1. Introduction
Since elliptical vibration cutting (EVC), equipment can improve machining accuracy and can be used to generate the surface micro-textures [
1]. It is widely used in the machining of different hard-to-cut materials, such as aerospace, illumination, micro-motion systems, etc. [
2,
3]. There are some advantages to using EVC technology compared to the conventional machining method, such as reduced cutting force and cutting heat and increased cutting tool life [
4]. According to the working mode of the EVC, it can be classified into the resonant and the non-resonant types. Compared with the resonant EVC device working at the specific resonant frequency, the non-resonant EVC device can output a larger displacement at various frequencies and can be easily controlled by an advanced control algorithm to improve its accuracy of output displacement. Flexible hinge mechanisms are generally used in the non-resonant EVC device, which is normally driven by piezoelectric stack actuators (PSAs) to realize displacement transmission and synthesize an elliptical trajectory [
5,
6,
7]. Since the PSA is directly connected to the flexible hinge mechanism, the output displacement of the PSA is equal to the input of the flexible hinge mechanism. However, the PSA has piezoelectric hysteresis nonlinearity characteristics, which significantly affect the accuracy of the generated elliptical trajectory of the EVC device. Therefore, many studies have focused on designing a control algorithm for improving the performance of the EVC device [
8]. Hysteretic nonlinearity usually refers to a system with a delayed response, in which changes in the system’s input are reflected in the output within a certain period. This means that the system’s output is not only related to the current input but also affected by the input history of the previous period.
There are many mathematical models proposed to explain the hysteresis phenomenon and used to fit and express the input and output relationships. The commonly used hysteresis models are the Duhem model, Bouc–Wen model, Preisach model, and PI model [
9,
10]. In 1897, Duhem proposed the Duhem hysteresis model, which uses a piecewise exponential curve to approximate the hysteresis characteristic. It is an easy method to be used to establish an inverse model. JinHyoung et al. [
11] proposed a rate-dependent Duhem model and obtained the auxiliary function of the Duhem model by polynomial approximation. The model shows an excellent fitting accuracy of the output displacement of PSA with frequency variation. The Bouc–Wen model was proposed by the German mathematician R. Boc in 1967 [
12]. Although the Bouc–Wen model has a simple structure, few parameters need to be determined and it is easy to implement in the controller. The accuracy of this model is greatly affected by the initial state. Fung et al. [
13,
14] used an adaptive differential evolution algorithm to identify the Bouc–Wen model. The experimental results show that the controller based on the model can effectively eliminate the effects of hysteresis and improve the motion accuracy of the positioning platform. In 1935, the German physicist Ferenc Preisach [
15] first proposed the Preisach model in the hysteresis effect research center based on ferromagnetic materials. Then, some researchers used this model in the hysteresis modeling of piezoelectric materials and achieved good results. Zhou et al. [
16] improved the classic Preisach model and identified the model weight function through the fast Fourier transform method, which improved the fast response capability of the system under frequency conversion signals. The PI (Prandtl-Ishlinskii) model was first proposed by Prandtl [
17] in 1928 to describe plastic elastic deformation, which is also a phenomenological hysteresis model based on the hysteresis operator. Duhem model and Bouc–Wen model are usually described by differential equations with complex model structures and difficult-to-determine parameters. The Preisach model and the PI model are obtained by weighted superposition of multiple basic hysteresis operators with fewer model parameters and can accurately describe the hysteresis phenomenon. Moreover, compared to the Preisach model, the expression of the PI model is more concise, and there is an analytical inverse model expression.
Based on the hysteresis models mentioned above, various control methods have been designed to eliminate the hysteresis nonlinearity and reduce the hysteresis error of the PSA. An effective control method is voltage feedforward compensation. The inverse model is solved by the established hysteresis model, and the controller is designed based on the inverse model to linearize the voltage input and displacement output. Mohammad et al. [
18] established the inverse of the rate-dependent PI model and applied it to the open-loop control of piezoelectric micro-positioning actuators. Galinaitis et al. [
19] proposed an improved inverse Preisach model to compensate for the rate-dependent hysteresis nonlinearity in piezoelectric ceramic actuators. Tang et al. [
20] used the Bouc–Wen inverse model to reduce the hysteretic nonlinearity of the system. Combined with the single-neuron PID feedback controller, the position-tracking accuracy of the piezoelectric ceramic platform was greatly improved. Fan et al. [
21] proposed a radial basis function neural network combined with a rate-dependent PI model, and a disturbance observer was designed for tracking control of PSAs with input frequencies from 1 to 100 Hz. Kang et al. [
22] proposed a new fractional normalized Bouc–Wen (FONBW) model. Compared with the classical Bouc–Wen model, the developed FONBW model has a relatively simple mathematical expression and fewer parameters and can characterize the asymmetric and rate-dependent hysteresis behavior of PSAs. Due to external interference, the feedforward control is hard to compensate for the error of the system output. Thus, the feedforward control method makes it difficult to achieve the ideal linearization. Therefore, feedback links are often added on the basis of feedforward control to improve system accuracy.
Kim. et al. [
23,
24] developed two non-resonant EVC devices in which the vibration amplitudes were used as feedback signals to design a PID control system. These controlled EVC devices have been used to machine micro-grooves, quadrangular pyramids, and other structures. Zhu et al. [
25] used two piezoelectric stack actuators arranged in parallel to drive flexure hinge mechanisms to obtain the elliptical vibration trajectory and then designed the controller with a fuzzy PID control method to generate a wedge shape with a frequency of 40 Hz. The micro-pit machining experiment showed that the non-resonant EVC device has stable displacement output performance. Ren et al. [
26] proposed a robust output feedback control model based on an uncertainty and disturbance estimator (UDE) without using a state observer for nonlinear single input single output (SISO) systems. The experimental results of piezoelectric nano-positioning show that the model can achieve high precision and high bandwidth trajectory tracking. Cheng et al. [
27] established an adaptive Takagi–Sugeno fuzzy model for the input–output relationship of stick-slip type piezoelectric actuators and realized the accurate control of the end effector. Unfortunately, the stability analysis of the system is based on several specific assumptions, but most of the actual systems can not meet these assumptions.
In summary, numerous works have been conducted by researchers relating to controlling the PSA. Each control method has its advantages and drawbacks. The feedforward control has high efficiency and a simple controller structure, making adjustments to the system before deviations, which meets the requirements for PSA controllers. However, the robustness of feedforward control is poor, and it is prone to under-compensation or overcompensation under external disturbances. Feedback control has good stability and high accuracy, and the system has strong anti-interference ability, but the structure is complex, and the calculation amount is large. The compensation of feedback control to the system always occurs after the deviation. Introducing feedback links based on feedforward control and combining the two methods can improve system control performance and motion accuracy. So, to improve the machining quality of the EVC device, a compound control method for trajectory error compensation is proposed in this paper. This paper is organized as follows: in
Section 2, the output trajectory characteristics of the non-resonant EVC device are studied, the dynamic hysteresis model of the PSA in each axis is given, and the hysteresis behavior of the PSA is described.
Section 3 shows the identification of the parameters of the model through PSO. The piezoelectric hysteresis model is obtained, and the controller is designed according to the established piezoelectric hysteresis model. In
Section 4, trajectory tracking experiments have been carried out. Finally, conclusions are provided in
Section 5.
3. Controller Design with Dynamic Hysteresis Compensation
The advantage of feedforward control lies in its predictability. However, in practical engineering applications, measuring all disturbances in advance and obtaining accurate predictive models is impossible. At this point, it is necessary to add feedback control, which can correct the deviation of the system in real time under any external interference. Considering the hysteresis characteristics of the output displacement of the PSA used in the non-resonant EVC device and the unpredictable disturbances (temperature, wear, etc.) in the processing process, this paper adopts a compound control method of feedforward and PID control. The inverse dynamic PI model is used to construct the feedforward controller for the feedforward control, while PID feedback is used to reduce the impact of insufficient model accuracy and potential interference and to improve the control accuracy.
The block diagram of the compound control is shown in
Figure 12. Where
represents the feedforward controller based on the inverse hysteresis model of the non-resonant EVC device. The controller calculates the reference displacement
to obtain the feedforward voltage
. The PID feedback controller uses the deviation
between the actual displacement
x and the reference displacement
to calculate the deviation voltage
. Then, the
and
are superimposed to obtain the output voltage signal
u to apply on the PSA to drive the EVC device.
The inverse dynamic PI model is used to design the feedforward controller. Because the PI model has an analytical inverse, its inverse model is still a PI model in expression. According to references [
33,
34,
35,
36], the inversion formula of the Prandtl-Ishlinskii model is also applicable to the segmented dynamic PI model proposed in this paper. The relationships of thresholds and weights between the dynamic PI model and its inverse model can be obtained directly as:
where,
is the expected displacement,
and
are the weight and operator threshold of the inverse PI model, respectively. Then, the inverse dynamic PI model is expressed as follows:
Based on the conventional PID control algorithm, the signal
is expressed as:
Since the feedforward control algorithm uses a discrete numerical calculation mode, the PID control algorithm needs to be discretized. After discretization, the PID algorithm in Equation (18) can be expressed as:
where
is the scale factor,
is the integral constant,
is the differential constant,
is the sampling period,
is the sampling number.
4. Trajectory Tracking of the Non-Resonant EVC Device
The experiments have been conducted to verify the control accuracy of the proposed compound control method on the non-resonant EVC device.
The control program is built into LabView 2018 software. A National Instruments data acquisition card (Model: NI USB–6361X, National Instruments, Austin, TX, USA) was used to acquire data. PSAs made by Suzhou Pante Company (Model: PTJ1500505202, Suzhou Pante Company, Suzhou, China) were used to drive the EVC device. Voltage signals used for PSAs were magnified by a Trek piezo amplifier (Model: PZD350, Advanced Energy Industries Inc., Denver, CO, USA). Vibrations of the non-resonant EVC device in Y- and Z- directions are independently controlled by the same program. Two Micro Sense capacitance sensors (Model: 5300, KLA Company, Milpitas, CA, USA) were orthogonally arranged to measure the tool vibration trajectories. The experimental setup is shown in
Figure 13.
The parameters required for the feedforward controller are listed in
Table 2,
Table 3,
Table 4 and
Table 5. The parameters
,
, and
required for the PID feedback control algorithm are obtained by the Cut and Try method. The identified parameters of
,
, and
in Y-direction are 50, 300, and 0.06, respectively, while the corresponding parameters in Z-direction are 70, 450, and 0.08, respectively.
During the experiments, two different cases were studied based on the ellipse inclination angle. The ellipse inclination angle is defined as the angle between the long axis of the ellipse and the Z-direction. For case 1, the ellipse inclination angle was set as 0° and depicted as a reference track in
Figure 14a. The vibration amplitude in the Y-direction was set as within 1–6 μm, while the vibration amplitude in the Z-direction was set within 1–11 μm. For case 2, the ellipse inclination angle was set as 20° and depicted as a reference track in
Figure 14b. The corresponding vibration amplitudes in Y- and Z-directions were set within 1–5 μm and 1–9 μm, respectively. For both cases, the vibration frequencies were set as 100 Hz. The measured elliptical trajectories are shown in
Figure 14, denoted as actual tracks (orange lines in the electronic version). The measured vibration displacements in Y- and Z-directions are demonstrated in
Figure 15 for both cases. In
Figure 15, the Y REF and Z REF (solid lines) represent the ideal displacements, while the Y ACTL and Z ACTL (dash lines) represent measured displacements. It can be seen from
Figure 15a that the maximum errors of displacements in Y- and Z-directions are 0.308 μm and 0.369 μm, respectively. From
Figure 15b, the maximum errors of displacements in Y- and Z-directions are 0.154 μm and 0.252 μm, respectively. For both cases, the relative error is less than 6.2%, which means the proposed compound control method has good control accuracy for this non-resonant EVC device. So, the non-resonant EVC device can output an elliptical vibration trajectory with a higher frequency range with the proposed compound control method.