In-situ tool wear condition monitoring during the end milling process based on dynamic mode and abnormal evaluation

Rapid tool wear conditions during the manufacturing process are crucial for the enhancement of product quality. As an extension of our recent works, in this research, a generic in-situ tool wear condition monitoring during the end milling process based on dynamic mode and abnormal evaluation is proposed. With the engagement of dynamic mode decomposition, the real-time response of the sensing physical quantity during the end milling process can be predicted. Besides, by constructing the graph structure of the time series and calculating the difference between the predicted signal and the real-time signal, the anomaly can be acquired. Meanwhile, the tool wear state during the end milling process can be successfully evaluated. The proposed method is validated in milling tool wear experiments and received positive results (the mean relative error is recorded as 0.0507). The research, therefore, paves a new way to realize the in-situ tool wear condition monitoring.

theoretical basis for system state identification.However, with the increasing complexity and systematization of mechanical equipment, the failure modes also become complex and variable, which leads to the instability of the proposed methods.In addition, traditional sparse measure optimization methods strongly rely on the prior knowledge of professional technicians and diagnostic experts in the diagnostic process (such as system structure, fault frequency, etc.) 11 , which restricts the applicability of these excellent methods in a wider range of engineering application scenarios.
With the rapid development of machine learning technology, artificial intelligence (AI) based fault diagnosis and prediction have increasingly become an important strategy for equipment safety and service monitoring 12 .Via related intelligent algorithms, the data-driven diagnostic method can adaptively identify equipment operation status information from existing data without the need of prior knowledge for professional technicians 13 .With an edge-labeling graph neural network method, Zhi et al. 14 propose a tool for wear condition monitoring using wear images which suitable for small sample conditions.Mishra et al. 15 developed a tool condition estimation method during the precision machining process with the unsupervised approach.However, data-driven methods are inevitably influenced by the distribution of training data, which may lead to data bias in the training model.Combining the sparrow search algorithm, Li et al. 16 developed a CNN-BiLSTM-based neural network to effectively predict sea level heights.Therefore, if the equipment status characteristics can be effectively recharacterized through a simple method, it is expected to overcome related shortcomings and achieve robust identification of its service performance.
Recently, time-domain-based CM-FD methods have been intensively investigated and successfully applied in some application scenarios 17 .Among them, the graph-based method enjoys the merits of anomalies quantitative evaluation and approximate shift-invariance 18,19 .However, it remains challenging to establish the adjacency matrix in a short time which might threaten the online evaluation reliability 20 .To overcome the potential drawbacks, as an extension of our works, Shiliang Feng et al. 21proposed a time-domain signal-driven mechanical system state description method and validated in some typical mechanical experiments.
During the manufacturing process, rapid tool wear might unpredictably occur, especially for hard-to-machining materials (e.g.nickel-based alloy or titanium alloy).The rapid tool wear will greatly affect the durability of cutting tools and the integrity of machining surfaces.To effectively trace the tool wear dynamic variation and avoid rapid deterioration of surface integrity, it is crucial to predict the short-term time-series response and estimate the tool wear status in advance.However, very little optimization work has been carried out on the dynamic evaluation of the tool wear state based on the predicted short-term time series.
Focus on the drawbacks and the existing research gap mentioned above, inspired by the time-domain-based CM-FD methods, in this research, a tool service state evaluation method that does not rely on any prior knowledge is proposed.With dynamic mode decomposition, the time series in the next snapshot can be predicted.Furthermore, based on graph structure, the anomalies of the tool wear state can be identified.The proposed strategy enjoys the merit of short-time time series prediction and offers exciting opportunities for rapid monitoring of tool wear state.The main structure of the manuscript is summarized as follows.A brief description of the proposed data prediction based on the dynamic mode decomposition method is listed in "Data prediction based on dynamic mode decomposition method" section."Anomalies identification for time-series" section presents the proposed anomaly identification for time series.The In-situ tool wear condition monitoring is summarized in "The proposed in-situ tool wear condition monitoring method" section.The validation experiment of the proposed tool wear estimation method is listed in "Experimental investigation" section.The conclusion of the manuscript is listed in "Conclusion" section.

Data prediction based on dynamic mode decomposition method
As a typical fluid dynamics analysis method, benefiting from the extraordinary spatiotemporal feature presentation ability (decomposing complex flow processes into low-rank spatiotemporal features), dynamic mode decomposition has lately received great attention.Because the decomposition does not rely on any given dynamic

Model solution
To decrease the model calculation complexity, intrinsic orthogonal decomposition methods (e.g.singular value decomposition) are widely employed to map the high-dimensional variables to low dimension.
With singular value decomposition, the matrix X 1 can be decomposed by 23 : where U is a m-order unitary matrix, V is a n-order unitary matrix, Σ is a non-negative real diagonal matrix with dimension of m × n.Generally, each eigenvector in V is called the right singular vector of M, each eigenvector in U is called the left singular vector, and the elements on the diagonal of D are called the singular values of M.
When the singular values are arranged in descending order, a unique D can be determined.If the matrix X 1 is truncated for singular value decomposition with a rank of r, the Koopman matrix A can be approximated using the following matrix: where matrix U r ∈ R M×r , V r ∈ R (T−1)×r , Σ r ∈ R r×r are the truncation matrixes of the unitary matrix U, the unitary matrix V, the non-negative real diagonal matrix, respectively.

Data prediction
If it is necessary to solve the modal of a matrix and analyze its spatiotemporal characteristics using it, the matrix can be decomposed into eigenvalues 24 : where is a diagonal matrix (the diagonal elements are the corresponding eigenvalues), the matrix Q is composed by the eigenvectors.Therefore, eigenvalues and eigenvectors can be used to analyze and predict the complex spatiotemporal characteristics of the system.
The mode of the dynamic can be defined as: Therefore, the dynamic prediction of data can be represented as: where the symbol † indicates the Moore Penrose generalized inverse operation. (1) (3) (5)

Graph structure description
In recent years, a novel abnormal health status of equipment evaluation method using time-domain signals has been proposed and aroused wide concerns in mechanical systems 25,26 .Based on the graph structure in computer science, the internal feature structure of signals can be evaluated, and the health state of the equipment can also be evaluated accordingly.
According to the graph theoretical, a graph structure can be expressed by G = {N, E}, where N is the set of nodes and E is the set of connections.Among them, the node set can be used to describe different sampling points, and the connection set E is used to describe the connection strength between different nodes.The connection strength between different nodes is reversible, so the set of connections is clearly a symmetric matrix.
For any temporal signal (as shown in Fig. 2a), the adjacency matrix of its data segments can be expressed as a symmetric matrix (as shown in Fig. 2c).In this study, the connection strength between different nodes is described by the Euclidean distance between nodes (as shown in Fig. 2b).Therefore, the collected temporal signals can be reprojected into a set of adjacency matrices (detail of the process can be seen in Ref. 25 ): where Xn is the n-th symmetric matrix that constitutes the set of adjacency matrices.

Anomalies identification
Based on the definition of graph structure, related researchers found that if the equipment state (i.e.operating state) is changed, the internal structure or parameters of its corresponding adjacency matrix will also change.Therefore, by evaluating the differences between the corresponding adjacency matrices, the dynamic characteristic of the equipment operation status can be monitored.The methods for anomaly identification can be summarized as follows.
The "standard template" X (in normal state) can be established from the sampled time series based on the traditional graph structure.The standard template matrix can be decomposed by 27 : where is the eigenvalue matrix (diagonal elements are eigenvalues), Ŵ is the eigenvector matrix (each column is the eigenvector of matrix X).
For a given test signal y, the corresponding adjacency matrix Y can be established accordingly.Similarity, the adjacency matrix Y can be decomposed as 28 : where the symbol diag [.] represents the diagonal elements of the adjacency matrix, and non-diag [.] represents the non-diagonal elements of the adjacency matrix.
By evaluating the non-diagonal components, the similarity between the two signals can be evaluated.For in-situ tool condition monitoring problems, the Frobeniu norm of the non-diagonal can be directly employed for the tool wear evaluation.
(10) The proposed in-situ tool wear condition monitoring method Combining the dynamic mode decomposition and real-time prediction signal anomaly identification, a method for evaluating the wear status of machining tools without relying on any given prior knowledge is proposed in this research.The main process of this method can be described in Fig. 3, the specific step is listed as follows: Step 1 Based on the sensor (near the cutting area) and the digital signal acquisition device, the dynamic signal which can reflect the tool service status information is recorded.
Step 2 By using the dynamic mode decomposition method established in "Data prediction based on dynamic mode decomposition method" section, the time-series signal in the next moment can be predicted.
Step 3 With the graph establishment approach (as shown in Eq. ( 10)), the graph structures of the current moment and the predicted signal can be constructed.
Step 4 Taking the acquired signal as the "standard template", as shown in Eq. ( 12), the graph structure of the predicted signal can be decomposed.
Step 5 The Frobeniu norm of the acquired non-diagonal elements can be used to evaluate the similarity (the tool wear state) between the two signals.

Experimental investigation
To verify the proposed method, an open-source database, and actual milling experiments are employed for the effectiveness verification of the proposed method.

Investigation in NASA database
As a typical dataset, the NASA Ames and UC Berkeley milling dataset is widely used in the research on the tool condition monitoring of general machining 29 .To investigate the applicability of the proposed method, the NASA dataset is employed in this section.As listed in Table 1, the acoustic emission signals acquired during the experiment (Mstsuura MC-510 V machine center is employed for the experiment) under the spindle speed of 826 rev/min, depth of cut is 0.75 mm, feed speed of 0.25 mm/rev.A 70 mm face mill with 6 inserts is employed for the processing.
According to the presentation above, appropriate graph structure construction is crucial for tool wear evaluation.The collected time-domain signals contain a significant amount of dynamic information, which can reflect the state of the machining process.Long sampling points can better preserve dynamic information but inevitably affect the timeliness of calculations.Too few sampling points result in the generated graph structure being unable to accurately describe the state information of the machining process.To investigate the performance of tool wear situation in NASA database, setting 800 as the data length of the research, the first sample (data length is 800) is considered as the reference signal (or healthy signal), other samples in the database can be considered as testing signals.Based on the adjacency establish equation before, the calculated reference adjacency is shown in Fig. 4a.Accordingly, the corresponding eigenvalue and eigenvectors (Fig. 4b), columns are the corresponding eigenvectors) are acquired after the diagonalization of the reference adjacency.
Based on the proposed method and the acquired eigenvectors, the fluctuation value of the corresponding signal can be calculated.To evaluate the performance quantitatively, the measured tool wear area and the anomaly are normalized as [0, 1].The normalization can be represented as:  where z indicates the tool wear area or the anomalies sequence, z n is the normalized sequence, min(⋅), max(⋅) are the minimum and maximum value of the sequence respectively.The evaluated tool wear condition is shown in Table 2.As illustrated in the table, in the whole 23 continuous samples, the corresponding error is ranging from 0 to 0.3280.Accordingly, the mean error between the evaluated tool wear state and the measured tool wear values is calculated as 0.1482.Figure 5 plots the normalized similarity (evaluated tool wear state based on the proposed method, red solid curve in the figure) and the normalized tool wear values (measured tool wear value, blue solid curve in the figure).As shown in Fig. 5, during the milling process, with the deterioration of the tool state, the Frobeniu norm (anomalies) also increased.The two variables had significant simultaneous change trends.

Experiment setup
The experiment setup is shown in Fig. 6.In this experiment, the end milling experiment is conducted on a vertical machining center (Dalian Machine Tool Group DMTG VDL 850A).During the experiment, a kind of uncoated tungsten steel end milling cutter (diameter of 10 mm, detail of the milling cutter can be seen in Fig. 7, Table 3)      www.nature.com/scientificreports/ was employed to cut the workpiece (45 steel, with dimension of 300 × 100 × 80 mm, the chemical properties of the workpiece material is shown in Table 4).Related literatures 30,31 have shown that there is a certain correlation between the sound pressure signals and the tool wear status during the manufacturing process.To minimize the impact of sensor installation on the machining environment, in this experiment, a non-contact sound pressure sensor is employed to collect dynamic signals during the milling process.During the whole process, the sound pressure sensor (GRAS 46AE, the sensitivity is 50 mV/Pa) is mounted on the table of the machining center near the workpiece (approximately 100 mm away from the workpiece) and used for the acquisition of sound signal.The dynamic signals in this experiment are recorded by a data acquisition instrument (Econ MI-7016 Avant) with a 12 kHz sampling frequency.
To accurately evaluate the actual tool wear state, a direct measuring instrument (Ksgaopin precision instrument GP-300C, as shown in Fig. 8) is employed for the estimation.Obviously, the three teeth are independent, the evaluation process of each tooth should perform separately.In the experiment process, the workpiece is manufactured layer upon layer.There are three forward and two backward cuts in each layer, as described in Fig. 9.After finishing one layer of the workpiece, the tool holder is taken off to evaluate the tool wear state.Generally, according to ISO3685-1977, the tool wear state is presented as the tool flank wear VB.However, as mentioned in related literature, the one-dimensional evaluation parameter cannot fully reflect the tool wear state.In this research, the mean flank wear area of the three flanks is indicated for the tool wear state evaluation.

Tool wear evaluation
In order to validate the reliability in actual milling experiments, two milling tests were conducted with the experimental setup.The experimental conditions are shown in Table 5.
Case I. Figure 10 plots the waveforms of the acquired acoustic signals via the sound sensor, as well as the frequency domain.The sampled signals in the 1st layer, 3rd layer, and 5th layer are described in Fig. 10a,c,e,    respectively.The corresponding frequency spectrums are listed in Fig. 10b,d,f, respectively.As can be seen in the figure, there is almost no obvious variation law or characteristics between the two samples either in time-domain waveforms or frequency-domain.The local magnification of the frequency spectrum (green dashed rectangle in Fig. 10f) of the 5th layer is shown in Fig. 10g.The spindle speed during the milling process is 2400 rev/min.Therefore, the spindle rotating frequency (40 Hz) and its harmonics can be observed (orange dashed lines).Caused by the distributed three teeth, the cutting frequency (120 Hz) and its harmonics can also be monitored (green dashed lines).
With the mentioned direct measuring instrument, the actual tool wear state during the manufacturing process can be monitored.Generally, the evolution of wear in tool experiments is a continuous process.The variation process of the wear area and its average value of three cutting edges is shown in Fig. 11, as well as the tool wear images.According to Fig. 11, caused by direct contact with the cutting material, generally, a triangular wear band will occur at the cutter edge tip.Besides, the maximum flank wear will also tend to appear around the tooltip.The specific information during the milling tool wear evolution process is shown in Table 6.As can be seen in Fig. 11 and Table 6, with the increase in cutting time, the tool wear area grew to mm 2 (mean tool wear area).
By considering the first sample (still setting the data length as 800) is considered as the reference signal (or healthy signal), the other samples can be considered as testing signals.Based on the adjacency establish equation before, the calculated reference adjacency is shown in Fig. 12a.Accordingly, the corresponding eigenvalue and eigenvectors (Fig. 12b, columns are the corresponding eigenvectors) are acquired after the diagonalization of the reference adjacency.
Based on the proposed method and the acquired eigenvectors, the fluctuation can be calculated for the similarity evaluation according to Eq. ( 12).The measured tool wear states, the anomalies, and their errors are summarized in Table 7.As illustrated in the table, in the whole 6 continuous samples, the corresponding error is ranging from 0 to 0.2350.Accordingly, the mean error between the evaluated tool wear state and the measured tool wear values is calculated as 0.0881.Figure 13    www.nature.com/scientificreports/Case II.The variation process of the wear area and its average value of three cutting edges is shown in Fig. 14, as well as the tool wear images.The specific information during the milling tool wear evolution process is shown in Table 8.As can be seen in Fig. 14 and Table 8, with the increase in cutting time, the tool wear area grew to mm 2 (mean tool wear area).
With the same investigation method mentioned before, the measured tool wear states, the anomalies, and their errors can be calculated, as listed in Table 9.As illustrated in the table, in the whole 6 continuous samples, the corresponding error ranges from 0 to 0.1266 (the measured tool wear values is calculated as 0.0484).Figure 15 plots the normalized similarity (evaluated tool wear state based on the proposed method, blue solid curve in the figure) and the normalized tool wear values (measured tool wear value, red solid curve in the figure).

Tool wear evaluation in different data length
As presented above, data length during the data processing process is crucial for the tool wear evaluation.To investigate the influence of data length in the evaluation process, repeat measuring experiments are conducted.The results in different data lengths are shown in Table 10.As can be conducted in the table, when the data length varies from 100 to 1000, the evaluation error distribution in the two experiments has no obvious change regular pattern.According to the results, the mean error for the two experiments reaches its minimum value when the data length is 800.Therefore, in this research, the data length is set as 800.

Conclusion
Aiming to the quantitative evaluation of tool wear state, in this research, authors proposed a two-stage method to estimate the tool running condition directly from the time series.In the prediction stage, with the engagement of dynamic mode decomposition, the real-time response of the end milling process can be predicted.In the estimation process, by constructing the graph structure of the time series and calculating the difference between the predicted signal and the real-time signal, the tool wear state during the end milling process can be successfully evaluated.To further confirm the effectiveness of the proposed method, investigations in an open source are presented and achieve a preferable effect.The results were also confirmed by an actual milling experiment from our laboratory.Accordingly, the combination of dynamic mode decomposition and anomalies evaluation method presents a wide range of possibilities for the further development of condition monitoring and fault detection techniques via time series.Besides, in this research, it is assumed that there is a linear relationship between the anomalies and the real tool wear.The nonlinear factors caused by environmental factors such as

Figure 3 .
Figure 3. Flowchart of the proposed method.

Figure 5 .
Figure 5. Tool wear evaluation of NASA signal.

Figure 6 .
Figure 6.Experiment setup.(a) Machine tool.(b) Magnification of the marked rectangle area.

Figure 7 .
Figure 7. Diagram of the milling cutter.

Figure 10 .
Figure 10.Signal samples in the milling process.(a) Time-domain signal of the 1st layer.(b) Frequency spectrum of the 1st layer.(c) Time-domain signal of the 3rd layer.(d) Frequency spectrum of the 3rd layer.(e) Time-domain signal of the 5th layer.(f) Frequency spectrum of the 5th layer.(g) Local magnification of (f).
plots the normalized similarity (evaluated tool wear state based on the proposed method, blue solid curve in the figure) and the normalized tool wear values (measured tool wear value, red solid curve in the figure).The slimier simultaneous trend indicates the potential mapping relationship between the Frobeniu norm (anomalies) and the tool wear.

Figure 11 .
Figure 11.Tool wear variation in milling Case I.

Figure 12 .
Figure 12.Signal diagonalization of the first sample in Case I. (a) Adjacency.(b) Eigenvector.

Figure 13 .
Figure 13.Tool wear evaluation in Case I.

Figure 14 .
Figure 14.Tool wear variation in milling Case II.

Figure 15 .
Figure 15.Tool wear evaluation in Case II.

Anomalies identification for time-series A
graph mechanism based on temporal signals is proposed to identify anomalies in temporal data.

Table 1 .
Experiment conditions in NASA database.

Table 2 .
Tool wear comparison in NASA database.

Table 3 .
Dimension of the milling cutter.
Parameter d/mm D/

Table 5 .
Experiment conditions in milling tests.

Table 6 .
Tool wear area in milling Case I.

Table 7 .
Tool wear comparison in Case I.

Table 8 .
Tool wear area in milling Case II.

Table 9 .
Tool wear comparison in Case II.

Table 10 .
Tool wear evaluation error in different length.