Design of Piezoelectric Inertial Actuator with Wedge‐shaped Foot Structure for Cell Manipulation

Aiming at the performance requirements of positioning accuracy and stability of piezoelectric inertial actuators in fields including micro‐operation and biomedical engineering, a piezoelectric inertial actuator with wedge‐shaped friction foot structure using bimorph films is proposed in this paper. The wedge‐shaped friction foot structure can adjust the friction force in different driving stages, which suppresses the backward motion and effectively improves the output performance of the actuator. And the bimorph films are processed by Micro Electromechanical Systems (MEMS) manufacturing process technology. It enables the actuator to move steadily with a small tip mass and facilitates the miniaturization of actuator. The simulation model is constructed based on the dynamic model of actuator and the prototype parameters are optimized from the results of simulation tests. Then a series of output performance experiments are carried out. Experimental results show that the proposed actuator has the advantages of high resolution, stability, and displacement linearity. And the highest resolution of 0.035 µm is achieved. Depending on above results, the cell drug injection simulation experiments are successfully conducted. The success of experiments shows that the prototype has good output performance and great application potential and value in the field of cell manipulation.


Introduction
In recent years, piezoelectric precision drive and control technology has attracted widespread attention of many fields of researchers. It has been applied in such fields which include biological engineering, [1][2][3][4][5][6][7][8] ultra-precision manufacturing, [9][10][11] optical scanning, [12][13][14] and so on. Among these applications, the core technology of the whole machine system is to get inertial mass to generate enough driving force. The large inertial mass not only increases the weight, and size of the system but also damages the positioning accuracy and motion stability. That is mainly because the large stiffness of the thick bimorph (∼0.5 mm) produced by the traditional sintering technique restricts the transmission of motion of the moving body. [27] Some piezoelectric actuators are proposed and designed for solving those deficiencies. Cheng et al. designed a piezoelectric inertial rotary actuator based on asymmetrical clamping structures. [28] The experimental results indicate that the stable minimum output stepping angle is 0.85 µrad under a square signal of 20 V, 6 Hz. Ma et al. proposed a piezoelectric inertia rotary actuator based on an asymmetric clamping mechanism. [29] The angular displacement resolution of 12 µrad is obtained with a square wave of 15 V, 4 Hz. These actuators use large inertial mass to increase the driving force produced by the bending of the piezoelectric vibrator, but it also increases the volume and mass of the actuator itself, which is not conducive to the integration and miniaturization of the actuator.
Therefore, based on the deficiencies mentioned above, this paper innovatively designs a wedge-shaped friction foot structure and uses bimorph films made by micro electromechanical systems (MEMS) process to realize directional driving. On one hand, the proposed wedge-shaped foot structure shows friction anisotropy in the motion process and partly realizes the matching control of the inertial impact force and the friction force in the driving process. It increases the output step displacement and displacement linearity of the prototype. On the other hand, reducing the stiffness of the bimorph is the primary object currently for the piezoelectric inertial actuators to achieve miniaturization, high resolution, and stability. The MEMS manufacturing process technology is used to fabricate the bimorph thick films used in this study, which results in a great reduction of thickness and stiffness of piezoelectric bimorphs. This method realizes the miniaturization design of the prototype and also improves the motion stability. The prototype based on above designs achieves stable linear displacement in a series of mechanical output performance experiments, while also showing good operation ability in the simulation experiments of cell drug injection. Compared with the traditional inertial impact actuator, the advantages of the actuator proposed in this paper are mainly embodied in two aspects. First of all, the size of the actuator design is small and compact, which is easy to help the actuator achieve miniaturization and integration. Thus, the designed miniaturized prototype is beneficial to practical application in cell manipulation and other fields. Secondly, the motion output performance of the designed actuator with high precision is stable. It has good stability and superior displacement linearity. The problem of low positioning precision caused by vibration and motion instability in traditional inertial impact actuators is reduced. Figure 1 shows the configuration of proposed piezoelectric actuator, which is based on the inertial impact principle. It is mainly composed of a moving body with two wedge-shaped feet, a printed circuit board, and two bimorph-thick films with tip mass. Figure 1a shows the overall three-dimensional structure of the prototype and the wedge-shaped foot structure is as shown in Figure 1b. In order to realize the ideal displacement output state of the proposed actuator, this structure which can change the friction force between the driving mechanism and the contact surface is added. The angle between the left side of the wedge-shaped foot mechanism and the ground is 90°, and the angle between the right side and the ground is less than 90°. There is plastic deformation under gravity between the wedge-shaped foot structure and the contact surface. Wedgeshaped foot mechanism has no furrow effect when moving to the right, but a furrow effect when moving to the left. The friction coefficient when moving to the left with the contact surface is greater than moving to the right, which can suppress the fallback motion of the proposed actuator. So as to improve the output performance during driving motion. With the asymmetrical clamping structure shown in Figure 1c, the effective length of the driving unit is different on two clamping sides, realizing the asymmetry of the stiffness. The components of the piezoelectric vibrator are depicted in detail in Figure 1d.

Structure of Proposed Actuator
In this study, MEMS processing technology is used to fabricate the bimorph of the piezoelectric vibrator for the miniaturization, lightweight, and high precision of the proposed actuator. Figure 2 illustrates the MEMS fabrication processes of the vibrator as follows.
Step I: the metal layer and piezoelectric layer are prepared as a supporting and functional layer, respectively.
Step II: both sides of beryllium bronze (thickness of 50 µm) and one side of PZT thick film (thickness of 400 µm) are polished to improve the bonding quality.
Step III: a 30/200 nm Cr/Au layer as a bottom electrode is sputtered on the polished side of PZT thick films by magnetron sputtering.
Step IV-V: the Cr/Au side of two PZT thick films is bonded together with the beryllium bronze on both sides by epoxy resin.
Step VI: the PZT layers are thinned down to ≈50 µm by mechanical lapping.
Step VII: a 30/200 nm Cr/Au layer as a top electrode is sputtered on the other side of PZT thick films by magnetron sputtering.
Step VIII: the designed overall dimensions of bimorph are obtained by laser cutting.
Step IX: ultrasonic cleaning is used to remove the stain from the manufacturing processes.
Step X: the tip masses are made of brass, and the wires are bonded to the bimorph to form the structure of the piezoelectric vibrator finally. Figure 3 shows the measurement results of the thickness dimension parameters and surface quality of the bimorph piezoelectric vibrator fabricated by the MEMS fabrication processes. According to the SEM images of the sectional view of the bimorph layers [ Figure 3b], the top and bottom thickness of PZT layers are respectively 46.3 µm and 48.2 µm, whereas the thickness of the beryllium bronze layer is 49.0 layers have been significantly improved after polishing, as shown in Figure 3c. The very thin (thickness of ≈150 µm) and smooth bimorph films consisting of PZT and beryllium bronze [ Figure 1c] are obtained for keeping the bending stiffness of the driving unit at a low level. Thus, small brass blocks are used as the inertial tip masses. www.advmatinterfaces.de

Working Principle of Proposed Actuator
Piezoelectric inertial actuators usually use the symmetrical square wave signal to achieve directional driving as shown in Figure 4a. For the proposed actuator, the two thin and smooth bimorph films generate a symmetric driving force by excitation of input voltage signal and the asymmetrical clamping structure makes the vibrator stiffness different when bimorph bends to both sides. The actuator has four motion states and finally obtains a step forward [ Figure 4b]. And Figure 4c illustrates the specific movement process over one cycle.
According to the motion posture of the actuator in a whole period, it can be divided into two stages: forward stage and backward stage. Take point "a" as the initial state and point "e" as the end state, the whole motion process of the proposed actuator is as follows: a. At the static stage I, the input signal varies from "a" to "b". Two bimorphs keep bended by constant voltage signal and the actuator remains stationary at the initial position P1. b. At the forward stage, the input voltage signal rapidly varies from "b" to "c". Both two bimorphs bend to the opposite side where the small clamping block is located and result in the resultant force F1 to the other side. The force moves the actuator forward with a displacement of D f and reaches position P2 as shown in Figure 4c. c. At the static stage II, the input signal varies from "c" to "d".
These two vibrators have no driving motion for constant voltage signal again and hold the previous state. The actuator remains stationary at previous position P2. d. At the backward stage, the input voltage signal quickly changes from "d" to "e". The two bimorphs bend to a side different from the previous one at the same time, it is also the side where the large clamping block is located. And deformation of bimorphs produces resultant force F 2 . Depending on previous research, [31,32] different clamping blocks can make bimorphs form different stiffness during deformation, resulting in different resultant forces. The small clamping block  can make the effective stiffness of bimorphs larger deformation and the larger generated force than the large clamping block. Thus, the resultant force F 2 is smaller than F 1 , the proposed actuator moves backward with a displacement of D b and arrives at position P3 as shown in Figure 4c.
The actuator achieves directional displacement ΔD = D f −D b in one cycle and realizes large strokes continuously by repeating the above process. Besides, the reasonable matching caused by wedge-shaped foot mechanism between the driving forces and the friction force makes the actuator realize continuous and stable linear motions.

Dynamic Model of Proposed Actuator
To further analyze the motion progress of proposed actuator, the simplified dynamic model is established as shown in Figure 5.

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According to the dynamic model of the actuator, the dynamic differential equations of the bimorphs and the main motion body are as follows: the deformation of bimorphs, ϕ 31 is the piezoelectric coefficient, f is the friction force between actuator and the contact surface, f F i is the vector indicating the magnitude and direction of friction. Due to the friction direction is always opposite to the direction of object motion, the relationship between f and F i can be described by Equation (3). Substituting (3) into (2), the final differential equation of the main motion body is as follows: In a complete motion cycle, the number i = 1 in the dynamic differential equation when proposed actuator is in the forward stage and i = 2 in the backward stage. Because of the characteristics of friction anisotropy caused by the wedge-shaped foot structure during the movement of the prototype, F f in the forward process is different from that in the backward process, and the friction force f 1 F in the forward process is less than the friction force f 2 F in the backward process. In order to clearly describe and obtain a relatively accurate value of the friction force F f received by proposed actuator in the process of motion, LuGre friction model is used for calculation. [33,34] The calculation formula of friction is as follows: where σ 0 , σ 1 and σ 2 are bristle stiffness, damping, and viscous friction relative velocity coefficient, z is the average deflection of the bristles on the contact surfaces, v is the relative velocity between the two surfaces. The parameters used in the model are shown in Table 1 and the simulation model is shown in Figure 6.

Experimental System and Prototype
To evaluate the performance of the proposed actuator, experiments using the prototype shown in Figure 1a were carried out in the experimental system as shown in Figure 7. The square signal generated from a signal generator (RIGOL, DG4162) was amplified by a power amplifier (Physical Instrument, E-472.20) to drive the actuator. The laser sensor (KEYENCE, LK-HD500) was used to measure the displacement of the prototype directly and further processed with LK-Navigator software. Subsequently, the software on the PC recorded and processed the data detected from the laser sensor.

Experiments of Actuator Characteristics
The transient displacement of the drive unit is tested as shown in Figure 8a. The experimental result shows that the two piezoelectric bimorphs produce different transient displacements when bending to both sides with the asymmetric clamping structure. The feasibility of the proposed prototype in working principle is proved. In order to further show the working principle of the proposed actuator intuitively, the phase change of the actuator operation in the time domain is recorded by the scanning vibrometer system which is shown in Figure 8b. The left side is the forward stage in the driving step cycle of proposed prototype, at which time the bimorphs are bending from the long clamping side to the short clamping side. And the right side indicates the backward stage in the step cycle, which moment the bimorphs are bending from the short clamping side to the long clamping side to the short clamping side. The alternation of these two stages makes the prototype produce the expected linear displacement.
The prototype is driven by the wedge-shaped foot structure to realize the different friction forces between the moving body and the contact surface in the reciprocating motion. The greater the difference in friction between the two reciprocating directions, the better the output performance of the actuator. In order to explore the influence of friction anisotropy caused by wedge-shaped foot structure on the output performance of  actuator, experiments with different materials and angles were carried out. The friction coefficients between different guide materials and wedge-shaped foot structures are measured by pin-on-disc friction and a wear tester. And the results are shown in Table 2. The results show that when the material is organic glass, the difference of friction coefficient between the wedge-shaped foot structure and the contact surface in the reciprocating direction is the largest. That is to say, the effect of friction anisotropy is the most obvious. Thus, in order to achieve the ideal output performance of the actuator, the material chosen should be organic glass. Then, the effects of different angles of the wedge-shaped foot structure on the output performance of actuator were experimented. And the experimental results are shown in Figure 9. The parameter β indicates the angle between right side of wedge-shaped foot mechanism and the contact surface. The mechanism is an ordinary driving foot when β is equal to 90°, and there is no furrow effect between it and the contact surface. The experimental results show that under the action of a wedgeshaped driving foot with a furrow effect, the average step output displacement of actuator is improved and the linearity between displacement and voltage is better than that without furrow effect. The output step displacement performance of proposed actuator can be improved by wedge-shaped foot mechanism. At the same time, by comparing the simulation results with the actual results, it can be found that the actual motion process is most consistent with the theoretical results when the angle is 45°. And when the angle is 90°, the actual motion process has a large deviation from the theoretical results. This further shows that the designed wedge-shaped drive foot structure has a beneficial impact on the output performance of the proposed actuator. Moreover, in order to obtain the most ideal output performance, the wedge-shaped foot structure angle design should adopt 45°.
The simulation and experiment accumulated displacements at different voltages are shown in Figure 10. It can be seen that the proposed actuator could obtain stable step motion at different voltages both in simulation and the experiment. The  www.advmatinterfaces.de cumulative displacement of the prototype increases when input voltage is rising. And the experimental results are in good agreement with the simulation results, which shows the feasibility of the proposed actuator scheme in theory.
The stepping performance within 80 periods of the proposed actuator at different driving voltages has been investigated in Figure 11. Figure 11a clearly shows the relationship between the accumulated displacements and the input voltage, which presents that the accumulated displacements increase with the rising of voltage. The total displacement of the proposed actuator is 2.83 µm in 80 cycles at the voltage of 5 V in Figure 11a. The highest resolution can be calculated by the following formula: 2.83 80 0.0354 = ÷ = S It clearly shows that the highest resolution reaches 0.035 µm under the control signal of 5 V. The driving voltages used in the proposed actuator are selected based on a consideration between linearity and other output performances required in application such as resolution, speed, and so on. And the driving voltage of 5 V is the starting voltage of the proposed actuator. Under the condition of starting voltage, the driving force and the friction force are close in magnitude. In other words, the nonlinear friction force has a great influence on the output performance of the proposed actuator. The displacement linearity is sacrificed in order to achieve high resolution at this voltage. Therefore, the correlation coefficient is smaller than 0.9 when the driving voltage is 5 V as shown in the fitting equation in Figure 11b. When the voltage ranges from 7 V to 19 V, the driving force is larger than the friction force in order of magnitude. The nonlinear friction force has little effect on the output performance of the proposed actuator. At above voltages, the correlation coefficients are greater than 0.99 as shown in Figure 11b. And the proposed actuator exhibits good displacement linearity. To sum up, these results obviously show that the output performance of the proposed actuator has good stability, linearity, and high resolution.
Stability and repeatability are important properties to evaluate an actuator, which is vital for the key applications of piezoelectric actuators in some fields. The acceleration durability test is carried out to test the stability of the prototype. The experimental results show that the driving force has good stability which the fluctuation range is in small variation. The linear displacement of proposed actuator before and after durability test is also been described in Figure 12.   The comparison of the linear displacement before and after the durability test in Figure 13a illustrates that the piezoelectric bimorph has good stability in multiple cycles of operation. As shown in Figure 13a,b, both the linear displacements and the actuating forces have little difference before the durability test and after durability. That is to say, the prototype exhibits an expected linear motion and actuating force after the durability test. This fact indicates that the proposed actuator can maintain good stability during the operation process.

Demonstration Experiment of Cell Manipulation
In order to test the application potential of the designed actuator in cell manipulation, a microprobe was fixed on the actuator to penetrate a zebrafish embryo (diameter: 1000 µm) to simulate cell injection. The experimental testing system configuration of this cell manipulation is shown in Figure 14. In the experiment, zebrafish embryos are clamped by parallel V-grooves, which are made of agarose gel in a petri dish. A microscope is used to locate the embryo and observe the movement of the microprobe. A high-speed camera is used to record the whole process of cell manipulation. Figure 15 shows the drug injection simulation processes using the zebrafish embryo in the demonstration experiment of cell manipulation.
The average one-cycle displacement of the proposed actuator under the action of 40 V voltage and 1 Hz frequency control signal is ≈12 µm. After 22.4s and 44.8s, the forward displacement of the actuator can be calculated as 154 µm and 124 µm. Under the action of the square wave signal, the actuator drives the glass microprobe to approach the zebrafish embryo step by step until the embryo is punctured. The experimental results show that the designed actuator has high stability and controllability in cell manipulation. And the success of cell demonstration experiment shows a fact that the designed actuator has great potential application value in the field of cell manipulation.

Conclusions
A novel piezoelectric inertial actuator using a pair of wedgeshaped foot structures and bimorph films was proposed. The wedge-shaped foot structure of proposed actuator could improve the output performance of proposed actuator by restraining the backward motion in driving process. And the bimorph films used by proposed actuator were fabricated by the MEMS process technology. The drive unit with these bimorph films can obtain low stiffness, which was able to benefit from miniaturization and integration of actuator. The prototype parameters were optimized by theoretical analysis and simulation experiments. Then the experimental test system of mechanical output performance of proposed actuator was established. The experiment results demonstrated the proposed actuator had high motion stability and displacement linearity, achieving the highest resolution of 0.035 µm. Furthermore, an experimental system for cell manipulation was also constructed to evaluate the application potential of actuator. The results of the cell drug injection simulation experiment also showed the stability and controllability of the actuator in terms of cell manipulation. Therefore, the designed piezoelectric inertial actuator could be served as a precision positioning and operation device for cell manipulation and have widespread application prospects in the field of cell manipulation.