## 1 Introduction
The logarithmic creep model and LTI creep model can be used in parallel or in series with the hysteresis model to comprehensively describe the hysteresis and creep phenomena of piezoelectric actuators. This combination allows hysteresis and creep models to operate independently with low coupling. For example, [PERSON] et al. proposed the concept of \"voltage creep\", assuming that constant displacement may cause input voltage creep, and developed a logarithmic creep model based on this to compensate for the nonlinear behavior of piezoelectric actuators [22]. [PERSON] et al. further designed a logarithmic creep controller combined with a PI model to achieve open-loop control of piezoelectric actuators [23]. [PERSON] et al. proposed hysteresis and logarithmic creep models, and used inverse feedforward control to simultaneously compensate for the hysteresis and creep of piezoelectric actuators [24]. In addition, there are studies combining the creep model with the hysteresis model. [PERSON] et al. extended an operator-based hysteresis model and provided a physical structure diagram including hysteresis and creep [25]. [PERSON] et al. developed a feedforward controller based on the LTI creep model and hysteresis operator to compensate for the hysteresis and creep of piezoelectric actuators [26]. [PERSON] et al. modeled the nonlinear effects of hysteresis and creep based on the superposition of basic operators, describing the creep operator as a linear first-order system [27]. [PERSON] et al. combined the PI model and LTI model to propose a compensator that integrates an adaptive controller and an online approximator to compensate for the creep and hysteresis of piezoelectric actuators [28]. Through the comparative analysis of existing research methods in Table 1, it can be seen that the existing research mainly focuses on hysteretic characteristics and nonlinearity caused by static creep, while ignoring the nonlinear influence caused by NCD. The traditional model is not enough to solve the mixed-shape nonlinear problem. Based on the above facts, in order to solve the mixed nonlinearity of piezoelectric actuators in multi-stage trajectories, the coupling mechanism between the hysteresis nonlinearity of piezoelectric actuators and NCDs in S-shaped curve trajectory is studied in this paper, and a hybrid feedforward control strategy is proposed to realize the preset trajectory shaping. The purpose is to separate NCD and hysteresis and to compensate for the coupling nonlinearity. The effectiveness of this strategy is verified by an experimental case study. This research helps to improve the applicability of S-shaped curve trajectory in piezoelectric drive systems and bridge the gap between theoretical motion planning and practical nonlinear control challenges. The main innovations of this paper include the following: a method of NCD separation is proposed, a hybrid feedforward control strategy is designed, and the lag problem is solved by using the clockwise operator. This hybrid control strategy applies not only to the S-trajectory drive voltage in PAMTs, but also can be further extended to other multi-stage trajectories. In addition, this control idea can also be used for reference for other piezoelectric actuating equipment to adapt to its specific working conditions. The main structure of this article is as follows: Section 2 introduces the typical structure of PAMTs, derives the general deformation law of piezoelectric actuators, and theoretically analyzes the complex nonlinearity of piezoelectric actuators under S-shaped trajectory driving voltage. Section 3 analyzes the mechanism of NCD, proposes an NCD separation method, and proposes a hybrid control strategy combining nonlinear compensation and trajectory shaping. Section 4 validates the proposed method through experiments. Section 5 provides a summary of the article.