## 3 Feedforward Control Considering Non-Uniform Creep Difference and Hysteresis Nonlinearity  
### NCD Mechanism Analysis and Separation Basis  
From the above analysis, it can be seen that the piezoelectric actuator under the S-shaped trajectory driving voltage is coupled by the inherent hysteresis characteristics of the material and the NCD caused by the continuous change of the driving voltage speed. Therefore, analyzing the change mechanism of NCD is helpful to compensate the nonlinearity of the piezoelectric actuator and trajectory shaping. Since the triangular wave voltage rate is constant, there is no NCD in the nonlinear characteristics of the piezoelectric actuator under the triangular wave voltage. Finding the points where the voltage change rate is the same as the triangular wave voltage change rate in the S-shaped trajectory driving voltage. These points with the same voltage speed are the theoretical basis for separating the NCD nonlinear effects under different states. Figure 3 shows, under the trajectory voltage with a maximum voltage of 120 V, the process of finding the point with the same triangular wave velocity as its corresponding point. Figure 2: Complex nonlinear characteristics of piezoelectric actuators under S-shaped trajectory driving voltage: (**a**) 0.25 Hz. (**b**) 0.5 Hz. (**c**) 1 Hz. (**d**) 2 Hz. These points are defined as Marker Point 1, Marker Point 2, Marker Point 3, and Marker Point 4. The four marker points divide the S-shaped trajectory driving voltage into six parts. By analyzing the speed changes of each road section, NCD can analyze the following:  
When the voltage is below Marker Point 1, the rising speed of the S-shaped trajectory driving voltage is slower than that of the triangular wave voltage, and due to its longer charging time, the NCD is more pronounced, resulting in a slightly larger displacement than that of the triangular wave voltage. However, since the electric field strength is low at this stage, the impact of NCD is limited, and thus the displacement difference between the two is small. When the voltage lies between Marker Point 1 and Marker Point 2, the rising speed of the S-shaped trajectory driving voltage is slightly higher than that of the triangular wave voltage, and the charging time is shorter, reducing the creep effect. It is even possible that the deformation of the piezoelectric ceramic under the S-shaped trajectory driving voltage is smaller than that under the triangular wave voltage, leading to a gradual narrowing of the nonlinear gap between the two. As the voltage further increases to the range between Marker Point 2 and the maximum voltage, the rising speed of the S-shaped trajectory driving voltage slows down, but due to sufficient charging, the NCD becomes significantly stronger, causing the displacement under the S-shaped trajectory driving voltage to be noticeably greater than that under the triangular wave voltage. Although the average speed of the S-shaped trajectory driving voltage is lower than that of the triangular wave voltage, at 120 V, its displacement still exceeds that of the triangular wave voltage. When the voltage exceeds the maximum voltage but remains below Marker Point 3, the rising speed of the S-shaped trajectory driving voltage further decreases, and the NCD becomes even more pronounced. At this point, the reduction in displacement is smaller than that of the triangular wave voltage, so at 120 V, the displacement under the S-shaped trajectory driving voltage remains greater than that under the triangular wave voltage, and the difference between the two gradually increases. Figure 3: S-shaped trajectory driving voltage and triangular wave driving voltage: (**a**) 0.25 Hz. (**b**) 0.5 Hz. (**c**) 1 Hz. (**d**) 2 Hz. When the voltage is between Marker Point 3 and Marker Point 4, the rising speed of the S-shaped trajectory driving voltage once again surpasses that of the triangular wave voltage, and the NCD is not significant.