1 Introduction

Batch precision machining stimulated by high productivity levels and inexpensive high-quality products is competitive in manufacturing industries. Cutting tools are important components of machining that affect processing efficiency and product quality. As such, the development of tool condition monitoring (TCM) systems is desirable so as to monitor the cutting process online and recognize tool wear state. With such systems, machining parameters can be timely adjusted to extend tool life and preserve product quality in mass production.

Over the last few decades, a few studies on machining process monitoring have been conducted [1, 2]. Two types of techniques, namely, direct and indirect sensing methods, can be used to measure tool wear. Direct sensing techniques use digital cameras (e.g., toolmaker’s microscope) to measure tool conditions (e.g., flank wear width). However, direct measurements can only be employed in intermittent operations. On the contrary, indirect sensing techniques can continuously measure auxiliary in-process signals (e.g., cutting force [3, 4], vibration [5, 6], acoustic emission [7], and motor current [8]) to infer tool wear because in-process signals can be measured in real time; thus, indirect sensing techniques are suitable for real-time monitoring [9]. In indirect methods, signal features extracted from the original collected data of measurable signals include the time domain (TD) [10], frequency domain (FD) [11], time–frequency domain (TFD) [12], and fusion features [13].

However, each signal feature is not always correlated with tool wear. A subset of features that best predicts tool wear must be selected instead. The feature selection methods for tool wear can be divided according to whether they are related to subsequent tool wear model features. Specifically, feature selection methods may be filter and wrapper or embedded methods. Filter methods are independent of the subsequent tool wear model. Such techniques also use the statistical performance of all training data to directly evaluate and select features. Principal component analysis is a common filter method [14]. Model-independent feature selection methods are computationally fast, but a large deviation exists between feature evaluation and the performance of the tool wear model. Therefore, such a performance may not be optimized. Features can be selected by wrapper or embedded methods that consider the tool wear model [15]. Wrapper approaches generally employ optimization algorithms (such as genetic algorithm (GA) [16] and ant colony algorithm [17]) to search for optimal features with high correlation with the tool model. In the present study, GA, which is generally effective for the rapid global search of large, nonlinear, and poorly understood spaces [18], is used for the further selection of input features that are relevant to tool wear states. Similarly, Kaya et al. [19] employed GA to reduce the dimensionality of a feature set by selecting the features that correlate best with tool conditions, and they also demonstrated efficacy experimentally.

Using selected signal characteristics as inputs, a predictive model can be built, and tool wear must be estimated in real time. Data-driven approaches use learning algorithms, and experimental data build predictive models so that an in-depth understanding of underlying physical processes is not a prerequisite. Artificial neural network (ANN) [20, 21] and support vector machine (SVM) [22, 23] are the most widely applied data-driven approaches for tool wear monitoring. Relative to ANN, SVM is a powerful learning model that is free from the three problems of training efficiency, testing efficiency, and overfitting [24,25,26]. Besides, researchers can apply these models on different products obtained in different batches because the insensitive region of SVM absorbs small-scale random fluctuations in random response [24]. The internal parameter values of SVM greatly affect predictive model accuracy. Certain optimization algorithms are used to optimize SVM parameters and thereby obtain the best parameter combination and reduce parameter adjustment time. García proposed a particle swarm optimization (PSO)–SVM-based model and successfully used it to predict tool flank wear [27]. Li and Zhang used GA to optimize internal SVM parameters. The experimental results indicated that GA–SVM can effectively track the tool wear trend and achieve tool wear and tool life monitoring [28, 29]. Zhang et al. proposed a pattern recognition approach for intelligent fault diagnosis of rotating machinery by using the SVM optimized by the ant colony algorithm. The simulation results revealed that the model has strong generalizability and high prediction accuracy [30]. The gray wolf optimization (GWO) algorithm is an emerging group intelligent optimization method that mimics the leadership level and hunting mechanism of gray wolves in nature. GWO also has a simple structure, easy implementation, and good global performance, all of which are beneficial to engineering implementation [31]. However, GWO algorithm has never been employed to optimize internal SVM parameters for tool wear monitoring through a complex and stochastic process. Therefore, SVM optimized by the GWO algorithm (GWO–SVM) is developed in the current work to build a prediction model of tool wear.

In this work, a TCM system is developed by using an indirect measurement method that is based on GWO–SVM. First, the cutting force signal of the machining process is collected via a cutting process test. Second, the main TD, FD, and wavelet domain features of the signal samples that must be evaluated are extracted. In addition, feature selection (dimensionality reduction) is performed by GA, which avoids the processing complexity of high-dimensional nonlinear feature data and weakens the noise component characteristics of the signal. GWO–SVM is employed to evaluate the dimensionality reduction features and obtain the tool wear grade. Moreover, the robustness of the tool wear monitoring system is further validated with the vibration signals from ref. [32].

2 Methodologies

The tool wear monitoring system used in this study is illustrated in Fig. 1. The input is the cutting force signal of a real-time milling process, and the output is the wear state of the milling cutter (initial, normal, and severe wear). The specific implementation process is divided into three modules: feature extraction, feature selection, and tool wear state classification modules. Their details are introduced in the following subsections.

Fig. 1
figure 1

Tool wear state recognition model

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2.1 Feature extraction module

During the process of machining, the dynamic cutting force data generated are always large. Such huge data can affect the calculation speed, and the online wear monitoring system must reflect the result quickly. To improve the calculation speed, the feature extraction module significantly reduces the dimension of raw data (in TD and FD). The module also aims to maintain the relevant information about tool wear conditions in the extracted features. Signal characteristics are extracted using TD analysis, FD analysis, and four-layer wavelet packet decomposition. Table 1 presents the 28 corresponding feature values extracted, of which S401–S416 represent the 16 corresponding proportions of wavelet energy.

Table 1 Statistical parameters and mathematical expressions in the TFD
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2.2 Feature selection module

Multiple types of features extracted through feature extraction module can result in many potential choices. Thus, combining them into a classification module is not efficient. For such a reason, GA is used for feature selection. The GA process includes replication, hybridization, mutation, competition, and selection. Its advantages are simplicity, randomness, and compatibility with other algorithms. Detailed descriptions of the GA feature selection can be found in [32]. The fitness function set in GA feature selection module is fitval = 1 − |r|, and the closer the value of “fitval” is to 0, the higher the correlation between the feature and the tool wear. r is the Pearson correlation coefficient, and the expression is as follows:

$$ r=\frac{\sum \limits_{i=1}^n\left({x}_i-\overline{x}\right)\left({y}_i-\overline{y}\right)}{\sqrt{\sum \limits_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}\sqrt{\sum \limits_{i=1}^n{\left({y}_i-\overline{y}\right)}^2}} $$
(1)

In Formula (1), r represents the correlation coefficient between variables x and y.

2.3 Tool wear state classification module

2.3.1 SVM classification algorithm

SVM is a common method of classification and pattern recognition. Proposed by Vladimir N. Vapnik and Alexey Ya, SVM aims to transform a signal into a high-dimensional feature space and then solve a binary problem in a hyper plane [33].

Taking the soft edge problem as an example to illustrate the SVM classification calculation method, this study provides the specific calculation formula. When the input data are linearly separable (hard edge problem), they can be solved by taking a small λ value. The specific calculation method is as follows:

$$ \left[\frac{1}{n}\sum \limits_{i=1}^n\max \left(0,1-{y}_i\left(\overrightarrow{w}\overrightarrow{x_i}-\mathrm{b}\right)\right)\right]+\uplambda {\left|\left|\overrightarrow{w}\right|\right|}^2. $$
(2)

The initial calculation is to change Formula (2) into a constrained optimization problem with a differentiable objective function. For i ϵ {1,  … , n}, variable ζi = max (0, 1 − yi(wxi − b)) is introduced. ζi satisfies yi(wxi − b) ≥ 1 − ζi and is the minimum non-negative number. Thus, the optimization problem becomes

$$ \operatorname{minimize}\frac{1}{n}\sum \limits_{i=1}^n{\upzeta}_i+\uplambda {\left|\left|\overrightarrow{w}\right|\right|}^2. $$
(3)

The given constraint is yi(wxi − b) ≥ 1 − ζi, and ζi ≥ 0.

The given Lagrangian dual problem is solved by the following simplification:

$$ {\displaystyle \begin{array}{c}\operatorname{minimize}\ \mathrm{f}\left({c}_1,\dots, {c}_n\right)=\\ {}\sum \limits_{i=1}^n{c}_i-\frac{1}{2}\sum \limits_{i=1}^n\sum \limits_{j=1}^n{y}_i{c}_i\left({x}_i{y}_j\right){y}_j{c}_j\\ {}\mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}\sum \limits_{i=1}^n{c}_i{y}_i=0\\ {}0\le {c}_i\le \frac{1}{2 n\lambda}\end{array}\right..\end{array}} $$
(4)

The dual maximization problem is a quadratic function problem of the ci linear constraint. This problem can be effectively solved by the quadratic programming algorithm. Thus, ci has the following definition:

$$ \overrightarrow{\mathrm{w}}=\sum \limits_{i=1}^n{c}_i{y}_i\overrightarrow{x_i}. $$
(5)

When \( \overrightarrow{x_i} \) is on the correct side of the edge line, ci = 0; when \( \overrightarrow{x_i\ } \) is on the edge line, 0 ≤ ci ≤ (2ηλ)−1. Given the definition, \( \overrightarrow{\mathrm{w}} \) can be written as a linear combination of support vectors. Offset b can also be calculated by \( \overrightarrow{x_i} \).

$$ {\mathrm{y}}_i\left(\overrightarrow{\mathrm{w}}\overrightarrow{x_i}-b\right)=1\leftrightarrow b=\overrightarrow{\mathrm{w}}\overrightarrow{x_i}-{\mathrm{y}}_i, $$
(6)

where yi =  ± 1 and yi−1 = yi.

Nonlinear problems can be transformed into linear problems in a high-dimensional space by nonlinear function mapping and processing. Suppose that a nonlinear classification model must be trained. After mapping point \( \overrightarrow{x_i} \) to a high-dimensional space on the model, the nonlinear classification model can be changed into a linear model, that is, \( \overrightarrow{x_i}\to \varphi \left(\overrightarrow{x_i}\right) \). Then, a kernel function can be defined.

$$ k\left(\overrightarrow{x_i},\overrightarrow{x_j}\right)=\varphi \overrightarrow{x_i}\varphi \overrightarrow{x_j} $$
(7)

The classification vector \( \overrightarrow{\mathrm{w}} \) in the converted space can be expressed as follows:

$$ \overrightarrow{\mathrm{w}}=\sum \limits_{i=1}^n{c}_i{y}_i\varphi \left(\overrightarrow{x_i}\right), $$
(8)

where ci can be solved by the following formula.

$$ \operatorname{minimize}\ \mathrm{f}\left({c}_1\dots {c}_n\right) $$
$$ {\displaystyle \begin{array}{c}=\sum \limits_{i=1}^n{c}_i-\frac{1}{2}\sum \limits_{i=1}^n\sum \limits_{j=1}^n{y}_i{c}_i\left(\varphi \left(\overrightarrow{x_i}\right)\varphi \left(\overrightarrow{x_j}\right)\right){y}_j{c}_j\\ {}=\sum \limits_{i=1}^n{c}_i-\frac{1}{2}\sum \limits_{i=1}^n\sum \limits_{j=1}^n{y}_i{c}_ik\left(\overrightarrow{x_i},\overrightarrow{x_j}\right){y}_j{c}_j\\ {}\mathrm{s}.\mathrm{t}.\Big\{\begin{array}{c}\sum \limits_{i=1}^n{c}_i{y}_i=0\\ {}0\le {c}_i\le \frac{1}{2 n\lambda}\end{array}\kern1.5em .\end{array}} $$
(9)

Coefficient ci can be solved according to the aforementioned method. b can also be solved according to the point \( \varphi \left(\overrightarrow{x_i}\right) \) on the edge line in the transformed space.

$$ {\displaystyle \begin{array}{c}b=\overrightarrow{\mathrm{w}}\varphi \left(\overrightarrow{x_i}\right)-{y}_i=\left[\sum \limits_{k=1}^n{c}_k{y}_k\varphi \left(\overrightarrow{x_k}\right)\varphi \left(\overrightarrow{x_i}\right)\right]-{y}_i\\ {}=\left[\sum \limits_{k=1}^n{c}_k{y}_kk\left(\overrightarrow{x_k},\overrightarrow{x_i}\right)\right]-{y}_i\ \end{array}} $$
(10)

The classification of the new points after the conversion space is calculated.

$$ {\displaystyle \begin{array}{c}\overrightarrow{\mathrm{z}}\to \mathit{\operatorname{sgn}}\left(\overrightarrow{\mathrm{w}}\varphi \left(\overrightarrow{z}\right)-b\right)\\ {}=\mathit{\operatorname{sgn}}\left(\left[\sum \limits_{i=1}^n{c}_i{y}_ik\left(\overrightarrow{x_i},z\right)\right]-b\right)\end{array}} $$
(11)

ci and b are two important SVM parameters. This study uses a radial basis function as a kernel function, which is presented below.

$$ k\left(\overrightarrow{x_i},\overrightarrow{x_j}\right)=\exp \left(\frac{-{\left\Vert \overrightarrow{x_i}-\overrightarrow{x_j}\right\Vert}^2}{2{g}^2}\right)\kern0.75em g>0 $$
(12)

This study focuses on the optimum combination of penalty factor c and nuclear parameter g of the final model [34].

2.3.2 GWO–SVM

To improve the prediction accuracy of SVM, this study uses the GWO algorithm in optimizing penalty factor c and nuclear parameter g in SVM. Mirjalili (2014) first proposed GWO, which is a relatively new algorithm [35]. The GWO algorithm simulates the hunting behavior process under the gray wolf hierarchy. The algorithm can be divided into three stages: encirclement, pursuit, and attack. The mathematical model is described as follows.

  1. (1)

    Surround

The first step in the hunting process is the encirclement of prey, which can be described as follows:

$$ {\displaystyle \begin{array}{c}\overrightarrow{D}=\mid \overrightarrow{C}\ast \overrightarrow{X_p}(t)-\overrightarrow{X}(t)\mid \\ {}\overrightarrow{X}\left(t+1\right)=\overrightarrow{X_p}(t)-\overrightarrow{A}\ast \overrightarrow{D},\end{array}} $$
(13)

where \( \overrightarrow{D} \) and \( \overrightarrow{A} \) are the weighting coefficients, \( \overrightarrow{X_p} \) is the position vector of the prey, t is the number of iterations, and \( \overrightarrow{X} \) is the position of the wolf.

In Formula (13), coefficients \( \overrightarrow{A} \) and \( \overrightarrow{C} \) are respectively updated as follows:

$$ {\displaystyle \begin{array}{c}\overrightarrow{A}=2\overrightarrow{a}\ast \overrightarrow{r_1}-\overrightarrow{a},\\ {}\overrightarrow{C}=2\overrightarrow{r_2},\end{array}} $$
(14)

where \( \mid \overrightarrow{a}\mid \) is a linearly decreasing constant and \( \mid \overrightarrow{a}\mid \in \left(2,0\right) \); r1,r2 are constants, and r1,r2 ∈ (0, 1).

  1. (2)

    Hunt

Gray wolves can identify the position of their prey and surround it. The hunting process is usually dominated by α, and β and δ are responsible for the assistance. In an abstract space, the best position of the prey is unknown. Thus, when simulating the hunting behavior of gray wolves in mathematics, the position of α is assumed to be the best candidate solution. Moreover, β and δ can further understand the potential of the prey. In terms of location, the three best solutions that are currently available should be prepared. Other search units (including ω) can also be asked to update their location by search. The formula is expressed as follows:

$$ {\displaystyle \begin{array}{c}\overrightarrow{D_{\alpha }}=\mid \overrightarrow{C}\ast \overrightarrow{X_{\alpha }}-\overrightarrow{X}\mid, \overrightarrow{D_{\upbeta}}=\mid \overrightarrow{C}\ast \overrightarrow{X_{\upbeta}}-\overrightarrow{X}\mid, \overrightarrow{D_{\updelta}}=\mid \overrightarrow{C}\ast \overrightarrow{X_{\updelta}}-\overrightarrow{X}\mid \\ {}\overrightarrow{X_1}=\overrightarrow{X_{\alpha }}-\overrightarrow{A_1}\ast \left(\overrightarrow{D_{\alpha }}\right),\overrightarrow{X_2}=\overrightarrow{X_{\upbeta}}-\overrightarrow{A_2}\ast \left(\overrightarrow{D_{\upbeta}}\right),\overrightarrow{X_3}=\overrightarrow{X_{\updelta}}-\overrightarrow{A_3}\ast \left(\overrightarrow{D_{\updelta}}\right)\\ {}\overrightarrow{X}\left(t+1\right)=\left(\overrightarrow{X_1}+\overrightarrow{X_2}+\overrightarrow{X_3}\right)/3,\end{array}} $$
(15)

where \( \overrightarrow{X}\left(t+1\right) \) is the optimal solution for the current iteration.

  1. (3)

    Attack

The attack is the last step in the hunting process. The purpose is to capture the target, that is, to complete GWO. According to the formula, \( \overrightarrow{A}=2\overrightarrow{a}\ast \overrightarrow{r_1}-\overrightarrow{a} \). When \( \overrightarrow{a} \) is linearly descending from 2 to 0, \( \overrightarrow{A} \) is decremented, that is, \( \overrightarrow{A}\in \left(-2\mathrm{a},2\mathrm{a}\right). \) The judgment condition is \( \mid \overrightarrow{A}\mid \). When \( \mid \overrightarrow{A}\mid \) < 1, the attack is initiated (GWO finds the optimal solution); when \( \left|\overrightarrow{A}\right| \) > 1, the wolf group diverges (seeking the new optimal solution).

The process of using GWO to optimize the SVM algorithm is depicted in Fig. 2.

Fig. 2
figure 2

GWO of SVM parameters

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The specific steps are as follows.

  1. Step 1

    Initialization: The parameter \( \overrightarrow{a} \) of the GWO algorithm and the constant vectors \( \overrightarrow{A} \) and \( \overrightarrow{C} \) are initially assigned. The range of values is also agreed upon.

  2. Step 2

    Parameters c (penalty factor) and g (nuclear parameter) of SVM are used as the 2D coordinates of the individual positions of the wolves.

  3. Step 3

    The SVM model is trained through the training set samples to calculate the fitness. The ranks of the wolves (α, β, δ, and ω) are classified according to the fitness values. The formula for calculating the fitness value is as follows:

$$ \mathrm{Fit}=1-\frac{y_t}{y_t+{y}_f}, $$
(16)

where yt is the correct number of classifications and yf is the number of classification errors.

  1. Step 4

    The position of the wolves is updated by the position update formula.

  2. Step 5

    The fitness update of individuals is calculated, and the optimal fitness value of the current algebra is recorded as Fit _ best. If Fitbest > Fitα (the fitness of α), then Fitα is updated to Fitbest, and the corresponding position is recorded. If Fitβ < Fitbest < Fitα, then Fitbest is assigned to β. The corresponding position is also updated to β. If Fitδ < Fitbest < Fitβ, then Fitbest and the corresponding position are updated to δ. The position of α is the optimal position for the population that must be iterated to the current algebra.

  3. Step 6

    Whether the number of iterations reaches the set threshold or the global optimal position meets the minimum limit is determined. If the condition is satisfied, then the loop is terminated, and the optimal parameters bestc and bestg are returned. Otherwise, the process returns to Step 4.

  4. Step 7

    SVM classification models are established through bestc and bestg.

3 Milling experiment and feature selection

3.1 Experimental design

This experiment uses VDL-600A CNC machining center, JR-YDCL-III05B piezoelectric three-measurement system, and a metallographic microscope to build a test platform (Fig. 3). The tool is a two-flute tungsten carbide-coated solid carbide milling cutter with a diameter of 10 mm. The material of the workpiece is carbon steel (AISI 1045). AISI 1045 has a good processing performance and wide application range. AISI 1045 is also mainly used to manufacture moving parts with high strength and high surface quality, such as turbine impellers and compressor pistons. The basic information of AISI 1045 is presented in Table 2.

Fig. 3
figure 3

Milling processing platform

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Table 2 Basic information of workpiece material
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Table 3 shows that the cutting parameters in the experiment are fixed and that the flank wear value (VB) is measured after a complete 200-mm cutting distance using a metallographic microscope. The average VB of the two blades of the milling cutter is considered the tool wear value. The sampling frequency of the force gauge is set to 20,000 Hz, and the cutting force data generated in the first 50 mm distance are collected during each pass. The cutting force signal includes three directions: X, Y, and Z. A total of 108 sets of data are obtained.

Table 3 Level of machining variable values
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According to the feature extraction method of the statistical signal and wavelet packet analysis in the TFD, the number of extracted features is up to 28 × 3 = 84 (the force has three directions). In addition, the extracted feature directly used as an input can cause a dimensional disaster, and the calculation amount is large. The obtained mathematical models often fail to meet the target requirements at the same time. Therefore, obtaining a wear state recognition model with good calculation performance requires the selection of the most relevant feature signals from the model.

3.2 Feature selection based on GA

This study uses MATLAB GA toolbox for feature selection. The specific GA parameters are set and are shown in Table 4.

Table 4 GA parameter settings
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According to the parameter settings in Table 4, the force signals in the three directions of x, y, and z are selected. When GA reaches the termination condition, the selected feature index value and the optimal and average fitness values are returned.

Figure 4a, b, and c represent the feature selection process of Fx, Fy, and Fz by GA, respectively. In Fig. 4a, the cycle is terminated when the iteration is 40. Here, the optimal and average fitness values are 0.295. The correlation between the selected features and the wear amount is high, and the absolute value of the Pearson coefficient is close to 0.7. Figure 4b illustrates that the average fitness value is 0.283 after 10 iterations from 0.56 and that the whole cycle is terminated when iterating to the 40th generation of the population. The final optimal fitness value is 0.283, which indicates that the Pearson coefficient value |r| between the selected Fy eigenvalues and the tool wear is also above 0.7. In Fig. 4c, the average and best fitness values converge to the optimal in the 27th generation. Both values equal to 0.217, that is, the correlation between the selected feature and the wear value is above 0.78.

Fig. 4
figure 4

a–c GA feature selection

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The feature selection program is repeated 30 times in the repeatability test to find the features related to tool wear and avoid large random errors in a single operation. The features with high frequency of occurrence are selected from Table 5. The number of signal features selected is 14, which is significantly lower than the previous number of 84 feature vectors.

Table 5 Selected features by GA (force signals)
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3.3 Feature correlation analysis

To verify the validity of the GA feature selection, this study plots the correlation between the feature and the wear value (Fig. 5). The abscissa is the tool wear value, and the ordinate is the corresponding feature value. Given the space limitations, only 3 of the 14 features selected on the basis of GA are used. The correlation of these features with wear is illustrated in Fig. 5a, c, and e. Figure 5b, d, and f present the correlations between the feature and wear values that are not selected by GA.

Fig. 5
figure 5

a–f Analysis of the correlation between the features and tool wear

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Among the characteristic signals of Fx, Fig. 5a is the correlation diagram between the characteristic Fx_σ^2 (variance of the Fx signal) and the wear value selected by GA. In the diagram, Fx_σ^2 increases significantly as the wear value increases. The correlation is evident. The graph in Fig. 5b presents the correlation curve between the Fx_max (maximum Fx signal) and the wear value that is not selected by GA. When the wear value increases, no evident variation law of Fx_max exists. The correlation of Fx_max with the wear value is not as great as that of Fx_σ^2 (selected feature).

The correlation between the eigenvalue of Fy signal and the wear value is illustrated in Fig. 5c and d. Figure 5c presents the selected feature Fy_s402 (the ratio of the energy of the s402 band to the total energy after reconstructing the wavelet packet decomposition of the Fy signal), which increases as the wear value increases. By contrast, Fig. 5d shows that the unselected feature Fy_FC (FC of the Fy signal) exhibits random fluctuations when the wear value changes. Such fluctuations are minimal.

Among the Fz signal characteristics, the selected and unselected features are displayed in Fig. 5e and f, respectively. The selected feature Fz_s406 has a significant positive correlation with wear value, whereas Fz_mean and Fz signal are average (not selected). The characteristic has no obvious regular change trend when wear value increases. Therefore, the correlation between Fz_mean and wear value is not large.

Comparing other selected features with the unselected features also follows the same rules, which explain the validity of the GA feature selection to a certain extent.

4 Experimental results and analysis

The tool wear state recognition model takes the signal feature extracted in Section 3.3 as the input. The output is the corresponding tool wear state.

During the milling process, 108 sets of data are obtained. The tool wear process is presented in Fig. 6. Different wear stages are divided by tool wear values. The first 16 sets are the initial wear stage, the 17–87 sets are the normal wear stage, and the remaining 21 sets are the severe wear stage. Figure 7 illustrates two marked points where the tool wear rate changes from the previous data.

Fig. 6
figure 6

Wear process of cutting tool

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Fig. 7
figure 7

Wear state division (108 sets)

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The training and test sets, including their data, are made separate. To compare the effectiveness of the GA feature selection method used in this study (Section 3 of this paper) and the optimized SVM effects of PSO and GWO, we analyzed the optimization process of the four algorithms and the accuracy of the classification results, as presented in Figs. 8 and 9, respectively. The relevant parameters of PSO and GWO are set, as presented in Table 6.

Fig. 8
figure 8

Iterative process comparison

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Fig. 9
figure 9

Classification accuracy for the test set

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Table 6 Parameter settings of the optimization algorithms
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In Fig. 8, PSO–SVM represents a classification algorithm that optimizes SVM parameters using PSO with all extracted features as inputs. GWO–SVM represents a classification algorithm that uses all extracted features as inputs and GWO to optimize SVM parameters. GA–PSO–SVM is a classification algorithm that optimizes SVM parameters by PSO and uses the features selected by GA as inputs. After GA feature selection, GA–GWO–SVM uses the features as inputs, and GWO optimizes the SVM parameters. The convergence speed and final fitness value of GA–GWO–SVM are better than those of GWO–SVM. However, the convergence speed of GA–PSO–SVM and the fitness value on the training set are worse than those of PSO–SVM. In addition, the optimization effect of the GWO algorithm is higher than that of PSO.

In Fig. 9, the classification accuracy of the four algorithms in the three wear stages of the test set is presented. The accuracy of the classification algorithm using GA feature selection is greater than that of the classification algorithm with all features as inputs, namely, GA–PSO–SVM > PSO–SVM, GA–GWO–SVM > GWO–SVM. GWO–SVM has a better effect than PSO–SVM and benefits from random walks and few parameters. The accuracy of PSO–SVM is 81.48%, that of GWO–SVM is 92.59%, and the accuracy of GA–PSO–SVM is 90.74%.

The classification accuracy of GA–GWO–SVM is 96.29%. Table 7 and Fig. 9 compare the classification effects of various algorithms on the test set; here, GS stands for grid search. SVM and GA–SVM divide the points of all test sets in the second category (normal abrasion), and the classification effect is extremely poor. Among other classification algorithms, GA–GWO–SVM has the best classification effect, and the classification accuracy rates in the three wear stages are 100%, 94.29%, and 100%. The final total classification accuracy is 96.30%, which can result in the superior effect of online tool wear recognition.

Table 7 Comparison of classification methods (force signals)
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5 Method validation with vibration signals

The milling dataset from Ref. [32] is implemented to validate the tool wear monitoring system. The data set has force, vibration, and current signals in the milling data set. The vibration signals are selected to verify the robustness of the proposed method. In the experiment, the stainless steel (HRC52) is machined by a two-flute ball nose cutter in a high speed milling machine (Röders Tech RFM760) with a spindle speed of up to 42,000 rpm. Three Kistler piezo accelerometers are mounted on the workpiece to measure the machine tool vibrations of the cutting process in the x, y, and z directions.

The 315 sets of tool wear values are divided into three stages, namely, Stage I, Stage II, and Stage III (Fig. 10). The tool wear rates differ in each stage. Based on the proposed tool wear monitoring system in this paper, the vibration signals are filtered by time domain, frequency domain, and Wavelet analysis, and the 84 features are obtained. Then 28 of the 84 features, which are show in Table 8, are selected by GA after running the feature selection program 30 times. Finally, the selected features are fed into GWO–SVM and other models to recognize tool wear states.

Fig. 10
figure 10

Wear state division (315 sets)

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Table 8 Selected features by GA (vibration signals)
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The results of tool wear states recognition based on vibration signals are presented in Table 9. It is seen from Table 9 that the classification accuracy of the models with feature selection by GA is higher than the models without feature selection, and the running time of models with feature is less than the models without feature selection except SVM model. SVM model saves time at the expense of accuracy because its parameters are not optimized. In addition, GA–GWO–SVM and GWO–SVM have the highest classification accuracy. Comprehensive consideration of classification accuracy and running time, the proposed GA–GWO–SVM outperforms other algorithms.

Table 9 Comparison of classification methods (vibration signals)
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6 Conclusions

Efficiently and accurately monitoring tool wear state during machining provides guidance for the adjustment of machining parameters to ensure the stability of machining quality. In this study, an automated tool wear condition monitoring scheme for the milling process is developed using a cutting force sensor. The scheme includes three modules, namely, feature extraction, feature selection, and prediction. Feature extraction is implemented by TD analysis, FD analysis, and wavelet packet decomposition to obtain comprehensive information. Feature selection employs GA to reduce the complexity of the prediction model. The SVM model optimized by GWO (i.e., GWO–SVM) for generalizability is proposed to predict tool wear state.

GA searches the correlation between signal and wear features and selects signal features that are highly correlated with wear features from the extracted features by TD analysis, FD analysis, and wavelet packet decomposition. GA greatly reduces the input dimension of the prediction model and improves the prediction accuracy. In Table 7, the comparison between the systems with and without feature selection based on GA indicates a 4% increase in system accuracy because of the use of GA. The results of the comparison of the proposed method and the SVM model optimized by certain common optimization techniques reveal the superiority of the GWO–SVM model (Table 7). Therefore, the GWO–SVM model outperforms other models (SVM, GA–GS–SVM, and GA–PSO–SVM) in terms of prediction accuracy and time.

Vibration signals taken from Ref. [32] are used to verify the robustness of the tool wear condition monitoring scheme. Table 9 shows that the proposed tool wear monitoring scheme maintains its superiority in prediction accuracy. Therefore, the proposed tool wear condition monitoring scheme based on GA–GWO–SVM technique is beneficial in improving recognition accuracy of tool wear state, and provides an effective guide for decision-making in the machining process.