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Abstract

A tool condition monitoring system based on support vector machine and differential evolution is proposed in this article. In this system, support vector machine is used to realize the mapping between the extracted features and the tool wear states. At the same time, two important parameters of the support vector machine which are called penalty parameter C and kernel parameter $σ^{2}$ are optimized simultaneously based on differential evolution algorithm. In order to verify the effectiveness of the proposed system, a multi-tooth milling experiment of titanium alloy was carried out. Cutting force signals related to different tool wear states were collected, and several time domain and frequency domain features were extracted to depict the dynamic characteristics of the milling process. Based on the extracted features, the differential evolution-support vector machine classifier is constructed to realize the tool wear classification. Moreover, to make a comparison, empirical selection method and four kinds of grid search algorithms are also used to select the support vector machine parameters. At the same time, cross validation is utilized to improve the robustness of the classifier evaluation. The results of analysis and comparisons show that the classification accuracy of differential evolution-support vector machine is higher than empirical selection-support vector machine. Moreover, the time consumption of differential evolution-support vector machine classifier is 5 to 12 times less than that of grid search-support vector machine.

Keywords

Support vector machine differential evolution tool condition monitoring

Introduction

Tool wear during machining process has a negative effect on the quality of work-piece. Researches indicate that 20% of downtime for machine tools can be attributed to the tool failure.¹ For decades, many pattern recognition methods, such as artificial neural network (ANN),^2–4 hidden Markov model (HMM),^5–7 have been proposed and utilized to realize tool wear monitoring. However, there are some shortcomings for both methods. ANN, which is based on empirical risk minimization principle, is easier to appear over fitting and low convergence. Additionally, the generalization ability of ANN is obviously lowered when the number of the training samples is limited.⁸ In terms of HMM, independence assumption for the data sequence is indispensable during the application of the algorithm. However, this condition is hard to be satisfied because of the continuity of the tool wear evolution, which lowered the recognition accuracy of the built classifier.⁹ In recent years, kernel function–based support vector machine (SVM), which is based on structure risk minimum principle, is presented to realize tool condition monitoring (TCM). Elangovan et al.¹⁰ applied SVM to realize wear monitoring of the carbide tipped tool during turning operation. By comparing various kernel functions, the authors claimed that the radial basis function (RBF) kernel gives higher classification efficiency. Shi and Gindy¹¹ presented a new tool wear predictive model by combining least squares support vector machines (LS-SVM) with principal component analysis (PCA) technique. The authors claimed that a good agreement could be found between the predicted and the measured wear value. Chen and Limchimchol¹² demonstrated the application of SVM in grinding process monitoring. The analysis results demonstrated that the performance of the grinding process can be identified accurately. All these applications show that SVM-based models are effective for tool wear classification, especially in the case of the limited training samples. At the same time, it can also be demonstrated that the kernel parameters $σ^{2}$ and penalty parameter C play crucial roles on the performance of SVM classifier. The parameter C determines the trade-offs between the minimization of the fitting error and the minimization of the model complexity. σ is the bandwidth of the RBF kernel, which affects the mapping between the transformation of data space and complexity degree of sample distribution in the higher dimensional feature space.¹³ It has been reported that the incorrect select of these two parameters may weaken the classification accuracy obviously.¹⁴ Recently, to select the optimal value of these two parameters, several methodologies have been proposed. One is to use grid search (GS) strategy to get the optimal combination of the two parameters. Rajpoot and Rajpoot¹⁵ utilized the GS algorithm to find the optimal parameters of SVM so as to realize classification of the tissue cell. Li et al.¹⁶ compared four kinds of different GS algorithms and claimed that the bilinear GS strategy is the optimal for choosing kernel parameters. The other method is based on empirical selection (ES) method. Cherkassky and Ma¹⁷ proposed an ES formula to determine the hyper-parameters in the SVM model so as to improve its generalization performance. Chalimourda et al.¹⁸ discussed the influence of C and $σ^{2}$ on the classification accuracy and obtained the optimal SVM parameters based on ES algorithm. These studies prove that the predetermined parameters are necessary for guaranteeing the performance of SVM classifier. However, it can also be found that some shortcomings exist for both methods. The parameters calculated based on ES formula are only suitable for specific samples and it is hard to adopt them as a universal criteria. For the GS strategy, a great deal of computation time needs to be consumed, which largely limit the application of SVM classifier in real industrial environment.

In this article, differential evolution (DE) algorithm is presented to realize the optimal selection of the kernel parameter $σ^{2}$ and penalty parameter C in SVM classifier. The DE algorithm, put forward by Storn and Price,¹⁹ is based on swarm intelligence theory. The main characteristic of DE optimization lies that its crossover constant and scaling factor do not require fine tuning, even though it is indispensable for many other evolutionary algorithms.²⁰ Therefore, it can be easily used to solve a wide variety of real valued problems especially in the noisy, multi-modal, multi-dimensional spaces. Up to now, DE has been used in many fields, such as manipulation of robotics,²¹ communication,²² and pattern recognition.²³ Many literatures have claimed that DE algorithm displays better results than other kinds of evolutionary algorithms.^24–27

Based on SVM and DE algorithm, a TCM system is constructed in this article to realize tool wear monitoring of milling process. In this system, SVM is used to build the nonlinear relationship between sensory information and tool wear category. At the same time, kernel parameter $σ^{2}$ and penalty parameter C are optimized to improve the classification performance by DE optimization. In order to verify the effectiveness of the proposed differential evolution-support vector machine (DE-SVM) classifier, multi-tooth milling experiment of Titanium alloy was carried out. The cutting force signals related to different tool wear states were collected by force dynamometer, and five time domain and four frequency domain features are extracted to depict the dynamic characteristic of the sensory information. Based on the extracted features, DE-SVM classifier is realized to recognize the tool wear states. To make a comparison, GS strategy and ES algorithm are also introduced to optimize these two parameters based on the same training and test samples. Moreover, cross validation is utilized to improve robustness of the classifier evaluation. The analysis and comparison results show that the accuracy rate of DE-SVM is higher than that of empirical selection-support vector machine (ES-SVM). Moreover, the time consumption of grid search-support vector machine (GS-SVM) classifier is 5 to 12 times as much as that of DE-SVM.

The rest of this article is organized as follows. First, the principle of the integrated DE-SVM classifier is presented. Then, a framework of TCM system is constructed and the principal of the feature extraction for the force signals is also depicted. In the following section, milling experiment is described and dynamic features are extracted from force signals. Also, in this section, DE-SVM classifier is constructed and compared with GS-SVM and ES-SVM methodologies by means of cross validation scheme. Some useful conclusions are summarized in the last section.

Principles of DE-SVM

SVM

SVM is mainly based on statistical learning theory and characterized by the application of nonlinear kernels, high generalization capability and the sparseness solution.²⁸ The basic concept of the SVM is to map the training samples which cannot be classified by linear function into a high-dimensional feature space by nonlinear function $φ (x)$ . The classifying principle of SVM²⁹ is shown in Figure 1, in which $x_{i}, x_{j}$ represent two training samples in sample set X and $ξ > 0$ is slack variable whose introduction permits the existence of error-classifying samples. H is the separating line; H₁ and H₂ pass through the samples which are nearest to H. Meanwhile, both H₁ and H₂ are parallel to H, and their distance is defined as the classification margin. The samples on this margin are called support vectors. Giving a set of training sample $(x_{i}, y_{i})$ , where, $i = 1, 2, \dots, n$ , $x_{i} \in R^{n}$ , n represents the dimension of the input space, and $y_{i} \in R$ is the category label. The classification function is defined as follows³⁰

f (x) = sgn {\sum_{i = 1}^{n} a_{i} * y_{i} K (x_{i} • x) + b *}

(1)

where $a_{i} *$ is the optimal solution of Lagrange coefficient corresponding to each sample, $K (x_{i} • x)$ is kernel function and $b *$ is classification threshold. $a_{i} *$ can be acquired by maximizing the Lagrangian dual objective function based on the following equation³¹

{\begin{matrix} max Q (a) = \sum_{i = 1}^{n} a_{i} - \frac{1}{2} \sum_{i, j = 1}^{n} a_{i} a_{j} y_{i} y_{j} K (x_{i} • x_{j}) \\ \sum_{i = 1}^{n} y_{i} a_{i} = 0 \\ a_{i} \geq 0, i = 1, 2, \dots, n \end{matrix}

(2)

Figure 1.

Classifying principle of SVM.

The classical definition of $K (x_{i} • x)$ is $K (x_{i}, x_{j}) = φ (x_{i}) \cdot φ (x_{j})$ . If all samples are ensured to be classified correctly, the following constraint condition should be satisfied

{\begin{matrix} y_{i} [ω \cdot φ (x) + b] - 1 + ξ_{i} \geq 0 \\ ξ_{i} \geq 0, i = 1, 2, \dots, n \end{matrix}

(3)

where ω is the weight vector and b is the bias term. ω can be calculated by minimizing the following objective function³²

min h (ω, ξ) = \frac{1}{2} ‖ ω ‖^{2} + C (\sum_{i = 1}^{n} ξ_{i})

(4)

where $(1 / 2) ‖ ω ‖^{2}$ is the regularization term and C > 0 is constant, which is called penalty parameter. The parameter C determines the trade-offs between the minimization of the fitting error and the minimization of the model complexity.

For SVM, RBF²⁸ is one of the most common used kernel functions whose mathematical expression is given as follows

K (x, x_{i}) = \exp {- \frac{{| x - x_{i} |}^{2}}{2 σ^{2}}}

(5)

where σ is the width of the RBF kernel function, which affects the mapping between the transformation of data space and complexity degree of sample distribution in the higher dimensional feature space.¹³

DE

DE is motivated by natural evolution of the species and has been successfully used to solve global optimization problem. DE algorithm works by cooperation and competition among individuals in certain population during evolution process. Each population comprises NP individuals, and each individual can be depicted by target vector e_ij (i = 1, 2,…, NP; j = 1, 2,…, D).³³ The basic operation of DE includes three processes: mutation, crossover and selection, which are executed and repeated until the terminal condition is satisfied. The detailed description of DE optimization process is given in the following.

Step 1: initialization

First, the population size NP, the set of objective parameters D and the maximum number of generation $g_{max}$ are initialized. In this article, NP is equal to 10D, where D denotes the number of the objective parameters.

Step 2: mutation

Mutation is crucial for maintaining diversity in a population. The most commonly used mutation strategy is DE/rand/1/bin,³⁴ whose mathematical equation is given as follows³⁵

v_{i}^{t + 1} = e_{r 1}^{t} + F * (e_{r 2}^{t} - e_{r 3}^{t})

(6)

where t represent the tth generation, $r_{1}, r_{2}, r_{3} \in {1, 2, \dots, NP}$ and $r 1 \neq r 2 \neq r 3 \neq i$ . $e_{r 1}^{t}, e_{r 2}^{t} and e_{r 3}^{t}$ are three random individuals which are selected to perform mutation and crossover operation. $F \in [0, 2]$ is called the scaling factor which controls the zoom of difference vector.³⁶

Step 3: crossover

The trial vector $u_{i}^{t + 1}$ is generated by mixing the mutated vector $v_{i}^{t + 1}$ with target vector $e_{i}^{t}$ using the following scheme³⁷

u_{ij}^{t + 1} = {\begin{matrix} v_{ij}^{t + 1}, rand (j) \leq C R, & or, j = randn (i) \\ e_{ij}^{t}, & otherwise \end{matrix}

(7)

where $rand (j) \in [0, 1]$ is a uniform random number, j represent the jth objective parameter, $CR \in [0, 1]$ is crossover constant. $randn (i) \in {1, 2, \dots, D}$ is a random index which is designed to ensure that at least one trial individual is different from target individual $e_{i}^{t}$ .

Step 4: selection

After the fitness value of each trial individual and its corresponding target individual are calculated, the selection process is realized by the following greedy algorithm³⁸

e_{i}^{t + 1} = {\begin{matrix} u_{i}^{t + 1}, G (u_{i}^{t + 1}) < G (e_{i}^{t}) \\ e_{i}^{t}, G (u_{i}^{t + 1}) \geq G (e_{i}^{t}) \end{matrix}

(8)

where G is the fitness function. Obviously, the individual with better fitness value is most likely to be selected into the new generation.

Step 5: termination

The optimization process is terminated only if the iteration times are larger than the predefined maximum generation number g_max and the values obtained in the last generation are taken as the optimal parameters.

DE-based SVM classifier

The flowchart of DE-SVM classifier is shown in Figure 2. First, the parameters of SVM classifier, such as the parameter set $(C, σ^{2})$ , the population size NP and so on, are initialized. Then, the training samples are input into the SVM model and the mutation and crossover operations are carried out correspondingly based on equations (6) and (7). After that, SVM training is performed and fitness value is calculated to evaluate effectiveness of each parameter combination in this generation. A new parameter set $(C, σ^{2})$ is produced by the selection operation based on equation (8) and the mutation and crossover are performed again. This process is repeated until the termination condition is satisfied. The final combination of $(C, σ^{2})$ corresponding to the highest fitness value is selected from the last generation and adopted as the optimal parameter set. Based on these parameters, the SVM classifier is constructed to realize the recognition of the tool wear states.

Figure 2.

Flowchart of DE-SVM classifier.

TCM system based on DE-SVM

Framework of TCM system

The framework of TCM system is shown in Figure 3. The realization of this system includes two stages. One is to construct DE-SVM model and the other is to utilize it for monitoring tool wear states in real application. In the first stage, force signals are collected and their corresponding tool wear values are measured by an optical microscope and recorded simultaneously. Under different tool wear categories, time and frequency domain features are extracted from the collected force signals during machining process and the corresponding training and test samples are organized correspondingly. Then, the prepared training samples were used to construct SVM model. The parameter C and $σ^{2}$ are optimized by DE algorithm and the corresponding SVM classifier which called DE-SVM will be obtained. During the second stage, the built optimal SVM classifier is used to realize online TCM by taking the time and frequency domain features as the input.

Figure 3.

Framework of DE-SVM–based TCM system.

Feature extraction

Effective features are necessary to depict the dynamic relationship between the sensory information and tool wear states. In this article, five time domain and four frequency domain features are extracted based on the force signals. The time domain features include skewness, kurtosis, root mean square, variance and peak-to-peak value. The frequency domain features include mean frequency, frequency domain skewness, frequency domain kurtosis and root mean square ratio. The mathematical equations of these features are shown in Table 1.

Table 1.

Mathematical equations of time and frequency domain features.

Feature	Mathematical expression	Feature	Mathematical expression
Skewness	$SKE = \frac{N \sum_{i = 1}^{N} {(x_{i} - \bar{x})}^{3}}{(N - 1) (N - 2) σ^{3}}$	Frequency domain skewness	$FDS = \frac{\sum_{i = 1}^{K} {(f_{i} - \bar{f})}^{3} S (f_{i})}{σ^{3} K}$
Kurtosis	$KUR = \frac{1}{N} \sum_{i = 1}^{N} {(\frac{x_{i} - \bar{x}}{σ_{t}})}^{4}$	Frequency domain kurtosis	$FDK = \frac{\sum_{i = 1}^{K} {(f_{i} - \bar{f})}^{4} S (f_{i})}{σ^{4} K}$
RMS	$RMS = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {x_{i}}^{2}}$	Root mean square ratio	$RMSR = \frac{\sum_{i = 1}^{K} \sqrt{(f_{i} - \bar{f})} S (f_{i})}{\sqrt{σ} K}$
Variance	$VAR = \frac{\sum_{i = 1}^{N} {(x_{i} - \bar{x})}^{2}}{N - 1}$	$\bar{f}$	$\frac{\sum_{i = 1}^{K} f_{i} S (f_{i})}{\sum_{i = 1}^{K} S (f_{i})}$
Peak-to-peak	$P_{k} = \max (x_{i}) - \min (x_{i})$	σ	$\sqrt{\frac{\sum_{i = 1}^{K} (f_{i} - \bar{f}) S (f_{i})}{K}}$
Mean frequency	$MF = \sqrt{\frac{\sum_{i = 1}^{K} {f_{i}}^{2} S (f_{i})}{\sum_{i = 1}^{K} S (f_{i})}}$

RMS: root mean square.

Monitoring and analysis

Experiment and data preparation

Experiments of titanium alloy were conducted to verify the effectiveness of the proposed method. The framework of the experimental setup is shown in Figure 4.

Figure 4.

Framework of the experimental setup.

The machine tools used in this experiment is MAKINO vertical milling center. Ti-6A1-4V was chosen as work-piece material and the size was set to be 150 × 100 × 20 mm. The cutting parameters were set as follows: rotating speed $v = 600 r / \min$ , feed rate $v_{f} = 0.1 mm / rev$ , cutting width $a_{e} = 8 mm$ , cutting depth $a_{p} = 10 mm$ , cutter diameter $d = 50 mm$ , teeth number $n_{t} = 4$ . A Kistler dynamometer was used to collect dynamic cutting force signals in the feeding direction. An optical microscope was utilized to measure the value of the tool wear. A multi-channel charge amplifier (type: Kistler 5070) was utilized to amplify the cutting force signals and a data acquisition card was applied to collect them with 2 kHz sampling rate. The data corresponding to four kinds of tool wear states were obtained and their tool wear scopes are given in Table 2. In the meantime, time and frequency domain features were extracted correspondingly and organized as the candidate dataset. For each tool wear state, 100 samples are obtained and the total number of samples in this dataset is 400.

Table 2.

Tool wear scopes for four kinds of tool wear states.

Tool states	New cutter	Small wear	Middle wear	Severe wear
Scope (mm)	(0, 0.05)	(0.1, 0.15)	(0.18, 0.25)	(0.45, 0.6)

The spatial distribution of several selected feature vectors under different tool wear states is shown in Figure 5. It can be observed that the distribution of these features is severely dispersed and overlapped, which casts some difficulties on the construction of classification decision boundary. In the following section, DE-SVM–based TCM system is utilized to realize the classification of the each tool wear state.

Figure 5.

Spatial distribution of the feature vectors. (a) The spatial distribution of kurtosis(KUR) and skewness(SKE) under different tool wear states. (b) The spatial distribution of variance(VAR) and root mean square value (RMS) under different tool wear states.

k-Fold cross validation

As a kind of statistics analysis tool, cross validation is widely utilized to evaluate the performance of classification accuracy.^39,40 In k-fold cross validation, the overall data are randomly partitioned into k sets of equal sizes. Each of the k subsets is used once for testing and the rest is utilized for training. The steps above are repeated k times, and the average accuracy of these k iterations is taken as the final result. The main advantage of this method is that each subset can be utilized for testing and training which make the final evaluation result robust. In this article, fourfold cross validation is applied to evaluate the performance of ES-SVM, DE-SVM and GS-SVM classifier.

Tool wear classification based on DE-SVM

DE-SVM classification

The initialization parameter of DE-SVM classifier is given as follows: penalty parameter $C = 0.65$ , kernel parameter $σ^{2} = 0.08$ , population size $NP = 20$ , the maximum generation number $g_{max} = 10$ , scaling factor $F = 1$ and crossover constant $CR = 0.5$ . Based on cross validation algorithm, four groups of optimal parameters are calculated and the corresponding DE-SVM classifiers are constructed. The classifying accuracy under each fold is illustrated in Figure 6. It can be seen that accuracy rate of DE-SVM can reach 99.86%, 99.97%, 99.89% and 99.96%, which prove that the DE-SVM classifier can recognize the tool wear states accurately.

Figure 6.

Comparison of the accuracy rate for DE-SVM and ES-SVM.

Comparison with ES-SVM

To make a comparison, ES algorithm is introduced so as to construct ES-SVM classifier based on the same training samples. Two empirical equations, which have been adopted in several articles, are used to select the optimal parameters in SVM. That is, $C = y_{max} • l$ and σ=0.2*range( X ),^17,18 where $y_{max}$ is the maximum output value, l is the number of training data and X is the sample data. For the given training samples, the optimal C is finally calculated as 100 and σ is equal to 0.08. On the basis of these two parameters, four SVM classifiers are trained and tested by cross validation scheme. The accuracy rate of each classifier is calculated and also illustrated in Figure 6.

It can be observed that the average accuracy rate of ES-SVM is only 48.5% while DE-SVM can reach 99.92%. This shows that although ES method can provide an intuitive and fast instruction for choosing SVM parameters, it is not realistic to be adopted as a universal algorithm.

Comparison with GS-SVM

In this section, DE-SVM classifier is further compared with GS-SVM. GS algorithm is a kind of enumeration strategy, whose principle is to evenly split the searching space into grids and try every parameter combination at the node. The optimal parameters are obtained by comparing the classification accuracy of all possible combinations and selecting one parameter set corresponding to the highest accuracy. In this article, considering that the scope of these two parameters C and σ is very large, the GS is carried out in their linear logarithmic space. Therefore, the scope of $\log C$ is from −4 to 5 and $\log σ^{2}$ is from −5 to 1. Within this scope, four kinds of searching density, that is, 20 × 20, 25 × 25, 28 × 28 and 30 × 30, are adopted. The combination of C and σ corresponding to the highest classification accuracy is selected and four classifiers, which are labeled as GS1-SVM, GS2-SVM, GS3-SVM and GS4-SVM, are built. The accuracy rates of these classifiers for the test samples are shown in Figure 7. Moreover, the time consumption of these GS-based classifiers and DE-SVM are shown simultaneously in Figure 8.

Figure 7.

Comparison of the accuracy rate for DE-SVM and GS-SVM.

Figure 8.

Time consumption comparison of DE-SVM and GS-SVM.

From Figure 7, it can be observed that the scope of the accuracy rates for four kinds of GS-SVM models varies from 96.58% to 97.83%. The denser grid can get higher classification accuracy. In contrast, the average accuracy rate of DE-SVM can reach 99.92%, which is a little higher than each of the GS-SVM model. In addition, it can be demonstrated from Figure 8 that the time consumption of DE-SVM is about 87.452 s. In contrast, the shortest computational time of GS-SVM is 445.647 s and the longest is 1127.928 s, which are 5 to 12 times as much as that of DE-SVM. These results show that DE-SVM performs better than GS-SVM in both classification accuracy and computational time.

Conclusion

In this article, an online TCM system–based DE-SVM is constructed. The main characteristic of this system is that the combined application of DE and SVM can meet the requirements of both accuracy and training speed. The milling experiment of Titanium alloy was carried out to verify the effectiveness of the proposed system. The cutting force signals related to different tool wear states were collected and several time domain and frequency domain features are extracted correspondingly. Based on that, ES-SVM, GS-SVM and DE-SVM models are constructed to recognize the tool wear states and make a comparison. Meanwhile, cross validation is applied to improve the robustness of classifier evaluation. The analysis and comparison results show that the accuracy rate of DE-SVM is higher than ES-SVM and GS-SVM. Moreover, the computational time of DE-SVM algorithm is far less than GS-SVM.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project is supported by National Science and Technology Major Projects of China (2014ZX04012-014), National Natural Science Foundation of China (51175371 and 51420105007) and Tianjin Science and Technology Support Program (13ZCZDGX04000 and Key Project of International Cooperation 51420105007.

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. Parallel multitask cross validation for support vector machine using GPU. J Parallel Distr Com 2013; 73: 293–302.

40.

Camacho

Ferrer

Cross-validation in PCA models with the element-wise k-fold (ekf) algorithm: practical aspects. Chemometr Intell Lab 2014; 131: 37–50.

Tool condition monitoring system based on support vector machine and differential evolution optimization