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

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.


Introduction
Advances in in situ nanoindentation testers have resulted in great interest in developing compact-size mechanical mechanisms with better performances. e 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. en, they continued to develop a new 1-dof stage with specifications of crystal silicon [6]. Moreover, Huang et al. [6] designed a 1dof 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].
is 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. ey 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. ey 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.
is paper aimed to develop a new compliant 2-dof stage which can be employed in locating biomaterials in in situ nanoindentation systems. e design details of the stage are developed based on the multiple lever amplification mechanism and series-parallel chain. e kinetostatics and dynamics are derived based on the deformable mechanics, the elastic beam theory, and the Lagrange approach.
e 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) e stiffness and amplification ratio are established. en, the dynamic equation of the 2-dof stage is formulated. e 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.

Conceptual Design of a Compliant 2-dof Stage
2.1. Design Scheme of the 1-dof Mechanism. Figure 1 describes the operating mechanism of a 1-dof stage which comprises an eight-lever displacement amplifier and a parallelogram mechanism. e 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.

Operation Scheme of the 2-dof Stage.
Based on the design and operation schemes of the 1-dof stage in Figure 1, the operation principle and design scheme of a new 2-dof stage are provided in Figure 2(a). e design of the 2-dof stage is a module-based scheme. e 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. e 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. e overall working operation of the 2-dof stage is mainly based on the deformations of the elliptical joints, right circular joints, and P-joints. e suggested stage is driven using a PZT actuator. e main dimensioning parameters of the proposed stage are illustrated in Figure 2(b). e values of the parameters are provided in Table 1.
In the proposed design, a double-lever amplification mechanism is widely used for many researches on flexurebased mechanisms [16]. Meanwhile, the symmetric eightlever displacement amplifier integrated into the 2-dof stage for locating biospecimens has been investigated. erefore, 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. e 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. erefore, inspired from the advantages of the compliant mechanisms, the authors will investigate to contribute for the new compliant structures.
As illustrated clearly in Figure 2(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.

Proposed Methodology
In this part, a modeling and dimensional optimization synthesis method is developed for the compliant 2-dof stage. e 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.

Modeling and Dimensional Optimization Synthesis.
e flowchart of the proposed methodology for modeling and dimensional optimization synthesis for the developed stage is shown in Figure 3. e main steps of the present methodology are briefly performed as follows:

Neural Network Algorithm.
Inspired from the artificial neural network (ANN), the neural network algorithm (NNA) was developed [17]. is 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. is algorithm is aimed to decrease the error among the targets and the predicted values, and achieve a global optimally best result. e flowchart of the NNA is given in Figure 4.
A pattern solution (P s ) is determined as A matrix of pattern solution (X) is defined by e i th cost of the function is defined as follows: A weight matrix is determined by  Computational Intelligence and Neuroscience  Computational Intelligence and Neuroscience   Computational Intelligence and Neuroscience e weight value is determined as A new matrix (X) is calculated by e weight matrix is updated by following formula: A bias means that the exploration influences a new solution and weighting. Considering rand > ψ (ψ is solutions percentage), the function operator is computed by In this work, an initial population of 20, max iterations of 500, and ψ of 1 are utilized for the NNA.

Modeling of Kinetostatics and Dynamics.
e kinetostatics and dynamics of the 2-dof stage are performed to evaluate its behaviors. e deformable mechanics theory, the elastic beam theory, and the Lagrange method are utilized for these analyses. In this work, the specifications of the 2dof stage are analyzed (i.e., ratio of displacement amplification, stiffness, stroke, and resonant frequency). Figure 5 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.

Modeling of Amplification Ratio and Input Stiffness.
A schematic diagram of the MDLA is provided in Figure 6. 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 (l 1 , l 2 , l 1 ′, l 2 ′, l 3 , l 4 , l 5 , and l 6 ) are also in Figure 6 in which O 1 , O 2 , O 3 , and O 4 are the rotation centers of the levers.
As shown in Figure 6, 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]: where δ in and δ out symbolize the input displacement and the output displacement, respectively. Next, the amplification ratio will be computed as follows: where A is the displacement amplification ratio. ree 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. Figure 7 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. e mechanism's input stiffness is investigated. e general spring with bending stiffness (K c δxFx ) is generated via the lateral force, and the torsion stiffness (K c θzMz ) is made via the torque. e linear stiffness (K c θyMy ) is created via the axial force for the flexure right circular hinge. Furthermore, the leaf beam's torsion and linear stiffness are denoted by K l θzMz and K l θyMy , respectively. Correspondingly, the flexure elliptical notched hinge is behaved with bending stiffness (K e δ X F X ) produced via the lateral force, torsion stiffness (K e θzMz ) produced via the torque, and linear stiffness (K e θyMy ) generated via the axial force [22]. Next, Figure 8 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.
Stiffness values of the leaf hinge are calculated with the following equations: Figure 9 shows the dimensions of the flexure elliptical hinge as well as the directions of the force and moment acting on the elliptical hinge.
Next, the following equations describe how to calculate the stiffness of the flexure elliptical hinge that bends in different directions.
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. e width and length of the leaf beam are represented by a and l, respectively. e dimensionless factor representing the elliptical hinge geometry a x is denoted by (β X ), and (a y ) is denoted by (β y ). e symbol (∈) is the multiplication factor of the ratio of major to minor axes. On the other hand, the symbols (f(β x )) and (f(β y )) are the dimensionless compliance factors according to (β X ) and (β y ), respectively. By subjecting an input displacement (∈) at the stage's input end, the corresponding input force (F in ) is determined via the following equation:  Computational Intelligence and Neuroscience where K in is the input stiffness of the micromanipulator. Figure 10 describes the initial state and working state of the MDLA. e 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. Figures 11-14 present the force analysis diagram of beam 4, beam 3, beam 2, and beam 1, respectively, in the lever amplification mechanism. e diagrams illustrate the details of the applied force, moment, rotation angle, as well as the deformation of the elastic joint.
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 Figure 10. As a result, the following equations are formed by using the mechanic equilibrium state of beam 4, as asserted in Figure 11. e following force equilibrium equations can be derived as Specifically, the bending moments of the points O 3 , G, and E are represented as M O3t, M Gt, , and M Et , respectively. ree forces F O3y , F Ey , and F Gy are acting at points O 3 , E, and G in the direction of y-axis when the structure operates, respectively. e displacement value at point G is a combination of two displacements including displacement due to hinges at O 3 rotating around its axis (l 5 θ 4 ) and displacement due to drift (δ 4 ). Furthermore, by multiplying stiffness by displacement, the forces F O3y and F Gy are determined. ey are calculated via the following equations: where K O3x is the lateral bending stiffness by the force F O3y , and K Gl is the G point's output stiffness of the MDLA. In addition, the rotational stiffnesses K O3t , K Gt , and K Et are the stiffnesses of the flexure hinges at points O 3 , G, and E, respectively. Substituting (30) and (31) into (28), the results are yielded as Substituting (31)-(34) into (29), the results are achieved as where F Ey , F O3y , and F Gy are the forces generated at points of E, O 3 , and G, respectively, when the structure operates.
Considering the equilibrium state of beam 3, the force and moment are computed as where M O4t , M It, , and M Ht are moments of bending of the points O 4 , I, and H. Moreover, F O4y , F Hy , and F Iy are forces which are acting at point O 4 , H, and I in the direction of yaxis, respectively. e displacement value at point I is a combination of two displacements including displacement due to hinges at O 4 rotating around its axis ((l 3 + l 4 ) θ 3 ) and displacement due to drift (δ 3 ).
Furthermore, the forces F O4y and F Iy can be calculated as where K O4x is the lateral bending stiffness by force F O4y , and K 3 is the I point's output stiffness. K O4t is the O 4 point's rotational stiffness, and K It , and K Ht are the rotational stiffness of I and H, respectively. Substituting (48) and (49) into (46), the results are obtained as Substituting (49)-(52) into (47), the results are yielded as Computational Intelligence and Neuroscience F Hy l 3 � K 3 l 3 + l 4 l 3 + l 4 θ 3 + δ 3 (55b) Substituting (54) into (55b) θ 3 and δ 3 can be derived as Considering beam 3, the ratio of the amplifier and the input stiffness are expressed by Substituting (57) and (58b) into (59) and (60), respectively, the following formulas can be obtained: (63) Figure 13 shows the force analysis diagram of beam 2.
In a similar calculation for beam 2, the forces and moments are computed by where M O2t , M Jt , and M Ct are bending moments at the O 2 , J, and C points, respectively. e forces acting at points O 2 , C, and J in the direction of y-axis are F O2y , F Cy , and F Jy , respectively. e displacement value at point C is a combination of two displacements including displacement due to hinges at O 2 rotating around its axis (l' 2 θ 2 ) and displacement due to drift (δ 2 ).
Moreover, the forces F O2y and F Cy are expressed as where K O2x is the lateral bending stiffness by force F O2y , and K 2 is the output stiffness of the E point. K O2t is the rotational stiffness of the O 2 point. K Jt and K Ct are the rotational stiffnesses of J, and C points, respectively. Substituting (66) and (67) into (64), Substituting (67)-(70) into (65), the following equations can be derived: Substituting (72) into (73b), the value of δ 2 can be derived as Similarly, the ratio of the amplifier and the input stiffness of beam 2 are determined by Substituting (75) and (76d) into (77) and (78), respectively, the following formulas can be obtained: (81) Figure 14 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: where M O1t is the bending moments of the O 1 point, while M At and M Bt are the bending moments of the A and B points, respectively. e forces acting at points O 1 , A, and B in the direction of y-axis are F O1y , F Ay , and F By , respectively. e displacement value at point B is a combination of two displacements including displacement due to hinges at O 1 rotating around its axis (l 2 θ 1 ) and displacement due to drift (δ 1 ).
Besides, the forces F O1y and F By are expressed by where K O1x is the lateral bending stiffness by force F O1y , and K 1 is the output stiffness of the B and A points, respectively. K O1t is the rotational stiffness of the O 1 point, while K At and K Bt are the rotational stiffness of A and B points, respectively. Similarly, as beam 2, the θ 1 and δ 1 can be derived as Regarding beam 1, the ratio of the amplifier and the input stiffness are expressed by Substituting (89) and (90) into (91) and (92), respectively, the results are yielded ass where K 1 is the output stiffness that combines A and B points, K 12 is the output stiffness that combines beam 1 and beam 2, and K 13 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 K y12 and K y13 , respectively.
where K l7 θzMz , K l8 θzMz , and K l9 θzMz are the stiffnesses which are created via the torque of the leaf hinges l 7 , l 8 , and l 9 , respectively. In addition, K b1 , K b2 , and K b3 are stiffness conversion of K l7 θzMz , K l8 θzMz , and K l9 θzMz , the purpose of this action is making the equations simpler. Finally, the linear stiffness is created via the axial force of the leaf hinge l 7 , also display by value K l7 θyMy . Finally, the total ratio of two-stage lever amplifier is computed by (112) It is found that F in � 2F Ay , and considering the symmetric amplifier, the input stiffness is expressed by A simplified stiffness diagram of the stage is given in Figure 15 when the platform is driven in the direction of yaxis. Figure 16 shows a simplified principle diagram of the 2dof stage.
Let d out be the output displacement of the stage and d in be the input displacement in the vertical direction. Besides, the rotational angles of the prismatic beams are symbolled as ω b1 , ω b2 , ω b3 , α 1 , β 1 , c 1 , and ε 1 , which can be calculated by Computational Intelligence and Neuroscience (120) e potential energy inside the flexure hinges is expressed by Substituting (114)-(120) into (121b), the stiffness K N can be achieved by  Figure 15: e stiffness of the output mechanism.
Next, the output stiffness K out and K Gl of the stage are determined by in which K a as shown in Figure 15 and calculated by where K N is the combines stiffness of the micromanipulator in one direction, K a is the stiffness of the beam K l7 θyMy , and K out 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 Table 3.

Modeling of Working Workspace.
Assuming that the A is the ratio of amplifier and S 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. ere is only the bending stress is considered. To sum up, the stress (σ r ) is computed by where σ y is the yield strength of the material and s is the safety factor. e maximal stress σ Max r is computed as where k c is concentration factor and f(β) is the compliance factor [22].
where β is the dimensionless geometry factor. Let δ Max in is the maximal input displacement and δ Max in is the output displacement of stage. e maximal deformation angle (θ max ) is computed by Substituting (126) and (131) into (127), the results are yielded as By using the values in Table 2, the (133) and let s of 1.8, the maximal input displacement is expressed by

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To sum up, the output displacement of the stage may  Figure 17: Symmetrical structure with two 2-stage amplifiers.
Solving (158), the natural frequency (f ) of the stage can be obtained as Table 4 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.

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. erefore, the first natural frequency is proposed to maximize in order to increase the response speed as well as avoid the resonance phenomena of the 2dof stage. In this research, the optimization problem is aimed to maximize the resonant frequency, which is briefly expressed as follows. Find design vector: S.t:  Computational Intelligence and Neuroscience 21 where f(x) denotes the resonant frequency. In addition, x 1 , x 2 , x 3 , x 4 , and x 5 are the dimensions of A, B, C, D, and E, respectively.

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.

Verification and Comparisons.
By using the optimal parameters, a 3D stage is created. e 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 Table 5. In addition, the error among the optimal value and FEA value is 4.785%, as given in Table 6 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.
Compared with the previous designs of the 2-dof stage, the present design proposed a higher frequency, as given in Table 7.
In 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 Figure 19. e 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. e safety factor is found about 1.86 (i.e., the resulting stress is much smaller than the yield strength of used material). e achieved optimal results satisfied the desirable technical specifications of the XY stage.

Conclusions
is study presented a new modeling and dimensional optimization synthesis of the 2-dof stage. e developed 2dof stage would be applied for positioning a biomaterial sample in in situ nanoindentation. e 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

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en, 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. e 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.
e 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.

Abbreviations
2-dof: Two degrees of freedom MDLA: Modified displacement lever amplifier LAM 1: Lever amplification mechanism of floor 1 LAM 2: Lever amplification mechanism of floor 2 LAM 3: Lever amplification mechanism of floor 3 l 1 : Length of segment AO 1 l 2 : Length e ratio of the amplifier of beam 1 k in1 : e input stiffness of beam 1 K l7 θzMz : e stiffness created via the torque of the leaf hinge l 7 K l7 θyMy : e linear stiffness created via the axial force of the leaf hinge l 7 K l8 θzMz : e stiffness created via the torque of the leaf hinge l 8 K l9 θzMz : e stiffness created via the torque of the leaf hinge l 9 K b3 : e output stiffness combines K l9 θzMz and double e dimensionless geometry factor of a flexure circular notched hinge δ Max in : e maximal input displacement θ max : e maximal deformation angle T: e kinetic energy of the whole stage T η 1 : e kinetic energy of y-axis T η 2 : e kinetic energy of x-axis T i : e kinetic energy of the element in the micromanipulator (i � m 1 to m b3 ) η .
1 : e coordinate used to describe the motions of yaxis η .
2 : e coordinate used to describe the motions of xaxis.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.