characteristic not only fully utilizes the nanoscale dynamic adjustment capability of piezoelectric actuators but also transforms their high-frequency response advantages into stable high-frequency micro-cutting functions, thereby achieving a synergistic improvement in surface quality and processing efficiency in ultra-precision machining. The piezoelectric actuator is affected by many factors, such as the polarization process, the doping preparation, the strong external electric field or the high stress [10; 11]. Nonlinear effects, such as hysteresis, creep, and rate-dependent dynamics, directly cause deviations between the expected S-shaped curvetrajectory and the actual output [12; 13]. In addition, because the S-shaped trajectory driving voltage couples multiple common motion characteristics, the speed changes inherent in such multistage trajectories introduce additional non-uniform creep differences (NCD), which not only reduce tracking accuracy, but also cause unnecessary oscillations that affect surface smoothness and tool life. Therefore, compensating the non-linearity of the hysteretic coupling NCD can effectively improve the accuracy of trajectory tracking and the output performance of PAMTs. For nonlinear compensation of piezoelectric actuators, most studies utilize traditional phenomenological models to describe hysteresis more accurately, and design inverse model controllers to compensate for nonlinearity [14; 15]. For example, [PERSON] et al. employ an improved Preisach hysteresis model and two-degree -of-freedom H-Infinity robust control to precisely control the motion of piezoelectric actuators. The results obtained by using this control strategy show that the lag modeling accuracy and tracking performance are significantly improved, and the mean root mean square error (RMSE) is 0.0107 \(\upmu\)m and 0.0212 \(\upmu\)m, respectively [16]. [PERSON] et al. investigated an improved Prandtl Ishlinskii (PI) model that utilizes a one-sided dead zone operator to overcome the proposed hysteresis asymmetry of the actuator. Comparing the improved PI model with the traditional PI model, the BoucWen model, and experimental data, the maximum relative error is only 5.70%, and the model fitting accuracy is greater than 0.99 [17]. [PERSON] et al. proposed a hyperplane probabilistic c-regression model (HPCRM) algorithm and established its hysterically nonlinear TS fuzzy model [18]. However, these studies only compensate for the single nonlinear feature of hysteresis, and do not take into account the coupled nonlinearity of factors such as creep. On the other hand, the creep behavior of piezoelectric actuators is usually described by combining hysteresis models, which are mainly divided into logarithmic creep models, linear time-invariant (LTI) creep models, and fractional-order models [19; 20; 21].