# Modeling and Optimization for a New Compliant 2-dof Stage for Locating Biomaterial Samples by an Efficient Approach of a Kinetostatic Analysis-Based Method and Neural Network Algorithm

Authors: Minh Phung Dang; Hieu Giang Le; Minh Nhut Van; Ngoc Le Chau; Thanh-Phong Dao
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

The nanoindentation technique is employed to characterize the behaviors of biomaterials. Nevertheless, there is a lack of development of a miniaturized precise positioner for in situ nanoindentation. Besides, modeling behaviors of the positioner are restricted due to its complex kinematic characteristics. Therefore, this paper proposes a novel compliant two degrees of freedom (dof) stage for positioning a biomaterial sample in in situ nanoindentation. In addition, a new modeling and dimensional optimization synthesis method of the stage is developed. The proposed effective methodology is developed based on a kinetostatic analysis-based calculation method, the Lagrange approach, and a neural network algorithm. With an increased advance in artificial intelligence, a neural network algorithm is proposed to extend the applicability of artificial neural networks in optimizing the parameters of the stage. First, the 2-dof stage is built via a combination of an eight-lever displacement amplifier and a symmetric parallelogram mechanism. Second, a chain of mathematical equations of the 2-dof stage is constructed using a kinetostatic analysis-based method to calculate the ratio of displacement amplification and the input stiffness of the 2-dof stage. Then, the Lagrange method is utilized to formulate the dynamic equation of the 2-dof stage. Finally, a neural network algorithm is adopted to maximize the natural first frequency of the proposed stage. The optimal results determined that the frequency of the stage can achieve a high value of 112.0995 Hz. Besides, the formed mathematical models were relatively precise by comprising the simulation verifications.


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## Body

## 1. Introduction

Advances in in situ nanoindentation testers have resulted in great interest in developing compact-size mechanical mechanisms with better performances. The goals of a compact-size mechanism are to obtain lightweight, low cost, and to able be integrated into nanoindentation devices in observing the online mechanical behaviors of a material sample [1]. Especially, in situ nanoindentation checking plays a vital role in testing for behaviors of biomaterials in implants (e.g., bone, teeth, femur, prosthetics, and so forth) [2, 3]. In a general in situ nanoindentation system, there are two main mechanisms/stages, including the one degree of freedom (dof) fine stage for bringing the indenter and the 2-dof fine stage for locating the material sample [4].With the aforementioned regards in mind, many researchers have made significant 2-dof mechanical stages in developing the in situ nanoindentation. Huang et al. [5] used the concept of flexure hinge-based compliant mechanisms to design a 1-dof stage for driving the indenter. Then, they continued to develop a new 1-dof stage with specifications of crystal silicon [6]. Moreover, Huang et al. [6] designed a 1-dof stage for characterizing behaviors of metallic glass. In these studies, they mainly employed the compliant mechanism in developing the 1-dof stage for utilizing the indenter driver. It is well known that compliant mechanisms offer nonassembled structure, zero friction, and simple manufacture which are advantages to overcome the limitations of rigid-counterparts in precise mechanical engineering. In the last decade, flexure hinge-based compliant mechanisms have been widely utilized for precise engineering and fine positioner with compact sizes such as a positioner [7, 8] and valve [6, 9].At present, many different compliant 2-dof stages have been developed for micro/nanomanipulations. Similar to these applications, a piezoelectric actuator (PZT) is often chosen due to its high precision and fast response. Nevertheless, one of the drawbacks of the PZTs is a limited output stroke [10]. This causes a limitation of providing a large input displacement for the 2-dof stages. As a result, a few common amplifiers were developed, such as lever, bridge, and hybrid lever-bridge types [11]. Considering the design of compliant 2-dof stages, many research groups have focused on developing the series-parallel kinematic chains in coupling with the PZTs to gain the high performances. For instance, Ling et al. [12] developed a 2-dof stage with good specifications. They used the pseudo-rigid-body model and the Lagrange method to analyze the kinetostatic and dynamic behaviors of the stage. Qu et al. [13] designed a 2-dof stage using the parallelogram scheme. They analyzed the kinematics, stiffness, and workspace of the 2-dof stage through the matrix method. More recently, Wang et al. [14] designed a 2-dof stage, and the compliance matrix was employed for characterizing the stiffness. Xiao et al. [15] developed a 2-dof stage, and the stiffness as well as kinematic models was formulated through the matrix displacement method. Although the previous 2-dof stages were well developed, they were almost tended for developing the micro/nanomanipulations. Based on the aforementioned studies, it can be seen that there is a lack of deep investigations in developing compliant 2-dof stages for use in locating biomaterials in the in situ nanoindentation tester industry.This paper aimed to develop a new compliant 2-dof stage which can be employed in locating biomaterials in in situ nanoindentation systems. The design details of the stage are developed based on the multiple lever amplification mechanism and series-parallel chain. The kinetostatics and dynamics are derived based on the deformable mechanics, the elastic beam theory, and the Lagrange approach.The main contributions of this paper are summarized as follows: (i) A new compliant 02-dof stage with great dynamic characteristics is designed for locating biospecimens in nanoindentation tester systems. (ii) The stiffness and amplification ratio are established. Then, the dynamic equation of the 2-dof stage is formulated. The correctness of the established mathematical models are verified through finite element simulations. (iii) Finally, the main parameters of the 2-dof stage are optimized using the neural network algorithm which benefits excellent behaviors of an artificial neural network.


## 2. Conceptual Design of a Compliant 2-dof Stage

### 2.1. Design Scheme of the 1-dof Mechanism

Figure1 describes the operating mechanism of a 1-dof stage which comprises an eight-lever displacement amplifier and a parallelogram mechanism. The amplifier is aimed to amplify the displacement, so-called stroke, of the proposed 1-dof stage. Meanwhile, the parallelogram mechanism is employed to generate good translation in the desirable motion and reduce the undesirable motions. It means that this design scheme ensures a good design with a large amplification ratio as well as reduces the parasitic motion error.Figure 1
Design scheme of the 1-dof symmetrical mechanism.


### 2.2. Operation Scheme of the 2-dof Stage

Based on the design and operation schemes of the 1-dof stage in Figure1, the operation principle and design scheme of a new 2-dof stage are provided in Figure 2(a). The design of the 2-dof stage is a module-based scheme. The module of the 1-dof stage is symmetrically arranged in the vertical and horizontal directions. It means that the 2-dof stage includes the four modules of the 1-dof stage. The positioning stage is employed to locate the biomaterials in an in situ nanoindentation system. As illustrated in Figure 2(a), the P-joint shows a leaf hinge, so-called as the prismatic joint, to generate a large deformation. Meanwhile, the remain positions are designed with the elliptical hinges to ensure good positions. The overall working operation of the 2-dof stage is mainly based on the deformations of the elliptical joints, right circular joints, and P-joints. The suggested stage is driven using a PZT actuator. The main dimensioning parameters of the proposed stage are illustrated in Figure 2(b). The values of the parameters are provided in Table 1.Figure 2
The proposed 2-dof stage: (a) design diagram and (b) main dimensioning parameters.
(a)(b)Table 1
Main geometric parameters of the proposed 2-dof stage.
FactorValueFactorValueUnita451k100mmb451m100mmc248p54mmd248A0.9 ≤ A ≤ 1.1mme20B0.7 ≤ B ≤ 0.9mmf248C0.6 ≤ C ≤ 0.8mmg248D0.6 ≤ D ≤ 0.8mmh54E0.6 ≤ E ≤ 0.7mmIn the proposed design, a double-lever amplification mechanism is widely used for many researches on flexure-based mechanisms [16]. Meanwhile, the symmetric eight-lever displacement amplifier integrated into the 2-dof stage for locating biospecimens has been investigated. Therefore, the proposed displacement amplifier is a new structure for positioning a biomaterial sample in in situ nanoindentation.Each component has an important task in overall nanoindentation. Specifically, the main task of the 2-dof stage is aimed to locate the fine position of the biospecimens. The only difference of the proposed mechanism design in nanoindentation/in situ nanoindentation in comparison with traditional mechanisms is a monolithic structure in order to reduce the mechanical component quantity as well as to reduce friction and wear of an assembly cluster. Therefore, inspired from the advantages of the compliant mechanisms, the authors will investigate to contribute for the new compliant structures.As illustrated clearly in Figure2(b) and Table 1, the parameters A, B, and C with different colors are the thicknesses of the elliptical hinges at lever floor 1, lever floor 2, and lever floors 3 and 4, respectively. In addition, D is the thickness of the right circular hinge at the end of lever floor 4, and E is the thickness of the leaf hinge of the parallel guiding mechanism. In addition, the technical specifications of the proposed stage are determined in Table 2.Table 2
The desirable technical specifications of the XY stage.
Technical specificationsDesirable resultsWorkspace size451 × 451 (mm)Desirable working stroke770 × 770 (μm)Desirable resonant frequency100 (Hz)Desirable displacement amplification ratio7Parasitic motion error/cross-axis coupling ratio0.04 (%)Safety factor1.7Range of the maximum load3000 N


## 2.1. Design Scheme of the 1-dof Mechanism

Figure1 describes the operating mechanism of a 1-dof stage which comprises an eight-lever displacement amplifier and a parallelogram mechanism. The amplifier is aimed to amplify the displacement, so-called stroke, of the proposed 1-dof stage. Meanwhile, the parallelogram mechanism is employed to generate good translation in the desirable motion and reduce the undesirable motions. It means that this design scheme ensures a good design with a large amplification ratio as well as reduces the parasitic motion error.Figure 1
Design scheme of the 1-dof symmetrical mechanism.


## 2.2. Operation Scheme of the 2-dof Stage

Based on the design and operation schemes of the 1-dof stage in Figure1, the operation principle and design scheme of a new 2-dof stage are provided in Figure 2(a). The design of the 2-dof stage is a module-based scheme. The module of the 1-dof stage is symmetrically arranged in the vertical and horizontal directions. It means that the 2-dof stage includes the four modules of the 1-dof stage. The positioning stage is employed to locate the biomaterials in an in situ nanoindentation system. As illustrated in Figure 2(a), the P-joint shows a leaf hinge, so-called as the prismatic joint, to generate a large deformation. Meanwhile, the remain positions are designed with the elliptical hinges to ensure good positions. The overall working operation of the 2-dof stage is mainly based on the deformations of the elliptical joints, right circular joints, and P-joints. The suggested stage is driven using a PZT actuator. The main dimensioning parameters of the proposed stage are illustrated in Figure 2(b). The values of the parameters are provided in Table 1.Figure 2
The proposed 2-dof stage: (a) design diagram and (b) main dimensioning parameters.
(a)(b)Table 1
Main geometric parameters of the proposed 2-dof stage.
FactorValueFactorValueUnita451k100mmb451m100mmc248p54mmd248A0.9 ≤ A ≤ 1.1mme20B0.7 ≤ B ≤ 0.9mmf248C0.6 ≤ C ≤ 0.8mmg248D0.6 ≤ D ≤ 0.8mmh54E0.6 ≤ E ≤ 0.7mmIn the proposed design, a double-lever amplification mechanism is widely used for many researches on flexure-based mechanisms [16]. Meanwhile, the symmetric eight-lever displacement amplifier integrated into the 2-dof stage for locating biospecimens has been investigated. Therefore, the proposed displacement amplifier is a new structure for positioning a biomaterial sample in in situ nanoindentation.Each component has an important task in overall nanoindentation. Specifically, the main task of the 2-dof stage is aimed to locate the fine position of the biospecimens. The only difference of the proposed mechanism design in nanoindentation/in situ nanoindentation in comparison with traditional mechanisms is a monolithic structure in order to reduce the mechanical component quantity as well as to reduce friction and wear of an assembly cluster. Therefore, inspired from the advantages of the compliant mechanisms, the authors will investigate to contribute for the new compliant structures.As illustrated clearly in Figure2(b) and Table 1, the parameters A, B, and C with different colors are the thicknesses of the elliptical hinges at lever floor 1, lever floor 2, and lever floors 3 and 4, respectively. In addition, D is the thickness of the right circular hinge at the end of lever floor 4, and E is the thickness of the leaf hinge of the parallel guiding mechanism. In addition, the technical specifications of the proposed stage are determined in Table 2.Table 2
The desirable technical specifications of the XY stage.
Technical specificationsDesirable resultsWorkspace size451 × 451 (mm)Desirable working stroke770 × 770 (μm)Desirable resonant frequency100 (Hz)Desirable displacement amplification ratio7Parasitic motion error/cross-axis coupling ratio0.04 (%)Safety factor1.7Range of the maximum load3000 N


## 3. Proposed Methodology

In this part, a modeling and dimensional optimization synthesis method is developed for the compliant 2-dof stage. The proposed methodology is formed based on the analytical methods and the metaheuristic algorithm. In this work, the deformable mechanics theory is coupled with the elastic beam theory and Lagrange method in establishing the kinetostatic and dymamic equations of the 2-dof stage.


### 3.1. Modeling and Dimensional Optimization Synthesis

The flowchart of the proposed methodology for modeling and dimensional optimization synthesis for the developed stage is shown in Figure3. The main steps of the present methodology are briefly performed as follows:(i)
Step 1: predetermine a conceptual design of the 2-dof stage, i.e., a built kinematic chain diagram.(ii)
Step 2: predetermine the specification’s requirements for the 2-dof stage.(iii)
Step 3: establish the kinetostatic and dynamic mathematical equations for the 2-dof stage by using the the deformable mechanics theory, the elastic beam theory, and the Lagrange approach.(iv)
Step 4: verify the corrections of mathematical equations through finite element analysis (FEA).(v)
Step 5: if the kinematic and dynamic equations are corrected, the calculation process moves to the further steps. Otherwise, the process goes back the step 1.(vi)
Step 6: define the design variables, objective, and constraint functions for the 2-dof stage.(vii)
Step 7: in order to optimize the parameters of the 2-dof stage, the neural network algorithm is employed in improving the dynamic performance of the stage.(viii)
Step 8: the optimal results are verified via FEA analysis. If it satisfied, a protoype of the stage will be made.(ix)
Step 9: the dynamic performance of the 2-dof stage is compared with that from the previous stages.Figure 3
Flowchart of the proposed methodology for modeling and optimization synthesis of the 2-dof stage.


### 3.2. Neural Network Algorithm

Inspired from the artificial neural network (ANN), the neural network algorithm (NNA) was developed [17]. This NNA optimizer is also considered as other metaheuristic optimizers based on the population differential evolution algorithm [18], particle swarm optimization [19], nondominated sorting genetic algorithm [20], and so on. This algorithm is aimed to decrease the error among the targets and the predicted values, and achieve a global optimally best result. The flowchart of the NNA is given in Figure 4.Figure 4
Schematic diagram of the neural network algorithm.A pattern solution (Ps) is determined as(1)Ps=x1,x2,…,xd,d:problem size.A matrix of pattern solution (X) is defined by(2)X=x11x21xD1⋮⋱⋮x1npopx2npopxDnpop,npop is the population number.Theith cost of the function is defined as follows:(3)Ci=fx1i,x2i,…,xDi.A weight matrix is determined by(4)Wt=W11W2i⋯WDnpop⋮⋱⋮Wnpop1Wnpopi⋯Wnpopnpop=W11⋯Wi1⋯Wnpop1⋮⋱⋮W1npopWinpop⋯Wnpopnpop.The weight value is determined as(5)∑j=1npopwijt=1,i∈npop,(6)wij∈U0,1,i,j∈npop,A new matrix (X) is calculated by(7)Xjnewt+1=∑t=1npopwijt×Xtt,j∈npop,(8)Xit+1=Xit+Xinewt+1,i∈npop.The weight matrix is updated by following formula:(9)Wtupdatet+1=Wit+2×rand×Wtargett−Wit,i∈nppop,A bias means that the exploration influences a new solution and weighting. Consideringrand > ψ (ψ is solutions percentage), the function operator is computed by(10)Xt∗t+1=Xit+1+2×rand×Xtargett−Xit+1,i∈nppop,In this work, an initial population of 20, max iterations of 500, andψ of 1 are utilized for the NNA.


## 3.1. Modeling and Dimensional Optimization Synthesis

The flowchart of the proposed methodology for modeling and dimensional optimization synthesis for the developed stage is shown in Figure3. The main steps of the present methodology are briefly performed as follows:(i)
Step 1: predetermine a conceptual design of the 2-dof stage, i.e., a built kinematic chain diagram.(ii)
Step 2: predetermine the specification’s requirements for the 2-dof stage.(iii)
Step 3: establish the kinetostatic and dynamic mathematical equations for the 2-dof stage by using the the deformable mechanics theory, the elastic beam theory, and the Lagrange approach.(iv)
Step 4: verify the corrections of mathematical equations through finite element analysis (FEA).(v)
Step 5: if the kinematic and dynamic equations are corrected, the calculation process moves to the further steps. Otherwise, the process goes back the step 1.(vi)
Step 6: define the design variables, objective, and constraint functions for the 2-dof stage.(vii)
Step 7: in order to optimize the parameters of the 2-dof stage, the neural network algorithm is employed in improving the dynamic performance of the stage.(viii)
Step 8: the optimal results are verified via FEA analysis. If it satisfied, a protoype of the stage will be made.(ix)
Step 9: the dynamic performance of the 2-dof stage is compared with that from the previous stages.Figure 3
Flowchart of the proposed methodology for modeling and optimization synthesis of the 2-dof stage.


## 3.2. Neural Network Algorithm

Inspired from the artificial neural network (ANN), the neural network algorithm (NNA) was developed [17]. This NNA optimizer is also considered as other metaheuristic optimizers based on the population differential evolution algorithm [18], particle swarm optimization [19], nondominated sorting genetic algorithm [20], and so on. This algorithm is aimed to decrease the error among the targets and the predicted values, and achieve a global optimally best result. The flowchart of the NNA is given in Figure 4.Figure 4
Schematic diagram of the neural network algorithm.A pattern solution (Ps) is determined as(1)Ps=x1,x2,…,xd,d:problem size.A matrix of pattern solution (X) is defined by(2)X=x11x21xD1⋮⋱⋮x1npopx2npopxDnpop,npop is the population number.Theith cost of the function is defined as follows:(3)Ci=fx1i,x2i,…,xDi.A weight matrix is determined by(4)Wt=W11W2i⋯WDnpop⋮⋱⋮Wnpop1Wnpopi⋯Wnpopnpop=W11⋯Wi1⋯Wnpop1⋮⋱⋮W1npopWinpop⋯Wnpopnpop.The weight value is determined as(5)∑j=1npopwijt=1,i∈npop,(6)wij∈U0,1,i,j∈npop,A new matrix (X) is calculated by(7)Xjnewt+1=∑t=1npopwijt×Xtt,j∈npop,(8)Xit+1=Xit+Xinewt+1,i∈npop.The weight matrix is updated by following formula:(9)Wtupdatet+1=Wit+2×rand×Wtargett−Wit,i∈nppop,A bias means that the exploration influences a new solution and weighting. Consideringrand > ψ (ψ is solutions percentage), the function operator is computed by(10)Xt∗t+1=Xit+1+2×rand×Xtargett−Xit+1,i∈nppop,In this work, an initial population of 20, max iterations of 500, andψ of 1 are utilized for the NNA.


## 4. Results and Discussion

### 4.1. Modeling of Kinetostatics and Dynamics

The kinetostatics and dynamics of the 2-dof stage are performed to evaluate its behaviors. The deformable mechanics theory, the elastic beam theory, and the Lagrange method are utilized for these analyses. In this work, the specifications of the 2-dof stage are analyzed (i.e., ratio of displacement amplification, stiffness, stroke, and resonant frequency).


#### 4.1.1. Modeling of Amplification Ratio and Input Stiffness

Figure5 demonstrates a variant of the modified displacement lever amplifier (MDLA), a so-called variant of a lever amplifier. It includes the main parameters of the lever lengths of the different amplification floors. Because of a symmetrical structure, a half of the displacement amplifier is chosen to analyze the quality characteristics of the proposed 2-dof stage.Figure 5
A variant of the symmetrical displacement lever amplifier.A schematic diagram of the MDLA is provided in Figure6. It consists of main levers, including the lever amplification mechanism #1 (LAM1), lever amplification mechanism #2 (LAM2), and lever amplification mechanism #3 (LAM3). Besides, the parameters of the lengths of levers (l1, l2, l1′, l2′, l3, l4, l5, and l6) are also in Figure 6 in which O1, O2, O3, and O4 are the rotation centers of the levers.Figure 6
Schematic diagram of the modified displacement lever amplifier.As shown in Figure6, the MDLA is operated based on three levers (LAM1, LAM2, and LAM3). It can be determined that the output displacement of the amplifier can be derived as follows [21]:(11)δout=l5l6l2l1+l4l3+1+l4l3+1δin,where δin and δout symbolize the input displacement and the output displacement, respectively. Next, the amplification ratio will be computed as follows:(12)A=l5l6l2l1+l4l3+1+l4l3+1,where A is the displacement amplification ratio.Three kinds of flexure joints are used at appropriate places in the original design, thus helping the design to achieve a large displacement and reduce undesirable motions of the stage. Figure7 demonstrates the main associated dimensions of a flexure circular notched hinge, and it shows the effect of force and torque that recovers when the mechanism operates.Figure 7
Diagram of a circular arc flexure hinge.The mechanism’s input stiffness is investigated. The general spring with bending stiffnessKδxFxc is generated via the lateral force, and the torsion stiffness KθzMzc is made via the torque. The linear stiffness KθyMyc is created via the axial force for the flexure right circular hinge. Furthermore, the leaf beam's torsion and linear stiffness are denoted by KθzMzl and KθyMyl, respectively. Correspondingly, the flexure elliptical notched hinge is behaved with bending stiffness KδXFXe produced via the lateral force, torsion stiffness KeθzMz produced via the torque, and linear stiffness KθyMye generated via the axial force [22].(13)KθzMzc=2Ebt2.59πr0.5,(14)KθyMyc=Ebt0.5πr0.5−0.5t0.5,(15)KδxFxc=2Ebt2.53πr1.53r+t,Next, Figure8 describes the simplified structure diagram of the flexure leaf hinge. In addition, the main parameters such as the length, width, and thickness are also displayed along with the direction of force and torque acting on the leaf hinge.Figure 8
Diagram of a leaf-type arc flexure hinge.Stiffness values of the leaf hinge are calculated with the following equations:(16)KθzMzl=Eba312l,(17)KθyMyl=Ebal,(18)MθzMzl=KθzMzlΔθ.Figure9 shows the dimensions of the flexure elliptical hinge as well as the directions of the force and moment acting on the elliptical hinge.Figure 9
Diagram of an elliptical arc flexure hinge.Next, the following equations describe how to calculate the stiffness of the flexure elliptical hinge that bends in different directions.(19)KeθzMz=2EbaX23∈fβy,(20)KθyMye=Eb312∈gβy,(21)KδXFXe=Eb∈gβy,(22)βX=t2aX,(23)βy=t2ay,(24)∈=aXay=βyβx,(25)fβy=f∈βx=12βy+βy23+4βy+2βy21+βy2βy+βy2+61+βy2βy+βy23tan−12+βyβy,(26)gβy=g∈βx=21+βy2βy+βy2tan−12+βyβy−π2,where b denotes the thickness of the flexure hinge, E denotes the elastic modulus of the manufacturing material, r is the radius of a flexure circular notched hinge, and t denotes the smallest width of the circular/elliptical hinge. The width and length of the leaf beam are represented by a and l, respectively. The dimensionless factor representing the elliptical hinge geometry ax is denoted by βX, and (ay) is denoted by βy. The symbol ∈ is the multiplication factor of the ratio of major to minor axes. On the other hand, the symbols fβx and fβy are the dimensionless compliance factors according to βX and βy, respectively.By subjecting an input displacement∈ at the stage’s input end, the corresponding input force Fin is determined via the following equation:(27)Fin=Kinδin,where Kin is the input stiffness of the micromanipulator.Figure10 describes the initial state and working state of the MDLA. The deformation angles formed from the position change of the multistage lever help to show the different degrees of deformation of the structure while it is working.Figure 10
Force and deformation diagram of a half of the MDLA.Figures11–14 present the force analysis diagram of beam 4, beam 3, beam 2, and beam 1, respectively, in the lever amplification mechanism. The diagrams illustrate the details of the applied force, moment, rotation angle, as well as the deformation of the elastic joint.Figure 11
Force analysis diagram of beam 4.Figure 12
Force analysis diagram of beam 3.Figure 13
Force analysis diagram of beam 2.Figure 14
Force analysis diagram of beam 1.Only half of the displacement amplification structure is investigated due to the symmetrical structure. A main force analysis scheme of the mechanical amplification mechanism is depicted in Figure10. As a result, the following equations are formed by using the mechanic equilibrium state of beam 4, as asserted in Figure 11.The following force equilibrium equations can be derived as(28)FO3y=FEy+FGy,(29)FEyl6=FGyl5+MO3t+MGt+MEt.Specifically, the bending moments of the points O3, G, and E are represented as MO3t,MGt,, and MEt, respectively. Three forces FO3y, FEy, and FGy are acting at points O3, E, and G in the direction of y-axis when the structure operates, respectively. The displacement value at point G is a combination of two displacements including displacement due to hinges at O3 rotating around its axis (l5θ4) and displacement due to drift (δ4). Furthermore, by multiplying stiffness by displacement, the forces FO3y and FGy are determined. They are calculated via the following equations:(30)FO3y=KO3xδ4,(31)FGy=KGyl5θ4−δ4,(32)MO3t=KO3tθ4,(33)MGt=KGtθ4,(34)MEt=KEtθ4,where KO3x is the lateral bending stiffness by the force FO3y, and KGl is the G point’s output stiffness of the MDLA. In addition, the rotational stiffnesses KO3t, KGt, and KEt are the stiffnesses of the flexure hinges at points O3, G, and E, respectively.Substituting (30) and (31) into (28), the results are yielded as(35)KO3xδ4=FEy+KGyl5θ4−δ4,(36)δ4=FEy+KGyl5θ4KO3x+KGy.Substituting (31)–(34) into (29), the results are achieved as(37a)FEyl6=KGyl5l5θ4−δ4+KO3tθ4+KGtθ4+KEtθ4,(37b)FEyl6+KGyl5δ4=KGyl52+KO3t+KGt+KEtθ4,where FEy, FO3y, and FGy are the forces generated at points of E, O3, and G, respectively, when the structure operates.Substituting (36) into (37b), the θ4 and δ4 can be derived as(38)FEyl6+KGyl5FEy+KGyl5θ4KO3x+KGy=KGyl52+KO3t+KGt+KEtθ4,(39)FEyl6KO3x+KGy+KGyl5FEy+KGyl5θ4=KO3x+KGyKGyl52+KO3t+KGt+KEtθ4,(40)θ4=l6KO3x+KGy+KGyl5KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEy,(41a)δ4=FEy+KGyl5l6KO3x+KGy+KGyl5/KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEyKO3x+KGy,(41b)δ4=KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52+KGyl5l6KO3x+KGy+KGyl5KO3x+KGyKO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEy,(41c)δ4=KGyl52+KO3t+KGt+KEt+KGyl5l6KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEy.Regarding beam 4, the ratio of the amplifier and the input stiffness are determined by(42)λ4=l5θ4−δ4l6θ4+δ4,(43)Kin4=FEyl6θ4+δ4.Substituting (40) and (40c) into (42) and (43), respectively, the following formulas can be obtained:(44a)λ4=l5l6KO3x+KGy+KGyl5−KGyl52+KO3t+KGt+KEt+KGyl5l6l6l6KO3x+KGy+KGyl5+KGyl52+KO3t+KGt+KEt+KGyl5l6,(44b)λ4=l5l6KO3x+KGy−KO3t+KGt+KEt+KGyl5l6l6KO3x+KGy+2KGyl5l6+KGyl52+KO3t+KGt+KEt,(44c)λ4=l5l6KO3x−KO3t−KGt−KEtKGyl5+l62+KO3xl62+KO3t+KGt+KEt,(45)Kin4=KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52KGyl5+l62+KO3xl62+KO3t+KGt+KEt.Figure12 presents the force analysis diagram of beam 3.Considering the equilibrium state of beam 3, the force and moment are computed as(46)FHy=FIy+FO4y,(47)FHyl3=FIyl3+l4+MIt+MHt+MO4t,where MO4t, MIt,, and MHt are moments of bending of the points O4, I, and H. Moreover, FO4y, FHy, and FIy are forces which are acting at point O4, H, and I in the direction of y-axis, respectively.The displacement value at point I is a combination of two displacements including displacement due to hinges at O4 rotating around its axis ((l3 + l4) θ3) and displacement due to drift (δ3).Furthermore, the forcesFO4y and FIy can be calculated as(48)FO4y=KO4xδ3,(49)FIy=K3l3+l4θ3+δ3,(50)MO4t=KO4tθ3,(51)MIt=KItθ3,(52)MHt=KHtθ3,where KO4x is the lateral bending stiffness by force FO4y, and K3 is the I point’s output stiffness. KO4t is the O4 point’s rotational stiffness, and KIt, and KHt are the rotational stiffness of I and H, respectively.Substituting (48) and (49) into (46), the results are obtained as(53)FHy=K3l3+l4θ3+δ3+KO4xδ3,(54)δ3=FHy−K3l3+l4θ3KO4x+K3.Substituting (49)–(52) into (47), the results are yielded as(55a)FHyl3=K3l3+l4l3+l4θ3+δ3+KO4tθ3+KItθ3+KHtθ3,(55b)FHyl3=K3l3+l4δ3+K3l3+l42+KO4t+KIt+KHtθ3.Substituting (54) into (55b) θ3 and δ3 can be derived as(56a)FHyl3=K3l3+l4FHy−K3l3+l4θ3KO4x+K3+K3l3+l42+KO4t+KIt+KHtθ3,(56b)FHyl3KO4x+K3−K3l3+l4FHy=KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42θ3,(57)θ3=l3KO4x+K3−K3l3+l4KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42FHy,(58a)δ3=FHy−K3l3+l4l3KO4x+K3−K3l3+l4/KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42FHyKO4x+K3,(58b)δ3=K3l3+l42−K3l3+KO4t+KIt+KHtKO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42FHy.Considering beam 3, the ratio of the amplifier and the input stiffness are expressed by(59)λ3=l3+l4θ3+δ3l3θ3+δ3,(60)Kin3=FHyl3θ3+δ3.Substituting (57) and (58b) into (59) and (60), respectively, the following formulas can be obtained:(61a)λ3=l3+l4l3KO4x+K3−K3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHtl3l3KO4x+K3−K3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(61b)λ3=l3KO4x+K3l3+l4−K3l3+l42+K3l3+l42−K3l3+KO4t+KIt+KHtl32KO4x+K3−K3l3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(61c)λ3=l3KO4x+K3l3+l4−K3l3+KO4t+KIt+KHtl32KO4x+K3−K3l3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(62)Kin3=KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42l32KO4x+K3−K3l3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(63)K3=KO3yKin4KO3y+Kin4.Figure13 shows the force analysis diagram of beam 2.In a similar calculation for beam 2, the forces and moments are computed by(64)FO2y=FCy+FJy,(65)FJyl1′=FCyl2′+MO2t+MJt+MCt,where MO2t, MJt, and MCt are bending moments at the O2, J, and C points, respectively. The forces acting at points O2, C, and J in the direction of y-axis are FO2y, FCy, and FJy, respectively.The displacement value at point C is a combination of two displacements including displacement due to hinges at O2 rotating around its axis (l’2θ2) and displacement due to drift (δ2).Moreover, the forcesFO2y and FCy are expressed as(66)FO2y=KO2xδ2,(67)FCy=K2l2′θ2−δ2,(68)MO2t=KO2tθ2,(69)MCt=KCtθ2,(70)MJt=KJtθ2,where KO2x is the lateral bending stiffness by force FO2y, and K2 is the output stiffness of the E point. KO2t is the rotational stiffness of the O2 point. KJt and KCt are the rotational stiffnesses of J, and C points, respectively. Substituting (66) and (67) into (64),(71)KO2xδ2=FJy+K2l2′θ2−δ2,(72)δ2=FJy+K2l2′θ2KO2x+K2.Substituting (67)–(70) into (65), the following equations can be derived:(73a)FJyl1′=K2l2′l2′θ2−δ2+KO2tθ2+KCtθ2+KJtθ2,(73b)FJyl1′+K2l2′δ2=K2l2′2+KO2t+KCt+KJtθ2.Substituting (72) into (73b), the value of δ2 can be derived as(74a)FJyl1′+K2l2′FJy+K2l2′θ2KO2x+K2=K2l2′2+KO2t+KCt+KJtθ2,(74b)FJyl1′KO2x+K2+K2l2′FJy+K2l2′θ2=KO2x+K2K2l2′2+KO2t+KCt+KJtθ2,(75)θ2=l1′KO2x+K2+K2l2′KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy,(76a)δ2=FJy+K2l2′l1′KO2x+K2+K2l2′/KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJyKO2x+K2,(76b)δ2=KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2+K2l2′l1′KO2x+K2+K2l2′KO2x+K2KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy,(76c)δ2=KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2+K22l2′2+K2l2′l1′KO2x+K2KO2x+K2KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy,(76d)δ2=K2l2′2+KO2t+KCt+KJt+K2l2′l1′KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy.Similarly, the ratio of the amplifier and the input stiffness of beam 2 are determined by(77)λ2=l2′θ2−δ2l1′θ2+δ2,(78)Kin2=FJyl1′θ2+δ2.Substituting (75) and (76d) into (77) and (78), respectively, the following formulas can be obtained:(79a)λ2=l2′l1′KO2x+K2+K2l2′−K2l2′2+KO2t+KCt+KJt+K2l2′l1′l1′l1′KO2x+K2+K2l2′+K2l2′2+KO2t+KCt+KJt+K2l2′l1′,(79b)λ2=l1′l2′KO2x+K2−KO2t+KCt+KJt+K2l2′l1′l1′2KO2x+K2+2K2l2′l1′+K2l2′2+KO2t+KCt+KJt,(79c)λ2=l1′l2′KO2x−KO2t+KCt+KJtl1′2KO2x+K2+2K2l2′l1′+K2l2′2+KO2t+KCt+KJt,(80)Kin2=KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2l1′2KO2x+K2+2K2l2′l1′+K2l2′2+KO2t+KCt+KJt,(81)K2=KEyKin4KEy+Kin4.Figure14 presents the force analysis diagram of beam 1.By a similar consideration to the equilibrium state of the beam 1, the following equations are achieved:(82)FO1y=FAy+FBy,(83)FAyl1=FByl2+MO1t+MBt+MAt,where MO1t is the bending moments of the O1 point, while MAt and MBt are the bending moments of the A and B points, respectively. The forces acting at points O1, A, and B in the direction of y-axis are FO1y, FAy, and FBy, respectively.The displacement value at point B is a combination of two displacements including displacement due to hinges at O1 rotating around its axis (l2θ1) and displacement due to drift (δ1).Besides, the forcesFO1y and FBy are expressed by(84)FBy=K1l2θ1−δ1,(85)FO1y=KO1xδ1,(86)MO1t=KO1tθ1,(87)MBt=KBtθ1,(88)MAt=KAtθ1,where KO1x is the lateral bending stiffness by force FO1y, and K1 is the output stiffness of the B and A points, respectively. KO1t is the rotational stiffness of the O1 point, while KAt and KBt are the rotational stiffness of A and B points, respectively.Similarly, as beam 2, theθ1 and δ1 can be derived as(89)θ1=l1KO1x+K1+K1l2KO1x+K1K1l22+KO1t+KBt+KAt−K12l22FAy,(90)δ1=K1l1l2+K1l22+KO1t+KBt+KAtKO1x+K1K1l22+KO1t+KBt+KAt−K12l22FAy.Regarding beam 1, the ratio of the amplifier and the input stiffness are expressed by(91)λ1=l2θ1−δ1l1θ1+δ1,(92)kin1=FAyl1θ1+δ1.Substituting (89) and (90) into (91) and (92), respectively, the results are yielded ass(93)λ1=l1l2KO1x−KO1t+KBt+KAtl12KO1x+K1+2K1l1l2+K1l22+KO1t+KBt+KAt,(94)Kin1=KO1x+K1K1l22+KO1t+KBt+KAt−K12l22l12KO1x+K1+2K1l1l2+K1l22+KO1t+KBt+KAt,(95)K1=K12+K13,(96)K12=Ky12Kin2Ky12+Kin2,(97)Ky12=KθyMye2+2KθyMye5=9KθyMye10,(98)K13=Ky13Kin3Ky13+Kin3,(99)Ky13=2KθyMye5,where K1 is the output stiffness that combines A and B points, K12 is the output stiffness that combines beam 1 and beam 2, and K13 is the output stiffness that combines beam 1 and beam 3. In addition, the intermediary stiffness between beam 1 and beam 2, and the intermediary stiffness between beam 1 and beam 3 are Ky12 and Ky13, respectively.(100)KO1t=KO2t=KO4t=KAt=KθzMze,(101)KBt=KCt=KDt=KIt=KO3t=KEt=KθzMze2,(102)KHt=KJt=KMt=KNt=2KθzMze5,(103)KGt=KθzMzc,(104)KO3y=KEy=KCy=KθyMye2,(105)KO1x=KO2x=KO4x=KδxFxe,(106)KO3x=KδxFxe2,(107)Kb2=KθzMzl7=Eba312l7,(108)KθyMyl7=Ebal7,(109)Kb1=KθzMzl8=Eba312l8,(110)KθzMzl9=Eba312l9,(111)Kb3=2KθzMzl9KθzMzcKθzMzc+4KθzMzl9,where KθzMzl7, KθzMzl8, and KθzMzl9 are the stiffnesses which are created via the torque of the leaf hinges l7, l8, and l9, respectively. In addition, Kb1, Kb2, and Kb3 are stiffness conversion of KθzMzl7, KθzMzl8, and KθzMzl9, the purpose of this action is making the equations simpler. Finally, the linear stiffness is created via the axial force of the leaf hinge l7, also display by value KθyMyl7.Finally, the total ratio of two-stage lever amplifier is computed by(112)A=λ4λ1+λ3+1+λ3+1.It is found thatFin = 2FAy, and considering the symmetric amplifier, the input stiffness is expressed by(113)Kin=Finl1θ1+δ1=2FAyl1θ1+δ1=2Kin1.A simplified stiffness diagram of the stage is given in Figure15 when the platform is driven in the direction of y-axis. Figure 16 shows a simplified principle diagram of the 2-dof stage.Figure 15
The stiffness of the output mechanism.Figure 16
Simplified principle diagram of the 2-dof stage.Letdout be the output displacement of the stage and din be the input displacement in the vertical direction. Besides, the rotational angles of the prismatic beams are symbolled as ωb1, ωb2, ωb3, α1, β1, γ1, and ε1, which can be calculated by(114)ωb1=doutl8,(115)ωb2=doutl7,(116)ωb3=doutl9,(117)α1=doutl5,(118)β1=doutl3+l4l6l5,(119)γ1=doutl2,l6l5,(120)ε1=doutl2l6l5.The potential energy inside the flexure hinges is expressed by(121a)Ep=12∑i=ABKiε21+∑i=JDKiγ21+∑i=O4IKiβ21+KGα21+12∑j=b1b3Kjω2j,(121b)Ep=12KAtε12+KO1tε12+KBtε12+KJtγ12+KO2tγ12+KDtγ12+KO4tβ12+KHtβ12+KItβ12+KGtα122+4Kb1ωb12+6Kb2ωb22+2Kb3ωb32.Substituting (114)–(120) into (121b), the stiffness KN can be achieved by(122)KN=2l6l2l52KAt+KO1t+KBt+l6l′2l52KJt+KO2t+KDt+l6l4+l3l52KO4t+KHt+KIt+1l52KGt+41l82Kb1+61l72Kb2+21l92Kb3.Next, the output stiffnessKout and KGl of the stage are determined by(123)Kout=KNKaKN+Ka,in which Ka as shown in Figure 15 and calculated by(124)Ka=3KθyMyl72,(125)KGl=0.5Kout,where KN is the combines stiffness of the micromanipulator in one direction, Ka is the stiffness of the beam KθyMyl7, and Kout is the output stiffness of the micromanipulator.For a simple calculation, the geometric parameters and of the AL7075-T6 properties of the stage are provided in Table3.Table 3
Geometric parameters and AL7075-T6 for the proposed stage.
l1l2l’1l’2l3l4l5l6l7l8l913.949.71246145261.8187376.965.9rt (circular)a (l8, l9)a (l7)axayt (elliptical)blE (GPa)σr (MPa)4.50.60.60.84.52.50.6516971.7503


#### 4.1.2. Modeling of Working Workspace

Assuming that the A is the ratio of amplifier andS is the stroke of the PZT, the workspace of the stage is calculated as AS x AS. I axial tension and shearing effects are ignored. There is only the bending stress is considered. To sum up, the stress (σr) is computed by(126)Maxσr=σys,where σy is the yield strength of the material and s is the safety factor.The maximal stressσrMax is computed as(127)σrMax=4Er2Kcfβt2θMax,where kc is concentration factor and fβ is the compliance factor [22].(128)β=t2r,(129)kc=1+β920,(130)fβ=12β+β23+4β+2β21+β2β+β2+61+β2β+β23tan−12+ββ,where β is the dimensionless geometry factor.LetδinMax is the maximal input displacement and δinMax is the output displacement of stage. The maximal deformation angle (θmax) is computed by(131)θMax=AδinMaxl5+λ3δinMaxl3+l4.Substituting (126) and (131) into (127), the results are yielded as(132)σys≥4Er2Kcfβt2AδinMaxl5+λ3δinMaxl3+l4,(133)δinMax≤l5l3+l4fβt2σy4Er2KcsAl3+l4+λ3l5.By using the values in Table2, the (133) and let s of 1.8, the maximal input displacement is expressed by(134)δinMax≤44.6286μm,To sum up, the output displacement of the stage may achieve up to 1363.8μm. Therefore, the workspace of the 2-dof stage is 1363.8 μm × 1363.8 μm.


#### 4.1.3. Modeling of Dynamics

The natural frequencies can be achieved by Lagrange’s method by calculating the kinetic energy (T) and potential energy (V). Figures 12 and 17 show the position of masses m in the proposed design. Based on the position and motion trajectory of the components, a motion classification step will be performed to assist the calculation of the natural frequency of the design.Figure 17
Symmetrical structure with two 2-stage amplifiers.The hingesm2, m3, m5, m6, m8, m9, m12, m13, mb1, mb2, mb3 represent the rotational and translational motions. The hinges m0, m1, m4, m7, m10, m11, m14, m15, m16 note the translations. The kinetic energy of the whole stage is computed as(135)T=Tη1+Tη2,(136)Tη1=Tη2=∑i=m2m16Ti+∑j=mb1mb3Tj,(137)Tm0=12m0vA2=12m0η1.A2=m0A22η1.2,(138)Tm1=m12η1.2,(139)Tm4=12m4vλ1222=m4λ124η1.2,(140)Tm7=m7λ224η1.2,(141)Tm10=m104η1.2,(142)Tm11=m114η1.2,(143)Tm15=m15A22η1.2,(144)Tm16=m16A22η1.2,(145)Tm2=12m2v222+12m2l123vl122=7m212v2=7m212η1.2,(146)Tm3=7m3λ1212η1.2,(147)Tm5=7m5λ2212η1.2,(148)Tm6=7m612η1.2,(149)Tm8=7m8A212η1.2,(150)Tm9=7m9A212η1.2,(151)Tm12=7m1212η1.2,(152)Tm13=7m13λ3212η1.2,(153)Tmb1=12mb1vA24+12mb1l823vAl824=8mb1A23v2=8mb1A23η1.2,(154)Tmb2=12mb2vA26+12mb2l723vAl726=4A2mb2η1.2,(155)Tmb3=12mb3vA22+12mb3l923vAl922=4A2mb33η1.2.Potential energy:(156)V=12k1η12+12k2η22.Lagrange’s equation: Lagrangian mechanics defines a mechanical system to be a pair (M, L) of a configuration space M and a smooth function L = L(q, v,t) called Lagrangian.(157)L=T−V,ddt∂T∂μi∗−∂T∂μi+∂V∂μi=Fi,Mdi∗∗+Kdi=Fi,where i = 1; 2 is the free vibration of horizontal and vertical directions of the stage. Fi is a nonconservative generalized force corresponding to the coordinate. Let the equivalent mass M=diagM and the stiffness K=diagk, a conservative system is expressed by(158)Mη∗∗+Kη=0,(159)ddt∂T∂μ∗i−∂T∂μi+∂V∂μi=0.Substituting (136)–(155) into (135), the results are yielded as(160)T=m0A22+m12+7m212+7m3λ1212+m4λ124+7m5λ2212+7m612+m7λ224+7m8A212+7m9A212+m104+m114+7m1212+7m13λ3212+m14λ324+m15A22+m16A22+8mb1A23+4A2mb2+4A2mb33η∗21η∗22,(161)∂T∂μ∗i=m0A2+m1+7m26+7m3λ126+m4λ122+7m5λ226+7m66+m7λ222+7m8A26+7m9A26+m102+m112+7m126+7m13λ326+m14λ322+m15A2+m16A2+16mb1A23+8A2mb2+8A2mb33η∗1η∗2,(162)ddt∂T∂μ∗i=m0A2+m1+7m26+7m3λ126+m4λ122+7m5λ226+7m66+m7λ222+7m8A26+7m9A26+m102+m112+7m126+7m13λ326+m14λ322+m15A2+m16A2+16mb1A23+8A2mb2+8A2mb33η∗∗1η∗&lowast;2,(163)∂T∂μi=0,(164)∂V∂μi=k1+k2η1η2.Substituting (162)–(164) into (159), values M and K can be expressed by the following formula:(165)M=m0A2+m1+7m26+7m3λ126+m4λ122+7m5λ226+7m66+m7λ222+7m8A26+7m9A26+m102+m112+7m126+7m13λ326+m14λ322+m15A2+m16A2+16mb1A23+8A2mb2+8A2mb33,(166)K=Kin=2Kin1.Solving (158), the natural frequency (f) of the stage can be obtained as(167)f=12πKM0.5.


### 4.2. Evaluation and Verifications of Mathematical Models

Table4 shows that an error between the theoretical result and FEA result is about 5.426%. It means that the proposed methodology based on the kinetostatic-based methods (i.e., defomable mechanic theory, elastic beam theory, and Lagrange method) is reliable and effective in modeling the statics and dynamics of the 2-dof stage.Table 4
Error between the theory and simulation result.
ResponseAnalytical resultFEA resultError (%)f (Hz)92.121987.3815.426


### 4.3. Structural Optimization

#### 4.3.1. Optimization Problem Description

To avoid the resonance phenomena between the motors, PZT actuators, and the compliant 2-dof stage, the first natural frequency modes are either as small as possible or as large as possible. In order to increase the rapid responsiveness of the positioners, the first natural frequency should be chosen as large as possible. In addition, the angular frequency is proportional to the natural frequency of the compliant stage. Therefore, the first natural frequency is proposed to maximize in order to increase the response speed as well as avoid the resonance phenomena of the 2-dof stage. In this research, the optimization problem is aimed to maximize the resonant frequency, which is briefly expressed as follows.Find design vector:x=x1,x2,x3,x4,x5Maximize(168)fx,S.t:(169)fx>100Hz,Design variables (unit: mm):(170)0.7≤x1≤0.90.6≤x2≤0.80.6≤x3≤0.750.6≤x4≤0.70.6≤x5≤0.7,where f(x) denotes the resonant frequency. In addition, x1, x2, x3, x4, and x5 are the dimensions of A, B, C, D, and E, respectively.


#### 4.3.2. Optimal Results

Based on (1)–(170), MATLAB R2017b was employed for developing the combination approach of the kinetostatic analysis-based method and the neural network algorithm. As a result, the optimal parameters of the stage were found at A = 0.9 mm, B = 0.8 mm, C = 0.7 mm, D = 0.7 mm, and E = 0.7 mm, and the first natural frequency was 112.0995 Hz.


### 4.4. Verification and Comparisons

By using the optimal parameters, a 3D stage is created. The FEA result in ANSYS software showed that the first natural frequency was 106.98 Hz. Compared with the initial design, the frequency of the 2-dof stage is improved by up to 21.69%, as given in Table5. In addition, the error among the optimal value and FEA value is 4.785%, as given in Table 6. Besides, the resonant frequency values of the six modes from mode 1 to mode 6 are 106.98 Hz, 121.75 Hz, 328.69 Hz, 335.46 Hz, 602.42 Hz, and 603.88 Hz, respectively. Figure 18 illustrates the first mode shape analysis of the resonant natural frequency. So as to prevent the damage of the optimal positioner due to resonances, the positioner should avoid the abovementioned resonant frequencies during operation.Table 5
Improvement between optimal result and initial design result.
ResponseOptimal resultInitial design resultImprovement (%)f (Hz)112.099592.121921.69Table 6
Error between optimal result and FEA result.
ResponseOptimal resultFEA resultError (%)f (Hz)112.0995106.984.785Figure 18
The first mode shape analysis of the resonant natural frequency of the optimal 2-dof stage.Compared with the previous designs of the 2-dof stage, the present design proposed a higher frequency, as given in Table7.Table 7
Comparison of the proposed 2-dof stage with the previous designs.
Design of the 2-dof stageDimensionFrequency (Hz)Zhu et al. [23]NA59.3Lee et al. [24]NA80Present design451 mm × 451 mm × 16 mm112.0995In addition, in order to evaluate the material strength of the optimal stage, the FEA result in ANSYS software showed that the equivalent stress was 270.81 MPa, as illustrated in Figure19. The output total deformation was 877.37 μm, as depicted in Figure 20. Besides, the displacement amplification ratio was 7.79, the output force was 3480.4 N, and the parasitic motion error, so-called as the cross-axis coupling ratio between the two axes, was around 0.02%. Specifically, the achieved working stroke of the XY stage was 787.63 μm × 794.5 μm. The safety factor is found about 1.86 (i.e., the resulting stress is much smaller than the yield strength of used material). The achieved optimal results satisfied the desirable technical specifications of the XY stage.Figure 19
The equivalent stress of the optimal 2-dof stage.Figure 20
The total deformation of the optimal 2-dof stage.


## 4.1. Modeling of Kinetostatics and Dynamics

The kinetostatics and dynamics of the 2-dof stage are performed to evaluate its behaviors. The deformable mechanics theory, the elastic beam theory, and the Lagrange method are utilized for these analyses. In this work, the specifications of the 2-dof stage are analyzed (i.e., ratio of displacement amplification, stiffness, stroke, and resonant frequency).


### 4.1.1. Modeling of Amplification Ratio and Input Stiffness

Figure5 demonstrates a variant of the modified displacement lever amplifier (MDLA), a so-called variant of a lever amplifier. It includes the main parameters of the lever lengths of the different amplification floors. Because of a symmetrical structure, a half of the displacement amplifier is chosen to analyze the quality characteristics of the proposed 2-dof stage.Figure 5
A variant of the symmetrical displacement lever amplifier.A schematic diagram of the MDLA is provided in Figure6. It consists of main levers, including the lever amplification mechanism #1 (LAM1), lever amplification mechanism #2 (LAM2), and lever amplification mechanism #3 (LAM3). Besides, the parameters of the lengths of levers (l1, l2, l1′, l2′, l3, l4, l5, and l6) are also in Figure 6 in which O1, O2, O3, and O4 are the rotation centers of the levers.Figure 6
Schematic diagram of the modified displacement lever amplifier.As shown in Figure6, the MDLA is operated based on three levers (LAM1, LAM2, and LAM3). It can be determined that the output displacement of the amplifier can be derived as follows [21]:(11)δout=l5l6l2l1+l4l3+1+l4l3+1δin,where δin and δout symbolize the input displacement and the output displacement, respectively. Next, the amplification ratio will be computed as follows:(12)A=l5l6l2l1+l4l3+1+l4l3+1,where A is the displacement amplification ratio.Three kinds of flexure joints are used at appropriate places in the original design, thus helping the design to achieve a large displacement and reduce undesirable motions of the stage. Figure7 demonstrates the main associated dimensions of a flexure circular notched hinge, and it shows the effect of force and torque that recovers when the mechanism operates.Figure 7
Diagram of a circular arc flexure hinge.The mechanism’s input stiffness is investigated. The general spring with bending stiffnessKδxFxc is generated via the lateral force, and the torsion stiffness KθzMzc is made via the torque. The linear stiffness KθyMyc is created via the axial force for the flexure right circular hinge. Furthermore, the leaf beam's torsion and linear stiffness are denoted by KθzMzl and KθyMyl, respectively. Correspondingly, the flexure elliptical notched hinge is behaved with bending stiffness KδXFXe produced via the lateral force, torsion stiffness KeθzMz produced via the torque, and linear stiffness KθyMye generated via the axial force [22].(13)KθzMzc=2Ebt2.59πr0.5,(14)KθyMyc=Ebt0.5πr0.5−0.5t0.5,(15)KδxFxc=2Ebt2.53πr1.53r+t,Next, Figure8 describes the simplified structure diagram of the flexure leaf hinge. In addition, the main parameters such as the length, width, and thickness are also displayed along with the direction of force and torque acting on the leaf hinge.Figure 8
Diagram of a leaf-type arc flexure hinge.Stiffness values of the leaf hinge are calculated with the following equations:(16)KθzMzl=Eba312l,(17)KθyMyl=Ebal,(18)MθzMzl=KθzMzlΔθ.Figure9 shows the dimensions of the flexure elliptical hinge as well as the directions of the force and moment acting on the elliptical hinge.Figure 9
Diagram of an elliptical arc flexure hinge.Next, the following equations describe how to calculate the stiffness of the flexure elliptical hinge that bends in different directions.(19)KeθzMz=2EbaX23∈fβy,(20)KθyMye=Eb312∈gβy,(21)KδXFXe=Eb∈gβy,(22)βX=t2aX,(23)βy=t2ay,(24)∈=aXay=βyβx,(25)fβy=f∈βx=12βy+βy23+4βy+2βy21+βy2βy+βy2+61+βy2βy+βy23tan−12+βyβy,(26)gβy=g∈βx=21+βy2βy+βy2tan−12+βyβy−π2,where b denotes the thickness of the flexure hinge, E denotes the elastic modulus of the manufacturing material, r is the radius of a flexure circular notched hinge, and t denotes the smallest width of the circular/elliptical hinge. The width and length of the leaf beam are represented by a and l, respectively. The dimensionless factor representing the elliptical hinge geometry ax is denoted by βX, and (ay) is denoted by βy. The symbol ∈ is the multiplication factor of the ratio of major to minor axes. On the other hand, the symbols fβx and fβy are the dimensionless compliance factors according to βX and βy, respectively.By subjecting an input displacement∈ at the stage’s input end, the corresponding input force Fin is determined via the following equation:(27)Fin=Kinδin,where Kin is the input stiffness of the micromanipulator.Figure10 describes the initial state and working state of the MDLA. The deformation angles formed from the position change of the multistage lever help to show the different degrees of deformation of the structure while it is working.Figure 10
Force and deformation diagram of a half of the MDLA.Figures11–14 present the force analysis diagram of beam 4, beam 3, beam 2, and beam 1, respectively, in the lever amplification mechanism. The diagrams illustrate the details of the applied force, moment, rotation angle, as well as the deformation of the elastic joint.Figure 11
Force analysis diagram of beam 4.Figure 12
Force analysis diagram of beam 3.Figure 13
Force analysis diagram of beam 2.Figure 14
Force analysis diagram of beam 1.Only half of the displacement amplification structure is investigated due to the symmetrical structure. A main force analysis scheme of the mechanical amplification mechanism is depicted in Figure10. As a result, the following equations are formed by using the mechanic equilibrium state of beam 4, as asserted in Figure 11.The following force equilibrium equations can be derived as(28)FO3y=FEy+FGy,(29)FEyl6=FGyl5+MO3t+MGt+MEt.Specifically, the bending moments of the points O3, G, and E are represented as MO3t,MGt,, and MEt, respectively. Three forces FO3y, FEy, and FGy are acting at points O3, E, and G in the direction of y-axis when the structure operates, respectively. The displacement value at point G is a combination of two displacements including displacement due to hinges at O3 rotating around its axis (l5θ4) and displacement due to drift (δ4). Furthermore, by multiplying stiffness by displacement, the forces FO3y and FGy are determined. They are calculated via the following equations:(30)FO3y=KO3xδ4,(31)FGy=KGyl5θ4−δ4,(32)MO3t=KO3tθ4,(33)MGt=KGtθ4,(34)MEt=KEtθ4,where KO3x is the lateral bending stiffness by the force FO3y, and KGl is the G point’s output stiffness of the MDLA. In addition, the rotational stiffnesses KO3t, KGt, and KEt are the stiffnesses of the flexure hinges at points O3, G, and E, respectively.Substituting (30) and (31) into (28), the results are yielded as(35)KO3xδ4=FEy+KGyl5θ4−δ4,(36)δ4=FEy+KGyl5θ4KO3x+KGy.Substituting (31)–(34) into (29), the results are achieved as(37a)FEyl6=KGyl5l5θ4−δ4+KO3tθ4+KGtθ4+KEtθ4,(37b)FEyl6+KGyl5δ4=KGyl52+KO3t+KGt+KEtθ4,where FEy, FO3y, and FGy are the forces generated at points of E, O3, and G, respectively, when the structure operates.Substituting (36) into (37b), the θ4 and δ4 can be derived as(38)FEyl6+KGyl5FEy+KGyl5θ4KO3x+KGy=KGyl52+KO3t+KGt+KEtθ4,(39)FEyl6KO3x+KGy+KGyl5FEy+KGyl5θ4=KO3x+KGyKGyl52+KO3t+KGt+KEtθ4,(40)θ4=l6KO3x+KGy+KGyl5KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEy,(41a)δ4=FEy+KGyl5l6KO3x+KGy+KGyl5/KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEyKO3x+KGy,(41b)δ4=KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52+KGyl5l6KO3x+KGy+KGyl5KO3x+KGyKO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEy,(41c)δ4=KGyl52+KO3t+KGt+KEt+KGyl5l6KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEy.Regarding beam 4, the ratio of the amplifier and the input stiffness are determined by(42)λ4=l5θ4−δ4l6θ4+δ4,(43)Kin4=FEyl6θ4+δ4.Substituting (40) and (40c) into (42) and (43), respectively, the following formulas can be obtained:(44a)λ4=l5l6KO3x+KGy+KGyl5−KGyl52+KO3t+KGt+KEt+KGyl5l6l6l6KO3x+KGy+KGyl5+KGyl52+KO3t+KGt+KEt+KGyl5l6,(44b)λ4=l5l6KO3x+KGy−KO3t+KGt+KEt+KGyl5l6l6KO3x+KGy+2KGyl5l6+KGyl52+KO3t+KGt+KEt,(44c)λ4=l5l6KO3x−KO3t−KGt−KEtKGyl5+l62+KO3xl62+KO3t+KGt+KEt,(45)Kin4=KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52KGyl5+l62+KO3xl62+KO3t+KGt+KEt.Figure12 presents the force analysis diagram of beam 3.Considering the equilibrium state of beam 3, the force and moment are computed as(46)FHy=FIy+FO4y,(47)FHyl3=FIyl3+l4+MIt+MHt+MO4t,where MO4t, MIt,, and MHt are moments of bending of the points O4, I, and H. Moreover, FO4y, FHy, and FIy are forces which are acting at point O4, H, and I in the direction of y-axis, respectively.The displacement value at point I is a combination of two displacements including displacement due to hinges at O4 rotating around its axis ((l3 + l4) θ3) and displacement due to drift (δ3).Furthermore, the forcesFO4y and FIy can be calculated as(48)FO4y=KO4xδ3,(49)FIy=K3l3+l4θ3+δ3,(50)MO4t=KO4tθ3,(51)MIt=KItθ3,(52)MHt=KHtθ3,where KO4x is the lateral bending stiffness by force FO4y, and K3 is the I point’s output stiffness. KO4t is the O4 point’s rotational stiffness, and KIt, and KHt are the rotational stiffness of I and H, respectively.Substituting (48) and (49) into (46), the results are obtained as(53)FHy=K3l3+l4θ3+δ3+KO4xδ3,(54)δ3=FHy−K3l3+l4θ3KO4x+K3.Substituting (49)–(52) into (47), the results are yielded as(55a)FHyl3=K3l3+l4l3+l4θ3+δ3+KO4tθ3+KItθ3+KHtθ3,(55b)FHyl3=K3l3+l4δ3+K3l3+l42+KO4t+KIt+KHtθ3.Substituting (54) into (55b) θ3 and δ3 can be derived as(56a)FHyl3=K3l3+l4FHy−K3l3+l4θ3KO4x+K3+K3l3+l42+KO4t+KIt+KHtθ3,(56b)FHyl3KO4x+K3−K3l3+l4FHy=KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42θ3,(57)θ3=l3KO4x+K3−K3l3+l4KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42FHy,(58a)δ3=FHy−K3l3+l4l3KO4x+K3−K3l3+l4/KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42FHyKO4x+K3,(58b)δ3=K3l3+l42−K3l3+KO4t+KIt+KHtKO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42FHy.Considering beam 3, the ratio of the amplifier and the input stiffness are expressed by(59)λ3=l3+l4θ3+δ3l3θ3+δ3,(60)Kin3=FHyl3θ3+δ3.Substituting (57) and (58b) into (59) and (60), respectively, the following formulas can be obtained:(61a)λ3=l3+l4l3KO4x+K3−K3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHtl3l3KO4x+K3−K3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(61b)λ3=l3KO4x+K3l3+l4−K3l3+l42+K3l3+l42−K3l3+KO4t+KIt+KHtl32KO4x+K3−K3l3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(61c)λ3=l3KO4x+K3l3+l4−K3l3+KO4t+KIt+KHtl32KO4x+K3−K3l3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(62)Kin3=KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42l32KO4x+K3−K3l3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(63)K3=KO3yKin4KO3y+Kin4.Figure13 shows the force analysis diagram of beam 2.In a similar calculation for beam 2, the forces and moments are computed by(64)FO2y=FCy+FJy,(65)FJyl1′=FCyl2′+MO2t+MJt+MCt,where MO2t, MJt, and MCt are bending moments at the O2, J, and C points, respectively. The forces acting at points O2, C, and J in the direction of y-axis are FO2y, FCy, and FJy, respectively.The displacement value at point C is a combination of two displacements including displacement due to hinges at O2 rotating around its axis (l’2θ2) and displacement due to drift (δ2).Moreover, the forcesFO2y and FCy are expressed as(66)FO2y=KO2xδ2,(67)FCy=K2l2′θ2−δ2,(68)MO2t=KO2tθ2,(69)MCt=KCtθ2,(70)MJt=KJtθ2,where KO2x is the lateral bending stiffness by force FO2y, and K2 is the output stiffness of the E point. KO2t is the rotational stiffness of the O2 point. KJt and KCt are the rotational stiffnesses of J, and C points, respectively. Substituting (66) and (67) into (64),(71)KO2xδ2=FJy+K2l2′θ2−δ2,(72)δ2=FJy+K2l2′θ2KO2x+K2.Substituting (67)–(70) into (65), the following equations can be derived:(73a)FJyl1′=K2l2′l2′θ2−δ2+KO2tθ2+KCtθ2+KJtθ2,(73b)FJyl1′+K2l2′δ2=K2l2′2+KO2t+KCt+KJtθ2.Substituting (72) into (73b), the value of δ2 can be derived as(74a)FJyl1′+K2l2′FJy+K2l2′θ2KO2x+K2=K2l2′2+KO2t+KCt+KJtθ2,(74b)FJyl1′KO2x+K2+K2l2′FJy+K2l2′θ2=KO2x+K2K2l2′2+KO2t+KCt+KJtθ2,(75)θ2=l1′KO2x+K2+K2l2′KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy,(76a)δ2=FJy+K2l2′l1′KO2x+K2+K2l2′/KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJyKO2x+K2,(76b)δ2=KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2+K2l2′l1′KO2x+K2+K2l2′KO2x+K2KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy,(76c)δ2=KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2+K22l2′2+K2l2′l1′KO2x+K2KO2x+K2KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy,(76d)δ2=K2l2′2+KO2t+KCt+KJt+K2l2′l1′KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy.Similarly, the ratio of the amplifier and the input stiffness of beam 2 are determined by(77)λ2=l2′θ2−δ2l1′θ2+δ2,(78)Kin2=FJyl1′θ2+δ2.Substituting (75) and (76d) into (77) and (78), respectively, the following formulas can be obtained:(79a)λ2=l2′l1′KO2x+K2+K2l2′−K2l2′2+KO2t+KCt+KJt+K2l2′l1′l1′l1′KO2x+K2+K2l2′+K2l2′2+KO2t+KCt+KJt+K2l2′l1′,(79b)λ2=l1′l2′KO2x+K2−KO2t+KCt+KJt+K2l2′l1′l1′2KO2x+K2+2K2l2′l1′+K2l2′2+KO2t+KCt+KJt,(79c)λ2=l1′l2′KO2x−KO2t+KCt+KJtl1′2KO2x+K2+2K2l2′l1′+K2l2′2+KO2t+KCt+KJt,(80)Kin2=KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2l1′2KO2x+K2+2K2l2′l1′+K2l2′2+KO2t+KCt+KJt,(81)K2=KEyKin4KEy+Kin4.Figure14 presents the force analysis diagram of beam 1.By a similar consideration to the equilibrium state of the beam 1, the following equations are achieved:(82)FO1y=FAy+FBy,(83)FAyl1=FByl2+MO1t+MBt+MAt,where MO1t is the bending moments of the O1 point, while MAt and MBt are the bending moments of the A and B points, respectively. The forces acting at points O1, A, and B in the direction of y-axis are FO1y, FAy, and FBy, respectively.The displacement value at point B is a combination of two displacements including displacement due to hinges at O1 rotating around its axis (l2θ1) and displacement due to drift (δ1).Besides, the forcesFO1y and FBy are expressed by(84)FBy=K1l2θ1−δ1,(85)FO1y=KO1xδ1,(86)MO1t=KO1tθ1,(87)MBt=KBtθ1,(88)MAt=KAtθ1,where KO1x is the lateral bending stiffness by force FO1y, and K1 is the output stiffness of the B and A points, respectively. KO1t is the rotational stiffness of the O1 point, while KAt and KBt are the rotational stiffness of A and B points, respectively.Similarly, as beam 2, theθ1 and δ1 can be derived as(89)θ1=l1KO1x+K1+K1l2KO1x+K1K1l22+KO1t+KBt+KAt−K12l22FAy,(90)δ1=K1l1l2+K1l22+KO1t+KBt+KAtKO1x+K1K1l22+KO1t+KBt+KAt−K12l22FAy.Regarding beam 1, the ratio of the amplifier and the input stiffness are expressed by(91)λ1=l2θ1−δ1l1θ1+δ1,(92)kin1=FAyl1θ1+δ1.Substituting (89) and (90) into (91) and (92), respectively, the results are yielded ass(93)λ1=l1l2KO1x−KO1t+KBt+KAtl12KO1x+K1+2K1l1l2+K1l22+KO1t+KBt+KAt,(94)Kin1=KO1x+K1K1l22+KO1t+KBt+KAt−K12l22l12KO1x+K1+2K1l1l2+K1l22+KO1t+KBt+KAt,(95)K1=K12+K13,(96)K12=Ky12Kin2Ky12+Kin2,(97)Ky12=KθyMye2+2KθyMye5=9KθyMye10,(98)K13=Ky13Kin3Ky13+Kin3,(99)Ky13=2KθyMye5,where K1 is the output stiffness that combines A and B points, K12 is the output stiffness that combines beam 1 and beam 2, and K13 is the output stiffness that combines beam 1 and beam 3. In addition, the intermediary stiffness between beam 1 and beam 2, and the intermediary stiffness between beam 1 and beam 3 are Ky12 and Ky13, respectively.(100)KO1t=KO2t=KO4t=KAt=KθzMze,(101)KBt=KCt=KDt=KIt=KO3t=KEt=KθzMze2,(102)KHt=KJt=KMt=KNt=2KθzMze5,(103)KGt=KθzMzc,(104)KO3y=KEy=KCy=KθyMye2,(105)KO1x=KO2x=KO4x=KδxFxe,(106)KO3x=KδxFxe2,(107)Kb2=KθzMzl7=Eba312l7,(108)KθyMyl7=Ebal7,(109)Kb1=KθzMzl8=Eba312l8,(110)KθzMzl9=Eba312l9,(111)Kb3=2KθzMzl9KθzMzcKθzMzc+4KθzMzl9,where KθzMzl7, KθzMzl8, and KθzMzl9 are the stiffnesses which are created via the torque of the leaf hinges l7, l8, and l9, respectively. In addition, Kb1, Kb2, and Kb3 are stiffness conversion of KθzMzl7, KθzMzl8, and KθzMzl9, the purpose of this action is making the equations simpler. Finally, the linear stiffness is created via the axial force of the leaf hinge l7, also display by value KθyMyl7.Finally, the total ratio of two-stage lever amplifier is computed by(112)A=λ4λ1+λ3+1+λ3+1.It is found thatFin = 2FAy, and considering the symmetric amplifier, the input stiffness is expressed by(113)Kin=Finl1θ1+δ1=2FAyl1θ1+δ1=2Kin1.A simplified stiffness diagram of the stage is given in Figure15 when the platform is driven in the direction of y-axis. Figure 16 shows a simplified principle diagram of the 2-dof stage.Figure 15
The stiffness of the output mechanism.Figure 16
Simplified principle diagram of the 2-dof stage.Letdout be the output displacement of the stage and din be the input displacement in the vertical direction. Besides, the rotational angles of the prismatic beams are symbolled as ωb1, ωb2, ωb3, α1, β1, γ1, and ε1, which can be calculated by(114)ωb1=doutl8,(115)ωb2=doutl7,(116)ωb3=doutl9,(117)α1=doutl5,(118)β1=doutl3+l4l6l5,(119)γ1=doutl2,l6l5,(120)ε1=doutl2l6l5.The potential energy inside the flexure hinges is expressed by(121a)Ep=12∑i=ABKiε21+∑i=JDKiγ21+∑i=O4IKiβ21+KGα21+12∑j=b1b3Kjω2j,(121b)Ep=12KAtε12+KO1tε12+KBtε12+KJtγ12+KO2tγ12+KDtγ12+KO4tβ12+KHtβ12+KItβ12+KGtα122+4Kb1ωb12+6Kb2ωb22+2Kb3ωb32.Substituting (114)–(120) into (121b), the stiffness KN can be achieved by(122)KN=2l6l2l52KAt+KO1t+KBt+l6l′2l52KJt+KO2t+KDt+l6l4+l3l52KO4t+KHt+KIt+1l52KGt+41l82Kb1+61l72Kb2+21l92Kb3.Next, the output stiffnessKout and KGl of the stage are determined by(123)Kout=KNKaKN+Ka,in which Ka as shown in Figure 15 and calculated by(124)Ka=3KθyMyl72,(125)KGl=0.5Kout,where KN is the combines stiffness of the micromanipulator in one direction, Ka is the stiffness of the beam KθyMyl7, and Kout is the output stiffness of the micromanipulator.For a simple calculation, the geometric parameters and of the AL7075-T6 properties of the stage are provided in Table3.Table 3
Geometric parameters and AL7075-T6 for the proposed stage.
l1l2l’1l’2l3l4l5l6l7l8l913.949.71246145261.8187376.965.9rt (circular)a (l8, l9)a (l7)axayt (elliptical)blE (GPa)σr (MPa)4.50.60.60.84.52.50.6516971.7503


### 4.1.2. Modeling of Working Workspace

Assuming that the A is the ratio of amplifier andS is the stroke of the PZT, the workspace of the stage is calculated as AS x AS. I axial tension and shearing effects are ignored. There is only the bending stress is considered. To sum up, the stress (σr) is computed by(126)Maxσr=σys,where σy is the yield strength of the material and s is the safety factor.The maximal stressσrMax is computed as(127)σrMax=4Er2Kcfβt2θMax,where kc is concentration factor and fβ is the compliance factor [22].(128)β=t2r,(129)kc=1+β920,(130)fβ=12β+β23+4β+2β21+β2β+β2+61+β2β+β23tan−12+ββ,where β is the dimensionless geometry factor.LetδinMax is the maximal input displacement and δinMax is the output displacement of stage. The maximal deformation angle (θmax) is computed by(131)θMax=AδinMaxl5+λ3δinMaxl3+l4.Substituting (126) and (131) into (127), the results are yielded as(132)σys≥4Er2Kcfβt2AδinMaxl5+λ3δinMaxl3+l4,(133)δinMax≤l5l3+l4fβt2σy4Er2KcsAl3+l4+λ3l5.By using the values in Table2, the (133) and let s of 1.8, the maximal input displacement is expressed by(134)δinMax≤44.6286μm,To sum up, the output displacement of the stage may achieve up to 1363.8μm. Therefore, the workspace of the 2-dof stage is 1363.8 μm × 1363.8 μm.


### 4.1.3. Modeling of Dynamics

The natural frequencies can be achieved by Lagrange’s method by calculating the kinetic energy (T) and potential energy (V). Figures 12 and 17 show the position of masses m in the proposed design. Based on the position and motion trajectory of the components, a motion classification step will be performed to assist the calculation of the natural frequency of the design.Figure 17
Symmetrical structure with two 2-stage amplifiers.The hingesm2, m3, m5, m6, m8, m9, m12, m13, mb1, mb2, mb3 represent the rotational and translational motions. The hinges m0, m1, m4, m7, m10, m11, m14, m15, m16 note the translations. The kinetic energy of the whole stage is computed as(135)T=Tη1+Tη2,(136)Tη1=Tη2=∑i=m2m16Ti+∑j=mb1mb3Tj,(137)Tm0=12m0vA2=12m0η1.A2=m0A22η1.2,(138)Tm1=m12η1.2,(139)Tm4=12m4vλ1222=m4λ124η1.2,(140)Tm7=m7λ224η1.2,(141)Tm10=m104η1.2,(142)Tm11=m114η1.2,(143)Tm15=m15A22η1.2,(144)Tm16=m16A22η1.2,(145)Tm2=12m2v222+12m2l123vl122=7m212v2=7m212η1.2,(146)Tm3=7m3λ1212η1.2,(147)Tm5=7m5λ2212η1.2,(148)Tm6=7m612η1.2,(149)Tm8=7m8A212η1.2,(150)Tm9=7m9A212η1.2,(151)Tm12=7m1212η1.2,(152)Tm13=7m13λ3212η1.2,(153)Tmb1=12mb1vA24+12mb1l823vAl824=8mb1A23v2=8mb1A23η1.2,(154)Tmb2=12mb2vA26+12mb2l723vAl726=4A2mb2η1.2,(155)Tmb3=12mb3vA22+12mb3l923vAl922=4A2mb33η1.2.Potential energy:(156)V=12k1η12+12k2η22.Lagrange’s equation: Lagrangian mechanics defines a mechanical system to be a pair (M, L) of a configuration space M and a smooth function L = L(q, v,t) called Lagrangian.(157)L=T−V,ddt∂T∂μi∗−∂T∂μi+∂V∂μi=Fi,Mdi∗∗+Kdi=Fi,where i = 1; 2 is the free vibration of horizontal and vertical directions of the stage. Fi is a nonconservative generalized force corresponding to the coordinate. Let the equivalent mass M=diagM and the stiffness K=diagk, a conservative system is expressed by(158)Mη∗∗+Kη=0,(159)ddt∂T∂μ∗i−∂T∂μi+∂V∂μi=0.Substituting (136)–(155) into (135), the results are yielded as(160)T=m0A22+m12+7m212+7m3λ1212+m4λ124+7m5λ2212+7m612+m7λ224+7m8A212+7m9A212+m104+m114+7m1212+7m13λ3212+m14λ324+m15A22+m16A22+8mb1A23+4A2mb2+4A2mb33η∗21η∗22,(161)∂T∂μ∗i=m0A2+m1+7m26+7m3λ126+m4λ122+7m5λ226+7m66+m7λ222+7m8A26+7m9A26+m102+m112+7m126+7m13λ326+m14λ322+m15A2+m16A2+16mb1A23+8A2mb2+8A2mb33η∗1η∗2,(162)ddt∂T∂μ∗i=m0A2+m1+7m26+7m3λ126+m4λ122+7m5λ226+7m66+m7λ222+7m8A26+7m9A26+m102+m112+7m126+7m13λ326+m14λ322+m15A2+m16A2+16mb1A23+8A2mb2+8A2mb33η∗∗1η∗&lowast;2,(163)∂T∂μi=0,(164)∂V∂μi=k1+k2η1η2.Substituting (162)–(164) into (159), values M and K can be expressed by the following formula:(165)M=m0A2+m1+7m26+7m3λ126+m4λ122+7m5λ226+7m66+m7λ222+7m8A26+7m9A26+m102+m112+7m126+7m13λ326+m14λ322+m15A2+m16A2+16mb1A23+8A2mb2+8A2mb33,(166)K=Kin=2Kin1.Solving (158), the natural frequency (f) of the stage can be obtained as(167)f=12πKM0.5.


## 4.1.1. Modeling of Amplification Ratio and Input Stiffness

Figure5 demonstrates a variant of the modified displacement lever amplifier (MDLA), a so-called variant of a lever amplifier. It includes the main parameters of the lever lengths of the different amplification floors. Because of a symmetrical structure, a half of the displacement amplifier is chosen to analyze the quality characteristics of the proposed 2-dof stage.Figure 5
A variant of the symmetrical displacement lever amplifier.A schematic diagram of the MDLA is provided in Figure6. It consists of main levers, including the lever amplification mechanism #1 (LAM1), lever amplification mechanism #2 (LAM2), and lever amplification mechanism #3 (LAM3). Besides, the parameters of the lengths of levers (l1, l2, l1′, l2′, l3, l4, l5, and l6) are also in Figure 6 in which O1, O2, O3, and O4 are the rotation centers of the levers.Figure 6
Schematic diagram of the modified displacement lever amplifier.As shown in Figure6, the MDLA is operated based on three levers (LAM1, LAM2, and LAM3). It can be determined that the output displacement of the amplifier can be derived as follows [21]:(11)δout=l5l6l2l1+l4l3+1+l4l3+1δin,where δin and δout symbolize the input displacement and the output displacement, respectively. Next, the amplification ratio will be computed as follows:(12)A=l5l6l2l1+l4l3+1+l4l3+1,where A is the displacement amplification ratio.Three kinds of flexure joints are used at appropriate places in the original design, thus helping the design to achieve a large displacement and reduce undesirable motions of the stage. Figure7 demonstrates the main associated dimensions of a flexure circular notched hinge, and it shows the effect of force and torque that recovers when the mechanism operates.Figure 7
Diagram of a circular arc flexure hinge.The mechanism’s input stiffness is investigated. The general spring with bending stiffnessKδxFxc is generated via the lateral force, and the torsion stiffness KθzMzc is made via the torque. The linear stiffness KθyMyc is created via the axial force for the flexure right circular hinge. Furthermore, the leaf beam's torsion and linear stiffness are denoted by KθzMzl and KθyMyl, respectively. Correspondingly, the flexure elliptical notched hinge is behaved with bending stiffness KδXFXe produced via the lateral force, torsion stiffness KeθzMz produced via the torque, and linear stiffness KθyMye generated via the axial force [22].(13)KθzMzc=2Ebt2.59πr0.5,(14)KθyMyc=Ebt0.5πr0.5−0.5t0.5,(15)KδxFxc=2Ebt2.53πr1.53r+t,Next, Figure8 describes the simplified structure diagram of the flexure leaf hinge. In addition, the main parameters such as the length, width, and thickness are also displayed along with the direction of force and torque acting on the leaf hinge.Figure 8
Diagram of a leaf-type arc flexure hinge.Stiffness values of the leaf hinge are calculated with the following equations:(16)KθzMzl=Eba312l,(17)KθyMyl=Ebal,(18)MθzMzl=KθzMzlΔθ.Figure9 shows the dimensions of the flexure elliptical hinge as well as the directions of the force and moment acting on the elliptical hinge.Figure 9
Diagram of an elliptical arc flexure hinge.Next, the following equations describe how to calculate the stiffness of the flexure elliptical hinge that bends in different directions.(19)KeθzMz=2EbaX23∈fβy,(20)KθyMye=Eb312∈gβy,(21)KδXFXe=Eb∈gβy,(22)βX=t2aX,(23)βy=t2ay,(24)∈=aXay=βyβx,(25)fβy=f∈βx=12βy+βy23+4βy+2βy21+βy2βy+βy2+61+βy2βy+βy23tan−12+βyβy,(26)gβy=g∈βx=21+βy2βy+βy2tan−12+βyβy−π2,where b denotes the thickness of the flexure hinge, E denotes the elastic modulus of the manufacturing material, r is the radius of a flexure circular notched hinge, and t denotes the smallest width of the circular/elliptical hinge. The width and length of the leaf beam are represented by a and l, respectively. The dimensionless factor representing the elliptical hinge geometry ax is denoted by βX, and (ay) is denoted by βy. The symbol ∈ is the multiplication factor of the ratio of major to minor axes. On the other hand, the symbols fβx and fβy are the dimensionless compliance factors according to βX and βy, respectively.By subjecting an input displacement∈ at the stage’s input end, the corresponding input force Fin is determined via the following equation:(27)Fin=Kinδin,where Kin is the input stiffness of the micromanipulator.Figure10 describes the initial state and working state of the MDLA. The deformation angles formed from the position change of the multistage lever help to show the different degrees of deformation of the structure while it is working.Figure 10
Force and deformation diagram of a half of the MDLA.Figures11–14 present the force analysis diagram of beam 4, beam 3, beam 2, and beam 1, respectively, in the lever amplification mechanism. The diagrams illustrate the details of the applied force, moment, rotation angle, as well as the deformation of the elastic joint.Figure 11
Force analysis diagram of beam 4.Figure 12
Force analysis diagram of beam 3.Figure 13
Force analysis diagram of beam 2.Figure 14
Force analysis diagram of beam 1.Only half of the displacement amplification structure is investigated due to the symmetrical structure. A main force analysis scheme of the mechanical amplification mechanism is depicted in Figure10. As a result, the following equations are formed by using the mechanic equilibrium state of beam 4, as asserted in Figure 11.The following force equilibrium equations can be derived as(28)FO3y=FEy+FGy,(29)FEyl6=FGyl5+MO3t+MGt+MEt.Specifically, the bending moments of the points O3, G, and E are represented as MO3t,MGt,, and MEt, respectively. Three forces FO3y, FEy, and FGy are acting at points O3, E, and G in the direction of y-axis when the structure operates, respectively. The displacement value at point G is a combination of two displacements including displacement due to hinges at O3 rotating around its axis (l5θ4) and displacement due to drift (δ4). Furthermore, by multiplying stiffness by displacement, the forces FO3y and FGy are determined. They are calculated via the following equations:(30)FO3y=KO3xδ4,(31)FGy=KGyl5θ4−δ4,(32)MO3t=KO3tθ4,(33)MGt=KGtθ4,(34)MEt=KEtθ4,where KO3x is the lateral bending stiffness by the force FO3y, and KGl is the G point’s output stiffness of the MDLA. In addition, the rotational stiffnesses KO3t, KGt, and KEt are the stiffnesses of the flexure hinges at points O3, G, and E, respectively.Substituting (30) and (31) into (28), the results are yielded as(35)KO3xδ4=FEy+KGyl5θ4−δ4,(36)δ4=FEy+KGyl5θ4KO3x+KGy.Substituting (31)–(34) into (29), the results are achieved as(37a)FEyl6=KGyl5l5θ4−δ4+KO3tθ4+KGtθ4+KEtθ4,(37b)FEyl6+KGyl5δ4=KGyl52+KO3t+KGt+KEtθ4,where FEy, FO3y, and FGy are the forces generated at points of E, O3, and G, respectively, when the structure operates.Substituting (36) into (37b), the θ4 and δ4 can be derived as(38)FEyl6+KGyl5FEy+KGyl5θ4KO3x+KGy=KGyl52+KO3t+KGt+KEtθ4,(39)FEyl6KO3x+KGy+KGyl5FEy+KGyl5θ4=KO3x+KGyKGyl52+KO3t+KGt+KEtθ4,(40)θ4=l6KO3x+KGy+KGyl5KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEy,(41a)δ4=FEy+KGyl5l6KO3x+KGy+KGyl5/KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEyKO3x+KGy,(41b)δ4=KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52+KGyl5l6KO3x+KGy+KGyl5KO3x+KGyKO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEy,(41c)δ4=KGyl52+KO3t+KGt+KEt+KGyl5l6KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52FEy.Regarding beam 4, the ratio of the amplifier and the input stiffness are determined by(42)λ4=l5θ4−δ4l6θ4+δ4,(43)Kin4=FEyl6θ4+δ4.Substituting (40) and (40c) into (42) and (43), respectively, the following formulas can be obtained:(44a)λ4=l5l6KO3x+KGy+KGyl5−KGyl52+KO3t+KGt+KEt+KGyl5l6l6l6KO3x+KGy+KGyl5+KGyl52+KO3t+KGt+KEt+KGyl5l6,(44b)λ4=l5l6KO3x+KGy−KO3t+KGt+KEt+KGyl5l6l6KO3x+KGy+2KGyl5l6+KGyl52+KO3t+KGt+KEt,(44c)λ4=l5l6KO3x−KO3t−KGt−KEtKGyl5+l62+KO3xl62+KO3t+KGt+KEt,(45)Kin4=KO3x+KGyKGyl52+KO3t+KGt+KEt−KGy2l52KGyl5+l62+KO3xl62+KO3t+KGt+KEt.Figure12 presents the force analysis diagram of beam 3.Considering the equilibrium state of beam 3, the force and moment are computed as(46)FHy=FIy+FO4y,(47)FHyl3=FIyl3+l4+MIt+MHt+MO4t,where MO4t, MIt,, and MHt are moments of bending of the points O4, I, and H. Moreover, FO4y, FHy, and FIy are forces which are acting at point O4, H, and I in the direction of y-axis, respectively.The displacement value at point I is a combination of two displacements including displacement due to hinges at O4 rotating around its axis ((l3 + l4) θ3) and displacement due to drift (δ3).Furthermore, the forcesFO4y and FIy can be calculated as(48)FO4y=KO4xδ3,(49)FIy=K3l3+l4θ3+δ3,(50)MO4t=KO4tθ3,(51)MIt=KItθ3,(52)MHt=KHtθ3,where KO4x is the lateral bending stiffness by force FO4y, and K3 is the I point’s output stiffness. KO4t is the O4 point’s rotational stiffness, and KIt, and KHt are the rotational stiffness of I and H, respectively.Substituting (48) and (49) into (46), the results are obtained as(53)FHy=K3l3+l4θ3+δ3+KO4xδ3,(54)δ3=FHy−K3l3+l4θ3KO4x+K3.Substituting (49)–(52) into (47), the results are yielded as(55a)FHyl3=K3l3+l4l3+l4θ3+δ3+KO4tθ3+KItθ3+KHtθ3,(55b)FHyl3=K3l3+l4δ3+K3l3+l42+KO4t+KIt+KHtθ3.Substituting (54) into (55b) θ3 and δ3 can be derived as(56a)FHyl3=K3l3+l4FHy−K3l3+l4θ3KO4x+K3+K3l3+l42+KO4t+KIt+KHtθ3,(56b)FHyl3KO4x+K3−K3l3+l4FHy=KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42θ3,(57)θ3=l3KO4x+K3−K3l3+l4KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42FHy,(58a)δ3=FHy−K3l3+l4l3KO4x+K3−K3l3+l4/KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42FHyKO4x+K3,(58b)δ3=K3l3+l42−K3l3+KO4t+KIt+KHtKO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42FHy.Considering beam 3, the ratio of the amplifier and the input stiffness are expressed by(59)λ3=l3+l4θ3+δ3l3θ3+δ3,(60)Kin3=FHyl3θ3+δ3.Substituting (57) and (58b) into (59) and (60), respectively, the following formulas can be obtained:(61a)λ3=l3+l4l3KO4x+K3−K3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHtl3l3KO4x+K3−K3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(61b)λ3=l3KO4x+K3l3+l4−K3l3+l42+K3l3+l42−K3l3+KO4t+KIt+KHtl32KO4x+K3−K3l3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(61c)λ3=l3KO4x+K3l3+l4−K3l3+KO4t+KIt+KHtl32KO4x+K3−K3l3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(62)Kin3=KO4x+K3K3l3+l42+KO4t+KIt+KHt−K32l3+l42l32KO4x+K3−K3l3l3+l4+K3l3+l42−K3l3+KO4t+KIt+KHt,(63)K3=KO3yKin4KO3y+Kin4.Figure13 shows the force analysis diagram of beam 2.In a similar calculation for beam 2, the forces and moments are computed by(64)FO2y=FCy+FJy,(65)FJyl1′=FCyl2′+MO2t+MJt+MCt,where MO2t, MJt, and MCt are bending moments at the O2, J, and C points, respectively. The forces acting at points O2, C, and J in the direction of y-axis are FO2y, FCy, and FJy, respectively.The displacement value at point C is a combination of two displacements including displacement due to hinges at O2 rotating around its axis (l’2θ2) and displacement due to drift (δ2).Moreover, the forcesFO2y and FCy are expressed as(66)FO2y=KO2xδ2,(67)FCy=K2l2′θ2−δ2,(68)MO2t=KO2tθ2,(69)MCt=KCtθ2,(70)MJt=KJtθ2,where KO2x is the lateral bending stiffness by force FO2y, and K2 is the output stiffness of the E point. KO2t is the rotational stiffness of the O2 point. KJt and KCt are the rotational stiffnesses of J, and C points, respectively. Substituting (66) and (67) into (64),(71)KO2xδ2=FJy+K2l2′θ2−δ2,(72)δ2=FJy+K2l2′θ2KO2x+K2.Substituting (67)–(70) into (65), the following equations can be derived:(73a)FJyl1′=K2l2′l2′θ2−δ2+KO2tθ2+KCtθ2+KJtθ2,(73b)FJyl1′+K2l2′δ2=K2l2′2+KO2t+KCt+KJtθ2.Substituting (72) into (73b), the value of δ2 can be derived as(74a)FJyl1′+K2l2′FJy+K2l2′θ2KO2x+K2=K2l2′2+KO2t+KCt+KJtθ2,(74b)FJyl1′KO2x+K2+K2l2′FJy+K2l2′θ2=KO2x+K2K2l2′2+KO2t+KCt+KJtθ2,(75)θ2=l1′KO2x+K2+K2l2′KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy,(76a)δ2=FJy+K2l2′l1′KO2x+K2+K2l2′/KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJyKO2x+K2,(76b)δ2=KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2+K2l2′l1′KO2x+K2+K2l2′KO2x+K2KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy,(76c)δ2=KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2+K22l2′2+K2l2′l1′KO2x+K2KO2x+K2KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy,(76d)δ2=K2l2′2+KO2t+KCt+KJt+K2l2′l1′KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2FJy.Similarly, the ratio of the amplifier and the input stiffness of beam 2 are determined by(77)λ2=l2′θ2−δ2l1′θ2+δ2,(78)Kin2=FJyl1′θ2+δ2.Substituting (75) and (76d) into (77) and (78), respectively, the following formulas can be obtained:(79a)λ2=l2′l1′KO2x+K2+K2l2′−K2l2′2+KO2t+KCt+KJt+K2l2′l1′l1′l1′KO2x+K2+K2l2′+K2l2′2+KO2t+KCt+KJt+K2l2′l1′,(79b)λ2=l1′l2′KO2x+K2−KO2t+KCt+KJt+K2l2′l1′l1′2KO2x+K2+2K2l2′l1′+K2l2′2+KO2t+KCt+KJt,(79c)λ2=l1′l2′KO2x−KO2t+KCt+KJtl1′2KO2x+K2+2K2l2′l1′+K2l2′2+KO2t+KCt+KJt,(80)Kin2=KO2x+K2K2l2′2+KO2t+KCt+KJt−K22l2′2l1′2KO2x+K2+2K2l2′l1′+K2l2′2+KO2t+KCt+KJt,(81)K2=KEyKin4KEy+Kin4.Figure14 presents the force analysis diagram of beam 1.By a similar consideration to the equilibrium state of the beam 1, the following equations are achieved:(82)FO1y=FAy+FBy,(83)FAyl1=FByl2+MO1t+MBt+MAt,where MO1t is the bending moments of the O1 point, while MAt and MBt are the bending moments of the A and B points, respectively. The forces acting at points O1, A, and B in the direction of y-axis are FO1y, FAy, and FBy, respectively.The displacement value at point B is a combination of two displacements including displacement due to hinges at O1 rotating around its axis (l2θ1) and displacement due to drift (δ1).Besides, the forcesFO1y and FBy are expressed by(84)FBy=K1l2θ1−δ1,(85)FO1y=KO1xδ1,(86)MO1t=KO1tθ1,(87)MBt=KBtθ1,(88)MAt=KAtθ1,where KO1x is the lateral bending stiffness by force FO1y, and K1 is the output stiffness of the B and A points, respectively. KO1t is the rotational stiffness of the O1 point, while KAt and KBt are the rotational stiffness of A and B points, respectively.Similarly, as beam 2, theθ1 and δ1 can be derived as(89)θ1=l1KO1x+K1+K1l2KO1x+K1K1l22+KO1t+KBt+KAt−K12l22FAy,(90)δ1=K1l1l2+K1l22+KO1t+KBt+KAtKO1x+K1K1l22+KO1t+KBt+KAt−K12l22FAy.Regarding beam 1, the ratio of the amplifier and the input stiffness are expressed by(91)λ1=l2θ1−δ1l1θ1+δ1,(92)kin1=FAyl1θ1+δ1.Substituting (89) and (90) into (91) and (92), respectively, the results are yielded ass(93)λ1=l1l2KO1x−KO1t+KBt+KAtl12KO1x+K1+2K1l1l2+K1l22+KO1t+KBt+KAt,(94)Kin1=KO1x+K1K1l22+KO1t+KBt+KAt−K12l22l12KO1x+K1+2K1l1l2+K1l22+KO1t+KBt+KAt,(95)K1=K12+K13,(96)K12=Ky12Kin2Ky12+Kin2,(97)Ky12=KθyMye2+2KθyMye5=9KθyMye10,(98)K13=Ky13Kin3Ky13+Kin3,(99)Ky13=2KθyMye5,where K1 is the output stiffness that combines A and B points, K12 is the output stiffness that combines beam 1 and beam 2, and K13 is the output stiffness that combines beam 1 and beam 3. In addition, the intermediary stiffness between beam 1 and beam 2, and the intermediary stiffness between beam 1 and beam 3 are Ky12 and Ky13, respectively.(100)KO1t=KO2t=KO4t=KAt=KθzMze,(101)KBt=KCt=KDt=KIt=KO3t=KEt=KθzMze2,(102)KHt=KJt=KMt=KNt=2KθzMze5,(103)KGt=KθzMzc,(104)KO3y=KEy=KCy=KθyMye2,(105)KO1x=KO2x=KO4x=KδxFxe,(106)KO3x=KδxFxe2,(107)Kb2=KθzMzl7=Eba312l7,(108)KθyMyl7=Ebal7,(109)Kb1=KθzMzl8=Eba312l8,(110)KθzMzl9=Eba312l9,(111)Kb3=2KθzMzl9KθzMzcKθzMzc+4KθzMzl9,where KθzMzl7, KθzMzl8, and KθzMzl9 are the stiffnesses which are created via the torque of the leaf hinges l7, l8, and l9, respectively. In addition, Kb1, Kb2, and Kb3 are stiffness conversion of KθzMzl7, KθzMzl8, and KθzMzl9, the purpose of this action is making the equations simpler. Finally, the linear stiffness is created via the axial force of the leaf hinge l7, also display by value KθyMyl7.Finally, the total ratio of two-stage lever amplifier is computed by(112)A=λ4λ1+λ3+1+λ3+1.It is found thatFin = 2FAy, and considering the symmetric amplifier, the input stiffness is expressed by(113)Kin=Finl1θ1+δ1=2FAyl1θ1+δ1=2Kin1.A simplified stiffness diagram of the stage is given in Figure15 when the platform is driven in the direction of y-axis. Figure 16 shows a simplified principle diagram of the 2-dof stage.Figure 15
The stiffness of the output mechanism.Figure 16
Simplified principle diagram of the 2-dof stage.Letdout be the output displacement of the stage and din be the input displacement in the vertical direction. Besides, the rotational angles of the prismatic beams are symbolled as ωb1, ωb2, ωb3, α1, β1, γ1, and ε1, which can be calculated by(114)ωb1=doutl8,(115)ωb2=doutl7,(116)ωb3=doutl9,(117)α1=doutl5,(118)β1=doutl3+l4l6l5,(119)γ1=doutl2,l6l5,(120)ε1=doutl2l6l5.The potential energy inside the flexure hinges is expressed by(121a)Ep=12∑i=ABKiε21+∑i=JDKiγ21+∑i=O4IKiβ21+KGα21+12∑j=b1b3Kjω2j,(121b)Ep=12KAtε12+KO1tε12+KBtε12+KJtγ12+KO2tγ12+KDtγ12+KO4tβ12+KHtβ12+KItβ12+KGtα122+4Kb1ωb12+6Kb2ωb22+2Kb3ωb32.Substituting (114)–(120) into (121b), the stiffness KN can be achieved by(122)KN=2l6l2l52KAt+KO1t+KBt+l6l′2l52KJt+KO2t+KDt+l6l4+l3l52KO4t+KHt+KIt+1l52KGt+41l82Kb1+61l72Kb2+21l92Kb3.Next, the output stiffnessKout and KGl of the stage are determined by(123)Kout=KNKaKN+Ka,in which Ka as shown in Figure 15 and calculated by(124)Ka=3KθyMyl72,(125)KGl=0.5Kout,where KN is the combines stiffness of the micromanipulator in one direction, Ka is the stiffness of the beam KθyMyl7, and Kout is the output stiffness of the micromanipulator.For a simple calculation, the geometric parameters and of the AL7075-T6 properties of the stage are provided in Table3.Table 3
Geometric parameters and AL7075-T6 for the proposed stage.
l1l2l’1l’2l3l4l5l6l7l8l913.949.71246145261.8187376.965.9rt (circular)a (l8, l9)a (l7)axayt (elliptical)blE (GPa)σr (MPa)4.50.60.60.84.52.50.6516971.7503


## 4.1.2. Modeling of Working Workspace

Assuming that the A is the ratio of amplifier andS is the stroke of the PZT, the workspace of the stage is calculated as AS x AS. I axial tension and shearing effects are ignored. There is only the bending stress is considered. To sum up, the stress (σr) is computed by(126)Maxσr=σys,where σy is the yield strength of the material and s is the safety factor.The maximal stressσrMax is computed as(127)σrMax=4Er2Kcfβt2θMax,where kc is concentration factor and fβ is the compliance factor [22].(128)β=t2r,(129)kc=1+β920,(130)fβ=12β+β23+4β+2β21+β2β+β2+61+β2β+β23tan−12+ββ,where β is the dimensionless geometry factor.LetδinMax is the maximal input displacement and δinMax is the output displacement of stage. The maximal deformation angle (θmax) is computed by(131)θMax=AδinMaxl5+λ3δinMaxl3+l4.Substituting (126) and (131) into (127), the results are yielded as(132)σys≥4Er2Kcfβt2AδinMaxl5+λ3δinMaxl3+l4,(133)δinMax≤l5l3+l4fβt2σy4Er2KcsAl3+l4+λ3l5.By using the values in Table2, the (133) and let s of 1.8, the maximal input displacement is expressed by(134)δinMax≤44.6286μm,To sum up, the output displacement of the stage may achieve up to 1363.8μm. Therefore, the workspace of the 2-dof stage is 1363.8 μm × 1363.8 μm.


## 4.1.3. Modeling of Dynamics

The natural frequencies can be achieved by Lagrange’s method by calculating the kinetic energy (T) and potential energy (V). Figures 12 and 17 show the position of masses m in the proposed design. Based on the position and motion trajectory of the components, a motion classification step will be performed to assist the calculation of the natural frequency of the design.Figure 17
Symmetrical structure with two 2-stage amplifiers.The hingesm2, m3, m5, m6, m8, m9, m12, m13, mb1, mb2, mb3 represent the rotational and translational motions. The hinges m0, m1, m4, m7, m10, m11, m14, m15, m16 note the translations. The kinetic energy of the whole stage is computed as(135)T=Tη1+Tη2,(136)Tη1=Tη2=∑i=m2m16Ti+∑j=mb1mb3Tj,(137)Tm0=12m0vA2=12m0η1.A2=m0A22η1.2,(138)Tm1=m12η1.2,(139)Tm4=12m4vλ1222=m4λ124η1.2,(140)Tm7=m7λ224η1.2,(141)Tm10=m104η1.2,(142)Tm11=m114η1.2,(143)Tm15=m15A22η1.2,(144)Tm16=m16A22η1.2,(145)Tm2=12m2v222+12m2l123vl122=7m212v2=7m212η1.2,(146)Tm3=7m3λ1212η1.2,(147)Tm5=7m5λ2212η1.2,(148)Tm6=7m612η1.2,(149)Tm8=7m8A212η1.2,(150)Tm9=7m9A212η1.2,(151)Tm12=7m1212η1.2,(152)Tm13=7m13λ3212η1.2,(153)Tmb1=12mb1vA24+12mb1l823vAl824=8mb1A23v2=8mb1A23η1.2,(154)Tmb2=12mb2vA26+12mb2l723vAl726=4A2mb2η1.2,(155)Tmb3=12mb3vA22+12mb3l923vAl922=4A2mb33η1.2.Potential energy:(156)V=12k1η12+12k2η22.Lagrange’s equation: Lagrangian mechanics defines a mechanical system to be a pair (M, L) of a configuration space M and a smooth function L = L(q, v,t) called Lagrangian.(157)L=T−V,ddt∂T∂μi∗−∂T∂μi+∂V∂μi=Fi,Mdi∗∗+Kdi=Fi,where i = 1; 2 is the free vibration of horizontal and vertical directions of the stage. Fi is a nonconservative generalized force corresponding to the coordinate. Let the equivalent mass M=diagM and the stiffness K=diagk, a conservative system is expressed by(158)Mη∗∗+Kη=0,(159)ddt∂T∂μ∗i−∂T∂μi+∂V∂μi=0.Substituting (136)–(155) into (135), the results are yielded as(160)T=m0A22+m12+7m212+7m3λ1212+m4λ124+7m5λ2212+7m612+m7λ224+7m8A212+7m9A212+m104+m114+7m1212+7m13λ3212+m14λ324+m15A22+m16A22+8mb1A23+4A2mb2+4A2mb33η∗21η∗22,(161)∂T∂μ∗i=m0A2+m1+7m26+7m3λ126+m4λ122+7m5λ226+7m66+m7λ222+7m8A26+7m9A26+m102+m112+7m126+7m13λ326+m14λ322+m15A2+m16A2+16mb1A23+8A2mb2+8A2mb33η∗1η∗2,(162)ddt∂T∂μ∗i=m0A2+m1+7m26+7m3λ126+m4λ122+7m5λ226+7m66+m7λ222+7m8A26+7m9A26+m102+m112+7m126+7m13λ326+m14λ322+m15A2+m16A2+16mb1A23+8A2mb2+8A2mb33η∗∗1η∗&lowast;2,(163)∂T∂μi=0,(164)∂V∂μi=k1+k2η1η2.Substituting (162)–(164) into (159), values M and K can be expressed by the following formula:(165)M=m0A2+m1+7m26+7m3λ126+m4λ122+7m5λ226+7m66+m7λ222+7m8A26+7m9A26+m102+m112+7m126+7m13λ326+m14λ322+m15A2+m16A2+16mb1A23+8A2mb2+8A2mb33,(166)K=Kin=2Kin1.Solving (158), the natural frequency (f) of the stage can be obtained as(167)f=12πKM0.5.


## 4.2. Evaluation and Verifications of Mathematical Models

Table4 shows that an error between the theoretical result and FEA result is about 5.426%. It means that the proposed methodology based on the kinetostatic-based methods (i.e., defomable mechanic theory, elastic beam theory, and Lagrange method) is reliable and effective in modeling the statics and dynamics of the 2-dof stage.Table 4
Error between the theory and simulation result.
ResponseAnalytical resultFEA resultError (%)f (Hz)92.121987.3815.426


## 4.3. Structural Optimization

### 4.3.1. Optimization Problem Description

To avoid the resonance phenomena between the motors, PZT actuators, and the compliant 2-dof stage, the first natural frequency modes are either as small as possible or as large as possible. In order to increase the rapid responsiveness of the positioners, the first natural frequency should be chosen as large as possible. In addition, the angular frequency is proportional to the natural frequency of the compliant stage. Therefore, the first natural frequency is proposed to maximize in order to increase the response speed as well as avoid the resonance phenomena of the 2-dof stage. In this research, the optimization problem is aimed to maximize the resonant frequency, which is briefly expressed as follows.Find design vector:x=x1,x2,x3,x4,x5Maximize(168)fx,S.t:(169)fx>100Hz,Design variables (unit: mm):(170)0.7≤x1≤0.90.6≤x2≤0.80.6≤x3≤0.750.6≤x4≤0.70.6≤x5≤0.7,where f(x) denotes the resonant frequency. In addition, x1, x2, x3, x4, and x5 are the dimensions of A, B, C, D, and E, respectively.


### 4.3.2. Optimal Results

Based on (1)–(170), MATLAB R2017b was employed for developing the combination approach of the kinetostatic analysis-based method and the neural network algorithm. As a result, the optimal parameters of the stage were found at A = 0.9 mm, B = 0.8 mm, C = 0.7 mm, D = 0.7 mm, and E = 0.7 mm, and the first natural frequency was 112.0995 Hz.


## 4.3.1. Optimization Problem Description

To avoid the resonance phenomena between the motors, PZT actuators, and the compliant 2-dof stage, the first natural frequency modes are either as small as possible or as large as possible. In order to increase the rapid responsiveness of the positioners, the first natural frequency should be chosen as large as possible. In addition, the angular frequency is proportional to the natural frequency of the compliant stage. Therefore, the first natural frequency is proposed to maximize in order to increase the response speed as well as avoid the resonance phenomena of the 2-dof stage. In this research, the optimization problem is aimed to maximize the resonant frequency, which is briefly expressed as follows.Find design vector:x=x1,x2,x3,x4,x5Maximize(168)fx,S.t:(169)fx>100Hz,Design variables (unit: mm):(170)0.7≤x1≤0.90.6≤x2≤0.80.6≤x3≤0.750.6≤x4≤0.70.6≤x5≤0.7,where f(x) denotes the resonant frequency. In addition, x1, x2, x3, x4, and x5 are the dimensions of A, B, C, D, and E, respectively.


## 4.3.2. Optimal Results

Based on (1)–(170), MATLAB R2017b was employed for developing the combination approach of the kinetostatic analysis-based method and the neural network algorithm. As a result, the optimal parameters of the stage were found at A = 0.9 mm, B = 0.8 mm, C = 0.7 mm, D = 0.7 mm, and E = 0.7 mm, and the first natural frequency was 112.0995 Hz.


## 4.4. Verification and Comparisons

By using the optimal parameters, a 3D stage is created. The FEA result in ANSYS software showed that the first natural frequency was 106.98 Hz. Compared with the initial design, the frequency of the 2-dof stage is improved by up to 21.69%, as given in Table5. In addition, the error among the optimal value and FEA value is 4.785%, as given in Table 6. Besides, the resonant frequency values of the six modes from mode 1 to mode 6 are 106.98 Hz, 121.75 Hz, 328.69 Hz, 335.46 Hz, 602.42 Hz, and 603.88 Hz, respectively. Figure 18 illustrates the first mode shape analysis of the resonant natural frequency. So as to prevent the damage of the optimal positioner due to resonances, the positioner should avoid the abovementioned resonant frequencies during operation.Table 5
Improvement between optimal result and initial design result.
ResponseOptimal resultInitial design resultImprovement (%)f (Hz)112.099592.121921.69Table 6
Error between optimal result and FEA result.
ResponseOptimal resultFEA resultError (%)f (Hz)112.0995106.984.785Figure 18
The first mode shape analysis of the resonant natural frequency of the optimal 2-dof stage.Compared with the previous designs of the 2-dof stage, the present design proposed a higher frequency, as given in Table7.Table 7
Comparison of the proposed 2-dof stage with the previous designs.
Design of the 2-dof stageDimensionFrequency (Hz)Zhu et al. [23]NA59.3Lee et al. [24]NA80Present design451 mm × 451 mm × 16 mm112.0995In addition, in order to evaluate the material strength of the optimal stage, the FEA result in ANSYS software showed that the equivalent stress was 270.81 MPa, as illustrated in Figure19. The output total deformation was 877.37 μm, as depicted in Figure 20. Besides, the displacement amplification ratio was 7.79, the output force was 3480.4 N, and the parasitic motion error, so-called as the cross-axis coupling ratio between the two axes, was around 0.02%. Specifically, the achieved working stroke of the XY stage was 787.63 μm × 794.5 μm. The safety factor is found about 1.86 (i.e., the resulting stress is much smaller than the yield strength of used material). The achieved optimal results satisfied the desirable technical specifications of the XY stage.Figure 19
The equivalent stress of the optimal 2-dof stage.Figure 20
The total deformation of the optimal 2-dof stage.


## 5. Conclusions

This study presented a new modeling and dimensional optimization synthesis of the 2-dof stage. The developed 2-dof stage would be applied for positioning a biomaterial sample in in situ nanoindentation. The proposed effective methodology was proposed according to the kinetostatic analysis-based calculation method, the Lagrange approach, and the neural network algorithm. First of all, the 2-dof stage was designed using a hybrid integration of an eight-lever displacement amplifier-integrated elliptical hinges and a symmetric parallelogram. Then, a chain of mathematical equations of the 2-dof stage were formulated using a kinetostatic analysis-based method in terms of the ratio of displacement amplification and the input stiffness. Subsequently, the Lagrange method was employed to form the dynamic equation of the 2-dof stage. Finally, the neural network algorithm was adopted to maximize the natural first frequency of the proposed stage. The optimal results indicated that the frequency of the stage achieved a high value of 112.0995 Hz. Besides, the mathematical models were relatively close to the simulation verifications. The designed stage has a faster response than that from a few previous designs.In the future study, a prototype of the 2-dof stage will be manufactured, and the analytical results will be verified by the experiments. In addition, the closed loop system will be utilized for verifying the analytical results as well as enhancing position accuracy of the 2-dof stage.



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*Source: 6709464-2022-08-16.xml*