# Prediction of Fatigue Life for a New 2-DOF Compliant Mechanism by Clustering-Based ANFIS Approach

Authors: Ngoc Thoai Tran; Thanh-Phong Dao; Thao Nguyen-Trang; Che-Ngoc Ha
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Two-degree-of-freedom (2-DOF) compliant mechanism has some outstanding characteristics in accurate positioning systems. Studying the fatigue life for the 2-DOF compliant mechanism is a meaningful task to ensure a long working. However, a study for fatigue life prediction of this mechanism has not been conducted so far. In this article, a method for fatigue life prediction of 2-DOF compliant mechanism is developed for the first time. This method is the combining of the differential evolution algorithm and the adaptive neuro-fuzzy inference system (ANFIS) with subtractive clustering. The numerical results on two case studies consisting of material steel A-36 and the material AL 6061-T6 show that the accuracy of the proposed method is very high. Compared to the actual fatigue life, the root mean square error of the proposed method lies in the range [1.7, 3.97] cycles for Case 1 and [2.03, 10.38] cycles for Case 2. The statistical test also indicates that the proposed method outperforms the traditional method using triangular membership function, bell-shape, and Gaussian membership function, with the significance level from 0.05 to 0.1. These results demonstrate the feasibility of the proposed approach in fatigue life prediction of 2-DOF compliant mechanism.


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## Body

## 1. Introduction

Two-degree-of-freedom (2-DOF) compliant mechanism is a monolithic structure that has been widely utilized in ultra-high precise engineering. Basically, the 2-DOF compliant mechanism inherits outstanding characteristics such as minimal positioning error, free friction, reduced assemble, and easy machining [1]. Several applications of this mechanism can be found in the nanopositioning system [2], micropositioning system [3–9], space pointing mechanism [10], and so on. Although the 2-DOF compliant mechanism proposes excellent advantages in precision engineering and manipulators, this mechanism still exists major drawbacks, including nonlinear kinematic behaviors and fatigue influences. Considering the first disadvantage, nonlinear kinematic behaviors can be modeled and analyzed by pseudorigid-body model [11], compliance matrix method [12], and pseudostatic model [13]. Regarding the second disadvantage, fatigue influences on the 2-DOF compliant mechanism can result in a failure under dynamic working conditions.In similarity to rigid-link mechanisms, fatigue life of the 2-DOF complaint mechanism is a term indicating how long an object will last before it fails due to concentrated stresses. Evaluation of the fatigue life is a critically important task in design process of compliant mechanisms so as to guarantee the working safety and reliability. Practically, compliant mechanisms are often subjected to cyclic and reversed loads. Under varying stress cycles, fatigue phenomena can appear in compliant mechanisms. So, it is necessary to predict the fatigue life to avoid any damage. Nevertheless, modeling of the fatigue life for the 2-DOF compliant mechanism is so complicated task because the 2-DOF compliant mechanism is a complex structure that has an integration of kinematic behaviors and mechanical ones. Therefore, the problem of fatigue prediction has been interested to research recently to prevent undesired failures. In [14], the artificial neural networks (ANNs) were used to predict the fatigue life of polyamide-6 polymer. The results showed that the ANN-based approach can achieve a better performance than conventional methods. The adaptive neuro-fuzzy inference system (ANFIS) was used to model the fatigue behavior of a multidirectional laminate under four different loading conditions [15]. In the above study, both grid partitioning and subtractive clustering were used to build the ANFIS models. Nevertheless, a method for setting subtractive clustering parameters has not been conducted. The research in [16] provided a review of fatigue life modeling, fatigue life prediction, and application to composite structure design. The research studies in [17, 18] proposed a method for solving the partial differential equations using the deep neural network (DNN). This approach was then utilized for many applications in the field of Computational Mechanics. Various soft computing methods for predicting the fatigue life of different material systems were also studied in [19–28]. However, it can be noted that a study predicting the fatigue life for 2-DOF compliant mechanism has not been conducted so far.To fulfill the mentioned research gap, this paper develops a method based on the adaptive neuro-fuzzy inference system (ANFIS) with subtractive clustering for fatigue life prediction of 2-DOF compliant mechanism. Furthermore, for searching the optimal radius parameters of the clustering algorithm, the 10-fold cross-validation and the differential evolution (DE) algorithm are used. Particularly, the average mean square error of 10-folds is evaluated and optimized using the DE. The proposed framework is applied to predict the fatigue life of two cases of 2-DOF compliant mechanism consisting of the material steel A-36 and the material AL 6061-T6 based on their design variables.The rest of the paper is organized as follows. The related works including the adaptive neuro-fuzzy inference system, the subtractive clustering, and the differential evolution algorithm are presented in Section2. The proposed framework is presented in Section 3. The new method is applied to predict fatigue life for 2-DOF compliant mechanism consisting of steel materials A-36 and AL 6061-T6. In the conclusion, the main contribution, the results, and a few further research directions are summarized and discussed.


## 2. Related Works

### 2.1. Fuzzy Set

Fuzzy set, which was introduced by La [29], is a well-known theory for modeling data uncertainty. A fuzzy set A on universe X is a set defined by the membership function μAx which is a mapping from X into the interval [0, 1]. If μAx=1, x completely belongs to the fuzzy set A. If μAx=0, x does not belong to the fuzzy set. If 0<μAx<1, x is a partial member of the fuzzy set A. There are several ways to define the membership function μAx. Some well-known membership functions are the trapezoidal membership function, the triangular membership function, and the Gaussian membership function.


### 2.2. Fuzzy Inference System

The fuzzy inference system (FIS) [30] is a system that uses fuzzy set theory to map inputs to output (independent variables to dependent variable in the case of regression). There are two main types of FIS including the Mamdani and the Sugeno systems. This paper concentrates on the Sugeno system which builds a separate subregression model corresponding to each rule and calculates the final output as the weight-mean of all subregression outputs. For example, let X be a script variable that can be mapped to two corresponding fuzzy sets A1 and A2, Y be a script variable that can be mapped to two corresponding fuzzy sets B1 and B2, and Z be the output or dependent variable. Z can be computed via the Sugeno fuzzy logic inference as follows:IfX is A1 and Y is B1, then z1=p1x+q1y+r1 with the degree of applicability is λ1IfX is A2 and Y is B2, then z2=p2x+q2y+r2 with the degree of applicability is λ2IfX is A1 and Y is B2, then z3=p3x+q3y+r3 with the degree of applicability is λ3Finally, the outputZ is computed using the formula z=λ1z1+λ2z2+λ3z3/λ1+λ2+λ3In the above formulas,zi is the component predicted value of fuzzy rule i, the predicted value z can be considered as the weight-mean of all component predicted values zi, and pi,qi,ri are user-defined. A disadvantage of the Sugeno FIS is that expert knowledge is required to determine the fuzzy set parameters and the coefficients p, q, and r; therefore, the FIS is a nonoptimal model and lacks learning ability. To improve the performance of FIS, [31] presented a hybrid model called ANFIS that combines the Sugeno FIS and the artificial neural networks. ANFIS can take advantage of the fuzzy reasoning of FIS and the learning capability of ANN. A brief review of ANFIS is presented as follows.


### 2.3. Adaptive Neuro-Fuzzy Inference System

Soft computing refers to an approach that attempts to model the behavior of a complex system so that an approximate solution of this system can be provided. By the approximation, soft computing not only can provide a relatively good result but also can significantly reduce the computational cost. Therefore, this approach has many applications in engineering problems, such as civil engineering [32–34], mining engineering [35–37], and agricultural and biological engineering [38]. The adaptive neuro-fuzzy inference system (ANFIS) proposed by [31] can take advantage of the fuzzy reasoning of FIS and the learning capability of ANN. Therefore, ANFIS has been successfully performed in classification and prediction tasks and has become one of the most well-known soft computing methods so far. A simple illustration of an ANFIS model is presented in Figure 1.Figure 1
An example of the ANFIS system.It can be seen from Figure1 that ANFIS consists of five layers that perform different functions. Let Oli be the output of the ith node of the lth layer. Assuming that the variable X can be partitioned into two fuzzy sets, A1 and A2, whereas the variable Y can be partitioned into two fuzzy sets B1 and B2. In Layer 1, each node calculates the membership function μAjx and μBjy, j=1,2. In other words, O1i=μAjx for i=1,2 and O1i=μBiy for i=3,4. Some well-known membership functions are, namely, Gaussian membership function μAx=exp−x−c2/2σ2, triangular membership function μAx=maxminx−1/b−a,c−x/c−b,0, and generalized bell membership function μAx=1/1+x−c/a2b, where a, b, and c are called as premise parameters. Each node in Layer 2 is the AND operator that calculates the joint membership degree of x,y in Aj and Bk, j, k = 1,2. Finding the product of individual membership degrees is a popular method modeling the AND operator; therefore, O2i=TPμAjx,μBkx=μAjxμBkx, j, k = 1,2. Another operator for calculating the joint membership degree is defined as TMμAjx,μBkx=minμAjx,μBkx. In fact, both TP and TM are some specific examples of a t-norm which is used to generalize the conjunction in fuzzy logic. For all a, b in [0, 1], TPa,b≤TMa,b; that is, the logical strength represented by TP is stronger than TM. For more details of t-norms, readers can refer to [39]. As shown in Figure 1, node 1 in Layer 2 calculates the membership degree of X is A1 and Y is B1, whereas node 2 in Layer 2 calculates the membership degree of X is A2 and Y is B2. Certainly, more combinations between Ai and Bk, for example, X is A1 and Y is B2, can be handled. For the sake of convenience, only two nodes are illustrated in Figure 1. Let wi be the ith rule membership or firing strength obtained in Layer 2. Layer 3 is utilized for the normalization purpose. In this layer, the output of node i is the ratio of the ith rule’s firing strength to the sum of all rules’ firing strengths. For example, from ANFIS architecture in Figure 1, it can be implied that O3i=wi¯=wi/w1+w2, i=1,2. Layer 4 calculates the reasoning result for each fuzzy rule using the formula O4i=wi¯fi=wi¯pix+qiy+ri where pi,qi and ri are called consequent parameters. Finally, Layer 5 calculates the overall output using the sum of all O4i, that is, f=O51=∑i=12wi¯fi.ANFIS can optimize all parameters using a hybrid learning method. In the forward pass, the least-squares method is utilized to identify the consequent parameters in Layer 4. In the backward pass, the gradient descent method is utilized to identify the premise parameters in Layer 1. Therefore, the ANFIS inherits both the fuzzy reasoning of FIS and the learning abilities of ANN.The remaining problem of ANFIS is how to identify the number of fuzzy sets for each variable. The conventional ANFIS uses the grid partitioning method that requires a given number of fuzzy sets defined by expert knowledge. Furthermore, this method causes the dimensional curse when applied to large-scale data. For example, given an input of 12 variables, if each variable can be partitioned into two fuzzy sets, the grid partitioning method results in 212 = 4096 rules. This number is very large for any practical learning method. Therefore, the conventional ANFIS with the grid partitioning method might be not completely objective and might be infeasible when dealing with large-scale data. An alternative method is using ANFIS with subtractive clustering technique, which is presented in the following section.


### 2.4. Subtractive Clustering

Cluster analysis is to discover the underlying structure of a dataset by partitioning the data into groups such that each obtained cluster (group) is a set of similar data points [40]. In other words, all the elements in a cluster share the same characteristic behavior of the system. Therefore, each cluster’s center can be considered as the basis of a rule that represents a system behavior. Because clustering can create a concise representation of the system behavior, it can reduce the number of fuzzy rules or prevent rule explosion. Let x1,x2,…,xN be N d-dimensional normalized data points. Subtractive clustering [41], which is often applied to ANFIS, is summarized in the following steps:Step 1: normalize the data set, setk = 1, and calculate the likelihood that each data point is a cluster center, using(1)Pi=∑j=1N∏m=1dexp−4rmxim−xjm2,wherePi is the likelihood of point xi and rmis the neighborhood radius in dimension m.Step 2: select the data point with the highest likelihood as thekth cluster’s center.Step 3: update the likelihood that each data point is a cluster center, using formula(2)Pi≔Pi−Pk∗∏m=1dexp−4ηrmxim−xkm∗2,wherexk∗ and Pk∗ denote the kth cluster center and its likelihood, respectively. η>1 is a constant.Using formula (2), the closer the distance between xi and xk∗ is, the more reduction of its likelihood is. Therefore, all points with an updated Pi of 0 are assigned to cluster k and are not considered in the next steps.Step 4: repeat Steps 2 and 3 until all points are assigned to their clusters.In the above algorithm, the value ofrm has an important effect on the clustering result. When rm tends to zero, the algorithm tends to result in N single clusters, each of which has only one element. In contrast, when rm tends to one, the algorithm tends to result in a single cluster of N elements. It is not easy to determine a suitable value for rm that can work well for all case studies. Therefore, an additional framework for optimizing the set of rm is developed. The developed framework is based on the differential evolution algorithm, which will be presented in the next section.


### 2.5. Differential Evolution Algorithm

The differential evolution algorithm, DE, which is a well-known global search method based on population, is designed to solve both continuous and discrete problems [42]. To clarify the used notation, this article refers to the minimization of the objective function fx, where x is a vector of d variables in the decision space S=xl,xu and xl and xu are the vectors of lower bound and upper bound, respectively. The DE seeks an optimal solution through generations (iterations). In each generation, the DE evolves a population P of size NP, P=x1,x2,…,xNP. Each element in this set or each feasible solution xi,i=1,NP¯ is called a chromosome, which is a vector of d variables, so-called genes. The four major operators of the DE algorithm, which include initialization, mutation, crossover, and selection, are briefly summarized as follows.


#### 2.5.1. Initialization

In this phase, an initial population ofNP chromosomes is generated through a random sampling technique. Specifically, each individual is represented as a chromosome containing N genes and is generated by(3)xi,j=xjl+r0×xju−xjli=1,2,…,NP;j=1,2,…,d,where R0∼U0,1, NP is the size of the population, and xjl and xju are the lower and upper limits of xj, respectively [43].


#### 2.5.2. Mutation

In this phase, each individualxi in the current population generates a mutant vector using some mutation operators listed as follows:rand/1:vi=xr1+F×xr2−xr3rand/2:vi=xr1+F×xr2−xr3+F×xr4−xr5best/1:vi=x∗+F×xr1−xr2best/2:vi=x∗+F×xr1−xr2+F×xr3−xr4current-to-best/1:vi=xi+F×x∗−xi+F×xr1−xr2wherer1, r2, r3, r4, and r5 are mutually exclusive integers randomly chosen in {1,2, …, NP}, F is the scale factor randomly selected in the range [0, 2], and x∗ is the best individual in the current population.


#### 2.5.3. Crossover

After completing mutation, each target vectorxi produces a trial vector ui by substituting some components of the vector xi by some components of the mutant vector vi through the following binomial crossover operation [43]:(4)uij=vijif rand0,1≤CR,xijotherwise,where i∈{1,2, …, NP}, j∈{1,2, …, d}, and CR is the crossover control parameter chosen in [0,1].


#### 2.5.4. Selection

In this phase, each trial vectorui created after the crossover phase will be evaluated and compared with the target vector xi to choose a better individual for the next generation.


### 2.6. Evaluating an ANFIS Model

Letyi and y^i be the ith actual and predictive data points, respectively. Some criteria for evaluating the ANFIS model are presented as follows:Mean absolute error:(5)MAE=1N∑i=1Nyi−y^i.Mean squared error:(6)MSE=1N∑i=1Nyi−y^i2.Mean absolute percentage error:(7)MAPE=1N∑i=1Nyi−y^iyi.100%.


### 2.7.K-Fold Cross-Validation Method

K-fold cross-validation is a typical process being often incorporated into the training process for avoiding the over-fitting problem [44–46]. In this procedure, the original training data set is randomly divided into k subsets. k−1 subsets are used as new training data sets to develop an ANFIS model. The remainder is used for evaluating the model using the criteria presented above. This process is repeated k times until every subset is used once for evaluating. Finally, the overall performance is calculated by taking the average of k individual measures. The ANFIS with the lowest average of MAE, MSE, and MAPE can be considered as the most suitable model. Additionally, it can be noted that the test data set is not used in the training and cross-validation process but is only used in the final validation when the optimized ANFIS has been already built. An illustration for k-fold cross-validation process is presented in Figure 2 where k is set as three and αis the ratio of the test set compared to the data set.Figure 2
Illustration fork-fold cross-validation with k = 3.


## 2.1. Fuzzy Set

Fuzzy set, which was introduced by La [29], is a well-known theory for modeling data uncertainty. A fuzzy set A on universe X is a set defined by the membership function μAx which is a mapping from X into the interval [0, 1]. If μAx=1, x completely belongs to the fuzzy set A. If μAx=0, x does not belong to the fuzzy set. If 0<μAx<1, x is a partial member of the fuzzy set A. There are several ways to define the membership function μAx. Some well-known membership functions are the trapezoidal membership function, the triangular membership function, and the Gaussian membership function.


## 2.2. Fuzzy Inference System

The fuzzy inference system (FIS) [30] is a system that uses fuzzy set theory to map inputs to output (independent variables to dependent variable in the case of regression). There are two main types of FIS including the Mamdani and the Sugeno systems. This paper concentrates on the Sugeno system which builds a separate subregression model corresponding to each rule and calculates the final output as the weight-mean of all subregression outputs. For example, let X be a script variable that can be mapped to two corresponding fuzzy sets A1 and A2, Y be a script variable that can be mapped to two corresponding fuzzy sets B1 and B2, and Z be the output or dependent variable. Z can be computed via the Sugeno fuzzy logic inference as follows:IfX is A1 and Y is B1, then z1=p1x+q1y+r1 with the degree of applicability is λ1IfX is A2 and Y is B2, then z2=p2x+q2y+r2 with the degree of applicability is λ2IfX is A1 and Y is B2, then z3=p3x+q3y+r3 with the degree of applicability is λ3Finally, the outputZ is computed using the formula z=λ1z1+λ2z2+λ3z3/λ1+λ2+λ3In the above formulas,zi is the component predicted value of fuzzy rule i, the predicted value z can be considered as the weight-mean of all component predicted values zi, and pi,qi,ri are user-defined. A disadvantage of the Sugeno FIS is that expert knowledge is required to determine the fuzzy set parameters and the coefficients p, q, and r; therefore, the FIS is a nonoptimal model and lacks learning ability. To improve the performance of FIS, [31] presented a hybrid model called ANFIS that combines the Sugeno FIS and the artificial neural networks. ANFIS can take advantage of the fuzzy reasoning of FIS and the learning capability of ANN. A brief review of ANFIS is presented as follows.


## 2.3. Adaptive Neuro-Fuzzy Inference System

Soft computing refers to an approach that attempts to model the behavior of a complex system so that an approximate solution of this system can be provided. By the approximation, soft computing not only can provide a relatively good result but also can significantly reduce the computational cost. Therefore, this approach has many applications in engineering problems, such as civil engineering [32–34], mining engineering [35–37], and agricultural and biological engineering [38]. The adaptive neuro-fuzzy inference system (ANFIS) proposed by [31] can take advantage of the fuzzy reasoning of FIS and the learning capability of ANN. Therefore, ANFIS has been successfully performed in classification and prediction tasks and has become one of the most well-known soft computing methods so far. A simple illustration of an ANFIS model is presented in Figure 1.Figure 1
An example of the ANFIS system.It can be seen from Figure1 that ANFIS consists of five layers that perform different functions. Let Oli be the output of the ith node of the lth layer. Assuming that the variable X can be partitioned into two fuzzy sets, A1 and A2, whereas the variable Y can be partitioned into two fuzzy sets B1 and B2. In Layer 1, each node calculates the membership function μAjx and μBjy, j=1,2. In other words, O1i=μAjx for i=1,2 and O1i=μBiy for i=3,4. Some well-known membership functions are, namely, Gaussian membership function μAx=exp−x−c2/2σ2, triangular membership function μAx=maxminx−1/b−a,c−x/c−b,0, and generalized bell membership function μAx=1/1+x−c/a2b, where a, b, and c are called as premise parameters. Each node in Layer 2 is the AND operator that calculates the joint membership degree of x,y in Aj and Bk, j, k = 1,2. Finding the product of individual membership degrees is a popular method modeling the AND operator; therefore, O2i=TPμAjx,μBkx=μAjxμBkx, j, k = 1,2. Another operator for calculating the joint membership degree is defined as TMμAjx,μBkx=minμAjx,μBkx. In fact, both TP and TM are some specific examples of a t-norm which is used to generalize the conjunction in fuzzy logic. For all a, b in [0, 1], TPa,b≤TMa,b; that is, the logical strength represented by TP is stronger than TM. For more details of t-norms, readers can refer to [39]. As shown in Figure 1, node 1 in Layer 2 calculates the membership degree of X is A1 and Y is B1, whereas node 2 in Layer 2 calculates the membership degree of X is A2 and Y is B2. Certainly, more combinations between Ai and Bk, for example, X is A1 and Y is B2, can be handled. For the sake of convenience, only two nodes are illustrated in Figure 1. Let wi be the ith rule membership or firing strength obtained in Layer 2. Layer 3 is utilized for the normalization purpose. In this layer, the output of node i is the ratio of the ith rule’s firing strength to the sum of all rules’ firing strengths. For example, from ANFIS architecture in Figure 1, it can be implied that O3i=wi¯=wi/w1+w2, i=1,2. Layer 4 calculates the reasoning result for each fuzzy rule using the formula O4i=wi¯fi=wi¯pix+qiy+ri where pi,qi and ri are called consequent parameters. Finally, Layer 5 calculates the overall output using the sum of all O4i, that is, f=O51=∑i=12wi¯fi.ANFIS can optimize all parameters using a hybrid learning method. In the forward pass, the least-squares method is utilized to identify the consequent parameters in Layer 4. In the backward pass, the gradient descent method is utilized to identify the premise parameters in Layer 1. Therefore, the ANFIS inherits both the fuzzy reasoning of FIS and the learning abilities of ANN.The remaining problem of ANFIS is how to identify the number of fuzzy sets for each variable. The conventional ANFIS uses the grid partitioning method that requires a given number of fuzzy sets defined by expert knowledge. Furthermore, this method causes the dimensional curse when applied to large-scale data. For example, given an input of 12 variables, if each variable can be partitioned into two fuzzy sets, the grid partitioning method results in 212 = 4096 rules. This number is very large for any practical learning method. Therefore, the conventional ANFIS with the grid partitioning method might be not completely objective and might be infeasible when dealing with large-scale data. An alternative method is using ANFIS with subtractive clustering technique, which is presented in the following section.


## 2.4. Subtractive Clustering

Cluster analysis is to discover the underlying structure of a dataset by partitioning the data into groups such that each obtained cluster (group) is a set of similar data points [40]. In other words, all the elements in a cluster share the same characteristic behavior of the system. Therefore, each cluster’s center can be considered as the basis of a rule that represents a system behavior. Because clustering can create a concise representation of the system behavior, it can reduce the number of fuzzy rules or prevent rule explosion. Let x1,x2,…,xN be N d-dimensional normalized data points. Subtractive clustering [41], which is often applied to ANFIS, is summarized in the following steps:Step 1: normalize the data set, setk = 1, and calculate the likelihood that each data point is a cluster center, using(1)Pi=∑j=1N∏m=1dexp−4rmxim−xjm2,wherePi is the likelihood of point xi and rmis the neighborhood radius in dimension m.Step 2: select the data point with the highest likelihood as thekth cluster’s center.Step 3: update the likelihood that each data point is a cluster center, using formula(2)Pi≔Pi−Pk∗∏m=1dexp−4ηrmxim−xkm∗2,wherexk∗ and Pk∗ denote the kth cluster center and its likelihood, respectively. η>1 is a constant.Using formula (2), the closer the distance between xi and xk∗ is, the more reduction of its likelihood is. Therefore, all points with an updated Pi of 0 are assigned to cluster k and are not considered in the next steps.Step 4: repeat Steps 2 and 3 until all points are assigned to their clusters.In the above algorithm, the value ofrm has an important effect on the clustering result. When rm tends to zero, the algorithm tends to result in N single clusters, each of which has only one element. In contrast, when rm tends to one, the algorithm tends to result in a single cluster of N elements. It is not easy to determine a suitable value for rm that can work well for all case studies. Therefore, an additional framework for optimizing the set of rm is developed. The developed framework is based on the differential evolution algorithm, which will be presented in the next section.


## 2.5. Differential Evolution Algorithm

The differential evolution algorithm, DE, which is a well-known global search method based on population, is designed to solve both continuous and discrete problems [42]. To clarify the used notation, this article refers to the minimization of the objective function fx, where x is a vector of d variables in the decision space S=xl,xu and xl and xu are the vectors of lower bound and upper bound, respectively. The DE seeks an optimal solution through generations (iterations). In each generation, the DE evolves a population P of size NP, P=x1,x2,…,xNP. Each element in this set or each feasible solution xi,i=1,NP¯ is called a chromosome, which is a vector of d variables, so-called genes. The four major operators of the DE algorithm, which include initialization, mutation, crossover, and selection, are briefly summarized as follows.


### 2.5.1. Initialization

In this phase, an initial population ofNP chromosomes is generated through a random sampling technique. Specifically, each individual is represented as a chromosome containing N genes and is generated by(3)xi,j=xjl+r0×xju−xjli=1,2,…,NP;j=1,2,…,d,where R0∼U0,1, NP is the size of the population, and xjl and xju are the lower and upper limits of xj, respectively [43].


### 2.5.2. Mutation

In this phase, each individualxi in the current population generates a mutant vector using some mutation operators listed as follows:rand/1:vi=xr1+F×xr2−xr3rand/2:vi=xr1+F×xr2−xr3+F×xr4−xr5best/1:vi=x∗+F×xr1−xr2best/2:vi=x∗+F×xr1−xr2+F×xr3−xr4current-to-best/1:vi=xi+F×x∗−xi+F×xr1−xr2wherer1, r2, r3, r4, and r5 are mutually exclusive integers randomly chosen in {1,2, …, NP}, F is the scale factor randomly selected in the range [0, 2], and x∗ is the best individual in the current population.


### 2.5.3. Crossover

After completing mutation, each target vectorxi produces a trial vector ui by substituting some components of the vector xi by some components of the mutant vector vi through the following binomial crossover operation [43]:(4)uij=vijif rand0,1≤CR,xijotherwise,where i∈{1,2, …, NP}, j∈{1,2, …, d}, and CR is the crossover control parameter chosen in [0,1].


### 2.5.4. Selection

In this phase, each trial vectorui created after the crossover phase will be evaluated and compared with the target vector xi to choose a better individual for the next generation.


## 2.5.1. Initialization

In this phase, an initial population ofNP chromosomes is generated through a random sampling technique. Specifically, each individual is represented as a chromosome containing N genes and is generated by(3)xi,j=xjl+r0×xju−xjli=1,2,…,NP;j=1,2,…,d,where R0∼U0,1, NP is the size of the population, and xjl and xju are the lower and upper limits of xj, respectively [43].


## 2.5.2. Mutation

In this phase, each individualxi in the current population generates a mutant vector using some mutation operators listed as follows:rand/1:vi=xr1+F×xr2−xr3rand/2:vi=xr1+F×xr2−xr3+F×xr4−xr5best/1:vi=x∗+F×xr1−xr2best/2:vi=x∗+F×xr1−xr2+F×xr3−xr4current-to-best/1:vi=xi+F×x∗−xi+F×xr1−xr2wherer1, r2, r3, r4, and r5 are mutually exclusive integers randomly chosen in {1,2, …, NP}, F is the scale factor randomly selected in the range [0, 2], and x∗ is the best individual in the current population.


## 2.5.3. Crossover

After completing mutation, each target vectorxi produces a trial vector ui by substituting some components of the vector xi by some components of the mutant vector vi through the following binomial crossover operation [43]:(4)uij=vijif rand0,1≤CR,xijotherwise,where i∈{1,2, …, NP}, j∈{1,2, …, d}, and CR is the crossover control parameter chosen in [0,1].


## 2.5.4. Selection

In this phase, each trial vectorui created after the crossover phase will be evaluated and compared with the target vector xi to choose a better individual for the next generation.


## 2.6. Evaluating an ANFIS Model

Letyi and y^i be the ith actual and predictive data points, respectively. Some criteria for evaluating the ANFIS model are presented as follows:Mean absolute error:(5)MAE=1N∑i=1Nyi−y^i.Mean squared error:(6)MSE=1N∑i=1Nyi−y^i2.Mean absolute percentage error:(7)MAPE=1N∑i=1Nyi−y^iyi.100%.


## 2.7.K-Fold Cross-Validation Method

K-fold cross-validation is a typical process being often incorporated into the training process for avoiding the over-fitting problem [44–46]. In this procedure, the original training data set is randomly divided into k subsets. k−1 subsets are used as new training data sets to develop an ANFIS model. The remainder is used for evaluating the model using the criteria presented above. This process is repeated k times until every subset is used once for evaluating. Finally, the overall performance is calculated by taking the average of k individual measures. The ANFIS with the lowest average of MAE, MSE, and MAPE can be considered as the most suitable model. Additionally, it can be noted that the test data set is not used in the training and cross-validation process but is only used in the final validation when the optimized ANFIS has been already built. An illustration for k-fold cross-validation process is presented in Figure 2 where k is set as three and αis the ratio of the test set compared to the data set.Figure 2
Illustration fork-fold cross-validation with k = 3.


## 3. The Proposed Method

This section develops a method for fatigue life prediction of 2-DOF compliant mechanism. The ANFIS model which to be estimated has the formY = f(X1, X2,…, Xd) where X1, X2,…, Xd are d design variables and Y is the dependent variable. Using the subtractive clustering technique, the model performance relies on the choice of radius parameters ri (0< = ri < = 1, i = 1, 2, …, d + 1). Therefore, the 10-fold cross-validation is used and the differential evolution algorithm is utilized for optimizing the mean square error (MSE) average over 10-folds. Because the DE defines a solution by a chromosome, each solution needs to be first encoded. The encoding phase is summarized in Figure 3. As shown in Figure 3, a possible solution is defined as a chromosome. Each chromosome contains d + 1 genes representing the corresponding radii of the design variables and the dependent variables. The MSE, whose values may be different for different chromosomes, can be considered as an implicit objective function of ri.Figure 3
Illustration for the encoding phase.After the encoding phase is completed, chromosomes or solutions can be handled with mutation, crossover, and selection operators. Through each iterationi, only a fixed number of solutions can be selected for the next iteration. After a given number of iterations, maxiter, a chromosome that provides the smallest MSE average over 10-folds can be considered as the best solution and can be used as the parameters of the ANFIS. The overall process of the proposed method is outlined in Figure 4.Figure 4
The flowchart of the entire method.


## 4. Numerical Example

### 4.1. The Data Set

Figure5 gives a design scheme of a new 2-DOF compliant mechanism. This mechanism includes a shuttle table which is located at the middle position. The shuttle table can move in the x-direction when a piezostack actuator (PSA-X) applies a force to an end of the table. Besides, when a force comes from the PSA-Y, the shuttle table is moved in the y-direction. The overall movement of the shuttle table is based on the elastic deflection of flexure hinges, including flexure hinge 1 (thickness T1 and length V1), flexure hinge 2 (thickness T2 and length H), and flexure hinge 3 (length V2). Overall dimensions of the mechanism are given, as in Figures 5(a) and 5(b). The width of the mechanism is b of 10 mm (see Figure 5(c)). It is noted that the motions of flexure hinges are repeated due to these hinges are worked in an elastic area. Therefore, the mechanism is easy to subject a fatigue failure. Predicting fatigue life for the mechanism is critically important to prevent failure and ensure a safety operation.Figure 5
2-DOF compliant mechanism: (a) 2D scheme, (b) dimension, and (c) 3D model (unit: mm).During the design phase, manufacturing material affects the fatigue life of the 2-DOF mechanism because each material has a specific limitation of strength. This section applies the proposed method for predicting the fatigue life of the new 2-DOF compliant mechanism in two cases: material steel A-36 and material AL 6061-T6. The physical properties, chemical components of steel A-36, and Al 6061-T6 are shown in Table1, whereas the alternative stresses of the two materials are shown in Table 2 [47, 48]. In both cases, a load, F, of 20 N is applied the shuttle table along the x-axis and fixed supports are located by screws at three holes, as depicted in Figure 5.Table 1
Physical properties and chemical components of steel A-36 and Al 6061-T6 (http://asm.matweb.com/search/SpecificMaterial.asp?bassnum=MA6061T6).
Physical propertiesSteel A-36Al 6061-T6Density (kg/m3)78502700Young’s modulus (MPa)20000068900Poisson’s ratio0.320.33Yield strength (MPa)250276Tensile ultimate strength (MPa)550310Chemical components(Wt.%)(Wt.%)Carbon, C: 0.29Aluminum, Al: 95.8–98.6Iron, Fe: 98Chromium, Cr: 0.04–0.35Manganese, Mn: 0.8–1.2Copper, Cu: 0.15–0.40Phosphorous, P: 0.04Iron, Fe: ≤ 0.70Silicon, Si: 0.15–0.4Magnesium, Mg: 0.80–1.2Sulfur, S: 0.05Manganese, Mn: ≤ 0.15Silicon, Si: 0.40–0.80Titanium, Ti: ≤ 0.15Zinc, Zn: ≤ 0.25Table 2
The alternative stress for two cases (unit: MPa).
Steel A-36AL 6061-T6CyclesAlternative stressCyclesAlternative stress1039991700275.82028275000241.350189634000206.810014131.4E + 5172.420010698E + 5137.920004412.4E + 6117.2100002625.5E + 789.63200002141E + 882.741E + 51382E + 51141E + 686.2After boundary conditions are applied for the model, the model is meshed. A 3D model of the 2-DoF mechanism is built for finite element analysis, as shown in Figure6. However, uncertainties and errors are critically important for any data. Because the datasets of the fatigue life are determined through finite element simulations in ANSYS software, the numerical datasets are strongly influenced by some controllable parameters. Datasets are therefore also affected by those parameters. First of all, a few main parameters of the meshing model, such as number of elements, number of nodes, and type of elements, are determined to permit a good analysis. Adjusting any parameter also leads to a difference result. So, before implementing the simulations, a meshing quality is evaluated to ensure a good predicting accuracy. The mechanism is meshed with 55376 nodes and 24884 elements. The type of ten-node tetrahedral elements is utilized for the meshing model. Meshing quality is evaluated, as shown in Figure 7. It is noted that a better meshing quality ensures a more simulation accuracy. The results indicated that the meshing distribution is relatively good with a metric average value of 0.2546 and a standard deviation of 0.1891.Figure 6
Meshed model: (a) mesh for the mechanism; (b) meshing refirement.Figure 7
Meshing quality.Datasets of the fatigue life are then determined by simulations in ANSYS software R19 and are summarized in Table3. It can be noted in Table 3 that the range of input variables is set by the user, but the range of output variable is based on the simulated results. Furthermore, the natural log transformation is applied to the output variable for producing smaller values. The data sets can be also downloaded at https://sites.google.com/tdtu.edu.vn/nguyentrangthao/home.Table 3
Summary of input/output variables.
InputOutputVariableT1T2V1V2HY: fatigue lifeRange (unit)[0.45–0.65] (mm)[0.50–0.70] (mm)[7–11] (mm)[8–12] (mm)[28–30] (mm)[105–108] (cycles)


## 4.1. The Data Set

Figure5 gives a design scheme of a new 2-DOF compliant mechanism. This mechanism includes a shuttle table which is located at the middle position. The shuttle table can move in the x-direction when a piezostack actuator (PSA-X) applies a force to an end of the table. Besides, when a force comes from the PSA-Y, the shuttle table is moved in the y-direction. The overall movement of the shuttle table is based on the elastic deflection of flexure hinges, including flexure hinge 1 (thickness T1 and length V1), flexure hinge 2 (thickness T2 and length H), and flexure hinge 3 (length V2). Overall dimensions of the mechanism are given, as in Figures 5(a) and 5(b). The width of the mechanism is b of 10 mm (see Figure 5(c)). It is noted that the motions of flexure hinges are repeated due to these hinges are worked in an elastic area. Therefore, the mechanism is easy to subject a fatigue failure. Predicting fatigue life for the mechanism is critically important to prevent failure and ensure a safety operation.Figure 5
2-DOF compliant mechanism: (a) 2D scheme, (b) dimension, and (c) 3D model (unit: mm).During the design phase, manufacturing material affects the fatigue life of the 2-DOF mechanism because each material has a specific limitation of strength. This section applies the proposed method for predicting the fatigue life of the new 2-DOF compliant mechanism in two cases: material steel A-36 and material AL 6061-T6. The physical properties, chemical components of steel A-36, and Al 6061-T6 are shown in Table1, whereas the alternative stresses of the two materials are shown in Table 2 [47, 48]. In both cases, a load, F, of 20 N is applied the shuttle table along the x-axis and fixed supports are located by screws at three holes, as depicted in Figure 5.Table 1
Physical properties and chemical components of steel A-36 and Al 6061-T6 (http://asm.matweb.com/search/SpecificMaterial.asp?bassnum=MA6061T6).
Physical propertiesSteel A-36Al 6061-T6Density (kg/m3)78502700Young’s modulus (MPa)20000068900Poisson’s ratio0.320.33Yield strength (MPa)250276Tensile ultimate strength (MPa)550310Chemical components(Wt.%)(Wt.%)Carbon, C: 0.29Aluminum, Al: 95.8–98.6Iron, Fe: 98Chromium, Cr: 0.04–0.35Manganese, Mn: 0.8–1.2Copper, Cu: 0.15–0.40Phosphorous, P: 0.04Iron, Fe: ≤ 0.70Silicon, Si: 0.15–0.4Magnesium, Mg: 0.80–1.2Sulfur, S: 0.05Manganese, Mn: ≤ 0.15Silicon, Si: 0.40–0.80Titanium, Ti: ≤ 0.15Zinc, Zn: ≤ 0.25Table 2
The alternative stress for two cases (unit: MPa).
Steel A-36AL 6061-T6CyclesAlternative stressCyclesAlternative stress1039991700275.82028275000241.350189634000206.810014131.4E + 5172.420010698E + 5137.920004412.4E + 6117.2100002625.5E + 789.63200002141E + 882.741E + 51382E + 51141E + 686.2After boundary conditions are applied for the model, the model is meshed. A 3D model of the 2-DoF mechanism is built for finite element analysis, as shown in Figure6. However, uncertainties and errors are critically important for any data. Because the datasets of the fatigue life are determined through finite element simulations in ANSYS software, the numerical datasets are strongly influenced by some controllable parameters. Datasets are therefore also affected by those parameters. First of all, a few main parameters of the meshing model, such as number of elements, number of nodes, and type of elements, are determined to permit a good analysis. Adjusting any parameter also leads to a difference result. So, before implementing the simulations, a meshing quality is evaluated to ensure a good predicting accuracy. The mechanism is meshed with 55376 nodes and 24884 elements. The type of ten-node tetrahedral elements is utilized for the meshing model. Meshing quality is evaluated, as shown in Figure 7. It is noted that a better meshing quality ensures a more simulation accuracy. The results indicated that the meshing distribution is relatively good with a metric average value of 0.2546 and a standard deviation of 0.1891.Figure 6
Meshed model: (a) mesh for the mechanism; (b) meshing refirement.Figure 7
Meshing quality.Datasets of the fatigue life are then determined by simulations in ANSYS software R19 and are summarized in Table3. It can be noted in Table 3 that the range of input variables is set by the user, but the range of output variable is based on the simulated results. Furthermore, the natural log transformation is applied to the output variable for producing smaller values. The data sets can be also downloaded at https://sites.google.com/tdtu.edu.vn/nguyentrangthao/home.Table 3
Summary of input/output variables.
InputOutputVariableT1T2V1V2HY: fatigue lifeRange (unit)[0.45–0.65] (mm)[0.50–0.70] (mm)[7–11] (mm)[8–12] (mm)[28–30] (mm)[105–108] (cycles)


## 5. Results and Discussion

The performance of the proposed method is compared (subtractive clustering + DE) with four different ANFIS structures consisting of grid partitioning + triangular membership function (Model 1), grid partitioning + bell-shaped membership function (Model 2), grid partitioning + Gaussian membership function (Model 3), and subtractive clustering + random range of influencer (Model 4). The considered problems have five design variables and one output variable. For the traditional ANFIS with the grid partitioning method, the number of fuzzy is set as 2, for each variable. Based on the above information, the parameters of Models 1, 2, and 3 can be determined. Particularly, the number of nodes, the number of linear parameters, and the number of fuzzy rules are, respectively, 92, 192, and 32, for all the three models; the number of nonlinear parameters is 30 for Model 1 and 2 and is 20 for Model 3. For Model 4 and the proposed method, the above parameters do not need to be prior determined but depend upon the value of radius r, which are randomly chosen in Model 4 and are optimized using DE in the proposed method. For the proposed method, the convergence behavior depends upon the hyperparameters of DE, such as mutation rate F, crossover rate CR, and stop criteria. Nevertheless, a comprehensive investigation for all above parameters is inefficient in practice. In this paper, firstly, some hyperparameters of DE are selected based on the investigations in [43, 49, 50]. Specifically, the size of population NP is 20, the mutation rate F is randomly chosen in the range [0.4, 1], crossover rate CR is randomly chosen in the range [0.7, 1], and the tolerance Tol is 10−6. Finally, the number of maximum iterations maxiter is investigated.Figure8 illustrates the convergence behavior of DE regarding the iterations, for some cases. It can be seen that the value of objective function sharply decreases at some first iterations. After 10–15 iterations, the value of the objective function still decreases but is not significant. In this study, the number of maximum iteration maxiter is set as 20 to trade off the quality of objective function and the computational cost. In other words, to design an efficient algorithm, a relative optimal solution rather than an exact optimal solution is chosen.Figure 8
The convergence behavior of DE.Letα be the ratio of the test set compared to the data set. For each material, the performance of the proposed method for different values of α will be evaluated. For each α, the training set is first divided into k-folds. Applying the DE algorithm to training data, the optimal ANFIS that minimizes the average MSE (AMSE) over the k-folds can be found. Table 4 shows the optimal results for different values of α and materials.Table 4
The optimal results for different values ofα, for each material.
Material steel A-36αAMSEr1r2r3r4r5r60.10.21580.5577170.5705780.9006980.5521920.6783050.9490260.20.34940.588750.7281220.9881320.722640.5939820.8423780.30.14350.0515940.4336070.3228440.5425230.7217560.9914620.40.16970.5930790.1891860.5717130.4506990.5659660.8388480.50.30960.2765780.3068190.6510950.4964690.2879180.3702140.60.16420.1805830.2090910.9859430.8476470.7085880.939774Material Al 6061-T6αAMSEr1r2r3r4r5r60.11.08770.8212160.3059640.9319950.529410.9063540.7170820.20.90760.493910.6360160.8452970.8427380.9905370.0126170.30.46040.6336210.2867780.6534420.6612260.8821890.4994950.40.48010.3249630.5340110.9132220.5476660.9511570.8318970.50.54030.8208380.4152530.9070880.7513430.7811490.3813710.60.36750.8086860.064510.0530430.2609610.5769020.96311Based on the obtained optimal parameters, the respective ANFIS networks can be built. The obtained models are next evaluated on the test data set. Their predictions are, respectively, shown in Figures9 and 10, for some typical cases. It can be observed from Figures 9 and 10 that the three ANFIS models with the grid partitioning method result in underfitting in some instances. Meanwhile, most of instances are quite well predicted using the two ANFIS models with the subtractive clustering method. Compared to the subtractive clustering with random values of r, the subtractive clustering with DE has a slightly better prediction (9/16 and 9/14 samples for Case 1 and Case 2, respectively).Figure 9
The predicted results of comparative methods on the test set (material steel A-36, alpha = 0.6).Figure 10
The predicted results of comparative methods on the test set (material Al 6061-T6, alpha = 0.5).The average MSE of the five comparative ANFIS models for differentα values is presented in Figures 11 and 12 (two cases of materials). It can be seen that the average MSE in terms of the natural logarithm of number cycles of the proposed method lies in the range [0.28, 1.92] for Case 1 and [0.50, 5.52] for Case 2; that is, the average RMSEs in terms of the number of cycles of the proposed method lie in the range [1.7, 3.97] for Case 1 and [2.03, 10.38] for Case 2. These results show that the accuracy of the proposed method is very high. Besides, it can be observed that the effect of the ratio of test data on the performances of two models with subtractive clustering is not so strong. As shown in Figures 11 and 12, in both cases, their average MSEs slightly fluctuate around small values even for high values of α. Meanwhile, it can be observed that the average MSEs of other ANFIS models will increase with either too high or too small α values.Figure 11
MSE of the methods as the ratio of test data increases (steel A-36).Figure 12
MSE of the methods as the ratio of test data increases (Al 6061-T6).The box plots in Figures13 and 14 provide some statistical visualization of the performance of all methods for two kinds of materials. Among the three ANFIS models using the grid partitioning method, it can be seen from Figures 13 and 14 that the model with triangular membership function has a better result for steel A-36 but not for Al 6061-T6. Hence, the most suitable membership function cannot be determined. It might depend on which material is considered. These figures also demonstrate the superiority of the two ANFIS models using subtractive clustering over the other models. In comparison with the subtractive clustering with the random values of r, the subtractive clustering with DE has a slightly better performance in both cases. As shown in Figures 13 and 14, the median of the subtractive clustering with DE is lower than those of subtractive clustering with random values of r. Besides, the boxes, which stand for the interquartile range (from 25th percentile to 75th percentile), of the subtractive clustering with DE are also lower than those of subtractive clustering with random values of r, for both cases.Figure 13
Summary of MSE of all methods on test sets (steel A-36).Figure 14
Summary of MSE of all methods on test sets (Al 6061-T6).Although Figures13 and 14 provide the first insight into the performance of all ANFIS models, it is more reliable if a statistical test could be conducted. For this purpose, the Wilcoxon signed-rank test with a statistical significance value of 0.1 is used. The null hypothesis H0 is built as “there is no difference between the performance of the method A and the performance of the method B,” where A is the subtractive clustering with DE and B is a comparative method (grid partitioning + triangular membership function, grid partitioning + bell-shaped membership function, grid partitioning + Gaussian membership function, and subtractive clustering + random range of influence r). According to Table 5, it can be concluded that the subtractive clustering with DE can improve the performance of ANFIS models using the grid partitioning method, with a significance level of 0.1. The difference between subtractive clustering with DE and the subtractive clustering with random r is not significant due to the following reasons. Firstly, the difference between the subtractive clustering with DE and the subtractive clustering with random r is not as large as the differences observed in comparison with other methods. Secondly, the sample size is not large enough for rejection of the null hypothesis H0. Therefore, the subtractive clustering with random r can be used for other cases in which the computational time is important. On the other hand, more experiments on other ratios of test data α and other materials are required to provide a comprehensive comparison. The results show an efficient application of clustering-based ANFIS in fatigue life prediction of 2-DOF compliant mechanism.Table 5
Wilcoxon signed-rank test.
Steel A-36Al 6061-T6P valueReject H0P valueReject H0Clustering (optimized r) vs Triangular0.0937510.093751Clustering (optimized r) vs bell-shape0.0312510.031251Clustering (optimized r) vs Gaussian0.0312510.031251Clustering (optimized r) vs clustering (randomr)100.56250In summary, the above experiments have demonstrated the potential benefit of ANFIS in fatigue life prediction of 2-DOF compliant mechanism. Particularly, the two ANFIS models using subtractive clustering provide better performance compared to the three ANFIS models using the grid partitioning method. In comparison with the subtractive clustering with randomr, the subtractive clustering using DE has slightly better performance, although the difference is not significant. Based on the obtained results, it can be expected that the proposed method can be well-applied for other case studies (materials). The root mean square error of the fatigue life prediction is expected to be smaller than 100 cycles or smaller than 22 in terms of mean square error (MSE) of natural log transformation. Using this criterion, careful users can evaluate the suitability of the proposed method after obtained the training model to decide whether the proposed method can be applied or not.


## 6. Conclusions

This paper for the first time develops a method for fatigue life prediction of 2-DOF compliant mechanism. The adaptive neuro-fuzzy inference system (ANFIS) is utilized for the prediction purpose. In addition to the conventional ANFIS model using the grid partitioning method, the ANFIS model using subtractive clustering is applied. Furthermore, the differential evolution algorithm is utilized for optimizing parameters of the clustering algorithm. Five ANFIS models are applied to two case studies consisting of the steel A-36 and the AL 6061-T6. The obtained results show that the average root mean square error in terms of the number of cycles lies in the range [1.7, 3.97] for Case 1 and [2.03, 10.38] for Case 2. Compared to the ANFIS models using the grid partitioning method, the proposed method achieves a better performance with a significance level from 0.05 to 0.1. Although the subtractive clustering using DE is slightly better than the subtractive clustering with randomr, the respective difference has no statistical significance. The experiments also demonstrate the potential application of ANFIS in fatigue life prediction of 2-DOF compliant mechanism. The results can be extended in predicting the fatigue life for other compliant mechanisms and related engineering applications. In future work, some physical prototypes can be fabricated to estimate the fatigue life for the proposed 2-DOF compliant mechanism.



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