# A Compact 2-DOF Piezoelectric-Driven Platform Based on “Z-Shaped” Flexure Hinges

^{1}

^{2}

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## Abstract

**:**

_{x}= 17.65 μm and l

_{y}= 15.45 μm, respectively. The step response time for x-stage and y-stage are t

_{x}= 1.7 ms and t

_{y}= 1.6 ms, respectively.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Calculation of the “Z-shaped” Flexure Hinge

_{a}O

_{a1}; the structure is L = 16 mm, w = 6 mm and t = 1 mm. Figure 1b illustrates the right-circle flexure hinge, and it is divided into three parts: part O

_{r}O

_{r1}(basic right-angle), part O

_{r1}O

_{r2}(basic right-circle) and part O

_{r2}O

_{r3}(basic right-angle); the structure is l

_{1}= l

_{2}= 5 mm, R = 3 mm. The “Z-shaped” flexure hinge is shown in Figure 1c, and it consists of three basic right-angle flexure hinges: part O

_{z}O

_{z1}, part O

_{z1}O

_{z2}and part O

_{z2}O

_{z3}; the size is l

_{3}= l

_{5}= 8 mm, l

_{4}= 6 mm. According to the references [27,28], the widely utilized compliance matrix C

_{ra}for the right-angle flexure hinge is gotten by the following:

_{rc}is shown in the following [29]:

_{x}, u

_{y}and u

_{z}are the linear deformation along the x, y and z directions; θ

_{x}, θ

_{y}and θ

_{z}are the rotational angles around the x, y and z directions; F

_{x}, F

_{y}, F

_{z}, M

_{x}, M

_{y}and M

_{z}are the applied forces and torques in the x, y and z directions.

_{ri}with respect to point O

_{r}, which are achieved by:

_{ori}(α

_{ori}) and S

_{ori}are the rotation matrix and position matrix of point O

_{ri}with respect to point O

_{r}, respectively; α

_{ori}is the rotation angle of point O

_{ri}relative to point O

_{r}; r

_{xri}, r

_{yri}and r

_{zri}are the relative positions between point O

_{ri}and point O

_{r}.

_{zi}with respect to point O

_{z}; R

_{ozi}(α

_{ozi}) and S

_{ozi}are the rotation matrix and position matrix of point O

_{zi}to point O

_{z}, respectively.

_{z}= 323.5 μm by the MCM method, while that from the FEM method is D

_{z’}= 340.1 μm; the error between the MCM and FEM methods is e

_{z}= 4.9% based on Equation (20), which confirms the application feasibility of the MCM method. The maximum deformation for the right-angle flexure hinge is D

_{ra}= 265.8 μm with an error of e

_{ra}= 1.7%. The error of the right-circle flexure hinge is e

_{rc}= 2.2%, considering the influence of the torque induced by input force F in Equation (10). The comparison of FEM and MCM methods under other forces can be obtained in Figure 2, which shows the great agreement. Figure 3 illustrates the working stress of these three flexure hinges by FEM method in the case that a displacement D

_{y}along y direction is applied on the right end surface of each flexure hinge. The left end of each flexure hinge is fixed to the ground without freedom. The FEM mesh size is 0.1 mm for these three flexure hinges. Since the working stroke of the utilized piezo stack is around 10 μm while the driving voltage is 100 V, the applied displacement is chosen as D

_{y}= 10 μm. It is shown that the maximum working stress of these three flexure hinges are σ

_{ra}= 10.9 MPa, σ

_{rc}= 46.5 MPa and σ

_{z}= 8.8 MPa. The working stress of the “Z-shaped” flexure hinge is the smallest among them, which is of great significance to extend the working lifetime of piezoelectric-driven platforms. Hence, the “Z-shaped” flexure hinge is utilized for the design of the proposed 2-DOF nano-positioning platform.

#### 2.2. Platform Design

_{r}= 50 N was also applied on the surface to nest the piezoelectric stack on y-stage. Figure 4 illustrates the results of FEM calculation. In the case that all the six-connecting-rods for y-stage are “Z-shaped” flexure hinges, the deformation of area A in y-stage is about 3 μm, as shown in Figure 4a. Since the piezoelectric stack for x direction is nested inside y-stage, the reaction force makes the y-stage generate this unwanted deformation, which should be avoided. Meanwhile, in the case that two right-angle flexure hinges are utilized to replace “Z-shaped” flexure hinges for y-stage, the unwanted deformation of area A is rather small since the stiffness of right-angle flexure hinge along x direction is higher than the “Z-shaped” flexure hinge. Additionally, the output force of the used piezoelectric stacks is around F

_{stack}= 800 N, which is large enough to reduce the influence of the increased stress for y-stage motion stroke. Therefore, both “Z-shaped” flexure hinges and right-angle flexure hinges are utilized in the 2-DOF piezoelectric-driven platform.

_{stack}= 800 N, and the motion stroke is about L

_{stack}= 17 ± 2 μm in the case of input voltage V = 150 V; they are working on the d

_{33}model and the value is d

_{33}= 720 × 10

^{−12}m/V. The x-stage is nested inside of y-stage, which means that the proposed nano-positioning platform is in serial type. This serial structure can make the proposed platform in a compact size and with small coupling errors. Two kinds of flexure hinges (“Z-shaped” and right-angle) are exploited to form a parallel-six-connecting-rods structure for y-stage; the flexure hinges in x-stage are all “Z-shaped” flexure hinges. The material of the proposed 2-DOF piezoelectric-driven platform is structural steel. Wire electrical discharge machining (WEDM) was utilized to manufacture the platform, so as to make the whole platform in an integrated structure. The size of the proposed 2-DOF piezoelectric-driven platform is l

_{1}× l

_{2}× l

_{3}= 130 mm × 150 mm × 8mm.

#### 2.3. Dynamic Calculation

_{k}and potential energy E

_{t}keeps the same during the motion. In the case that force F

_{a}is applied on y-stage, the platform moves a distance s. Kinetic energy E

_{k}of the proposed platform is divided into two parts: kinetic energy E

_{k1}induced by the linear motion, kinetic energy E

_{k2}induced by the rotation motion of flexure hinges. Kinetic energy E

_{k}is achieved by the following equation [27]:

_{1}is the mass of the “Z-shaped” flexure hinge; m

_{2}is the mass of the right-angle flexure hinge; m

_{3}is the mass of y-stage; m

_{4}is the mass of x-stage; s is the motion displacement of y-stage; θ = arctan (s/L) ≈ s/L is the rotation angular displacement of flexure hinge; J

_{z}is the inertia moment of the “Z-shaped” flexure hinge; J

_{r}is the inertia moment of the right-angle flexure hinge.

_{t}is mainly induced by the six flexure hinges for y-stage, and it is obtained by the following equation:

_{θz}is the rotational stiffness of the “Z-shaped” flexure hinge; k

_{θr}is the rotational stiffness of the right-angle flexure hinge.

_{0}keeps the same and it can be achieved by the following equation:

_{1}= 678 Hz, f

_{2}= 965 Hz, f

_{3}= 1297 Hz, f

_{4}= 1553 Hz, f

_{5}= 1832 Hz and f

_{6}= 2002 Hz. It can be seen that the undesired transverse motion (z and θ

_{x}) is introduced in high frequency since the use of “Z-shaped” flexure hinges, which is not good for the 2-DOF nano-positioning platform. Generally, the proposed 2-DOF nano-positioning platform works under several dozens of Hertz; hence, the resonance phenomenon could be avoided. We are still working on this platform, and hope to overcome this disadvantage in future work.

## 3. Results and Discussion

#### 3.1. Experimental System

_{HPV}= ±150 V with a voltage resolution of e

_{HPV}= 1%; the maximum output current is I

_{HPV}= 0.5 A. The laser reflectors are attached to the x- and y- stages to reflect the laser signal. Laser sensor LK-G10 (KEYENCE COMPANY) is utilized to generate the laser signal to record the motion of the laser reflector. The laser head is placed in line with the motion direction of x-/y- stage which has the attached laser reflector. The measurement range of LK-G10 is L

_{LK}= ±1 mm with a resolution of e

_{LK}= 10 nm which is high enough for the proposed piezoelectric-driven platform. All data is processed and changed by the AD/DA card which is integrated inside the laser sensor LK-G10, and then saved by the IPC. The commercial software from KEYENCE COMPANY is utilized to control the laser sensor and save the data with a sampling time of t

_{sa}= 0.1 ms. The software from BOSHI COMPANY is exploited to control the signal producer. All this equipment is placed on the air-floating vibration isolation platform to isolate the vibration out of the experimental system.

#### 3.2. Output Performance

_{x}is illustrated in Figure 8a. The input voltage V

_{x}goes up from V

_{x}= 0 V to V

_{x}= 150 V, and then falls down to V

_{x}= 0 V. The motion displacement l

_{x}of x-stage increases to l

_{x}= 17.65 μm in the case of input voltage V

_{x}= 150 V. This experiment is repeated three times, and the repeatability is found excellent. The maximum displacement error between the loading (up) and unloading (down) curves is e

_{x}= 0.36 μm, which is thought to be induced by the hysteresis and system errors. Compared with the maximum motion displacement, the maximum relative displacement error is e

_{xr}= 2.03%. Many researchers have paid attention to reduce the hysteresis of piezoelectric actuators by closed-loop control systems and compensation algorithms. However, these methods make the whole system complicated and very costly. The requirement of the resolution in 3D cellular bio-assembly systems is not so high, since the size of cells is almost around 10 μm. Therefore, the influence of hysteresis is not discussed deeply here.

_{y}. The input voltage V

_{y}also increases from V

_{y}= 0 V to V

_{y}= 150 V, and then decreases to V

_{y}= 0 V. The maximum motion displacement of y-stage is l

_{y}= 15.45 μm in the case that the input voltage V

_{y}= 150 V. This experiment is also repeated three times, and the maximum displacement error between the loading (up) and unloading (down) curves is e

_{y}= 0.56 μm, which is also thought to be induced by the hysteresis and system errors. Hence, the maximum relative displacement error is about e

_{yr}= 3.62%, which is a little larger than that in x-stage. The difference between the repeated three groups is thought to be caused by the assembly errors of the piezoelectric stacks and the system errors.

_{cy}and e

_{cx}are achieved from Figure 9. In the case that the input voltage V

_{x}is applied to the x piezo stack, the laser sensor is utilized to measure the displacement of y-stage in y direction. Hence, the displacement coupling error e

_{cy}is shown in Figure 9a. It can be seen that the displacement coupling error e

_{cy}increases with input voltage V

_{x}. The maximum displacement coupling error for y-stage is e

_{cy}= 0.76 μm, in the case that the input voltage V

_{x}= 138 V. The relationship between the input voltage V

_{x}and the displacement coupling error e

_{cy}is simplified by the following equation with a correlation coefficient of R

^{2}= 0.95:

_{cx}for x-stage is obtained in the case that input voltage V

_{y}is applied for y-piezo stack (Figure 9b). The maximum displacement error e

_{cx}for x-stage is e

_{cx}= 0.51 μm, in the case of input voltage V

_{y}= 150 V. The relationship between e

_{cx}and V

_{y}is nonlinear by the polynomial equation:

_{y}of y-stage keeps almost the same, in the case that stand weight increases from m = 0 g to m = 1000 g, which indicates that the proposed 2-DOF piezoelectric-driven platform works stably with high output force.

_{x}= 1.7 ms; the stable displacement is about l

_{x}= 12.84 μm; therefore, the response velocity is v

_{x}= 7.84 × 10

^{−3}m/s. The step response time for y-stage is around t

_{y}= 1.6 ms; the stable displacement is about l

_{y}= 10 μm; therefore, the response velocity is v

_{y}= 6.25 × 10

^{−3}m/s.

#### 3.3. Discussion

_{cx}and e

_{cy}. However, displacement sensors are needed for the closed-loop control system, which makes the whole system complicated and expensive. In order to get high positioning accuracy with open-loop control system, we have to analyze the motion displacement and the coupling error. For example, the real motion displacement L

_{y}of y-stage contains two parts: one is the displacement l

_{y}caused by input voltage V

_{y}, the other part is the coupling error e

_{cy}induced by input voltage V

_{x}. Therefore, the real motion displacement L

_{y}is achieved by the following:

_{y}and V

_{y}is obtained from Figure 8a.

_{y}: firstly, the input voltage V

_{x}should be known to calculate the error e

_{cy}; next, wanted l

_{y}is got by using wanted L

_{y}subtracts the error e

_{cy}; finally, the need V

_{y}is obtained by Equation (28). Figure 12 shows the needed input voltage V

_{y}under different input voltage V

_{x}. It is illustrated that, in order to get high accuracy, different V

_{y}should be given under different V

_{x}. The largest voltage error of the applied V

_{y}is e

_{vy}= 5.39 V, in the case that V

_{x}= 0 V and 100 V. By applying the needed V

_{y}from the Equation (19), the wanted real motion displacement L

_{y}is obtained by this open-loop method.

## 4. Conclusions

_{x}= 17.65 μm in the case of input voltage V

_{x}= 150 V; the maximum motion displacement of y-stage is l

_{y}= 15.45 μm in the case of input voltage V

_{y}= 150 V. The step response time for x-stage is about t

_{x}= 1.7 ms, while that for y-stage is t

_{y}= 1.6 ms. All the experimental data indicate that “Z-shaped” flexure hinges are quite suitable for the design of piezoelectric-driven platforms to obtain submicron accuracy and microsecond response time. This study is meaningful for the application of piezoelectric-driven platform in 3D cellular bio-assembly system.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Structure of three kinds of flexure hinges: (

**a**) right-angle; (

**b**) right-circle; (

**c**) “Z-shaped”.

**Figure 3.**FEM results of different flexure hinges: (

**a**) right-angle; (

**b**) right-circle; (

**c**) “Z-shaped”.

**Figure 4.**FEM static results of the platform: (

**a**) only “Z-shaped” flexure hinges; (

**b**) with right-angle flexure hinges.

**Figure 9.**Displacement coupling errors: (

**a**) coupling error for y-stage; (

**b**) coupling error for x-tage.

**Figure 12.**Relationship between the real motion displacement L

_{y}and the needed input voltage V

_{y}.

Method | Right-Angle | Right-Circular | Z-Shaped |
---|---|---|---|

MCM (Matrix-based compliance modeling) | 265.8 μm | 28.0 μm | 323.5 μm |

FEM (Fine element method) | 261.4 μm | 27.4 μm | 340.1 μm |

Error | 1.7% | 2.2% | 4.9% |

Modal Number | Frequency (Hz) | Resonance Direction |
---|---|---|

First | 678 | y |

Second | 965 | z |

Third | 1297 | θ_{x} |

Fourth | 1553 | x |

Fifth | 1832 | θ_{z} |

Sixth | 2002 | θ_{y} |

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## Share and Cite

**MDPI and ACS Style**

Li, J.; Liu, H.; Zhao, H.
A Compact 2-DOF Piezoelectric-Driven Platform Based on “Z-Shaped” Flexure Hinges. *Micromachines* **2017**, *8*, 245.
https://doi.org/10.3390/mi8080245

**AMA Style**

Li J, Liu H, Zhao H.
A Compact 2-DOF Piezoelectric-Driven Platform Based on “Z-Shaped” Flexure Hinges. *Micromachines*. 2017; 8(8):245.
https://doi.org/10.3390/mi8080245

**Chicago/Turabian Style**

Li, Jianping, Hui Liu, and Hongwei Zhao.
2017. "A Compact 2-DOF Piezoelectric-Driven Platform Based on “Z-Shaped” Flexure Hinges" *Micromachines* 8, no. 8: 245.
https://doi.org/10.3390/mi8080245