Tool Condition Monitoring Based on Nonlinear Output Frequency Response Functions and Multivariate Control Chart

Tool condition monitoring (TCM) is a key technology for intelligent manufacturing. The objective is to monitor the tool operation status and detect tool breakage so that the tool can be changed in time to avoid significant damage to workpieces and reduce manufacturing costs. Recently, an innovative TCM approach based on sensor data modelling and model frequency analysis has been proposed. Different from traditional signal feature-based monitoring, the data from sensors are utilized to build a dynamic process model. Then the nonlinear output frequency response functions, a concept which extends the linear system frequency response function to the nonlinear case, over the frequency range of the tooth passing frequency of the machining process are extracted to reveal tool health conditions. In order to extend the novel sensor data modelling and model frequency analysis to unsupervised condition monitoring of cutting tools, in the present study, a multivariate control chart is proposed for TCM based on the frequency-domain properties of machining processes derived from the innovative sensor data modelling and model frequency analysis. The feature dimension is reduced by principal component analysis first. Then the moving average strategy is exploited to generate monitoring variables and overcome the effects of noises. The milling experiments of titanium alloys are conducted to verify the effectiveness of the proposed approach in detecting excessive flank wears of solid carbide end mills. The results demonstrate the advantages of the new approach over conventional TCM techniques and its potential in industrial applications.


Introduction
Tool condition monitoring (TCM) is important in advanced manufacturing as cutting tool anomalies often compromise product quality and machining efficiency [1], [2].With the application of the Internet of Things (IoT) techniques in industry, many researchers are dedicated to develop indirect TCM methods, which monitor tool conditions by analyzing sensor signals such as vibration, force, acoustic, sound, etc [3]- [5].A typical indirect TCM system consists of data collection, feature extraction, and tool condition identification [6].The features that are associated with tool conditions can be obtained in the time domain, frequency domain, and timefrequency domain.However, a signal feature-based method has limited adaptability to variable machining environments.And it is even more complicated to determine what features should be used among numerous candidates.
Recently, Liu et al. [7] have proposed a novel TCM approach based on an innovative idea known as sensor data modelling and model frequency analysis.Instead of extracting features from the sensor signals directly, this approach builds nonlinear models that represent a dynamic relationship between two different vibration sensor signals.Then, the model's frequency domain properties are analyzed and used as the features for tool condition monitoring.Analysis shows that compared to conventional signal feature-based tool condition monitoring, this approach can better adapt to external changes including tool replacement and process parameter variations and have better generalization capability.Condition monitoring based on sensor data modelling and model frequency analysis have been widely investigated in beam crack detection [8], structure health monitoring [9] and rotor system fault diagnosis [10].The current study presents another practical application of this strategy and demonstrates, for the first time, how to apply this strategy in unsupervised tool condition monitoring in advanced manufacturing.
In order to implement the proposed TCM system, multivariate statistical process control (MSPC) charts such as, Hotelling's 2 T chart [11] are applied.This method is easy to implement in a manufacturing setting and its performance in TCM based on signal features has been widely reported [5], [12]- [14].However, as signal features are over-sensitive to external noises, it is often challenging to reliably identify anomalies induced by worn cutting tools.The proposed TCM system can resolve this problem as,

Sensor Data Modelling
Sensor data modelling here is also known as system identification, which is a data-driven modelling technique aiming to find a dynamic relationship between the input and output of a process or system.Let As the matrix D usually contains redundant terms, the term selection is a key step in NARX model-based nonlinear system identification [15].In this study, we use the Forward Regression with Orthogonal Least Squares (FROLS) algorithm to select the most important model terms from the dictionary matrix D [9].At each step, the term with the strongest capability to represent the output y is selected.
where  is the penalty parameter and I is an 0 0 M M  identity matrix.The penalty parameter controls the trade-off between bias and variance in the estimator.An evolutionary algorithm is exploited to tune  such that the built model meets the stability, robustness and accuracy requirements.Details about the parameter tuning procedure can be found in [7].

Model Frequency Analysis
After a NARX model has been identified, the NOFRFs (Nonlinear Output Frequency Response Functions) of the built NARX model can be extracted to investigate the frequency behaviors of the system [8].
Because the choice of  guarantees the identified model is stable at zero equilibrium, the system can be described by the Volterra series in the discrete-time domain, see [9].The corresponding frequency domain representation can be written as [16] 1 where ( ) represents Fourier transform and  is the frequency variable.

Y j y t   F and ( ) ( ( ))
n n U j u t   F are the n th order output and input frequency spectrum, respectively. ( )n G j is the n th order NOFRFs, which allows the system n th order output frequency response ( ) n Y j to be described in a manner similar to the description for the output frequency response of linear systems.
In this study, a recently-proposed Generalized Associated Linear Equations (GALEs) method is employed to calculate NOFRFs [17].This method decomposes the NARX model into a series of linear difference equations such that the NOFRFs can be evaluated from the first order to an arbitrarily high order.Consider the general form of the polynomial NARX model [18] 1 where , J L    , p q j   , and , 1 ( ,..., ) p q p q c l l  represents the model coefficient.
The GALEs of the NARX model is defined as

( ) n p n p i p n i p i n n y t y t l y t y t y t l
With the GALEs obtained, the NOFRFs can be calculated by evaluating the n th order system output response * ( ) n y t to a specified input signal * ( ) u t .Then, the n th order NOFRFs of the system, * ( ) n G j , under the input excitation * ( ) u t can be calculated as where n  indicates the frequency range of * ( ) In this study, the data collected from different tool conditions are used to build corresponding NARX models.The same input excitation signal is used to evaluate the NOFRFs of these NARX models.It is expected that the changes in tool conditions can be reflected by the evaluated NOFRFs.As a result, the NOFRFs * n G , n=1,2,… can be used as representative features for tool condition monitoring.

NOFRFs Feature Dimension Reduction
In the resulting NOFRFs ,..., where the dimension of X is 1 P R  .
The multivariate process monitoring scheme assumes that the samples from in-control processes follow a multivariate normal distribution.The process behavior is reflected by the shift in the mean of these variables.Thus the control chats' capability of timely detecting mean shifts will decrease if the number of variables is very large [11].
Besides, the existence of multi-collinearity sometimes can lead to numerical instability.It is, therefore, necessary to reduce the dimension of variables and eliminate multicollinearity before implementing the control chart.
To achieve this, in the present study, the principal component analysis (PCA) is applied to X [20].Firstly, X is standardized to obtain  X with zero-mean and one-standard deviation variables.Then the covariance matrix is computed as For any subsequent feature vector p x , the transformed vector is calculated by p p   t x P , with

Tool Condition Monitoring
The multivariate control chart-based condition monitoring involves both training and monitoring stages.The training stage aims to construct a baseline description of in-control processes.The mean vector and covariance matrix are estimated based on "normal" samples.The control limits are determined as well.Then the subsequent process is monitored.The control limit determined in the training stage is used to judge if any operation condition falls outside these limit, which could indicate an out-ofcontrol process.
Machining is a very complicated process involving variations induced by inconsistent material properties, changing process dynamics, increasing tool wear, and so on.All these factors result in uncertainty in the collected signals, reflected by the extracted features.Hence, a monitoring solution purely relying on these features will inevitably suffer from either false alarms or missing faults.Even though, the long-term shift of feature vectors is still dominated by the increase of tool wear.A nature way to grasp the trend of a series of data and overcome local variations is to use the moving window strategy.
In this study, we adopt a subgroup observations-based Hotelling's 2 T control chart [11].The subgroup contains the samples within a moving window.The mean of the samples in each subgroup is used to calculate the statistics of the control chart such that the influence of external noises is eliminated.To be specific, the mean and covariance matrix of the k th subgroup samples are calculated by where w denotes the window length.As the training data set contains 1 P samples, the mean and covariance matrix representing incontrol processes are given as Therefore, the Hotelling's 2 T statistic of k z is calculated by Regarding the determination of control limits, if the assumption of multivariate normality of the measurements is true, the control limits can be determined theoretically as the statistics should follow a standard distribution.However, in practice, it is more reliable to determine the control limit according to the distribution of available statistic values.This study uses the kernel density estimation (KDE) to estimate the control limit [5].This approach treats the 2 T statistic as a random variable and estimate the probability density function where [0,1]   and 1   is the significance level, indicating the probability of the 2 T statistic falling beyond the upper control limit when the process is "in control".
The control limit is obtained in the training stage.In the monitoring stage, the condition is monitored by comparing 2  T statistic with the limit 2 UCL T .Even though the moving window method has been used to cope with local variations, this system cannot fully avoid false alarms.To further improve the reliability of the detection result, in the decision-making step, we set the alarm triggering condition as the occurrence of several successive samples exceeding the control limit.T statistic.In offline training, the mean, covariance matrix, and control limit are obtained to represent the normal process conditions.In the monitoring stage, the 2  T statistics of new samples are calculated and compared with the control limit.The system will trigger alarms when several successive samples are identified as out-of-control processes.Otherwise, the machining process continues.To validate the effectiveness of the proposed TCM method in tool condition monitoring, a run-to-failure experiment has been carried out at the Advanced Manufacturing Research Centre (AMRC), University of Sheffield.The proposed method is applied to the collected vibration signals.As a comparison, traditional signal features are extracted and used for tool condition monitoring.The experimental details and results are presented in this section.

Experimental Setup
The dynamic milling strategy was adopted in the experiment.As shown in Fig. 2 (a), the machining was run on a 5-axis milling machine (DMG MORI's DMU 40evo).The material of the workpiece is TC4 titanium alloy.The type of cutting tool is Sandvik CoroMill Plura 1630 solid carbide square shoulder end mill with a diameter of 16 mm.This cutting tool has 4 flutes.In the experiment, we used three cutters in total (referred to as T1, T2, and T3).Fig. 2 (b) shows the machining process.The milling of one cylinder workpiece was conducted layer by layer.Each layer consists of 6 round cuts (referred to as R1~R6) from outer to inner.To coincide with the machining condition in a real manufacturing process, the rotational speed was set as 2586.3rpm, the feed rate was 1055.2 mm/min, and the radial and axial cutting depth were 1.6 mm and 20 mm, respectively.
Two accelerometers were mounted on the spindle and workpiece supporting base to collect vibration signals, which were used as input and output for sensor data modelling.This setting coincides with the transmission of power from the spindle to the workpiece.Thus, the built models are able to represent the dynamics of milling processes.Table 1 lists the sensor types and specifications.A relatively-high-sensitivity sensor was mounted on the workpiece supporting base so as to record the base vibration better.The signals were acquired by NI CompactDAQ systems with a sampling rate of 51200 Hz.
After each round cut, the tool wear was measured by a microscope as shown in the left of Fig. 2 (c).As the experiment was a run-to-failure one, each tool was used to complete one and a half workpieces, i.e., six layers (referred to as L1~L6).As shown in Fig. 2 (c), the maximum flank wear of all tools reached 395.3 µm, 421.1 µm and 489.9 µm finally.In practice, 300 µm is typically used as the threshold for excessive tool wear.During the experiment, the flank wear of all three tools exceeded 300 µm at Layer 5.

Monitoring Results
From the signals collected for each round cut, a 1-second snapshot is taken every 5 seconds.After pre-processing, the signal segments are used for sensor data modelling.
Considering the tooth passing frequency is 172.41Hz (= 2586.27/60×4), to investigate the nonlinear characteristics at this frequency, an input excitation with the frequency range from 167 Hz to 177 Hz is designed to evaluate the corresponding NOFRFs of each identified NARX model.The designed input excitation is where 1 1 t    .Therefore, the features of the NOFRFs under input (15) are determined to evaluate the status of cutting tools.The magnitudes of the first three order  , covering 41, 81, and 121 frequency points, respectively, are used as the NOFRF features.The frequency range is determined according to the new frequency generation phenomenon occurring with a non-linear system [16].Hence, the dimension of one NOFRF feature vector is 1 243  .Since each tool has completed 36 round cuts and 30 NARX models are identified from each round, the total number of feature vectors over the tool's lifecycle is 1080.Tool condition monitoring was carried out using both NOFRFs-based features and signal features for a comparison.In the proposed TCM framework, the data collected from the initial stage when the tool is not severely worn was used for training.As the number of finished layers is 6 during the experiment, it is reasonable to select the data from Layers 1 and 2 for training and the remaining for monitoring.It should be noted that, the selection of training samples is more difficult when the lifecycle of one tool is unknown.In this case, a priori knowledge or expert experience is necessary, which is out of scope of this paper.
For the proposed NOFRFs-based method, in the dimension reduction step, the components whose cumulative variance percentage exceeds 99% are determined as principal ones.To create subgroup samples, the length of the moving window is set to 20.The significance level in KDE-based control limit determination is 0.1%, which means there is a very tiny possibility for the 2  T statistic of an in-control state exceeding the control limit.If 7 successive samples exceed the control limit, the system will trigger alarms.
For the signal features-based method, 8 statistics are extracted from one vibration signal, including average, variance, skewness, kurtosis, entropy, median, range, and crest factor.The dimension reduction is not performed in this case as the dimension of signal feature vectors is relatively small.Except that, the subsequent procedures such as control chart construction and parameter setting are the same as those applied for the NOFRFs feature based condition monitoring.Fig. 4-6 present the monitoring results for the three cutters.As can be seen, the statistic of NOFRFs is overall stationary in the initial monitoring stage and tends to increase when the tool wear reaches a value around 300 µm.In other words, the proposed NOFRFs features do not change a lot when the tool is slightly worn, compared with the distribution of features in the training stage.The alarms are triggered at the last several round cuts of Layer 4 when tool wear is between 268 µm and 285 µm.However, when using the signal feature-based control chart, the monitoring statistic is always increasing, making it difficult to identify whether tool wear exceeds the critical value.Hence, the alarms are triggered in a very early stage during monitoring.Especially for T2 and T3, after the first round cut of Layer 3, the tool condition is identified as worn.The result shows that the signal features are too sensitive to the process change, tending to cause more false alarms, and is cannot make reliable decision in practice.To further illustrate the effect of moving window, Fig. 7 shows the control charts based on NOFRFs features with individual samples, which means the transformed feature vectors p t are directly used to calculate monitoring statistics.Most of the statistics are randomly scattered within a range.Although there is an increasing trend as tool wear reaches 300 µm, the system does not trigger alarms as there are no consecutive samples exceeding the control limit.The use of moving window highlights the trend of these statistics, leading to success detection results.One issue with this method is how to select a reasonable window length.The consideration involves the data sampling rate, the pattern of machining process, and the wearing rate of tools.A further discussion will be given in future studies.show that the NOFRFs features are more stationary when the tool is not severely worn and can trigger alarms just in time when the tool wear reaches a critical threshold.This makes the NOFRFs a better choice for designing a TCM system in practice.Besides, in theory, the proposed framework can be extended to condition monitoring of a wide range of industrial processes due to its advantages of low costs and high reliability.
The potential applications deserve more research.
However, the NOFRFs feature-based method is more time-consuming and requires a significant amount of computation to guarantee the accuracy of the identified model.In our experiment, 6 to 10 seconds are needed to yield a detection result from raw data, which is acceptable for tool wear monitoring, but may not be acceptable for scenarios requiring emergency response [21].
As the most time-consuming step, the parameter estimation for building the NARX model can be implemented in parallel, in the future, we will try to improve the computational efficiency using parallel computing techniques.

Fig. 1
Fig. 1 shows the flowchart of the proposed TCM system.The whole procedure contains two stages.The feature extraction processes are the same for the training and monitoring stages, including NARX model identification, NOFRFs extraction, and PCA dimension reduction.Then a moving window is used to create subgroup samples, whose mean vector is used to determine the monitoring 2T statistic.In offline training, the mean, covariance matrix, and control limit are obtained to represent the normal process conditions.In the monitoring stage, the2  T statistics of new samples are calculated and compared with the control limit.The system will trigger alarms when several successive samples are identified as out-of-control processes.Otherwise, the machining process continues.

Fig. 1 .
Fig. 1.The flowchart of the proposed TCM system based on NOFRFs and multivariate control chart.

Fig. 2 .
Fig. 2. (a) Experimental setup, (b) illustration of the machining process, (c) measurement of tool wear (right: the images of flutes before and after machining).

Table 1 .
List of sensors and specifications

Table 2
summarizes the TCM results in the experimental study.For all the three tools, the detected tool wear is closer to 300 µm when the NOFRFs-based features are applied.Besides, the consistent monitoring results demonstrate that the proposed NOFRFs based method is able to overcome the impact of varying working environments, such as workpiece and tool material properties and coolant concentration variations.These demonstrate the advantage of the proposed NOFRFs-based tool condition monitoring over traditional signal features based approaches.

Table 2 .
Tool condition monitoring results using NOFRFs and signal features, respectively.In this study, a novel tool condition monitoring method based on NOFRFs and the multivariate control chart is proposed.The vibration signals measured from the spindle and workpiece supporting base are collected and used for identifying a NARX model that reveals the dynamic relationship between the vibration signals.From the identified NARX model, the NOFRFs are calculated to provide physically meaningful features of the milling process.Then the PCA technique is applied to reduce the dimension of the NOFRFs features, and the multivariate control chart with a moving window of observations is applied to monitor tool wear conditions.An industrial scale milling experiment is carried out to validate the performance of the proposed method.Traditional signal features are also extracted and used for tool condition monitoring for a comparison.The results