Cutting tool wear prediction based on the multi-stage Wiener process

Cutting tools are one type of critical component of modern computer numerical control (CNC) machining systems. They wear out continuously during the machining process until they fail, and cutting tool failure can lead to the collapse of the entire system and even cause substantial losses. Therefore, it is of great importance to study the method for tool wear prediction. A new model for wear prediction of cutting tools is established based on a multi-stage Wiener process, where the degradation rates of cutting tools are considered to change in three stages based on the typical cutting tool wear curve model. Firstly, the degradation processes of cutting tools are divided into three stages. Secondly, the parameter estimation for each stage of the degradation processes of cutting tools is completed, respectively, by utilizing the EM (expectation–maximization) algorithm. Then, the wear of cutting tools is predicted, and the reliability of cutting tools is analyzed by using a numerical integration simulation method based on the Monte Carlo algorithm. Finally, the proposed model is illustrated and verified via the flank wear data of cutting tools, and the prediction accuracy is measured by mean squared error (MSE) and the coefficient of determination (R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^{2}$$\end{document}). The prediction results show that the proposed model enables us to make more economical maintenance by delaying the tool replacement time with fewer degradation data. Compared to the traditional methods based on machine learning (ML), the proposed model can complete the wear prediction and reliability analysis more accurately.


Introduction
Cutting tools are a very important type of component of modern machining and manufacturing systems.The prognostic and health management (PHM) [1,2] of cutting tools can directly affect the production quality and costs [3].The flank wear width is one of the most important performance characteristics showing the health condition of cutting tools [4].Accurate wear prediction of cutting tools can help engineers make moderate maintenance and prevent serious accidents properly [5].In engineering practice, overprotection strategies are now widely used, and cutting tools are replaced prematurely without an empirical theoretical basis.The unnecessary replacement results in inefficient machining operations and insufficient use of cutting tools [6].Therefore, accurate wear prediction of cutting tools based on the degradation process for moderate replacement strategy has drawn the attention of many researchers.The wear prediction methods for cutting tools can be divided into three categories: methods based on physical models, machine learning (ML), and stochastic models [7].
Physical model-based approaches describe the degradation paths of cutting tools considering failure and damage mechanisms.Iliescu et al. [8] established a wear prediction model for cutting tools with the consideration of thrust and drilling parameters.Nouri et al. [9] presented a real-time monitoring method for the wear of end mills by tracking the coefficients of cutting forces.An et al. [10] proposed an accurate wear prediction model for cutting tools by describing the degradation process of cutting tools based on the Paris crack growth law.Hu et al. [11] established a three-dimensional model for the cutting tool turning process based on the finite element analysis simulation and predicted the cutting tool life with the consideration of turning speed, cutting depth, and feed.Attanasio et al. [12,13] developed a finite element analysis software subroutine that took into account the geometry update of cutting tools to simulate cutting tool wear, and good agreement was achieved.However, understanding the failure mechanism of cutting tools requires complicated work, including thermodynamic analysis, finite element analysis, and specific experiments, which limit the application of physical model-based methods.These methods rely on predefined cutting parameters and cutting tool geometry, which cannot be updated with real-time data from each cutting tool.Besides, the cutting processes are also affected by many random factors, and the degradation processes are stochastic [14], which obviously cannot be taken into account by the physical model approaches.
Different from the method based on physical models, ML-based methods can predict the wear and reliability of cutting tools according to the monitoring data of machining processes [15].Wang et al. [16] proposed a novel physics-guided neural network model for cutting tool wear prediction, which solved the problem that the traditional physical models could not consider the cutting conditions and dynamic changes of physical parameters in the machining process.Wu et al. [17] presented a random forest (RF) based cutting tool wear prediction method and compared the performance of RFs with the methods based on feed forward back propagation artificial neural network (FFBP-ANNs) and support vector regression (SVR).Experimental results showed that compared with FFBP ANNs and SVR with a single hidden layer, RFs could produce more accurate prediction results.ML-based methods have been widely used in machining systems with more complex failure mechanisms, and the advantage of these methods is that it does not need to study the mechanism of cutting tool wear in depth or establish a complex physical model of tool cutting.However, ML-based methods require a large amount of historical sample data, and the training and prediction samples need to be independently and identically distributed [18].More and more systems are designed to be highly reliable, and it is difficult and costly to obtain sufficient failure data.Sometimes, even no failure data could be obtained even with the help of accelerated life testing [19,20].To solve this problem, models for cutting tool wear prediction are proposed based on the theories of the stochastic degradation process.
Different from the methods based on physical models and ML models, which mainly focus on the establishment of deterministic models and are less likely to consider random factors, the methods based on stochastic processes are presented by using the degradation data of cutting tools.Different from the time-to-failure data obtained from destructive experiments, the degradation data of cutting tools can be obtained when the machining system is working by monitoring performance characteristics, such as crack growth and vibration.When utilizing stochastic models, we do not need to fully understand the degradation mechanism of cutting tools or go through training a large amount of historical empirical data to complete the wear prediction.Instead, the reliability of cutting tools can be obtained in the form of a probability density function (PDF) with Bayesian inference, and the uncertainty in the cutting tool wear process can be dealt with by random coefficients or filtering algorithms.Wiener process [21], gamma process [22], and inverse Gaussian process [23] are three commonly used stochastic processes that are very effective in dealing with mechanical degradation problems.These stochastic models allow good prediction of reliability and wear of cutting tools based on degradation data when time-tofailure data are not available.
In this paper, a multi-stage Wiener process is applied to model the degradation process of cutting tools, in which the degradation rates are considered to change at certain time points.The remainder of this paper is organized as follows: Section 2 presents the degradation model based on the Wiener process.Section 3 uses the EM algorithm to estimate the parameters in the Wiener process and derives a formula for tool reliability.Section 4 illustrates the theoretical basis for modeling the threestage Wiener process combined with the typical cutting tool wear curve, and a flow chart is drawn to summarize the research methodology of this paper after proposing a suitable prediction accuracy assessment algorithm.In Section 5, the proposed method is illustrated by numerical examples to check the validity and accuracy, and the calculation results are compared with classical ML-based wear prediction methods of FFBP ANNs, SVR, and RFs.The reliability of cutting tools is then calculated, and the result corroborates with the predictions obtained from the three-stage Wiener process model.Section 6 concludes the paper and provides an outlook for future work.

Degradation models based on Wiener processes
The Wiener process model is one of the most widely used stochastic models because the first hitting time of the Wiener process can be expressed as an inverse Gaussian distribution [24], which is convenient to give the analytical solution of the reliability function.For a complicated dynamic system, stochastic behavior is inevitable due to multiple sources of variability, which contribute to the uncertainty of the reliability estimation.Therefore, the effect of these uncertainties and different kinds of variability should be incorporated into a wear model to improve the accuracy of the reliability estimation [25].Due to the existence of the aforementioned uncertainties, combined with measurement errors in actual machining, the degradation path of the cutting tool is often not strictly monotonous.Sun et al. [26] modeled the tool wear process of a cutting tool with the Wiener process, considering the measurement variability, and then estimated the RUL of cutting tools.The gamma and inverse Gaussian processes require the degradation data to be strictly monotonic.Hence, the Wiener process model is selected to model the degradation data of cutting tools.Let X i (t) denote the observed degradation of the ith cutting tool at time t; the Wiener process model is used to model the cutting tools degradation process as follows.
where the parameters i and i are modeled as independent Gaussian random variables, that is, i ∼ N( , 2 ) , i ∼ N( , 2 ).k is a diffusion parameter, B(t) is a stand- ard Brownian motion, i represents the ith cutting tool, and I = 1,2…m.m represents the number of cutting tools.Furthermore, the degradation increments of the ith cutting tool under the jth observation can be expressed as: where X i (t j ) denotes the amount of degradation of the ith cutting tool at the jth measurement time, j = 1,2…n.tj denotes the moment of the jth measurement of the cutting tool wear, and it is assumed that the detection times are consistent for the same type of components.The increment of time t j − t j−1 is denoted by Δt .The distribution of the degradation incre- ments for the ith cutting tool is as follows.
The PDF for the jth degradation increment can then be expressed as: The joint probability density function for the random parameter Θ � i = i , i can be expressed as: (1) 3 The EM algorithm for estimating the parameters of stochastic processes Commonly used parameter estimation methods include the maximum likelihood estimation (MLE) method [27], the Bayesian theory-based methods [28], the Kalman filter (KF) algorithm [29], and the particle filter algorithm (PF) [30].
The EM algorithm [31,32] is an algorithm that can be used to maximize the likelihood function in the presence of missing data and/or latent variables.In this paper, we study the degradation data of the cutting tool in segments, and each segment of the degradation data is not a set of complete failure data.So, we can maximize the likelihood function by using the EM algorithm, which provides a computational advantage over utilizing the MLE approach directly.

Overview of the EM algorithm
To apply the Wiener process model to the cutting tool degradation process for wear prediction and reliability calculations, we need to estimate the individual parameters of the cutting tool degradation process at each stage based on the observed degradation data.In this paper, degradation observations X i (t) for cutting tool i are collected, and the measure- ment error is ignored.In addition, the degradation data set X i (t) collected is often incomplete.As for cutting tool wear experiments, if the cutting tool is tested for damage, it could lead to serious safety incidents, such as machining system failure.In contrast, this paper examines a complete degradation prediction process from cutting tool wear to failure and scrap.Therefore, to handle incomplete data [33], the EM algorithm is introduced into parameter estimation in this section.The two steps of the EM algorithm can be expressed as follows.
1. E-step: Calculate the likelihood function Q for the random parameters as shown in Eq. (7).
where q denotes the number of iterations and I denotes the complete dataset, including both observed and missing data, that is I = I obs , I miss .The observed data- set is obtained from the measured degradation data, d e n o t e d a s is the jth degradation increment of the ith cutting tool.The missing dataset is the set of ( 7) random parameter values and can be expressed as Taking the Wiener process as an example, it is assumed that the amount of degradation of the ith cutting tool follows the Wiener process, that is X i (t) = i + i t + 2 k B(t) where i and i are random parameters and i ∼ N( , 2 ), i ∼ N( , 2 ) ; the random parameter matr ix is expressed as Θ� i = i , i ., , 2 , 2 , and 2 k are deterministic parameters; the deterministic parameter matrix is then expressed as Θ = , , 2 , 2 , 2 k .

M-step:
Update Θ (q+1) with the function argmax Θ Q(Θ|Θ (q) ) .Let Q(Θ|Θ (q) )∕ Θ = 0 , then deter- ministic parameters in the q + 1 step are estimated, and the details of the calculation are described in the second subsection of this section.

The specific calculation process of the EM algorithm
The computational procedure for the estimation of parameters based on the Wiener process, utilizing the EM algorithm, is shown below.
The conditional expectations of the parameters i , i , i i , 2  i , 2 i involved in Eq. ( 9) above are shown in ( 8) Eq. ( 10) below, and the values of the unknown parameters in the expectations can be found in Eq. ( 11) [34].
Repeat M-steps according to the iterative formula until the difference between the last two values of the Q function is less , at which point the parameter values converge.The value of the deterministic parameter matrix Θ (q+1) = (q+1) , (q+1) , (q+1)2 , (q+1)2 , (q+1)2 k can then be obtained by using Eq. ( 12).The results of the parameter estimates are used in the next subsection to calculate the reliability of cutting tools.

Cutting tool reliability analysis based on Monte Carlo
Let denote the failure threshold of cutting tools in the Wiener process model.According to the concept of first hitting time [35], the product life T is defined as the moment when the amount of degradation of the ith cutting (10) tool X i (t) exceeds the failure threshold for the first time, that is T = inf t > 0|X i (t) > |X i (0) <  .The first hit- ting time of the Wiener process model follows inverse Gaussian distribution; the PDF and cumulative density function (CDF) of the first hitting time can be expressed as: Equation ( 15) for the conditional reliability of cutting tools is derived from Eqs. ( 13) and ( 14): The equation for the unconditional reliability of cutting tools is further derived as: Since Eq. ( 16) involves a complex double integration operation, it is not easy to obtain an analytical expression for cutting tool reliability, so a numerical integration simulation algorithm based on the Monte Carlo method is used, and the following Eq.( 17) is derived as follows.
where i_u 1 ∼ N( , 2 ) , i_v 1 ∼ N( , 2 ) , u 1 , and v 1 are the sampling metrics and then u and v are the sampling sizes for the parameters i and i , which are set to 5000 for the Monte Carlo simulations in this paper.The sample size can be larger if higher accuracy is desired, but this makes the calculation take longer time, and the sample size chosen in this paper was chosen after considering various factors.The cutting tool reliability R(t) can be calculated by substituting the deterministic parameters Θ = , , 2 , 2 , 2 k and the failure threshold of cutting tools into Eq.( 17). ( 13) The theoretical model of the proposed method in this paper Sections 2 and 3 describe the Wiener process and the parameter estimation based on the EM algorithm, and the reliability of cutting tools is estimated by the Monte Carlo algorithm.The next section describes the theoretical model of the proposed three-stage Wiener process, and the complete theoretical procedure is presented by a flow chart.Accuracy evaluation indicators are also given for subsequent comparison with other methods.

The degradation process of cutting tools
The degradation process (e.g., rotating bearings, lithiumion batteries) exhibits multiple stages characteristics in practice due to the environmental working conditions, internal materials, and so on [36,37].As the cutting time goes on, the amount of tool flank wear increases.Based on the cutting experiments, a typical cutting tool wear curve can be obtained as shown in Fig. 1.Cutting time and cutting tool flank wear are selected as the horizontal and vertical coordinates, respectively.The wear process of cutting tools can be divided into three stages: initial wear stage, normal wear stage, and rapid wear stage [38].
In this subsection, a theoretical basis for the segmentation of the degradation process of the cutting tool is presented.

Initial wear stage and early failure stage
In the early stage of use, the product has a high degradation rate and is characterized by rapid degradation.The main failures of products at this stage originate from defects in design and manufacture, such as improper design, material defects, machining defects, and improper installation and adjustment.The flank of a newly sharpened cutting tool has defects such as rough unevenness and micro-cracks, and the cutting edge has a small contact area with the machined surface.The projections of the back face at this stage are quickly smoothed out, and the cutting tool wears out faster.In addition, the general initial wear is 50-100 μm, and its size is directly related to the quality of tool sharpening.For example, the initial wear of a ground cutting tool is smaller.

Normal wear stage and accidental failure stage
After the product is put into use for some time, the degradation rate of the product can be reduced to a lower level and is basically in a stable state, which can be approximated as a constant degradation rate.This stage is the accidental failure stage; during this stage, the failure of the product is mainly caused by accidental factors.The rough surface has been smoothed, and the cutting tool enters the normal wear stage after the initial wear.The wear of cutting tools at this stage is relatively slow and uniform, and the amount of wear increases approximately proportionally with the extended cutting time.The normal wear stage is the main working stage of cutting tools.

Rapid wear stage and wear out failure stage
After the product is put into use for a considerable period of time, it enters the wear-out failure stage.In this stage, a large increase in product failures soon occurs until the product is finally scrapped.Failures in this stage are mainly caused by depletion factors such as ageing, fatigue, wear, and corrosion.The start point of the rapid wear stage can be determined by analyzing the degradation data of the product.Before it reaches that point, we should stop using the product and give preventive maintenance, which will extend the product's service life.When the width of the wear zone increases to a specific limit, the roughness of the machined surface increases and the cutting force and cutting temperature rise rapidly.As a result, the wear rate increases so quickly that cutting tools are damaged.This stage is expected to be avoided, and based on the current strategy of over-protection used in engineering practice, it is important to change the cutting tool or replace the cutting edge with a new one before this stage is reached.

Indicators for evaluating the accuracy of model predictions
This paper models the flank wear data of cutting tools based on the multi-stage Wiener process and fits the flank wear of cutting tools in three stages for cutting tool wear prediction.The mean squared error (MSE) and the coefficient of determination ( R 2 ) were chosen as evaluation indicators.
The MSE reflects the degree of difference between the true and estimated values obtained from the experiment.The smaller the value of the MSE is, the more accurate the cutting tool wear prediction is.R 2 is the coefficient of determination, which characterizes the fitness of models.Equation (19) shows that the normal range of values for the coefficient of determination is [0, 1] ; the closer to 1, the more accurate it is in predicting wear.In Eqs. ( 18) and (19), Xi is the true value measured experimentally, Xi is the estimated value generated by fitting the model, X i is the average of the raw data, and n is the total number of cutting tool flank wear observations.( 18)

Summary of the proposed three-stage Wiener process model
The proposed cutting tool wear prediction model is described in detail through Sections 2, 3, and 4 of this paper.In this subsection, the proposed model is summarized through the flow chart shown in Fig. 2.
1.The data obtained is divided into three groups.Three stages of cutting tool degradation are the initial wear stage, the normal wear stage, and the rapid wear stage.2. The Wiener process in Section 2 is used to model the data of each stage.It is important to note that when modeling each stage of data, time t must start at 1.This involves a process of temporal translation, where the starting point of time for the second and third stages is translated by a corresponding amount to the position of 1 according to the position of the segmentation point.3.After obtaining the Wiener process model for the three stages of the degradation process, the EM algorithm pro-posed in Section 3 is applied to it to obtain the required deterministic parameter sets , a n d for the random parameter values ( 1 , 1 ),( 2 , 2 ) , and ( 3 , 3 ) , respectively.4. Once the parameters are obtained, the time translations made in ( 2) are shifted back, and the three cutting tool degradation processes become integral again.Equations ( 3) and ( 4) are used to back-propagate the estimation of cutting tool wear.The accuracy evaluation metrics square mean error (MSE) and coefficient of determination ( R 2 ) are used to evaluate the effectiveness of the proposed wear prediction method.5.In addition to the cutting tool wear data prediction, the PDF of cutting tool reliability can be obtained after parameter estimation.However, as the PDF of cutting tool reliability involves complex double integration operations, the Monte Carlo method, a numerical inte-Fig.2 Flow chart of the cutting tool wear prediction method gration simulation algorithm, is used to find the approximate solution for the cutting tool reliability R(t).

Numerical example
In this section, to model the cutting tool flank wear data based on the three-stage Wiener process, which facilitates the prediction of cutting tool wear and reliability, a practical example is presented.The data were obtained from the 2010 PHM competition (Prognostics and Health Management Society 2010) [39], and some details of the experiment are presented in this section.The reliability of cutting tools is calculated while completing the fitting of the cutting tool wear data.Then, the failure time of cutting tools is estimated, and the significance of the proposed model in engineering practice is discussed.Finally, the accuracy and superiority of the proposed method are verified by accuracy evaluation indicators and compared with traditional ML-based methods in three aspects.

Basic introduction of the experiment
Three hundred fifteen cutting tests were conducted on a Röders Tech RFM760 CNC milling machine.High-speed steel and stainless steel were chosen as the material for cutting tools and workpieces, respectively.Other mainly operating conditions are shown in Table 1.To describe the degradation process of cutting tools, the width of the rear cutting tool face wear was chosen as the degradation characteristic.A digital microscope (Leica MZ12) was utilized to measure wear values off-line without considering the effects of measurement errors.In addition, the failure threshold is determined as 220 μm based on the material of cutting tools, which means that cutting tools fail when the amount of flank wear reaches 220 μm.
As shown in Fig. 3, during each cutting test (which took about 15 s), seven signal channels were monitored in real time, including cutting force (in x, y, and z-axis), vibration (in x, y, and z-axis), and acoustic emission data.A fixed dynamometer mounted on the CNC machine table was used to measure cutting forces.Three piezoelectric accelerometers mounted on the workpiece were used to measure vibration.Acoustic emission (AE) sensors mounted on the workpiece were used to monitor high-frequency stress wave that occurs spontaneously within metals due to crack formation or plastic deformation.When the microstructure of the material is rearranged, the release of strain energy results in acoustic emission.Cutting tool wear was measured offline using a microscope (Leica MZ12) after each cutting test.The output of these sensors was adjusted by charge amplifiers and summarized to the computer by a data collection device.The 6 sub-datasets of PHM2010 obtained in this experiment are C1, C2, C3, C4, C5, and C6, respectively.Due to the low reproducibility of the data obtained from the cutting tool wear experiments, in this numerical example, we only study the wear data of one cutting edge.And the data of flute_1 in C6 is selected as the research data in this paper.

Cutting tool wear data modeling based on the three-stage Wiener process
The model proposed in this paper can be used to model the process in which the rate of degradation changes abruptly at a certain point, and the determination of the specific segmentation point is determined based on the actual situation of the chosen dataset.As shown in Fig. 4, the degradation process of the selected cutting tool can be divided into three stages based on the typical cutting tool wear curve in Section 3.For the dataset chosen in this paper, we take 0.05 as the boundary.This is because the variation of cutting tool wear in the second segment is very small and close to zero.When ΔX ij goes from greater than 0.05 to less than 0.05, j  is the first point of the second segment, and when ΔX ij goes from less than 0.05 to greater than 0.05, j is the first point of the third segment.The segmentation points for the cutting tool wear process are thus determined to be 100 and 171.The specific segmentation of the tool degradation data is as follows.First, there is the initial wear stage from the 1st to the 99th observation, then there is the normal wear stage from the 100th to the 170th observation, and finally, the degradation process after the 171th observation is divided into the rapid wear stage.
The proposed model is utilized to model the obtained observational data, and the details are as follows.The Wiener process is first used to obtain the stochastic parameters Then, to find the deterministic set of parameters for the stochastic parameters, the EM algorithm is utilized for iteration.The iterative process is exemplified by the initial wear stage, where the initial values of the deterministic parameters are taken from historical empirical data, as shown in Table 2.The iteration starts from the initial value until the convergence condition | | Q(Θ|Θ (q+1) ) − Q(Θ|Θ (q) ) | | < 10 −6 is reached, and then the iteration stops, as shown in Fig. 5.The convergence figures for each parameter in the deterministic parameter matrix , 2 k1 are obtained, as shown in Fig. 6, and the convergence results are shown in Table 2.The process of convergence of the parameters of the Wiener process in the initial wear stage is shown above.The normal wear stage and the rapid wear stage are based on the same method and are therefore omitted here.All deterministic parameters at each stage converge, and the initial values and convergence results of Θ 2 = 2 , 2 , 2 2 , 2 2 , 2 k2 and Θ 3 = 3 , 3 , 2 3 , 2 3 , 2 k3 are shown in Tables 3 and 4.After obtaining the convergence values of the deterministic parameters set by the EM algorithm, the estimated value  3) and ( 4).The predicted degradation path obtained by the proposed method is compared with the true degradation path.
As shown in Fig. 7, the predicted degradation path matches the real degradation path obtained from cutting tool wear experiments.In other words, the proposed method is capable of accurately fitting the cutting tool degradation path.
In this paper, a cutting tool wear prediction method is presented to replace or repair cutting tools before reaching the failure threshold, preventing property damage and machining inefficiencies.There is also a need to improve the current engineering practice of over-protecting cutting tools by changing them when they are just entering the rapid wear stage.As shown in Fig. 8, the model proposed in this paper can be utilized when the cutting tool wear has just reached the rapid wear stage.The predicted failure time of the cutting tool is obtained by the intersection point of the solid black line and the solid red line at around the 380th cut.According to the prediction result of the proposed method, cutting tools can be replaced or repaired before the 380th cutting observation.Then, cutting tools are less likely to be suddenly damaged with serious consequences, and cutting tool resources are not excessively wasted.Based on the current overprotection strategy, cutting tools have to be replaced after the 171st cut when the cutting tool has just entered the rapid wear stage.This wastes nearly half of the life of the cutting tool, resulting in metal resource waste and machining efficiency reduction.It can be seen that the proposed method is of great significance in engineering practice.

Reliability calculation based on the Monte Carlo algorithm
In this paper, a numerical integration simulation algorithm based on Monte Carlo is presented to predict the reliability of cutting tools.By substituting the corresponding deterministic parameters in Tables 2, 3, and 4 and threshold   into Eq.( 17), the reliability of cutting tools can be obtained.
As cutting tools degrade slowly in the initial wear stage and normal wear stage, the condition of cutting tools is better at this time, and the reliability is calculated to be 1, as shown in Fig. 9. Until the 380th cut that cutting tools start to wear sharply, and the reliability starts to decrease significantly.At the 388th observation, the cutting tool reliability is calculated to be zero.At this point, cutting tools are completely damaged.The reliability prediction result is consistent with the result obtained in Fig. 8, which illustrates the accuracy of reliability prediction.As an extension to engineering practice, cutting tools are replaced when the reliability begins to decline dramatically.So, cutting tools can be replaced around the 380th cut.In conclusion, the method provides an accurate prediction of the failure time and the reliability of cutting tools and can be applied to engineering practice.
Fig. 7 The comparison between the true degradation path the predicted degradation path Fig. 8 The prediction of failure time with a small amount of degradation data

Comparison with ML-based cutting tool wear prediction methods
Common cutting tool wear prediction methods based on ML include FFBP ANNs, SVR, and RFs.Wu et al. [17] used the same data source from PHM2010 as in this paper to predict the cutting tool wear.The results showed that the RF method outperformed the FFBP ANNs and SVR.In this subsection, to address the advantages of the proposed cutting tool wear prediction model based on the three-stage Wiener process, the prediction results of the proposed method and the RFs method are compared in three aspects.
As shown in Fig. 10, the proposed wear prediction method fits all stages and does not have many points with large offsets as those seen in the traditional ML method.But in general, both methods have good prediction results, and a comparison from this perspective alone is not sufficient to demonstrate the superiority of the proposed method.So, further comparisons of the two methods are presented in the next part of this subsection.
As shown in Fig. 11, the observed and predicted values of cutting tool wear are visually compared, and the smaller the black dotted trace deviates from the red mean line, the more accurate the prediction.The predicted and observed values of the proposed method are very close and stable throughout the wear process.This shows that the proposed wear prediction method is very accurate and effective, and this comparison illustrates the superiority of the proposed method more visually.
In order to compare the analysis process more rigorously, the wear prediction accuracy indicators MSE and R 2 are used next to quantify the superiority of the proposed method.Various cases that performed better in terms of accuracy indicators among the various cutting tool wear prediction methods based on ML methods in [17] are selected for comparison with the proposed method.All the calculation results are shown in Table 5.It can be seen that compared to FFBP ANN and SVR methods, the proposed method in this paper has the smallest MSE and the closest R 2 to 1. Besides, the results indicate that the wear of cutting tools can be accurately predicted by both models, no matter it is based on the proposed three-stage Wiener process model or the RF methods.As shown in Table 5, compared with the RF methods, the proposed model can complete the cutting tool wear prediction at the same or even a higher accuracy level.Moreover, the proposed model can predict the failure time of cutting tools with less data, and the reliability is calculated, which is of great significance in engineering practice.These illustrate the superiority of the proposed method.

Conclusion
In this paper, a new cutting tool wear prediction method based on a multi-stage Wiener process is proposed.The wear of cutting tools is predicted after completing the parameter estimation of the three stages by utilizing the EM algorithm, and the wear prediction results are Fig. 9 The prediction result of cutting tool reliability Fig. 10 Cutting tool wear prediction using the proposed method Fig. 11 Comparison of observed and predicted cutting tool wear using the proposed method compared with the data obtained from PHM2010.Then, a numerical integration simulation algorithm based on Monte Carlo is used to give an analytical solution for the reliability of cutting tools.The main contributions of this paper are as follows.Firstly, a general theoretical method of cutting tool wear prediction is proposed.Combining the typical cutting tool wear curve, the degradation processes of cutting tools can be approximately treated as three linear degradation stages.The Wiener process is used to model the data of each stage separately.This provides more theoretical ideas for cutting tool wear data processing.Secondly, the EM algorithm is used to iteratively solve for the stochastic parameters of the Wiener process.The EM algorithm can handle the truncated data well, and the parameters obtained can be used for cutting tool wear prediction and reliability calculation.Thirdly, a numerical integration simulation method based on the Monte Carlo algorithm is used to calculate the reliability of cutting tools.By utilizing the Monte Carlo algorithm, complex double integration operations are avoided, and the accuracy of the proposed method for predicting the failure time of the cutting tools is verified indirectly.Finally, the proposed method is applied to the prediction of the degradation path of cutting tools by a numerical example.The significance of the proposed method in engineering practice is described in detail.Then, the wear prediction methods based on ML are compared with the proposed method, and the superiority of the proposed method is compared in three aspects.
The proposed method can be widely used to predict cutting tool wear, but the theoretical basis of the degradation process segmentation is based on the typical cutting tool wear curve.In many practical cases, the degradation processes of cutting tools do not follow this theoretical basis and cannot be divided into three stages.Therefore, further improvement of the method needs to be considered in the future to perfect the cutting tool wear prediction model based on the three-stage Wiener process.To better solve practical engineering problems, the models considering more relevant details, such as the random noise of the obtained degradation data and the repairments of the cutting tools, are worthy of being further explored.And the proposed method will be applied to more research objects in the future to achieve its expansion.

Funding
The study was financially supported by the National Natural Science Foundation of China (51975110, U22B2087) and Applied Basic Research Program of Liaoning Province (2023JH2/101300160).

Fig. 3
Fig. 3 The monitoring process of cutting tools

Fig. 6
Fig. 6 Convergence results of deterministic parameters in the initial wear stage

Table 1
Operating conditions of cutting tool wear experimentsParameter Value

Table 3
Parameter estimation of the normal wear stage

Table 4
Parameter estimation of the rapid wear stage

Table 5
Accuracy of ML-based methods and the proposed method