Displacement Amplifier Research Articles

Nonlinear design, analysis, and testing of a single-stage compliant orthogonal displacement amplifier with a single input force for microgrippers

To achieve dexterous and stable micro/nanomanipulation, a large grasping stroke, compact design, and parallel grasping are required for microgrippers; thus, a single-stage compliant orthogonal displacement amplifier (CODA) with a single input force would be an ideal transmission mechanism. However, the existing small-deflection-based design schemes cannot adapt to large deflections or shearing effect, thereby affecting the orthogonal movement transformation accuracy. This study proposed, analyzed, and experimentally investigated a nonlinear design scheme for a single-stage CODA with a single input force. First, the nonlinear design principle is described qualitatively. By combining closed-form analytical modelling, finite element analysis, and numerical fitting, the nonlinear extent of a pre-set variable cross-sectional beam in the CODA is formulated. By utilizing the beam constraint model and small-deflection-based modelling, the nonlinear extent of the undetermined uniform straight beam in the CODA is derived. Based on the design principle and nonlinear models, a nonlinear design scheme is proposed quantitatively. Finite element simulations and experimental tests are conducted to verify the proposed scheme, and the limitations of our previous study are revealed.

Analysis and Optimization of a Novel Compact Compliant 2-DOF Positioner for Positioning to Assess Bio-Specimen Characteristics

A novel compact 2-DOF compliant positioner is developed for the purpose of achieving good characteristics such as high natural frequency, high displacement amplification ratio, good linear motion, and compact structure based on its symmetrical structure. To be specific, the developed stage is proposed according to an advanced six-lever displacement amplifier arranged at an inclination angle of the rigid bar utilizing right circular hinges and a parallel guiding mechanism with integrated flexure leaf hinges to attain the above-mentioned characteristics and reduce the decoupling mobility error. First, to quickly assess the initial quality response, an integration method of kinetostatic analysis, the Lagrange method, and finite element analysis was applied to evaluate and verify the quality characteristic of the stage. The experimental result showed that the error between the analytical method and the FEA method was 1.3%, which was relatively small and reliable for quickly assessing the primary quality response of the proposed positioner. Next, to boost the important output characteristics of the developed positioner, the integration approach of the response surface method and NSGA-II algorithm was utilized to find the optimal design variables. Finally, a prototype was manufactured based on the CNC milling method to validate the experimental and FEA analysis results. The attained results show that the optimal results of safety factor and output displacement were 2.4025 and 248.9 µm. Moreover, the FEA verification results were 2.4989 and 242.16 µm, with errors for safety factor and output displacement between the optimal result and the FEA result of 3.86% and 2.78%, respectively. In addition, the simulation and experimental results of the first natural frequency were 371.83 Hz and 329.59 Hz, respectively, and the error between the FEA result and experimental result for the first natural frequency was 11.36%. Furthermore, the achieved results show that the relationship between input displacement and output displacement of the experimental result and the FEA result of the developed structure achieved a good linear connection. These results suggest that the proposed positioner will be a potential structure employed in precise positioning systems and nanoindentation testing positioning systems for checking bio-specimens’ behaviors.

Kinetostatic Modeling of an Asymmetrical Double-Stepped Beam for Displacement Amplification

Abstract A kinetostatic model of an asymmetrical double-stepped beam under axial loading is developed. The beam is composed of two thick segments and three thin segments where a thick segment is between two thin segments, and the longitudinal axis of the thick segment is not colinear with that of the thin segment. The kinetostatic model based on the beam constraint model (BCM) is capable of predicting accurate bending and near-buckling behaviors of the beam. A virtual rigid link neighboring the noncolinear segments is introduced in the BCM to broaden the applicability of the BCM. An analytical formula to represent the critical load of a symmetrical double-stepped beam under axial loading is derived and the value calculated by the formula agrees with the limit load of the asymmetrical double-stepped beam within the elastic range. The investigated beam has potential applications in displacement amplifiers and robotic grippers.

A piezoelectric transducer for bone conduction implants designed using finite element analysis

AbstractThis study describes the design of a transducer for bone conduction implants that combines a piezoelectric element and a displacement amplifier. To develop a displacement amplifier with the maximum possible amplification ratio, theoretical analysis was performed to derive the parameters that affect the amplification ratio. Parametric sweep analysis was conducted to calculate the amplification ratio according to changes in the various parameters. The resulting optimal displacement amplifier afforded the maximum amplification ratio when the beam thickness, depth, inclination angle, and length were 0.15 mm, 1.2 mm, 6.5°, and 4 mm, respectively. The displacement amplifier with these optimal parameter values amplified the piezoelectric element displacement by approximately 7.7‐fold. Frequency characteristic analysis by weight was performed to derive a mechanical resonance location with frequency characteristics that appropriately compensated for hearing loss. As a result of this analysis, when a weight of 0.7 g was applied to the displacement amplifier, the mechanical resonance was that required for bone conduction implant transducers.

A novel F-shaped linear guiding mechanism based compliant positioning stage with restricted parasitic motion

Positioning stages are significant in precision technologies under higher demands of operation and manufacturing. The performance of the selected guiding mechanisms is a decisive factor affecting the positioning accuracy of the stage. To restrict the parasitic motions along the degrees of constraint during motion transmission, a piezo-actuated compliant positioning stage with novel F-shaped linear guiding mechanisms (FLGM) is developed in this paper to enable a precision motion. With a simple and compact structure, a single FLGM can generate an approximate straight-line motion and enhance the transverse stiffness. Four FLGMs are arranged symmetrically in parallel to theoretically eliminate the parasitic motions, bringing in the merits of a low coupling rate and a wide bandwidth in the working direction. A stack piezoelectric actuator embedded in a forward bridge-type displacement amplifier is utilized to drive the guided stage. Analytical modeling of its kinematic, static, and dynamic characteristics is mathematically established by utilizing the pseudo-rigid-body method, compliance matrix method, and Lagrange’s equation. Finite element analysis is conducted and a fabricated prototype is experimentally tested, validating the veracity of the analytical models. Experimental results show that the parasitic motion is constrained by the guiding mechanism effectively with a coupling rate below 0.95%, and a high natural frequency of 856.9 Hz is achieved simultaneously.

A nano-positioning system based on an optimal two-stage linear displacement amplifier

Nano-positioning systems with large motion ranges often employ displacement amplifiers with poor linearity and a large footprint. Here, we propose a compact nano-positioning system comprising an optimally designed two-stage displacement amplifier with linear input-output relationship over a large motion range. The first stage is a truncated bridge amplifier whose size is 25% of a conventional bridge amplifier, while the second stage is a modified lever-type displacement amplifier whose amplified motion direction can be chosen as desired. It is proposed to exploit the compliance of the structure to improve its linearity by operating the truncated bridge amplifier at a specific optimal angle. A simple reduced-order model is proposed and is shown to agree with finite element analysis to within 1%. Further, the variation in the amplification ratio at various payloads are analysed as function of input displacement. Subsequently, a two-stage amplifier with amplification 28 demonstrating a linear range of 940 µm is fabricated using acrylic. This amplifier’s nonlinearity is less than 5.2%, representing about 10-fold improvement compared to a conventional bridge amplifier. Lastly, a nano-positioner employing an optimal amplifier is fabricated using spring steel and demonstrated to possess nonlinearity less than 1.5% over a range of 360 µm and a natural frequency of 210 Hz with a footprint of 21 cm2, which represents a substantial improvement in range-to-footprint ratio compared to other amplifiers of similar natural frequencies.

Rigid-Compliant Hybrid Cellular Expansion Mechanisms With Motion Amplification and Superposition

Abstract Motivated by heat dissipation, the rigid-compliant hybrid cellular expansion mechanisms with motion amplification and superposition are proposed in this paper. Compared with existing studies, the expansion mechanism is not only easy to realize the plane tessellation via cellular design due to its regular polygon structure but also has the ability of motion amplification and superposition due to its compliant displacement amplifier and rigid scissors. First, the scheme of expansion mechanisms, especially the working principle of motion amplification and superposition, is introduced. The configuration design of a family of expansion mechanisms is presented, including varying number of edges, concave/convex property, and inner/outer layout. Second, the constraint condition and analytical modeling of relations between output performances of expansion mechanisms and dimensional parameters are carried out. Third, the displacement amplification ratio of expansion mechanisms and output performances of several typical expansion mechanisms when they act as cells to tessellate a plane with a constrained area are analyzed. Finally, the output performances of expansion mechanisms are verified via the finite element analysis. The results show that proposed cellular expansion mechanisms are beneficial for realizing plane tessellation and offer motion amplification and superposition, which provide prospects in the field of mechanism design such as metamaterials.

Time-Delay Control Scheme with Adaptive Fixed-Time Convergent Super-Twisting Fractional-Order Nonsingular Terminal Sliding Mode for Piezoelectric Displacement Amplifier

Piezoelectric displacement amplifiers (PDAs) have been widely used in precision positioning fields. However, the inherent hysteresis and creep nonlinear effect of piezoelectric actuators (PEAs) and time-varying lumped disturbances bring extreme challenges to the precise motion control of PDAs. Although various control schemes based on PEAs have been developed and have shown significant results. However, due to the high sensitivity of precision positioning to environmental variations, the development and identification of accurate models and the control timeliness often become obstacles in engineering. To realize precise motion control of PDAs under complex lumped disturbances, a new time-delay control scheme (AFSTA-FONTSM) using an adaptive fixed-time convergent super-twisting algorithm (AFSTA) and a fractional-order nonsingular terminal sliding mode (FONTSM) is proposed. Specifically, the time-delay information obtained by time-delay estimation technology is used to estimate the lumped dynamic characteristic of the system, thus establishing a simple control framework without a system dynamic model. FONSTM is constructed as a sliding mode manifold, and satisfactory error dynamic characteristic is obtained. A new AFSTA is designed as the reaching law in the sliding mode phase. AFSTA has fixed-time convergence when the upper bound of lumped disturbances exists, which ensures the control timeliness. Benefiting from the newly designed adaptive algorithm, the upper bound value of lumped disturbances is no longer needed to determine the control gains, which effectively prevents overestimation of the control gains. Correspondingly, the convergence time of AFSTA is estimated, and the stability of the closed-loop system is analyzed by the Lyapunov theory. Three existing time-delay control schemes, namely MSTA-FONTSM, AMSTA-FONTSM, and ASTA-FONTSM are selected, and four scenes are designed for comparative experiments. The experimental results show that MSTA-FONTSM has the worst control performance among the four control schemes. For the step, and continuous cosine trajectories with periods of T=1s and T=2s, the root-mean-square error of the proposed AFSTA-FONTSM is reduced by 56.86%, 54.03%, and 50.24% compared with MSTA-FONTSM. For disturbance experiments under different loads, the control performance of the proposed AFSTA-FONTSM is still superior to the other three control schemes without load.

High efficiency and wideband wave energy capture of a bistable wave energy converter with a displacement amplifier

Ocean wave energy is a kind of renewable energy with huge reserves, which can play an important role in the future construction of smart oceans. The low capture efficiency and narrow bandwidth of traditional wave energy capture devices seriously restrict the large-scale application of wave energy. To address these issues, a rack-pinion-lever-spring (RPLS) adaptive bistable Power Take-Off (PTO) mechanism with a time-varying potential barrier is proposed and applied to a point absorber wave energy converter (WEC). In the proposed nonlinear PTO system, a rack-pinion-lever (RPL) mechanism, as a displacement amplifier, has an effect on enhancing the poor amplitude sensitivity issue of the negative stiffness mechanism. Based on linear wave theory, a special nonlinear one-and-half-degree of freedom dynamic model is developed to describe the adaptive bistable wave energy converter (AB WEC). In comparison with the traditional bistable WEC (TB WEC) and equivalent linear WEC, the numerical results show that the AB WEC can greatly enhance the capture efficiency of lower frequency energy, especially at small wave excitations. The energy capture efficiency of the AB WEC is close to that of the TB WEC even at large wave amplitudes and is higher than that of the equivalent linear WEC. Finally, the wave energy capture characteristics of the three kinds of systems are studied based on the wave distribution diagram of a certain sea area in the South China Sea. The results show that the AB WEC has a large wave height adaptability in the low frequency region. The AB PTO mechanism with a displacement amplifier proposed in this paper can provide a technical reserve for the highly efficient and wide bandwidth capture of low frequency ocean wave energy.

Design and characterization of non-linear electromagnetic energy harvester with enhanced energy output

This paper presents design, analysis, and characterization of a compliant non-linear electromagnetic energy harvester operating on the principle of bi-stability incorporated with a displacement amplification mechanism. The energy harvester engenders energy from ambient vibrations at multiple modes and with frequencies ranging from 80 to 120 Hz. The energy harvester is analytically modeled using the mechanic’s elastic beam theory. Finite Element Analysis (FEA) has been carried out using commercial Finite Element Method (FEM) based software to perform static, fatigue, dynamic and electromagnetic analysis. Bi-stable mechanism has been used to harness energy at a wider frequency bandwidth while lever mechanism is acting as a displacement amplifier for enhancing low frequency vibration amplitude with the amplification factor of 6.24. With such amplification factor, the energy harvester can be deployed to extract vibrations of diminutive level and recast them into electrical energy in the form of generated voltage. Spring steel material, owing to its structural sensitivity, stiffness and higher value of ultimate stress, was used for its fabrication. The use of Neodymium (N52) magnets, copper coils and shaker were put into practice for testing of this device. The coil resistance is 2.5 Ω with number of turns kept at 350. The maximum output voltage of 1.86 V was generated at 106 Hz with root mean square value of 548 mV. Testing results are in accordance with the results obtained through simulations and thus useful electrical power of 11.1 mW can be generated from ambient vibrations at a wider bandwidth of 18 Hz with operating frequency ranging from 99 to 117 Hz.

Resonant-Type Piezoelectric Pump Driven by Piezoelectric Stacks and a Rhombic Micro Displacement Amplifier.

To obtain a high flow rate, a resonant-type piezoelectric pump is designed, fabricated, and studied in this paper. The pump consists of four parts: a piezoelectric vibrator, a pump chamber, a check valve and a compressible space. The designed piezoelectric vibrator is composed of a rhombic micro displacement amplifier, counterweight blocks and two piezoelectric stacks with low-voltage drive and a large output displacement. ANSYS software (Workbench 19.0) simulation results show that at the natural frequency of 946 Hz, the designed piezoelectric vibrator will produce the maximum output displacement. The bilateral deformation is symmetrical, and the phase difference is zero. Frequency, voltage, and backpressure characteristics of the piezoelectric pump are investigated. The experimental results show that at a certain operating frequency, the flow rate and the backpressure of the piezoelectric pump both increase with the increase in voltage. When the applied voltage is 150 Vpp, the flow rate reaches a peak of 367.48 mL/min at 720 Hz for one diaphragm pump, and reaches a peak of 700.15 mL/min at 716 Hz for two diaphragm pumps.

Design and research of a valve-based piezoelectric pump with low forward resistance and high reverse shutoff

This paper proposes, designs, manufactures and experimentally studies a wedge valve piezoelectric pump driven by a double displacement amplifier vibrator (DDAV). A novel wedge valve is designed, manufactured and analyzed, and the static performance of the wedge valve is tested. The test results show that it has less forward resistance and high reverse shutoff. The DDAV and pump body are designed, and the test experimental platform is built. Results indicate that the static opening pressure of the wedge valve is 0.27 kPa, and the reverse shutoff pressure can reach 200 kPa. When the driving voltage is 700 Vpp and the driving frequency is 425 Hz, the maximum flow rate is 491.2 ml min−1 and the maximum output pressure is 160 kPa.

Design and analysis of a 2-DOF compliant serial micropositioner based on "S-shaped" flexure hinge

A serial 2D planar compliant mechanism has been proposed based on novel "S-shaped" flexure hinges for atomic force microscopy applications. The bridge-lever type mechanism driven by piezoelectric actuators serves as a displacement amplifier. The optimal geometrical parameters were found to maximize the workspace and natural frequency through the Monte-Carlo search algorithm. Also, the performance of the developed system was investigated by Matrix-based Compliance Modeling (MCM), Finite Element Analysis (FEA), and experiments. All models indicate that the mechanism has an approximately linear force-deflection relationship, high safety factor, and a reasonable amplification ratio of about 4.5 and 5.5 for the inner and outer stages in the analytical approach, 4.4 and 5.3 in FEA. Finally, experiments on a fabricated open-loop controlled prototype revealed 4.2 and 5.1 amplification ratio for stages that resulted in a 94 × 124 μm rectangular workspace and a natural frequency of 224.6 Hz, which is about 5% lower than the results predicted using FEA.

Performance enhancement of snap-through vibration energy harvester with displacement amplifier

Vibration energy harvesters (VEH) based on bistable and snap-through mechanisms are being widely employed to harvest energy from different vibration sources. However, under weak ambient excitation, the performance of these systems is marginal. Multi-stability and enhancement techniques, such as displacement amplifier, are being used with bistable VEH to improve its performance. Similar to this, multi-stable snap-through VEH has also been developed; however, the usage of a displacement amplifier in the context of snap-through VEH has not yet been investigated. In this novel study, a displacement amplifier/ displacement amplification mechanism (DAM) is integrated into a snap-through VEH, and the enhanced performance under weak excitation is investigated. A generalized transformation is applied to develop a reduced-order model of the proposed system, and its dynamics and performance under harmonic and random excitation are investigated analytically and numerically. The frequency–amplitude relationship is derived under harmonic excitation, and a detailed investigation of the influence of system parameters on the frequency and force responses is performed. It is observed that by adjusting the mass and stiffness ratio between the VEH and DAM, it is possible to harvest more energy under harmonic excitation of low intensity. In the case of Gaussian white noise excitation, effective potential, and stochastic averaging methods are used to obtain marginal, joint, and stationary probability density functions and mean square power. The influence of noise intensity and other system parameters on the aforementioned measures is thoroughly investigated. It is observed that for specific parameter values, the proposed VEH can harvest 3.3 times more power under harmonic excitation and four times more power under random excitation compared to the snap-through VEH without DAM.

Bidirectional motivated bimodal isothermal strand displacement amplifier with a table tennis-like movement for the ultrasensitive fluorescent and colorimetric detection of depression-related microRNA.

An increasing number of studies have highlighted the potential of microRNAs (miRNAs) as physiological indicators of major depressive disorder (MDD). Herein, we developed a bidirectional-motivated bimodal isothermal strand displacement amplifier (BB-ISDA) for the ultrasensitive fluorescent and colorimetric detection of MDD-related miRNA-132. In the BB-ISDA system, a pair of functionalized hairpin probes (HP1 and HP2) with nicking recognition sites are designed to recognize target miRNA. The recognition of target miRNA by HP1 (or HP2) generates copious numbers of nicked triggers by HP1 (or HP2)-based ISDA to recognize HP2 (or HP1) by autonomous strand polymerization, cleavage, and displacement, which in turn induces the subsequent generation of copious numbers of nicked G-quadruplex triggers by HP2 (or HP1)-based ISDA to recognize HP1 (or HP2) along a same line. After many cycles, this bidirectional motivated table-tennis-like movement amplifies the fluorescent signal from HP1 and the colorimetric signal from HP2, simultaneously. The dual-signal output pattern was cross-validated for sensing miRNA-132. Each of the detection modal shows the capability for qualitative and quantitative detection of miRNA-132 with high sensitivity and specificity. The adaptability of the bimodal mechanism was verified via the detection of target miRNA-132 from clinical human blood samples. We envision that this BB-ISDA with dual-signal output for accurate and reliable analysis of miRNA is promising for the molecular diagnosis of human mental diseases.

Design and Analysis of a Hybrid Displacement Amplifier Supporting a High-Performance Piezo Jet Dispenser

In this study, a compliant amplifier powered by a piezoelectric stack is designed to meet high-performance dispensing operation requirements. By studying the issue of low frequency bandwidth on the traditional bridge-type amplifier mechanism, we propose a displacement amplifier mechanism, hybrid bridge-lever-bridge (HBLB), that enhances its dynamic performance by combining the traditional bridge-type and lever mechanism. A guiding beam is added to further improve its output stiffness with a guaranteed large amplification ratio. An analytical model has been developed to describe the full elastic deformation behavior of the HBLB mechanism that considers the lateral displacement loss of the input end, followed by a verification through a finite element analysis (FEA). Results revealed that the working principle of the HBLB optimizes the structural parameters using the finite element method. Finally, a prototype of the displacement amplifier was fabricated for performance tests. Static and dynamic test results revealed that the proposed mechanism can reach a travel range of 223.2 μm, and the frequency bandwidth is 1.184 kHz, which meets the requirements of a high-performance piezo jet dispenser.

A new composite controller for trajectory tracking control of a piezo-driven bridge displacement amplifier

In this paper, a new composite controller (feedforward + feedback) is proposed for the trajectory tracking control of a piezo-driven bridge displacement amplifier. A hysteresis model considering dynamic effects is established based on the modified Bouc–Wen (MBW) model, which can describe the asymmetry and frequency-dependence behaviors. The effectiveness of the modified model is verified by the experimental comparison. A feedforward controller is designed to compensate for hysteresis nonlinearity based on the inverse multiplicative structure of the modified model. A feedback controller with the Kalman observer is designed based on a modified model predictive control (MMPC). The proportional term and integral term of feedback error are introduced into the MPC control law to improve the response speed and reduce the steady-state error. The proposed MBW + MMPC is compared with MBW + proportional–integral–derivative (PID), and MBW + MPC. The experimental tracking results of continuously differentiable signals and non-differentiable signals verify MBW + MMPC feasibility and show that it can improve tracking performance.

Design and Research of a Novel Piezostack-Driven Jetting Dispenser With a Diamond Spring

A novel piezoelectric jetting dispenser is proposed for microelectronic packaging in this article. It adopts a piezoelectric drive, a lever displacement amplifier, and a recovery system with double springs in a staggered position. The automatic dynamic analysis of mechanical systems (ADAMSs) simulation model of the dispenser is established and first applied to the study of the high-frequency response performance. The displacement curve and the velocity curve of the impactor are obtained at the frequency of 200 and 500 Hz. The test results prove that the ADAMS model is feasible. Based on dispensing experiments, the volume and the volume error of a drop at different dispensing frequencies are studied. The volume of a drop is 8.5 nL at the frequency of 800 Hz. The maximum relative error of the droplet volume is less than ±7.2% at the frequency of 200, 300, and 400 Hz.

Design and Optimization for a New XYZ Micropositioner with Embedded Displacement Sensor for Biomaterial Sample Probing Application.

An XYZ compliant micropositioner has been widely mentioned in precision engineering, but the displacements in the X, Y, and Z directions are often not the same. In this study, a design and optimization for a new XYZ micropositioner are developed to obtain three same displacements in three axes. The proposed micropositioner is a planar mechanism whose advantage is a generation of three motions with only two actuators. In the design strategy, the proposed micropositioner is designed by a combination of a symmetrical four-lever displacement amplifier, a symmetrical parallel guiding mechanism, and a symmetrical parallel redirection mechanism. The Z-shaped hinges are used to gain motion in the Z-axis displacement. Four flexure right-circular hinges are combined with two rigid joints and two flexure leaf hinges to permit two large X-and-Y displacements. The symmetrical four-lever displacement amplifier is designed to increase the micropositioner's travel. The displacement sensor is built by embedding the strain gauges on the hinges of the micropositioner, which is developed to measure the travel of the micropositioner. The behaviors and performances of the micropositioner are modeled by using the Taguchi-based response surface methodology. Additionally, the geometrical factors of the XYZ micropositioner are optimized by teaching-learning-based optimization. The optimized design parameters are defined with an A of 0.9 mm, a B of 0.8 mm, a C of 0.57 mm, and a D of 0.7 mm. The safety factor gains 1.85, while the displacement achieves 515.7278 µm. The developed micropositioner is a potential option for biomedical sample testing in a nanoindentation system.

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.