Abstract
This paper devotes a new method in modeling and optimizing to handle the optimization of the XY positioning mechanism. The fitness functions and constraints of the mechanism are formulated via proposing a combination of artificial neural network (ANN) and particle swarm optimization (PSO) methods. Next, the PSO is hybridized with the grey wolf optimization, namely PSOGWO, which is applied to three scenarios in handling the single objective function. In order to search the multiple functions for the mechanism, the multiobjective optimization genetic algorithm (MOGA) is applied to the last scenario. The achieved results showed that the fitness functions are wellformulated using the PSObased ANN method. In the scenario 1, the stroke achieved by the PSOGWO (1852.9842 μm) is better than that gained from the GWO (1802.8087 μm). In the scenarios 2, the stress gained from the PSOGWO (243.3183 MPa) is lower than that achieved from the GWO (245.0401 MPa). In the scenario 3, the safety factor retrieved from the PSOGWO (1.9767) is greater than that achieved from the GWO (1.9278). In the scenario 4, by using MOGA, the optimal results found that the stroke is about (1741.3 μm) and the safety factor is 1.8929. The prediction results are wellfitted with the numerical and experimental verifications. The results of this paper are expected to facilitate the synthesis and analysis of compliant mechanisms and related engineering designs.
1. Introduction
An advancement of micro/nanoscience and technology has a critically important significance in reducing the cost of products and increasing accurate working ability. In order to handle micro/nanoobjects, micro/nanomanipulators has become an essential demand. Micro/nanotechnologies have attended in various fields such as micromanipulation [1], microelectromechanical system [2], micro/nanoindentation [3].
Especially in the field of micromanipulation, ultrahigh positioners [4, 5] and grippers [6, 7] are primary applications. The positioners are often utilized to transfer a force from an actuator (e.g., piezoelectric actuator [8], electromagnetic actuator [9], electrostatic actuator [10]) toward a mechanical system. A sample is often mounted on the movable platform of the positioner, and the main assignment of the positioner is to locate the sample from an initial position to another one with micro/nanodisplacement steps [11]. Meanwhile, the grippers also perform a gripping task of a microobject to a desirable location. A basic application of the positioners can be found in the indentation system, and the grippers are searched in DC assemble [12]. Moreover, a 3DOF micromanipulation stage was developed to achieve a large rotational displacement [13]. In this study, the lever amplifier was utilized to enlarge the working stroke, and then the pseudorigid body method was applied to formulate the mathematical formulas of stiffness, displacement amplification ratio, and kinematics of the stage. Meanwhile, the dynamic performance of the stage was modeled through Lagrange’s method. The precision and accuracy in transferring the motions of micromanipulations are directly affected by mechanical part systems. Basically, the rigid mechanical systems often use kinematic joints in connecting rigid links, as well as actuators and sensors in transforming the force/moment and sensing positions. Such rigid systems exist backlash and need more lubrications. So, the precision and accuracy of the positioner and grippers are decreased.
Unlike the rigidbased mechanical systems, the positioners based on compliant mechanisms and flexure hinges can achieve to a higher precision because compliant mechanism utilizes a single structure without backlash and friction. The operator of compliant mechanisms is replied on flexure hinges. The manufacturing of these mechanisms are benefits in cost and maintenance, since they can be fabricated by advanced techniques such as wire electrical discharged machining or additive manufacturing technology.
The most important aspect of developing compliant mechanismbased positioners is concentrated on modeling their behaviors (e.g., displacement/stroke, fatigue, parasitic motion, stress, and so forth). Until now, many analytical modeling methods have been proposed, e.g., pseudorigidbody model (PRBM) [14], dynamic stiffness matrix [15], and beamconstraint model [16]. More generally, according to the survey on the compliant mechanisms [17], many design synthesis and analysis methods were wellsummarized, including PRBM, Castigliano’s theory, compliance method, beam theory, and Ryu’s method. These techniques are still helpful in the design synthesis of compliant mechanisms. However, a large deflection and complex configurations are mainly restricted problems in modeling procedures. More practically, the deformations of compliant mechanisms are often large. The analytical methods can analyze the large deflections, but they may be limited for analyzing a complex structure or irregular shape. In comparison with the analytical methods, finite element method (FEM) was proved as an efficient technique in analyzing a large deflection [18]. For solving a complex structure, i.e., parallelconnected series mechanisms, curvilinear structure, or irregular shape, FEM was an effective tool for compliant mechanism [19]. In modeling the nonlinear characteristics/large deformations of compliant mechanismbased positioner, a new intelligent modeling approach is proposed herein as an alternative for the analytical modeling techniques in this article.
In improving the characteristics of compliant mechanismbased positioners, two popular methods are usually used, including the experimental optimization method and metaheuristic optimization techniques. In the first type, traditional optimization methods, socalled nonmetaheuristic algorithms, gradient descent, Newton’s method, and Taguchi method, response surface method, grey relational analysis, utility method [20, 21]. However, these nonmetaheuristics lead to a local optimum set. In the second type, the metaheuristic optimization methods, such as genetic algorithm (GA) [22, 23], particle swarm optimization (PSO) [24], cuckoo search algorithm [25], Dingo optimizer [26], and so on are required as an alternative choice to achieve a global optimum solution. More recently, a few efficient approaches suggested for optimizing in this field. A multiobjective genetic algorithm was proposed to optimize the flexure constantforce module [27]. The optimal parameters of a spatial constantforce endeffector were optimized through particle swarm optimization [28]. To improve the convergence speed and ability of searching the global optimum values, the PSO is coupled with the grey wolf optimization (GWO) [29]. The main target of the PSOhybridized GWO optimizer is to get the capability of exploitation in the PSO and the capability of exploration in GWO in finding the global solutions. In recent years, many natureinspired algorithms have suggested enhancing the capacity of optimizers in balancing the exploration and exploitation, such as Monarch butterfly optimization [30], Slime mould algorithm [31], Moth search algorithm [32], Hunger games search [33], Runge–Kutta method [34], colony predation algorithm [35], weighted mean of vectors [36], and Harris hawks optimization [37] Among these optimizers, the PSOhybridized GWO is a relatively effective optimizer in solving the global optimum problems. Besides, the metaheuristic optimization algorithms have been used to improve the regression ability of artificial neural network (ANN), such as GAANN [38] and PSOANN [39].
In the literature, there are proposals in designing for compliant micro/nanopositioners, but a new design synthesis method is still essential. To overcome the limitations of the analytical methods and the local optimization methods, this paper proposes a new modeling and optimizing method in designing the compliantbased XY positioning mechanism. The main contribution of this study is to introduce the artificial neural network into the dimension synthesis of compliant mechanisms which is an assistant tool for further developing intelligent algorithms into the analysis and synthesis of compliant mechanisms.
First of all, a new structure of the mechanism is designed. And then, the fitness functions describing input design parameters and the output characteristics of the mechanism are modeled via using a combination of artificial neural network and particle swarm optimization. After that, the grey wolf optimization is hybridized with the particle swarm optimization to find some optimization scenarios for the mechanism. The results are also verified by simulations and experiments. The following organization of the paper are included. Section 2 gives an introduction of mechanical structure and optimization problems for the mechanism. Section 3 presents the research method. Section 4 describes the achieved results. At last, Section 5 concludes the paper with the further study.
2. Structural Optimization Problem of XY Monolithic Mechanism
2.1. Mechanical Design
The new compliant XY stage was based on fourlever displacement amplifier integrated elliptical hinges and parallel guiding according to zigzagbased flexure springs and leaf hinges, as illustrated in Figure 1(a). Through many initial simulations, it is noted that the zigzagbased flexure springs offer the deflections better than that of leaf hinges. Additionally, the zigzag type is combined with leaf hinges so as to eliminate the parasitic motion error. Therefore, the zigzag type is chosen for designing the mechanism. In order to generate a large workspace for the XY stage, the lever amplifier is chosen because this amplifier has a simple structure and easy manufacture. In addition, two levers are arranged in symmetrical architecture so as to guarantee a translation motion in x axis or y axis with a larger stroke. In this study, the certain angle to the transverse and axial directions in the levertype compliant amplifiers are aimed to reduce the mass of the amplifier, i.e., decreasing in unessential cross section of lever amplifier but still ensure a large amplification ratio. The XY stage is usefully potential for locating the specimens in a nanoindentation tester system. The material Al 7075 is selected for the proposed stage. This material possesses a few good properties such as a yield strength of 503 MPa, Young’s modulus of 71700 MPa, density of 2810 kg/m^{3}, and Poisson’s ratio of 0.33. By checking initial input displacement through the FEM simulations, specification of XY stage is supposed that input displacement is 120 μm. The parameters of XY stage is proposed in Figure 1(b). It consists of elements as follows: (i) nineteen holes are utilized to fix the stage on an unvibration table, (ii) a piezoelectric actuator (PZT) is combined with translational crew to generate the input displacement for the stage. The symmetric fourlever amplifier is integrated with elliptical hinges to generate a large stroke. The entire dimension of the stage is approximately 328 mm × 328 mm × 10 mm. In this research, elliptical hinges are proposed to integrate into XY stage because its benefits (e.g., minimal rotation axis shift, good safety factor, and large angular deflection) [40]. Besides, the zigzagbased flexure hinges and leaf hinges are developed and integrated in the stage in order to reduce parasitic motion and obtain more large displacement. The main geometric parameters of the proposed XY stage are provided in Table 1. Therefore, some main parameters of the stage should be optimized in improving the quality characteristics of the XY stage.
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2.2. Formulation of Optimization Problems
Based on the proposed scheme of the XY monolithic mechanism (see in Figure 1), the initial simulations determine that there are five main design parameters affecting the entire performances of the mechanism. The first parameter is the thickness of hinge A being located at the first lever amplifier. The second parameter is the thickness of hinge B which is utilized to transfer the motions from the first lever amplifier to the second lever amplifier. The third parameter is the thickness of hinge C locating at the second lever amplifier. The fourth parameter is the length E of the second lever amplifier. The fifth parameter is the thickness of hinge D which is employed to transfer the motion from the zigzag driving mechanism to the middle shuttle platform. The ratio of length of the level is much sensitive to the static and dynamic performances of compliant mechanisms. If the length is increased, the entire size of the system is also increased. Hence, it is assumed that the length ignored the length of lever during the optimization because this work is aimed to design a compact size for the mechanism which is tended to be integrated into the insitu nanoindentation tester.
It is observed that the proposed XY mechanism should possess multiple excellent performances, namely long fatigue life, large stroke, high safety factor, small stress, high resonant frequency, minimal parasitic motion, and so on. In the scope of this article, the three main performances are considered, including the large stroke, the high safety factor, and the minimal stress. For many different aspects in a few practical applications, the singleobjective optimization concerning scenarios #1, 2, 3 are desired for the mechanism while the scenario #4 handles multiperformances for the mechanism, simultaneously. In this paper, the particle swarm optimization (PSO) [41] is combined with Grey Wolf Optimizer (GWO) [42] optimizer, socalled PSOGWO [43], is employed to solve the single objective optimization problems for scenario 1–3. Three numerical problems are considered to demonstrate the efficiency of the proposed optimizer as follows.
2.2.1. Scenario #1
Find design vector:
Range of design variables (unit: mm):
2.2.2. Scenario #2
Find design vector:
Range of design variables (unit: mm):
2.2.3. Scenario #3
Find design vector:
Range of design variables (unit: mm):where x_{1}, x_{2}, x_{3}, x_{4}, and x_{5} are the design parameters corresponding to A, B, C, D, and E. The considered objective functions consist of the stroke F_{1}(x), the stress F_{2}(x), and the safety factor F_{3}(x).
In a real product, multiple functions of the mechanism are simultaneously required. Besides, the stroke is often conflicted with the safety factor or the stress. Therefore, the multiobjective optimization genetic algorithm (MOGA) optimizer [44] is utilized to solve a tradeoff between the stroke and the safety factor. The multipleobjective optimization problem for the mechanism is defined by scenario #4.
2.2.4. Scenario #4
Find design vector:
Subject to constraints:
Range of design variables (unit: mm):
3. Research Method
This part provides a new approach in optimizing the performances of the XY positioning mechanism. In this context, the proposed method is developed with three subphases. In the first subphase, the simulations for the mechanism are carried out by constructing a finite element model, setup parametric variables for inputs and outputs, perform finite element simulations, and collect datasets. In the second subphase, the data are put input into the ANN program, and the particle swarm optimization (PSO) algorithm is employed to train the artificial neural network (ANN). In the last subphase, the output performances are optimized by the PSOhybridized with grey wolf optimization (GWO) algorithm. The flowchart of modeling and optimization for the mechanism is given in Figure 2. Details of three subphases are presented below.
The present study is aimed to recommend the ANNbased PSO into the dimension synthesis of compliant mechanisms. The presented method can be the efficient assistance for further developing intelligent algorithms into the analysis and synthesis of engineering design.
3.1. Simulation of XY Monolithic Mechanism
In order to collect the performances of the mechanism, this section simulates the mechanism through the following detailed multiplesteps, as shown in Figure 3.(i)The mechanical model of the XY positioning mechanism is initialized.(ii)The key design variables (A, B, C, D, E), output performances and constraints (stroke, safety factor, and stress), are determined.(iii)Regarding material, Al7075 T73 is employed.(iv)The boundaries conditions are setup, and a load of N from the piezoelectric actuator is applied to the input port of the mechanism (see Figure 1).(v)The simulation environments are setup regarding the nonlinear characteristics in simulations. The nonlinearity means a large deformation of hinges.(vi)The variables (inputs, outputs, and constraints) are parametric.(vii)The 27 numerical experiments are planned, and the datasets are collected.(viii)If the performances are satisfied with the initial requirements, the process is ended herein. Otherwise, it goes back to proceed the parametric variables.
3.2. Artificial Neural Network Algorithm
The performances of the mechanism are modeled using the artificial neural network (ANN) method [45, 46]. The ANN is often utilized as modeling technique based on the reasoning of human brain. The ANN is considered as a basic type of deep learning method. A basic ANN includes neuron nodes, layers (input, hidden, and output), weights, bias, and activation functions. A primary formula of ANN is formulated as follows.where z is the output value of k^{th} neuron node, b is bias, W is weight, and x is input node. N is number of input nodes.
In this work, there are five main inputs, the design parameters of the mechanism (A, B, C, D, E), the number of nodes (n_{node}) in a hidden layer is computed by following formula.
In this study, the feedforward algorithm is employed for training the ANN architecture. An ANN architecture is primarily illustrated as in Figure 4.
3.3. Modeling the Performances Using the PSOANN Algorithm
The effectiveness and accuracy of an ANN architecture depends on several components such as training algorithm, activation function, number of neurons, number of hidden layers, weight, and bias. In the limitation of this context, the bias and weight of the feedforward ANN algorithm are optimized by the PSO optimizer [47]. The mean squared error (MSE) is considered as the objective function of the PSO algorithm, which is described by the formula as.where p is the simulated value and is the estimated value, and m is input size.
The PSO optimizer is aimed to update the weights and bias to determine the best ANN architecture. This method goes several steps in pseudocode of Algorithm 1.

3.4. Optimization for XY Positioning Mechanism by PSOHybridized GWO
In order to solve the four scenarios of the XY positioning mechanism (see in section 2), the three first scenarios in equations (1)–(3) are handled by the PSOhybridized GWO algorithm [43], as depicted in Figure 6. The last scenario in (4) is solved by the MOGA optimizer [44], as demonstrated in Figure 7.
The pseudocodes for the PSOGWO algorithm and the MOGA are shown in Algorithm 2 and Algorithm 3, respectively.


4. Results and Discussion
4.1. Simulation Results of XY Positioning Mechanism
In this part, the numerical simulations are conducted to achieve the datasets which are employed for the modeling and optimizing of the mechanism. In this article, there are five main parameters (A, B, C, D, E) directly affecting the stroke, stress, and safety factor of the proposed mechanism. Based on the number of design variables, the twentyseven numerical design points are planned. An input displacement of 120 μm from the actuator is applied to the input part, the output performances (stroke, stress, and safety factor) are collected. Considering the finite element simulations, a nonlinear analysis is employed. The boundary conditions and loads are recalled in Figure 1. Tennode tetrahedral elements are employed for the mesh. The results found that there are about 149242 elements and 275268 nodes. The mesh metrics is measured through Skewness criterion to ensure the analysis convergence. The entire inputs and outputs are resolved as parametric variables. The datasets are given in Table 2.
The results of Table 2 remarked that the safety factor is increased when the stress is correspondingly decreased. It is also noted that the stroke is always conflicted with the stress. The stress is considered as equivalent von Mises stress.
4.2. Modeling Results by PSOBased ANN
First of all, the data in Table 1 are embedded into the feedforward ANN program. The data are separated into the training, testing, and validating subsets. In this article, a hidden layer is suggested for the ANN architecture. By using the (6), the number of nodes of hidden layers is equal to 11. Subsequently, the formulated ANN program is transferred to the PSO optimizer to make a hybrid PSOANN algorithm. The main parameters of the PSO operator consist of population size of 50, inertia weight of 1, inertia weight damping ratio of 0.9, personal learning coefficient of 1.5, global learning coefficient of 2, and max iteration of 200. The PSO is applied to minimize the MSE value of the ANN and determine the optimal values of weights and bias. Summarily, the hybrid PSOANN algorithm is extended to build the fitness functions and constraints for the XY positioning mechanism. The proposed PSOANN code is performed in MATLAB R2019b.
Figure 8 presents the modeling results for the stroke. It includes the training state, performance, error histogram, and training process in Figures 7(d), respectively. The results found that the correlation coefficients (R) of the stroke’s ANN modeling for the entire data, training, testing, and validating are approximately 1. This means that the built ANN architecture is a reliable and accurate tool in modeling the stroke.
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The second step is focused on developing the ANN structure for the stress. The PSOANN hybridization is also applied to model the stress. The training, performance, error histogram, and correlation coefficient are provided in Figures 9(a)–9(d). It showed that the ANN structure has a high efficiency in modeling the stress with the correlation coefficients being close to 1.
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Lastly, the safety factor is modeled via using the PSOANN algorithm. The results indicated that the modeling of safety factor has a high accuracy with good correlation coefficients of entire data, training, testing, and validating being nearly 1, as depicted in Figure 10.
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Besides, the mean squared error (MSE), root mean squared error (RMSE), and coefficient of determination (R^{2}) for each model are calculated and given in Table 3. The results note that the R^{2} values are almost close to one. To sum up, this section establishes the fitness function and constraints with a high accuracy for the XY positioning mechanism.
4.3. Optimization Results
This part presents the optimization process and the optimal solutions. The proposed PSOGWO hybridization is carried out to solve the single objective function in three first scenarios (1–3) in MATLAB R2019b. The key parameters of this optimizer include an initial population of 30, c_{1} of 0.5, c_{2} of 0.5, and c_{3} of 0.5. The whole of the fitness functions from the established PSOANN are embedded into the PSOGWO. Regarding the optimization of the scenario 1 in equation (1), a max iteration of 200 is set up for optimizing the stroke. For the scenario 2 in equation (2), a maximum iteration of 500 is set for minimizing the stress. For the scenario 3 in equation (3), a max iteration of 1500 is set for maximizing the safety factor. The results are summarized in Table 4.
Considering the scenario 1 (i.e., maximize the stroke), the stroke achieved from the PSOGWO hybridization (1852.9842 μm) is a little better than the one gained from the GWO (1802.8087 μm). Regarding the scenarios 2 (i.e., minimize the stress), the stress got from the PSOGWO hybridization (243.3183 MPa) is smaller than that got from the GWO (245.0401 MPa). Considering the scenario 3 (i.e., maximize the safety factor), the safety factor gained from the PSOGWO (1.9767) hybridization is greater than that achieved from the GWO (1.9278). Besides, it found that the PSOGWO improved optimization strategy is superior to the original GWO in terms of computational efficiency (i.e., fast convergence speed) and better results.
In order to solve the scenario 4 (i.e., maximize the stroke and maximize the safety factor simultaneously), the MOGA optimizer is applied. The optimal results found that the stroke is about (1741.3 μm) and the safety factor is equal to 1.8929. Finally, the optimal design variables are found A = 0.9 mm, B = 0.5 mm, C = 0.5 mm, D = 0.5 mm, E = 73 mm, as given in Table 4.
The convergent histories of the scenario 1 (stroke), the scenario 2 (stress), and the scenario 3 (safety factor) achieved from the PSOGWO hybridization and GWO are provided in Figure 11 (1c), respectively.
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To sum up, the optimal results in the four scenarios are totally satisfied with the initial design targets of the XY positioning mechanism.
4.4. Verification Results
First of all, the optimal results are verified by using finite element analysis in ANYS 2019R2 software. The comparison results showed that the optimal results achieved from the proposed optimizers are wellagreed with the verification results. In comparison with the initial design, the optimal performances of the XY positioning mechanism are largely improved when the MOGA optimizer is applied, as provided in Table 5.
Compared with the previous designs of XY mechanisms, the present design proposed a larger stroke than that of previous studies [48–50]. It is found that the entire dimension of the presented XY mechanism is bigger than that of the other designs, as given in Table 6. However, the presented XY mechanism has a large enough dimension for locating the testing material samples in nanoindentation devices.
Finally, by using the optimal parameters (A = 0.9 mm, B = 0.5 mm, C = 0.5 mm, D = 0.5 mm, E = 73 mm), the prototype of the XY mechanism is fabricated by wire electrical discharged machining. The experiments are conducted to measure the stroke of the mechanism, as provided in Figure 12. The experimental results found that the stroke is about 1632 μm. The error between the experimental result and optimal result is 6.27%. Therefore, this result is close to the predicted stroke from the MOGA (1741.3 μm).
5. Conclusions
This paper has proposed a new modeling and optimizing approach applied to solve the optimization of the XY positioning mechanism. The fitness functions and constraints of the mechanism are built via proposing a combination of ANN and PSO method. The PSO helps to improve the accuracy of the ANN by finding the optimal weights and bias. Next, the PSO is hybridized with the GWO, namely PSOGWO, is employed to three scenarios in solving the single objective function. In order to search the multiple functions for the mechanism, the MOGA method is applied to the scenario 4. The achieved results from the PSOANN, PSOGWO, and MOGA methods are drawn as follows:(i)The fitness functions are wellbuilt through the PSObased ANN method with the good metrics (MSE, RMSE, and R^{2}).(ii)In the scenario 1, the stroke achieved by the PSOGWO hybridization (1852.9842 (μm)) is a little better than the one gained from the GWO (1802.8087 (μm)).(iii)In the scenarios 2, the stress gained from the PSOGWO hybridization (243.3183 (MPa)) is smaller than that achieved from the GWO (245.0401 (MPa)).(iv)In the scenario 3, the safety factor got from the PSOGWO (1.9767) hybridization is greater than that achieved from the GWO (1.9278).(v)In the scenario 4, the MOGA optimizer is applied. The optimal results found that the stroke is about (1741.3 μm) and the safety factor is equal to 1.8929.(vi)The prediction by the proposed methods are relatively fit with the numerical and experimental verifications.
From the modeling and optimizing results, the proposed techniques proved a good ability to handle the design problems for different compliant mechanisms and other engineering problems in the future studies.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work belongs to the project (grant No: T202287) funded by Ho Chi Minh City University of Technology and Education, Vietnam.