1 Introduction

Nanoindentation tester is designed for the measurement of hardness, elastic modulus and creep. The system can be adopted to define characteristics of organic, inorganic, hard and soft materials. An indentation measurement can be carried out in less than 3 min without waiting for thermal stabilization with the unique top surface referencing technique. Therefore, a positioning process has to be high precisely. Materials can be checked, including hard and soft types from tissue, biological cell, semiconductor nanomaterial, optics, material science, biomechanics, micro-electromechanical systems, and electronics (Hu et al. 2013; O’Brien et al. 2005; Nohava et al. 2009). During the indentation process, multiple microscopes are used to record the image of sample before and after indenting test to characterize the curve of displacement versus load while a material sample is bring to in front of microscope. In order to obtain a good image quality, a precise positioning platform is essential to locate the sample material. It means that the positioning platform is an important mechanism for the nanoindentation system. In commercialization, the entire system is difficult to gain high position accuracy due to using servo motors, ball screws, and rigid joint. This leads unfavorable disadvantages such as backlash, friction, and wear. To overcome the disadvantages of traditional technologies, a compliant positioning platform is proposed to improve the resolution of system because of essential merits such as no wear, no backlash, free friction, monolithic structure, light weight, low cost, no lubricant, high precision and compact structure (Xu 2014; Lu et al. 2014; Dao et al. 2017a, b; Polit and Dong 2011; Song et al. 2010; Dao et al. 2017a, b; Fung and Lin 2009). Similar to different applications, the indentation system also requires a large working travel with a high safety factor in order to conduct various positioning tasks.

A few recent years, regarding a large working travel, many studies have been conducted. Some concentrated on the use of various amplifiers (Ling et al. 2016). A large working travel of positioning stages were designed (Dao et al. 2017a, b). Yong et al. (2009) developed the amplification levers for a fast nanoscale positioning. Kang et al. (2005) proposed displacement amplification mechanism integrated platform-type parallel mechanisms using flexure hinges. Kim et al. (2012) utilized a double displacement amplifier for a 3-DOFs positioning system. Xu and Li (2011) presented an optimization of an integration bridge amplification mechanism. Le Zhu et al. (2018) designed an amplifier integrated a Scott-Russell mechanism and a half-bridge mechanism for 2DOFs nanopositioning stage. Unlike previous studies, inspired from a high flexibility in beetle’s motion, the proposed platform is designed so as to reach a small parasitic motion. Subsequently, a new multi-lever displacement amplifier is integrated inside the platform so as to generate a large working stroke. However, the platform is difficult to obtain a large displacement and ensure a good strength, simultaneously through a design process. As known, two quality responses are conflicted together. Therefore, a multi-objective optimization problem is needed to balance between them.

Regarding an multi-objective optimization process, Huang and Dao (2016a, b) optimized a compliant XY positioning stage utilizing the finite element analysis (FEA) and response surface methodology (RSM). Xiao et al. (2011) optimized the stiffness, working travel and safety factor using the radius basis functional network combined genetic algorithm. More recent years, Dao et al. (2017a, b) developed a hybrid Taguchi-cuckoo search algorithm to optimize overall the quality responses, simultaneously for a flexure-based focus locating stage. A hybrid approach of the grey relational analysis, Taguchi, RSM and entropy measurement was combined (Huang and Dao 2016a, b). RSM was integrated with the GA (Álvarez et al. 2009; Bahloul et al. 2013), ANFIS and Jaya (Suraj et al. 2016), GA and ant colony optimization (Zukhri and Paputungan 2013). In addition, Ling et al. (2019) proposed a kinematics-based optimization method for 2-DOF millimeter-range monolithic flexure mechanism. Generally, population-based optimization algorithms often require initially controllable parameters while the teaching learning based optimization (TLBO) could optimize rapidly multi-criteria with less parameters. So, the TLBO is proposed to conduct multi-objective optimization problem for the proposed platform. Prior to implement the TLBO programing, the mathematical equations or surrogate models for both displacement and safety factor should be established well. Because the compliant platform is a non-traditional type of mechanical engineering, the mathematical models are complicated to be established. If these equations may be wrong, the further optimized solutions are not true.

For a real compliant positioning platform, a good relation between design variables and quality responses are difficult to establish precise equations. Therefore, it leads an imprecisely optimized solution. As known, the RSM is considered as a surrogate model and can approximate a regression model but the prediction accuracy of the RSM is strongly dependent on the coefficient of determination. To cover come this limitation, artificial intelligences have been developed for various fields in which adaptive neuro-fuzzy inference system (ANFIS) is one of the most effective approaches. ANFIS is considered as an effective black box via an integration of fuzzy logic and neural network (Wei 2016; Cheng and Wei 2009). So, ANFIS is further adopted for the proposed platform. Even though the ANFIS is still useful for various problems but a basic ANFIS structure are largely dependent following four main controllable parameters as: (1) the number of input membership functions (MFs) of the fuzzy logic, (2) types of input MFs, (3) the training method such as hybrid procedure or back propagation, and (4) type of output MFs. However, most previous studies used these parameters as default and there have no considerations in identifying the suitable parameters of the ANFIS structure for a specific problem. In order to determine the appropriate parameters for ANFIS, the Taguchi method (TM) is utilized to optimize the root mean squared error, which is a performance metric of ANFIS. Based on the optimized parameters of ANFIS, a surrogate model is well established for the platform. In addition, for a multi-objective optimization problem, the weight factor (WF) for each response is different. The WF is often chosen based on the experiences or customer’s demands. Unlike previous studies, the WF can determined exactly via establishing the set of statistical equations because the WF depends on the sensitivity of design variables to responses.

The main aims of this study is to develop a new integration approach to conduct a multi-criteria optimization design for the beetle-inspired platform. A multi-lever displacement is designed to amplify the working travel. Sensitivity of design variables is analysed. The optimization principle consists of the following phases. First of all, the numerical data are collected by integrating FEA and RSM. Next, the WFs are computed accurately based on statistical equations. And then, the suitable controllable elements for a ANFIS structure is optimized by the TM. After improving the ANFIS structure, the relationships between the design variables and quality objectives are mapped by the improved ANFIS model. Finally, TLBO algorithm is adopted for solving the multi-objective optimization. A prototype was fabricated and its behaviors are tested to verify the predicted results.

2 Conceptual design

2.1 Compliant positioning platform

Inspired from the beetle’s movement, a new compliant positioning platform (CPP) was designed so as to reach the linear output displacement as well as decline parasitic motion, as depicted as Fig. 1a, b. Moreover, the body and leg movements of beetles are almost quite in the same plane and the high flexibility of their legs. Hence, beetle-like mechanism is suitable to construct a planar motion mechanism. Figure 1a is an image of a real beetle. Based on the shape of each beetle leg, one of its legs can be proposed to divide into five sections. This geometry can be imitated to develop a mechanism for a linear motion generation. Therefore, a model of the proposed CPP with five segments was designed, as shown in Fig. 1b. It included rigid links and flexure hinges. Its motion was replied on the flexure hinges. As a result, the geometric parameters of flexure hinges contributes largely on the performances of the CPP.

Fig. 1
figure 1

Model of beetle-inspired platform: a beetle animal, b compliant positioning platform

Full size image

The beetle-liked mechanism ensures a linear motion along the vertical direction and suppresses a parasitic motion along the horizontal direction. Overall performances of the CPP was assessed regarding a large displacement, a small parasitic motion and a high safety factor. A small parasitic motion could achieve by a direct design but two remain responses have to carry out a further multi-objective optimization problem.

2.2 Displacement amplification mechanism

Traditionally, a lever mechanism is used to amplify the value of force or displacement. Structure of the lever includes of a beam or rigid rod placed on a fixed hinge, or fulcrum, as illustrated in Fig. 2a. Point O is a fixed hinge that is the rotation center of the lever, point A is the input and point B is the output location. The operation principle of lever mechanism is illustrated as follows: when locating a vertical displacement \(\Delta l_{1}\) on the input point A, the lever will rotate an angle \(\gamma\) relative to the z-axis. As a result, the point B moves to B’ and the output displacement \(\Delta l_{2}\) can gain in the vertical direction (y-axis), as illustrated in Fig. 2b. First of all, one lever mechanism can be used to amplify output displacement. However, this mechanism creates easily large parasitic motion. So, a two-lever mechanism was designed to amplify the output displacement and ensure small parasitic motion based on its symmetric structure, as shown in Fig. 3b. Moreover, in order to amplify more large output displacement, four-lever displacement amplifier was proposed in this paper, as depicted in Fig. 4.

Fig. 2
figure 2

Diagram: a operation principle of lever mechanism, b amplification ratio

Full size image
Fig. 3
figure 3

Model of lever amplifier: a one lever mechanism, b two lever mechanism

Full size image
Fig. 4
figure 4

Model of proposed four-lever displacement amplifier

Full size image

Based on a schematic diagram (see in Fig. 2b), the amplification ratio of a lever can be approximately achieved as:

$$r_{lever} = {{\Delta l_{2} } \mathord{\left/ {\vphantom {{\Delta l_{2} } {\Delta l_{1} }}} \right. \kern-0pt} {\Delta l_{1} }} = {{l_{2} } \mathord{\left/ {\vphantom {{l_{2} } {l_{1} }}} \right. \kern-0pt} {l_{1} }}$$
(1)

The equation of multi-lever displacement amplifier is assumed as follows:

$$r_{T} = {{l_{2} } \mathord{\left/ {\vphantom {{l_{2} } {l_{1} }}} \right. \kern-0pt} {l_{1} }} \times {{l_{4} } \mathord{\left/ {\vphantom {{l_{4} } {l_{3} }}} \right. \kern-0pt} {l_{3} }}$$
(2)

where \(r_{T}\) is amplification ratio of four-lever displacement amplifier.

2.3 Beetle-liked platform with amplifier

The operating principle of CPP was based on the elastic deformation of the flexure hinges. In this study, Al T73-7075 was chosen as material for the stage due to its high yield strength of 503 MPa, Young’s modulus of E = 71,700 MPa, light density of 2810 kg/m3 and Poisson’s ratio of 0.33. Specification of linear stage was assumed that input displacement was 0.13 mm, as given in Table 1. As seen in Fig. 5, it consisted of elements as follows: (1) twelve fixed holes were utilized to locate the platform on an un-vibration table so as to test its characteristics, (2) a translational screw (locating at the input displacement position) was used to create the input displacement for the platform by connecting directly with the beetle-liked mechanism. The total dimension of the model was approximately 230 mm × 222 mm × 6 mm. The geometrical parameters of the platform were given in Table 1. According to a good static performances for positioning system, the proposed platform was required to create a large displacement and a high safety factor, simultaneously.

Table 1 Material and structure parameters of the beetle-liked platform
Full size table
Fig. 5
figure 5

Model: a beetle-liked platform, b design parameters

Full size image

Based on the design domain and ability of wire electrical discharged machining (WEDM) method as well as designer’s experience, the initial design parameters of the structure were proposed in order to ensure to achieve a high output displacement and be suitable for fabricating by WEDM method. In this study, the geometric parameters (the length and the width as well as radius) of the flexure hinges also affect to the linear motion of the structure. However, compared to other geometric parameters of the beetle leg mechanism, the thickness of flexure hinge is the most significant effect to linear motion and output displacement according to the Euler–Bernoulli beam theory and theory of compliant mechanism. So, this study considered the thickness of flexure hinge of beetle leg mechanism as a main parameter for establishing design variables and ignored the other geometric parameters.

2.4 A basic application for nanoindentation tester

The working principle of proposed CPP was based on the elastic deformation of the material. It was complied with the Hook’s law. Therefore, the CPP was operated within the elastic limitation. It was intended to apply in nanoindentaion tester. The CPP was utilized for positioning the material sample during nanoindentation testing process. Before indenter translated downward to check the characteristics of the material sample, the CPP would move linearly along the horizontal direction so as to locate the sample to an initially precise position. At the same time, the multiple microscopes were used to take photos of the sample monitoring the initial characteristics of the sample. And then, the indenter would be indented a depth into the testing sample. Next, the CPP would come back a reference position in order to prepare for testing next times. A basic application for nanoindentation tester was proposed in Fig. 6.

Fig. 6
figure 6

Nanoindentaion testing instrument

Full size image

3 Analysis of initial performances and parasitic motion error

In this section, a 3D model of platform was designed by Inventor Professional 2018, and then it was simulated via using the finite element analysis in software ANSYS 18.2. The purpose was to forecast the y-axis output displacement and safety factor of the CPP. To perform this analysis, the automatic meshing method was applied, and then flexure hinges was refined to achieve a good analysis precision, as illustrated in Fig. 7. The boundary conditions were proposed as follows: the platform was fixed at the holes. After a draft platform was designed, the output displacement of the model was investigated by varying the input displacement of translational screw from 190 to 220 µm, respectively in order to check the limitation of static failure and limit maximal input displacement for initial positioning stage.

Fig. 7
figure 7

Mesh generation diagram for the CPP

Full size image

In order to check the limitation of plastic deformation failure and determine the range of input displacement for the proposed stage, the input displacement from translational screw was varied from 190 to 220 µm. The corresponding output displacement was recorded. The results of Table 2 found that the output displacement approaches 3.8193 mm with respect to input displacement of 220 µm. This case, the safety factor was lower than 1 and the resulting stress was about 503.13 MPa which was greater than the critical stress or yield strength of the material (503 MPa), the platform could lead a plastic deformation failure. Therefore, the maximal input displacement should be smaller than 220 µm. Meanwhile, when the input displacement of 190 µm, the resulting stress was approximately 434.52 MPa which was lower than the yield strength of material. The safety factor was about 1.15. In this study, a safety factor over 1.5 was desired so as to ensure a good working life for the platform. As a result, the input displacement was proposed within the range from 10 to 180 µm. By varying the input displacement from 10 to 180 µm, the output displacement in the X-and-Y directions and the parasitic error motion were calculated using Eq. (3). As given in Table 3, the parasitic error motion was relatively small. The results also depicted that the equivalent stress was 297.31 MPa, safety factor was 1.6969, amplification ratio (AR) was 17.3608. However, all results do not meet practical requirements (AR is more than 18). Therefore, an optimization process would be conducted later.

Table 2 The output displacement
Full size table
Table 3 Results of output displacement and parasitic error motion
Full size table

When the platform moved along the y-axis, there was also a motion occurs along the x-axis and at the same time. The undesired movement along the x-axis was called as parasitic motion error. A similar analysis, the numerical simulation results indicated that parasitic motion error is very small with 0.01515% (lower than 1%), as illustrated in Table 3. It could be concluded that the parasitic motion error does not affect significantly to the output displacement accuracy and the proposed platform approaches a good linear movement. The parasitic motion error was defined as follows:

$$e = \frac{x}{y}100\%$$
(3)

where e represents the parasitic error, x and y are the output displacements along the x-axis and y-axis, respectively.

4 Statement of optimization problem

Because the CPP was worked according to elastic deformation of flexure-based joints. As a result, it was affected significantly by geometrical dimensions of flexure-based joints. These dimensions were defined as length of first lever, distance between two levers and thicknesses of flexure hinges. Therefore, in order to enhance the quality responses of the CPP, the main geometrical parameters of the platform should be optimized. The CPP should fulfill the following essential responses as:

  1. a.

    A small parastitic motion to increase the accuracy in linear motion;

  2. b.

    A large output displacement to expand broad positioning capacity of the platform for checking characteristics of material sample;

  3. c.

    A high safety factor to ensure long life of the platform.

A small parasitic error could be achieved via a good design of structure, as described in Sect. 3. However, the displacement and safety factor were difficult to satisfy through a design process. So, it should be enhanced through a further optimization. As known, the safety factor was conflicted with the output displacement. Therefore, so as to trade-off them simultaneously, a new efficient hybrid approach was developed in this study.

4.1 Design variables

Through a few FEA simulations to check initial characteristics, the result showed that the geometric dimensions of flexure hinges were significantly contributed to the displacement and safety factor. In this research, four design variables were considered, consisting of the length of first lever, distance between two levers as well as thicknesses of flexure hinges. The vector of design variables was described as \({\mathbf{X}} = \left[ {t,h,b,k} \right]^{T}\). The limit conditions for the design variables were established based on specialized knowledge and design engineer’s professional skills, which were expressed by:

$$\begin{aligned} 48{\text{ mm }} & \le \, b \, \le {\text{ 53 mm}} \\ 19{\text{ mm }} & \le \, h \, \le { 22} . 5 {\text{ mm,}} \\ 0.9{\text{ mm }} & \le \, t \, \le { 1} . 1\;{\text{mm}} \\ 0.67{\text{ mm }} & \le k \, \le { 0} . 7 4\;{\text{mm}} \\ \end{aligned}$$
(4)

where t, h, b, k are thickness of flexure hinge (amplification mechanism), distance between two levers, the length of first lever, thickness of flexure hinge (beetle’s leg mechanism), respectively.

4.2 Objective functions

The following multiple-criteria functions were considered: (1) the first objective function, \(y_{1} \left( {\mathbf{X}} \right)\), the high safety factor is required to ensure the strength of the platform, requested as high as possible. (2) The second objective function,\(y_{2} \left( {\mathbf{X}} \right)\), the output displacement along the y-axis, desired as large as possible in order to expand working travel capacity of the platform for checking characteristics of material sample. In conclusion, the optimization trouble was depicted as follows.

Design variables was regarded as \({\mathbf{X}} = \left[ {t,h,b,k} \right]^{T}\).

$${\text{Max }}y_{1} \left( {\mathbf{X}} \right)$$
(5)
$${\text{Max }}y_{2} \left( {\mathbf{X}} \right)$$
(6)

In order to conduct the multi-criteria optimal problem, two objective responses are conveyed into a single criterion function. Then, two objective functions were multiplied with a weight factor. The integrated objective function was declared as follows:

$$y\left( {\mathbf{X}} \right) = - w_{1} \times y_{1} \left( {\mathbf{X}} \right) - w_{2} \times y_{2} \left( {\mathbf{X}} \right),$$
(7)

where w1 and w2 are the weight factors of two quality responses, respectively.

4.3 Constraints

The proposed platform was operating in a elastic bound of proposed material, it was required to limitations by:

$$g\left( x \right) = \sigma \le \frac{{\sigma_{y} }}{SF},$$
(8)

where σy is the yield strength of proposed material; and SF is the safety factor.

The safety factor was expected to larger than 1.5 as:on the established

$$\, y_{1} \left( {\mathbf{X}} \right) \ge 1.5.$$
(9)

The output displacement was desired to more than 2.3 mm as:

$$\, y_{2} \left( {\mathbf{X}} \right) \ge 2.3\;{\text{mm}}$$
(10)

where y1 and y2 represent the safety factor and displacement, respectively.

5 Proposed hybrid approach


Considering to solve the multi-criteria optimization problem as mentioned, an effective hybrid approach of the RSM, the TM, improved ANFIS and TLBO was developed. Firstly, the experimental data was collected by RSM. Later on, the improved ANFIS was utilized to define relationship between design variables and quality responses. The improved ANFIS was achieved by using the TM. Based on minimization of the RSME, the TM was used to seek the suitable controllable parameters for ANFIS structure. Those parameters included as: (1) the number of input membership functions (MFs) of the fuzzy logic, (2) types of input MFs, (3) the training method such as hybrid procedure or backpropagation, and (4) type of output MFs. Details about ANFIS, the readers can refer to (Cheng and Wei 2009; Linh and Dao 2018; Chau et al. 2018). Finally, based on the established surrogate model from the improved ANFIS, TLBO algorithm was adopted to conduct multi-criteria optimization, simultaneously. The programing was implemented in Matlab R2017. A flowchart for optimal process of the CPP was illustrated in Fig. 8. The optimization process is divided into key phases as follows:

Fig. 8
figure 8

Flowchart of multi-criteria optimization for compliant positioning platform

Full size image

Phase 1: computer aided engineering design

Step 1: Define optimization problem

Multi-objective optimization problem for the proposed CPP was proposed to enhance two mentioned quality responses, simultaneously.

Step 2: Define design variables and objective functions

The length of first lever, distance between two levers and thickness of compliant joints were determined as input variables. The quality characteristics were considered, including a large output y-axis displacement and a high safety factor.

Step 3: Create 3D model and simulations

A 3D model is designed by using finite element method (FEM). During the analysis, the relationship between stress and strain was calculated as follows:

$$\sigma = E\varepsilon ,$$
(11)

where σ, E, ε are stress, Young’s modulus and strain, respectively.

Step 4: Test initial quality characteristics by FEM simulation

Initial quality characteristics were checked many times in order to limit suitable upper and lower bounds of design variables and two quality output responeses before building design of experiments.

Step 5: Build design of experiments and collect data by integrating of RSM and FEM


The numerical experimentations were collected based on RSM and FEM. Firstly, based on a initial checked 3D model was designed and checked in the FEM in the step 3 and 4. Then, the number of numerical experiments were established via using the central composite design integrated with RSM. Later on, the estimated results for the quality responses were retrieved. The number of necessary experiments was determined by the following equation:

$$N = 2^{(k - f)} + 2k + n_{c} ,$$
(12)

where N was the total of the design points, k was the number of design variables, f was the factorial number (f = 0), and nc = 1 was the number of replicates at the central point of the design space. In this research, k = 4, the factorial number for this circumstance of four design variables is f = 0. Replacing these values into Eq. (12), overall 25 design points are needed for the design of experiments.


Phase 2: define the weight factor

The two quality responses were conflicted. In addition, they had different important weights. Therefore, each objective function can established a WF. So far, the WF was defined according to a design experiences and specialized knowledge. So, in order to calculate the WF more accurately, we followed the methods of Dao et al. (2017a, b) to compute the WF for each response. Details about the WF computing equations, the readers can refer to Dao et al. (2017a, b). Every response function is normalized in the range [0, 1]. The WF is computed as follows:

  1. (1)

    Compute signal-to-noise ratio (S/N)

The S/N ratio is established by the TM and it illustrates a better ability corresponding to a larger S/N ratio value. The-larger-the-better is utilized for two reponses in this study, which was expressed by:

$$\eta = - 10\log \left( {\frac{1}{n}\sum\limits_{i = 1}^{n} {\frac{1}{{y_{i}^{2} }}} } \right),$$
(13)

where n is the number of repetitions of experiment ith, y is the quality feature

  1. (2)

    Normalize the S/N ratio

The normalized S/N ratios of each grade of each response function are computed by:

$$z_{i} = \frac{{\eta_{j} - \hbox{min} \eta_{j} }}{{\hbox{max} \eta_{j} - \hbox{min} \eta_{j} }},$$
(14)

where \(\eta_{i}\) illustrates the S/N value, \(\hbox{max} \eta_{i}\) and \(\hbox{min} \eta_{i}\) are the highest and smallest values of \(\eta_{i}\), respectively, \(z_{i}\) is the normalized value of S/N for the experiment ith of response function (j = 1, 2, …, k), k is the number of response functions.

  1. (3)

    Compute the mean value of normalized S/N ratios

The mean value of normalized S/N ratios at every grade for design variables were calculated for response functions. The S/N ratio values are normalized and the mean value of normalized S/N ratios can be defined as follows:

$$a_{Li} = \frac{1}{{N_{Li} }}\sum\limits_{i}^{m} {z_{ij} } ,$$
(15)

where NLji is number of repetition of grade ith. \(z_{ij}\) is the value of S/N ratio of grade ith of response function jth. aLi is the mean value of S/N ratio of grade ith of every design variable of every response function.

  1. (4)

    Calculate the range of each input variable

The range of the every grade of every input variable was defined as following equation:

$$r_{ij} = \hbox{max} \left\{ {z_{i,j,1} ,z_{i,j,2} , \ldots ,z_{i,j,m} } \right\} - \hbox{min} \left\{ {z_{i,j,1} ,z_{i,j,2} , \ldots ,z_{i,j,m} } \right\},$$
(16)

where \(r_{ij}\) is the range (max–min) of the S/N value for every grade of every parameter,\(j = 1,2, \ldots q\), \(q\) is the number of input variables, \(m = 1,2, \ldots l\), \(l\) is the number of experimental grade of every input variable.

  1. (5)

    Define the weight factor

The weight factor (WF) is computed by:

$$w_{i} = \frac{{\sum\nolimits_{j = 1}^{q} {r_{ij} } }}{{\sum\nolimits_{i = 1}^{{m_{OF} }} {\sum\nolimits_{j = 1}^{q} {r_{ij} } } }},$$
(17)

where \(w_{i}\) (\(w_{i} \ge 0\)) is the WF of each response function ith.

mOF is the number of response functions.


Generally, total of weight factor for total response functions are amounting to 1 and the equation is expressed by:

$$\sum\limits_{i = 1}^{{m_{OF} }} {w_{i} } = 1,$$
(18)

Phase 3: adaptive neuro-fuzzy inference system

ANFIS is a multi-layer feed-forward network which utilizes integration of neural network and fuzzy reasoning rules based on the merits of both fuzzy logic and artificial neural networks. Fuzzy logic is based on the consideration that people do determinations according to unaccurate and non-numerical information (Tsai et al. 2015; Bhattacharyya et al. 2015, Chau et al. 2018). The implementation of ANFIS architecture based on a first order Sugeno fuzzy inference system is more effective and rapid than the traditional methods as the error back propagation algorithm. In this paper, the fuzzy if–then rules can be described as:

$${\text{if }}x_{1} {\text{ is }}A_{1} {\text{ and }}x_{2} {\text{ is }}A_{2} ,{\text{ then }}y = p \times x_{1} + q \times x_{2} + r$$
(19)

where x1 and x2 are inputs corresponding A1 and A2 term set, y is output, p, q, r are constant.

The architecture of a representative ANFIS comprises of five layers, which conduct various operations in the ANFIS and are detailed below.

  1. (1)

    Layer 1: This layer conducts a fuzzification to generate the membership grades of the inputs. The outputs of this layer are identified by:

    $$M_{1}^{i} = \mu A_{i} \left( x \right)$$
    (20)

    where x is input to node i, and Ai is linguistic label associated with this node function. \(M_{1}^{i}\) is the membership function of Ai.

  2. (2)

    Layer 2: Each node in this layer is fixed node, marked by a circle and labeled П. Each node output can be expressed by:

    $$w_{i} = \mu A_{i} \left( x \right) \times \mu A_{i} \left( y \right) \ldots i = 1 , { 2, 3, } \ldots ,N$$
    (21)

    where \(w_{i}\) is the firing strength of a rule.

  3. (3)

    Layer 3: Each node in this layer is fixed node, marked by a circle and labeled N. The node function is normalized the firing strength by computing the ratio of the ith node firing strength to the sum of all rules’ firing strength.

    $$\bar{w} = \frac{{w_{i} }}{{w_{1} + w_{2} + w_{3} , \ldots , + w_{n} }}, \, i = 1,{ 2, 3, } \ldots , { }N$$
    (22)

    where \(\bar{w}\) is the normalized firing strength of rules.

  4. (4)

    Layer 4: Each node in this layer is an adaptive node, symbolized by a square. Each node i is expressed by:

    $$M_{1}^{i} = \bar{w}_{i}\, f_{i} \left( {px + qy, \ldots ,r} \right), \, i = 1,{ 2, 3, } \ldots , { }N$$
    (23)

    where \(\bar{w}_{i}\) is the output of layer 3 and p, q, r are the parameters set, referred to as the consequent parameters.

  5. (5)

    Layer 5: Each node in this layer is a fixed node marked \(\sum {}\), which calculates the total output as the sum of all incoming signals by:

    $$M_{5}^{i} = \sum\limits_{i}^{{}} {\bar{w}_{i}\, f_{i} } \frac{{\sum\limits_{i}^{{}} {w_{i}\, f_{i} } }}{{w_{i} }}$$
    (24)

The hybrid learning algorithm of the ANFIS integrate the gradient algorithm and the least squares method in order to update the parameters. Learning and modifying these adjustable parameters is the two-step process, which comprises a forward pass and a backward pass. In the forward pass, the premise parameters are hold fixed, node outputs go forward to layer 4. The consequent parameters are defined by the least squares.

Moreover, the four design parameters, including length of first lever, distance between two levers and thickness of compliant joints, \(t,h,b,k\) contributed the highest impact on two quality characteristics, comprising the y-axis displacement and the safety factor. Therefore, the architecture of ANFIS for the CPP is expanded, as depicted shown in Fig. 9.

Fig. 9
figure 9

Proposed ANFIS for the compliant positioning platform

Full size image

As depicted in Fig. 9, five network layers are utilized by ANFIS to conduct the following fuzzy inference steps: (1) input fuzzification, (2) fuzzy set database construction, (3) fuzzy rule base construction, (4) decision making, and (5) output defuzzification.


Phase 4: TLBO algorithm

Based on the established surrogate models for objective functions by the improved ANFIS, the optimization process was implemented by TLBO algorithm. Operation principle of this algorithm mimics the teaching–learning ability of the trainer and learners in a classroom. The expected learning outcome of TLBO was the marks results of students depended on the ability of trainer. It forecasted that a good quality student is because of a perfect training based on a excellent trainer. Moreover, besides learning from the trainer, the students can study other intellectuality from various members in class to enhance their marks. Courses were regarded as design variables and the student’s results were same the fitness value of the optimal process. The algorithm includes two phases: (1) trainer phase where candidates were distributed at random over the search space and the best solution was determined and (2) student phase where solutions tried to get new intellectuality from communicating with various learners. Details of the TLBO could be found in Rao and Patel (2013), Rao et al. (2011, 2012), Kumar and Singh (2018), Tawhid and Savsani (2018) and Singh et al. (2018).

6 Results and discussion

6.1 Collection of numerical data

First of all, a 3D model of the CPP was built via using FEM. A model was chosen for analysis and optimization later. Because there were four design variables, the number of experiments were 25, which was generated by the central composite design through the RSM. Later on, the boundary conditions and input displacement were assigned for the proposed platform. Eventually, by integrating of FEM and RSM, the values of two quality characteristics were retrieved, as depicted in Table 4.

Table 4 Design of number of experiments and computational results
Full size table

6.2 Calculation of weight factor

Later on, the experimental data were conveyed to the S/N ratios utilizing Eq. (13), as shown in Table 5. The S/N ratios were then normalized utilizing Eq. (14), as illustrated in Table 6. In this Table, the normalized S/N ratios for safety factor (η1) and the displacement (η2) were Z1, and Z2, respectively. Utilizing the Eqs. (16) and 17, the WF for the displacement and safety factor were caculated, as shown in Tables 7 and 8, respectively. The WF of the safety factor was 0.4416 and the WF of displacement was 0.5584. Sum of two WFs were equal to one. Generally, the WF for each characteristic was assigned 0.5 but this value was not correct. This would lead unaccurate optimized solution. As a result, this paper suggested an effective approach to determine the WFs.

Table 5 The values of S/N ratios
Full size table
Table 6 The normalized S/N ratios (zi)
Full size table
Table 7 The weight factor for the safety factor
Full size table
Table 8 The weight factor for the y-axis displacement
Full size table

6.3 Establishment of ANFIS model

ANFIS is same to a good black box or surrogate model where maps multiple inputs and outputs. In this paper, four design variables affected significantly to the two quality responses and the mathematical equations between them were complicated to set up. Therefore, ANFIS was an suitable tool for this research.

Basically, the controllable parameters of ANFIS could be used as default but some were suitable and others were not appropriate for the proposed CPP. For example, the MFs have a few common types such as trimf, trapmf, gbellmf, gaussmf, sigmf, and pimf. This simulation takes a lot of time but the retrieving solutions may be not reliable. To suppress this effect and make a good result, the suitable MFs were determined by optimizing the RMSE which was used as a performance criterion of ANFIS structure. Moreover, the ANFIS algorithm also depends on the training method, the number of MFs, and the type of output MFs.

To do an optimization for the RMSE, the controllable parameters of proposed ANFIS were identified and divided into their levels based on the specialized intellectuality and design engineer’s experiences, as depicted in Table 9. The number of input MFs were divided into levels as 3, 4, 5, 6 MFs. Moreover, the trimf, trapmf, gbellmf, and gaussmf were used as levels of the types of input MFs. The optimal learning method for the proposed ANFIS structure was divided into as hybrid learning procedure and backpropagation. Final sensitive factor of ANFIS structure was the types of output MFs that separated into constant or linear value. It was difficult to say which ones were the better for the ANFIS algorithm. As a result, an appropriate approach to determine these sensitive parameters of ANFIS was based on statistics and the TM.

Table 9 The levels of controllable factors for the RMSE
Full size table

Based on four parameters and their levels, an orthogonal array design L16 (4^2 & 2^2) was made to construct the 16 different training experiments for the ANFIS. The data of the y-axis displacement in the Table 4 would be conducted as training data and these data were trained in the ANFIS algorithm. And then, the RMSE values were determined and S/N ratios were computed using the TM. The results of RMSE and S/N ratios for the safety factor were calculated using the TM, as shown in Table 10.

Table 10 Orthogonal array design L16 for the RMSE of the safety factor
Full size table

To identify the optimal RMSE of the safety factor for the further ANFIS structure, the mean response of S/N ratios and main diagram of effects of each parameters were solved. From the results in Table 11 and Fig. 10, the optimal parameters were determined at the number of input MFs of 4, trapmf, hybrid learning method, and linear output MFs. The results also found that the optimal S/N ratio of RMSE is 134.833, as given in Table 12. This value was largest value compared with the values in Table 10. It proved that a larger S/N ratio is corresponding to a better solution.

Table 11 Mean response for S/N ratios for the RMSE of the safety factor
Full size table
Fig. 10
figure 10

Response diagram of the RSME of the safety factor

Full size image
Table 12 Prediction of optimal RMSE of of the safety factor
Full size table

Similarly, the optimizing process for the RMSE of displacement was carried out. The RMSE results for the displacement were given in Table 13. Based on the mean response for S/N ratios for the RMSE of safety factor and main diagram of effect in Table 14 and Fig. 11, the optimal controllable parameters for further ANFIS structure were found at the number of input MFs of 4, the trapmf, hybrid learning method, and linear output MFs. Also in Table 15, the optimal results depicted that the optimal RMSE is approximately 132.1082 being higher than the S/N ratio values in Table 13. This means that the optimal RMSE is accepted.

Table 13 Orthogonal array design L16 for the RMSE of the displacement
Full size table
Table 14 Mean response for S/N ratios for the RMSE of the displacement
Full size table
Fig. 11
figure 11

Response diagram of the RSME of the y-axis displacement

Full size image
Table 15 Prediction of optimal RMSE of of the y-axis displacement
Full size table

The ANFIS regularly uses the typical membership functions such as Gaussian, sigmoidal, triangular, trapezoidal, and bell types. Therefore, it is difficult to seek the suitable appropriate MFs. In this study, based on the results on Tables 9, 10, 11, 12, 13, 14 and 15, the trapezoidal was suitable for an improved ANFIS structure for modeling the CPP. The trapezoidal membership function was expressed by:

$$\mu_{A} \left( {x,a,l,r,b} \right) = \left\{ {\begin{array}{*{20}l} {\frac{{\left( {x - a} \right)}}{{\left( {l - a} \right)}}} \hfill & {\quad {\text{a}} \le {\text{x}} \le l} \hfill \\ 1 \hfill & {\quad l < x < r} \hfill \\ {\frac{{\left( {x - b} \right)}}{{\left( {r - b} \right)}}} \hfill & {\quad r \le x \le b} \hfill \\ \end{array} } \right.,$$
(25)

where μA depicts the MFs of the fuzzy set; a, b, l and r represent parameters; and x is variable. The shape of trapezoidal MFs were given, as illustrated in Fig. 12.

Fig. 12
figure 12

Shape of trapezoidal membership function

Full size image

After the suitable controllable parameters of ANFIS were determined based on the TM, the improved ANFIS structure was generated, as illustrated in Fig. 13. The number of linear parameters were 1280, as seen in this Table 16. By contrast, the number of nonlinear parameters were 48. If a analytical approach was utilized, many parameters can not determine accurately. In conclusion, ANFIS was the suitable method to apply for the suggested CPP.

Fig. 13
figure 13

Model for improved ANFIS structure

Full size image
Table 16 ANFIS parameters
Full size table

6.4 Optimal results

After gaining the WFs and the surrogate models for two objective functions, the TLBO was utilized to optimize the CPP. The optimization process was conducted utilizing Matlab 2017. The initial parameters including population size of 30 as well as tolerance of 10−6 were utilized for the TLBO. The optimized results were produced at the generation of 79. The optimization input variables were detected at xval = [0.67 0.9 19 53] and fval = − 2.01241668736275. The optimized solutions were corresponding to t = 0.9 mm, h = 19 mm, b = 53 mm, k = 0.67 mm, y1= 1.514097 and y2 = 2.406501242 mm.

6.5 Sensitivity analysis

Statistical technique was utilized to identify the effect grade of variables on the quality responses. As seen in Figs. 14 and 15, factor t illustrated that in the range from 0.9 to 1 mm, this parameter affected to y1 and y2 in increasing gradually, but in the range from 1 to 1.1 m, it caused gradually to y1 and y2 a reduction. Moreover, factor h showed that in the range from 19 to 20.75 mm, it affected a sharp rise to y1 and caused a slight rise to y2, however, from 20.75 to 22.5 mm, there was a gradual reduce to y1 and a sharp reduce to y2.

Fig. 14
figure 14

Effect diagram of t and h on the safety factor

Full size image
Fig. 15
figure 15

Effect diagram of t and h on the output displacement

Full size image

As drawn in Figs. 16 and 17, factor b indicated that in the range from 48 to 50.5 mm, it affected a gradual rise to y1 and y2, but, from 50.5 to 53 mm, there was a both gradual reduce to y1 and y2. Eventually, factor D illustrated that in the range from 0.67 to 0.705 mm, it affected gradual rise to y1 and caused a slight rise to y2, however, from 0.6 to 0.7 mm, there was a both gradual decrease to y1 and y2.

Fig. 16
figure 16

Effect diagram of b and k on the safety factor

Full size image
Fig. 17
figure 17

Effect diagram of b and k on the displacement

Full size image

In conclusion, overall effects of design variables were showed, as in Fig. 18. It reveals an increase and decrease range in each factor. Therefore, the designers could control the factors so as to obtain a best structure for the proposed CPP.

Fig. 18
figure 18

Sensitivity diagram of the parameters on the responses

Full size image

7 Verifications

To verify the optimized characteristics of the suggested CPP, the numerical simulations and experiments were carried out. Using the optimal parameters (t = 0.9 mm, h = 19 mm, b = 53 mm, k = 0.67 mm), a 3D model was designed in Inventor software and imported into ANSYS software in order to conduct simulations. The meshes of flexure hinges were refined to enhance the accuracy of analysis quality. And then, using the optimal design variables, the prototype of the CPP platform was fabricated by wire electrical discharged machining. As shown in Fig. 19, the experiments were implemented as follows: an input displacement was driven translational screw mechanism measured displacement by Digital dial indicator 1 (High precision 0.001 mm, 543-390B, Mitutoyo Japan). In addition, output displacement was measured by Digital dial indicator 2 (High precision 0.001 mm, 543-390B, Mitutoyo Japan) and Digital dial indicator mounted Magnetic base (Mitutoyo Japan).

Fig. 19
figure 19

Experimental installation for the prototype

Full size image

As seen in Table 17, the maximum of the y-axis probed deformation was 2.427 mm and the minimum of safety factor was 1.526. Based on the FEA result, the error between predicted result of optimal result is small: the error of safety factor is 0.786%. In addition, Table 17 illustrates that an error between optimal result and FEA test is also small: error of the probed displacement between optimal result and FEA result is 0.824% and between experimental result and FEA result is 7.581%. It means that there is a good agreement between the forecasted values and validations. Meanwhile, there were also some sources such as meshing, fabricating and material errors that could reduce these errors. Therefore, the suggested integrated optimal algorithm was reliable and effectual to optimize the CPP. Therefore, the proposed hybrid approach of FEM-based RSM, TM, improved ANFIS, and TLBO is enough reliable to search the optimal solution in this study. The experimental process was conducted 10 times and computed average values for experimental results.

Table 17 Comparison among the optimization, FEA, and experimental results
Full size table

The strong abilities of the positioning platform are summarized by: it allowed to gain working travel approximately 2406 µm, a displacement amplification ratio of more than 18 times. Compared with some previous researches, the proposed platform had a better working stroke. In conclusion, the results demonstrated that two referred regards are precise as the supposed theory. It can conclude the highlight performances as follows: (a) proposed platform allowed to achieve high displacement while safety factor is also more than 1.5 in order to ensure the platform working in the elastic condition and (b) the output displacement of the positioning platform is more than 18 times the input displacement. Figure 20 indicates relationship between input displacement and output displacement for FEA results and experiment results, respectively.

Fig. 20
figure 20

Relationship plot of the input displacement vs the ouput displacement

Full size image

8 Conclusions

This paper proposed an efficient hybrid optimization approach for the compliant positioning platform. The CPP was expected to locate the sample during nanoindentation tester. The proposed CPP was integrated the four-lever mechanism amplifier and beetle-liked structure. The platform imitated the biomechanical behavior of beetle in order to reach a linear displacement and reduce parasitic motion.

To enhance the output displacement and safety factor simultaneously, the main geometric parameters of proposed CPP were optimized by an integration of the RSM, FEM, TM, improved ANFIS and TLBO. The RMSE was optimized by the TM to find suitable controllable parameters for the ANFIS structure. And then, the weight factor of both responses were calculated by establishing the sets of statistical-based equations. The results found that the WFs of safety factor and displacement are 0.4416 (44.16%) and 0.5584 (55.84%), respectively. The WF’s values were assigned into the TLBO algorithm to conduct the multi-criteria optimal trouble. The sensitivity analysis as well as ANOVA were conducted to define the effects and significant contributions of design variables on the two quality characteristics.

The results illustrated that the optimal parameters were found at t = 0.9 mm, h = 19 mm, b = 53 mm, k = 0.67 mm. In addition, the results indicated that the optimal safety factor is 1.5141 and the optimal displacement is approximately 2.4065 mm. Moreover, the results depicted that the errors between the optimal results and the FEA validations for the safety factor and output displacement are 0.786% and 0.824%, respectively. The error between experimental result and the forecasted result was about 7.581%. The proposed hybrid approach is useful and effective to carry out the multi-criteria optimal trouble for complex design.