#### Abstract

Compliant mechanisms with flexure hinges have been widely applied for positioners, bioengineering, and aerospace. In this study, a new optimized design method for the mobile microrobotic platform was developed for the polishing robot system. A metaheuristic-based machine learning technique in combination with finite element analysis (FEA) was developed. The designed platform allows three degrees of freedom with two *x*-and-*y* translations and one *z*-axis rotation. A new hybrid displacement amplification mechanism was also developed using Scott-Russell and two-lever mechanisms to magnify the workspace of the platform. The leaf hinges were employed due to their large rotation, and the right circular hinges were adopted because of their high accuracy. In modeling the behaviors of the developed platform, the artificial neural network is formulated in combination with the teaching-learning-based optimization (TLBO) method. The ANN architecture was optimized through TLBO to a better approximation. And then, three optimized case studies were conducted by the TLBO. The data is collected through FEA simulation. The modeling results from the TLBO-based ANN were well established with excellent metrics of *R*, *R*^{2}, and MSE. The optimized results found that the proposed MPM platform achieves a max-y stroke of 1568.1 *μ*m, max-*x* stroke of 735.55 *μ*m, and max-*θ* rotation angle of 2.26 degrees. The proposed MPM platform can operate at a high displacement amplification ratio of over 9.

#### 1. Introduction

Compliant mechanisms play a vital role in ultrahigh precision engineering, such as stable switch [1, 2], vibration-assisted cutting [3], manipulations/microgrippers [4], fast servo in precision machining, energy harvester [5], alignment of optics [6], robotics [7], and so on. Compared with rigid-link counterparts, compliant mechanisms can propose a high resolution with precise smooth motion due to the excellent advantages such as without backlash, no friction, reduced assembly, cheap manufacture, and monolithic structure.

Currently, many planar compliant mechanisms from one degree of freedom (DOF) to three-DOF motions have been developed by using series architecture, parallel chain, or hybrid series-parallel type. The one DOF mechanisms often have a high accuracy with a minimal parasitic motion, but these mechanisms have still limited in some applications, e.g., positioners [8]. Then, two DOF mechanisms have been designed to propose more complicated applications, i.e., scanner [9]. The two DOF mechanisms possess a decoupled property. Although one or two DOF mechanisms can achieve a wide stroke, simple control, and high accuracy but their applications are still limited. Therefore, three DOF mechanisms have been developed as alternatives for many planar applications as a positioner, manipulation, and so forth [10]. However, the workspace of two translations and one rotation of the existing three DOF mechanisms are still small. To overcome such drawbacks, a kinematic structure with better properties is needed to provide a high load capacity, large stroke, high safety factor, and high stiffness. Hence, three DOF mechanisms have attracted much attention and become a hot topic for researchers.

Generally speaking, complaint mechanisms, which are acted by piezoelectrical actuators (PZT), have limited workspace. To overcome this drawback, many displacement amplification mechanisms were proposed to amplify the stroke of PZTs, such as Scott–Russell mechanism, lever and bridge types [11]. In addition, a lot of other researchers have also designed many different types of three-DOF compliant positioning platforms with desired characteristics. A micropositioning stage with 3-DOF was designed [12]. In this study, the compliance matrix and finite element method were utilized to build the stiffness and the input coupling ratio of the stage. Besides, the parameters of the stage were optimized to minimize the input coupling ratio. A 3-DOF spatial precision manipulation was designed and analyzed [13]. The translational and angular displacements were analyzed in this article. Besides, a 3-DOF translational mechanism was proposed, and it was analyzed via the pseudo-rigid-body model (PRB) method [14]. By using the PRB technique, another 3-DOF mechanism with two translations and one rotation was designed and analyzed [15]. This type for nanopositioning application was analyzed by a compliance matrix [16].

Although the discussed 3-DOF stages have been designed with multiple excellent characteristics, but the structure is still complicated. Moreover, the workspaces are still limited. Considering an application of 3-DOF compliant mechanisms in the robots, a planar micropositioning platform was designed, and the manufacturing error was analyzed [17]. Almost the behavior analysis of the previous stages employed some popular analytical techniques, such as PRB and compliance matrix. With high nonlinear characteristic behavior, modeling of them has a large error. This causes a large manufacturing error, decreasing the practical positioning ability. To overcome this obstacle, a new approach based on machine-learning-based methods and metaheuristics is devoted in the present article. The artificial neural network (ANN) is combined with the teaching-learning-based optimization algorithm (TLBO) in modeling the behaviors of a new *XYθ* mobile positioning microrobotic platform. The developed microrobotic platform can basically be applied for vibration-based polishing robot applications.

Motivated by the gaps between the existing studies, this paper presents an optimized design method for a three-DOF mobile microrobotic platform for use in polishing robot application. The developed platform is able to provide a large workspace in the *x*-and-*y* translations and rotation around the *z*-axis. In modeling the behaviors of the proposed microrobotic platform, artificial neural network is adopted to resolve the stroke and safety factor. To overcome the ANN limitations, the TLBO algorithm was extended to optimize the ANN approximate accuracy. Then, the geometrical factors of the proposed microrobotic platform were optimized by adopting the TLBO algorithm. Finally, three case studies are considered to confirm the accuracy and effectiveness of the proposed methodology.

#### 2. Conceptual Design of *XYθ* Mobile Positioning Microrobotic Platform

A basic application of the *XYθ* mobile positioning microrobotic (MPM) platform is used for manipulations and precise sample positioning from sub-micrometer to hundreds of micrometer scales. Figure 1 illustrates a design scheme of the MPM platform. The proposed MPM platform utilizes three piezoelectric stack actuators (PZT) to actuate an input displacement to three corresponding robotic legs (robotic leg #1, robotic leg #2, and robotic leg #3).

By arranging three robotic legs around a circle with 120 degrees and three PZTs located in a tripedalism, so-called tripedal topology, the MPM platform can generate a locomotion in three DOF on a planar surface. It means that the platform includes three main motions, such as two translations along the *x*-and-*y* axes and one rotation (*θ*_{z}) around the *z*-axis.

Overall, the MPM platform was manufactured with a monolithic flexure-based mechanism. The fabrication will be carried out via wire electrical discharged machining (WEDM). Each robotic leg was also a flexure structure that consists of a hybrid displacement amplification mechanism (HDAM) in combination with a leaf hinge. The robotic leg #1 was defined in a local coordinate of *O*_{1}*X*_{1}*Y*_{1}. The robotic leg #2 and the robotic leg #3 were defined in a local coordinate of *O*_{2}*X*_{2}*Y*_{2} and *O*_{3}*X*_{3}*Y*_{3}, respectively. More details of the HDAM are presented in next section. Under actuating the three PZTs simultaneously, the mobile platform of the microrobot makes two translations *δx*_{1} and *δy*_{1}, and a rotation *θ*_{1}.

Technical requirements and specifications of the MPM platform in the design phase are expected to achieve large strokes in the translations over 1000 (*μ*m) or higher than 1 mm and a wide rotation. Furthermore, a high safety factor of over 1.8 is required. The mentioned importantly technical specifications of the MPM platform can fulfil the practical applications. In addition, Al 7075-T651 is chosen to manufacture the microrobotic platform. The properties of Al 7075-T651 are listed, including a density of 2810 kg/m^{3}, Poisson ratio of 0.33, yield stress of 503 MPa, and Young’s modulus of 71.7 GPa.

Figure 2 illustrates the assembly scheme of *XYθ* mobile positioning microrobotic platform.

As shown in Figure 2, it includes the following key components:(1)Preload crew,(2)PZT mounting plate,(3)PZT actuator,(4)Intermediate plate,(5)Prototype,(6)Anti-vibration fixing plate,(7)Fixed hole.

As depicted in Figure 2, the prototype of the proposed microrobotic platform was mounted on the intermediate plate. The PZTs were fixed on the PZT mounting plate, and the preload screw was employed to adjust the PZT in contact with the input port of the platform. Finally, the whole of the system was put on the anti-vibration table.

A basic application of the proposed MPM platform is able to be employed for polishing robot system, as given in Figure 3. The proposed platform is mounted on the station. The polished sample is located on the mobile platform through fixing screws while the end-effector of the robotic arm brings the polishing tool.

When three PZTs act, the platform causes a micro-vibration for the sample. The micro-vibration is aimed to reduce the friction between the sample and the polishing tool. This leads to improvement of the surface roughness of the final workpiece. This machining process is considered as a vibration-assisted polishing process.

The dimensional scheme of the proposed MPM platform is provided in Figure 4, and the main dimensions are given in Table 1. The thickness of the platform in the out plane (*z*-axis) is 8 mm.

##### 2.1. Analysis of Hybrid Displacement Amplification Module

Figure 5 provides a new hybrid displacement amplifier. The suggested HDAM is built by a combination of Scott–Russell mechanism (SRM) amplifier with a two-lever displacement (TLD) amplifier. The hybrid amplifier is moved based on the deformation of right circular hinges. In the beginning, an input displacement of 135 *μ*m along the *x*-axis is acted to the SRM amplifier, and this displacement amplifier is rotated around the fulcrum (1) and then, the output motion of the SRM is transformed to the input port of the TLD amplifier, and this mechanism is rotated about the fulcrum (2) the output displacement is collected along the *y*-axis. Finally, the output motion of the proposed HDMA is kept to transfer to the leaf hinge (see Figure 1) so that the MPM platform is moved.

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To illustrate the amplification ratio of the proposed HDMA, the proposed HDMA is meshed and simulated by finite element analysis (FEA) ANSYS 2019R1 software. The number of nodes and elements are about 29047 and 16867, respectively. The quality of the mesh is measured by the Skewness technique with an average value of 0.44906. The results of the HDMA are provided in Table 2.

The results of Table 2 indicates that the amplification ratio of the proposed hybrid amplifier is about 12.43 with a high safety factor (SF) over 1.4 when the input displacement is from 90 *μ*m to 145 *μ*m. Besides, the stress is still lower than the yield stress of the material (503 MPa).

##### 2.2. Initial Evaluation of Static and Dynamic Behavior of Microrobotic Platform

In order to evaluate the initial specifications of the proposed MPM platform, the static and dynamic behaviors are simulated by ANSYS software. The three PZTs are employed simultaneously with 135 *μ*m, and the output stroke/displacement of the robotic leg #1 is measured. Figure 6(a) shows the boundary conditions for simulating the platform. The number of nodes is 71202, and the number of elements is 41045. Skewness average value is about 0.4877, as given in Figure 6(b).

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Figure 7 depicts the stress concentration. It is found that the high stress appeared on the surfaces of leaf hinges and right circular hinge.

The deformation of the MPM platform is provided in Figure 8.

The initial evaluation showed that the amplification ratio of the proposed MPM platform is about 9.85, with a high safety factor (SF) over 1.7 when the input displacement is from 90 *μ*m to 145 *μ*m. Besides, the stress is still smaller than the yield stress of material (503 MPa), as depicted in Table 3.

The dynamic behavior is achieved by FEA simulations. The four natural frequencies for the first mode shapes include 102.036 Hz, 113.81 Hz, 113.9 Hz, and 154.84 Hz, respectively, as provided in Table 4. Considering a resonance of the proposed MPM platform with the PZTs and others, the first mode shape is a *z*-axis translation. The second mode shape is the *x*-axis translation. The third mode shape is the *z*-axis translation. Finally, the fourth mode shape is the *z*-axis rotation.

##### 2.3. Formulation of Optimization Problems

The characteristics of the proposed MPM platform are desirable to gain the two main design targets, including a large stroke (*δy*_{1}) and a high safety factor.

When the stroke is enhanced, the rotation of the platform (*θ*_{1}) is also improved. A good SF over 1.8 can ensure a long working time. Based on the initial evaluations in the previous parts, it determined that the performances of the proposed MPM platform are strongly affected by varying the thickness values of right circular hinges (*A*, *B*, *C*, *D*) and the thickness of the leaf hinges (*E*).

Three optimization problems of the proposed MPM platform are considered as follows. Case #1.: maximize the stroke Find design variables: Bounds of design variables (unit: mm): Case #2.: maximize the safety factor Find design variables: Bounds of design variables (unit: mm): Case #3.: maximize the stroke and the safety factor simultaneously (multi-objective optimization problem) Find design variables: Bounds of design variables (unit: mm): where *X* is a vector of design variables. Parameters *A*, *B*, *C*, and *D* are the thickness of right circular hinges. Parameter *E* is the thickness of leaf hinges. The stroke and safety factor are represented as *f*_{1}(X) and *f*_{2}(X), respectively.

#### 3. Proposed Modeling and Optimization Method

As designed in Figure 1, the proposed MPM platform is a monolithic architecture with three robotic legs. The translations and rotation motions of the platform are totally based on the elastic motions of the leaf hinges and right circular hinges.

Because the MPM platform is built using the concept of flexure-based mechanism, so-called compliant mechanism, it inherits many excellent properties such as low weight, reduced assemble, simple fabrication, and without kinematic joints in comparison with rigid-link counterparts. Nevertheless, mathematical equations in modeling of the static behaviors of the MPM platform is difficult to exactly formulate because it has not kinematic joints. Therefore, the leaf hinges and right circular hinges are treated as virtual joints.

As a result, a modeling method based on ANN is chosen in approximating the stroke and the safety factor. In order to enhance the prediction ability of the ANN, the TLBO algorithm is employed. And then, the TLBO is extended to handle the three optimization cases of the MPM platform. The flowchart of the proposed modeling and optimization techniques is provided in Figure 9.

##### 3.1. Simulation Technique for Microrobotic Platform

In order to collect the data of the performances of the MPM platform, the FEA implements are carried out, as seen in Figure 10. With five design variables, twenty-seven experimental samples are made.(i)Build 3D model of the proposed MMP platform.(ii)Design variables (*A*, *B*, *C*, *D*, and *E*) and output performances (stroke and safety factor) are parameterized.(iii)Define properties of material Al 7075-T651.(iv)Determine boundary conditions and a load/input displacement from PZT.(v)Simulate the MPM platform by finite element method (FEM).(vi)Collect the data.(vii)If the data sets are not satisfied, it will return to adjust the range of variables.

##### 3.2. ANN Optimization by TLBO

In this study, feedforward-learning ANN technique is selected to formulate the modeling of stroke and safety factor for the proposed MPM platform. Basically, ANN is operated based on human brain [18]. In the reasoning of ANN, the geometrical parameters and output responses of the MPM platform are embedded into the programming. An ANN programming includes three main signals such as input, hidden, and output layer. To effectively operate, the learning rate, momentum rate, bias, minimum error, and activation function should be appropriately defined. Operation of the ANN can gain a high effectiveness when it can ensure a minimal training error. This can be well done when the weight and bias are reasonably updated.

Although the ANN can build nonlinear behavior modeling but the accuracy is still strongly dependent on its controllable factors. To solve this limitation, the TLBO [19] is applied to optimize the ANN architecture. One of the most problems is how to define exactly the number of hidden nodes in hidden layer. The following equation is utilized to resolve this problem.

With five design variables corresponding to one output performance, the hidden layer is 11 nodes. An optimization of ANN by TLBO is provided in Figure 11.

In the optimization problem, the objective function is mean square error (MSE) which is defined as below:where, *t* is the measured target and is the predicted target, and *k* is the dimension of inputs, so-called the number of data points.

Additionally, the coefficient of determination (*R*^{2}) is computed to estimate the regression model:where *t* is the actual target, is the predicted target, and is the average target.

##### 3.3. Optimization of Microrobotic Platform by TLBO Method

According to the TLBO algorithm, a good teacher can train a better learner. The task of teachers in a classroom is critically important [19]. The leaner is a population where a vector of design is a course vector. The two main strategies of the TLBO include teaching and learning.

###### 3.3.1. Teaching Strategy

The teacher strategy proposes some key ideals as follows.(i)Search the teacher with best solution from the population.(ii)Determine the mean results of learners (*M*_{j},_{i}) with respect to a specific subject.(iii)The teacher’s ability affects the quality of students by following equation.where, *Dm*_{j,k,i} is the increased mean value. *X*_{j,kbest,i} is the best learner (i.e., teacher) in *j*th subject. *T*_{F} is the teaching factor. *r*_{j,i} is a random value in [0, 1]. The *T*_{F} value is either 1 or 2. The *T*_{F} value is randomly determined by the following formula:

After that, the existing solution is updated by the following equation in the teacher strategy.where, is the updated value of . If the results of this phase are satisfied, and then, they are considered as inputs for the learner strategy.

###### 3.3.2. Learning Strategy

The learners can study somethings from other students in a classroom. At any iteration *i*, a learner is compared with the other learners. Specifically, *U* and *V* are two learners which are compared together () by following formula. is accepted when the value of objective function is better. Flowchart of the TLBO method is given in Figure 12.

#### 4. Results and Discussion

In this part, modeling behaviors of the MPM platform is provided. Besides, the optimization problems of the proposed platform are performed. The optimized results are validated.

##### 4.1. Setup of Simulations and Data Collection

From Figure 5, the boundary conditions are seen. Three input displacements from three PZTs are acted simultaneously. The stroke (*δ*_{y1}) along the y-axis is measured. Besides, the safety factor is calculated. AL 7075-T651 is employed for the platform. The results of 27 experiments are given in Table 5.

##### 4.2. Parametric Evaluation

To assess the associations of the geometrical parameters to the behaviors of the proposed MPM platform, analysis of variance (ANOVA) is adopted to solve this issue. The ANOVA results of stroke are given in Table 6. Moreover, the sensitive plot of whole inputs to the stroke is illustrated in Figure 13. The results indicated that the contributions of the parameters are listed as follows: C (37.65%), D (12.47%), E (7.20%), B (0.61%), and A (0.47%).

As shown in Table 7, the contributions of the input parameters on the safety factor are ordered as follows. The highest contribution is C (29.35%), A (5.78%), E (1.43%), B (1.98%), and D (0.06%), as provided in Figure 14.

##### 4.3. Modeling Behaviors of Microrobotic Platform by ANN-Based TLBO

Modeling behaviors of the MPM platform is carried out through the ANN. To improve the effectiveness of the ANN technique, the TLBO is embedded into the ANN programming. Firstly, the collected data in Table 5 comprised of training, testing, and validating. The optimized ANN architecture can find the best weights and biases. The modeling accuracy of the optimized ANN is assessed by metric computation of the MSE and *R*^{2}.

Furthermore, the correlation coefficients (*R*) are also computed. The modeling results of the stroke and safety factor achieved very well with high *R* values, as plotted in Figures 15 and 16(a), respectively. The best performance, the prediction error, and the difference among the prediction and numerical values are provided, as seen in Figures 15, 16(c), and 16(d), respectively.

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As depicted in Figures 15 and 16, the proposed artificial intelligent technique had better performances than those achieved from the linear regression.

##### 4.4. Parameter Optimization

In this part, the TLBO algorithm is initialized with a population of 50 and iterations of 5000. The optimization programming is implemented MATLAB R2019 environment. The optimized results for the three case studies are provided in Table 6. From Figure 17, the rotation angle (*θ*_{z}) around the O point of the proposed MPM platform is measured by FEA ANSYS software. From Table 8, the y-stroke is the displacement along the *y*-axis at *O*_{1} point. The *y*-stroke is the optimized displacement which is predicted from the proposed metaheuristic-intelligent method (ANN-TLBO). The *x-*stroke is the displacement along the *x*-axis at O_{1} point. The *x*-stroke, the stress and the rotation angle are calculated from the FEA ANSYS software.

From the achieved results of Table 8, it revealed that the optimized strokes in the *y*-axis of the MPM platform can obtain 1555.6763 *μ*m, 1300.6 *μ*m, and 1568 *μ*m for case #1, case #2, and case #3, respectively. Besides, the *x*-axis strokes of the platform are 266.4 *μ*m, 735.55 *μ*m, and 714 *μ*m for case #1, case #2, and case #3, respectively. The safety factor of the platform is over 1.5. Meanwhile, the stress appeared in three case studies is always lower than the yield stress (503 MPa) of AL 7075-T651. This guarantees a long working strength for the platform. The stress is calculated by the following equation.where, *S* represents the stress of the MPM platform. *S*_{yield} is the yield stress of AL 7075-T651. *SF* is the safety factor.

Based on the output stroke of the proposed MPM platform, the displacement amplification ratio can be calculated by following formula.where, *A*_{R} is the displacement amplification ratio. The *O*_{S} and *I*_{S} note the output y-stroke and input stroke.

By using equation (15), the *A*_{R} values are about 11.54 for case study #1, 9.63 for case study #2, and 11.61 for case study #3.

##### 4.5. Validations of Optimized Results

By using the optimized design parameters, the prototypes are built in Inventor software, and then, the simulations are performed to verify the optimized results. As given in Table 9, the errors between the proposed method and the simulation method are under 9%. The proposed method is reliable optimization technique in modeling and optimizing the MPM platform.

#### 5. Conclusions

This article has presented an optimized design method for the mobile microrobotic platform. The proposed MPM platform was built via using two combined modules, including the hybrid displacement amplification mechanism and leaf hinges. The developed HDAM was created by combination of Scott–Russell mechanism and two-double lever amplification mechanism. The new proposed HDAM amplifier could allow a large amplification ratio. With such a high amplification value, it ensured a large output stroke for the MPM platform. The developed MPM platform was able to be employed for locating the sample in the polishing robot system. The platform could achieve three motions, including two translations and one rotation.

In modeling the stroke and safety factor of the MPM platform, the ANN was used in combination with the TLBO method. By using the TLBO, the ANN architecture was optimized to a better approximation. And then, three optimized case studies were studied by the TLBO to improve the stroke and safety factor. Moreover, the case studies also demonstrated the effectiveness of the methodology. In this study, the FEM data was combined with ANN, TLBO for modeling process. The results of this paper could be listed as follows. The modeling results from the TLBO-based ANN were well established. The metrics were relatively good with the values of *R* and *R*^{2} being near 1 while the values of MSE were very small. The established intelligent predictors were better than the linear regression. The predicted values from the TLBO-ANN were close to the measured values. In case study #1, the optimized platform could operate with the *y*-axis stroke over 1558.6763 *μ*m and a safety factor of 1.58. In case study #2, the optimized platform could achieve a large *y*-axis of 1300 *μ*m and a safety factor of 2.3. In case study #3, the optimized platform could displace a large *y*-axis of 1568.1 *μ*m and a safety factor of 2.04. In summary, the proposed MPM platform could achieve a max-y stroke of 1568.1 *μ*m, max-*x* stroke of 735.55 *μ*m, and max-*θ* rotation angle of 2.26 degrees. The stress of three cases were still lower than the yield stress of Al 7075-T651. The proposed MPM platform could achieve a high displacement amplification ratio at least of 9.

In upcoming study, the real prototypes will be manufactured by WEDM. The physical verifications will be carried out. The polishing experiments will be conducted.

#### 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: T2021-11TĐ and was funded by Ho Chi Minh City University of Technology and Education, Vietnam.