CNC Machine Tool’s wear diagnostic and prognostic by

using dynamic bayesian networks.

Diego Tobon-Mejia, Kamal Medjaher, Noureddine Zerhouni

To cite this version:

Diego Tobon-Mejia, Kamal Medjaher, Noureddine Zerhouni. CNC Machine Tool’s wear diagnostic
and prognostic by using dynamic bayesian networks.. Mechanical Systems and Signal Processing,
2012, 28, pp.167-182. �10.1016/j.ymssp.2011.10.018�. �hal-00672204�

HAL Id: hal-00672204

https://hal.science/hal-00672204
Submitted on 20 Feb 2012

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est

archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents
entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non,
lished or not. The documents may come from émanant des établissements d’enseignement et de
teaching and research institutions in France or recherche français ou étrangers, des laboratoires
abroad, or from public or private research centers. publics ou privés.
 CNC Machine Tool’s Wear Diagnostic and Prognostic
by Using Dynamic Bayesian Networks
D.A. Tobon-Mejiaa,b , K. Medjahera,∗, N. Zerhounia
a
FEMTO-ST Institute, UMR CNRS 6174 - UFC / ENSMM / UTBM Automatic
Control and Micro-Mechatronic Systems Department
24, rue Alain Savary, 25000 Besançon, France
b
ALSTOM Transport, 7, avenue De Lattre De Tassigny, BP 49, 25290 Ornans, France

Abstract
The failure of critical components in industrial systems may have negative
consequences on the availability, the productivity, the security and the en-
vironment. To avoid such situations, the health condition of the physical
system, and particularly of its critical components, can be constantly as-
sessed by using the monitoring data to perform on-line system diagnostics
and prognostics.
The present paper is a contribution on the assessment of the health condition
of a Computer Numerical Control (CNC) tool machine and the estimation of
its Remaining Useful Life (RUL). The proposed method relies on two main
phases: an off-line phase and an on-line phase. During the first phase, the
raw data provided by the sensors are processed to extract reliable features.
These latter are used as inputs of learning algorithms in order to generate
the models that represent the wear’s behavior of the cutting tool. Then, in
the second phase, which is an assessment one, the constructed models are
exploited to identify the tool’s current health state, predict its RUL and the
associated confidence bounds. The proposed method is applied on a bench-
mark of condition monitoring data gathered during several cuts of a CNC
tool. Simulation results are obtained and discussed at the end of the paper.
Keywords: Diagnostic, Prognostic, Remaining Useful Life, Condition
Based Maintenance, Hidden Markov Models, Monitoring data, Tool wear.

∗
Corresponding author. Tel.: +33 03 81 40 27 96
Email address: kamal.medjaher@ens2m.fr (K. Medjaher)

Preprint submitted to Mechanical Systems and Signal Processing October 5, 2011

1. Introduction
The maintenance activity plays a major role in industrial systems as it
permits to improve the availability, reliability and security, while reducing
the life cycle cost. There exist several types of maintenance, which can be
classified in two main categories, namely: curative and preventive mainte-
nances [1, 2]. In the first case, the interventions are done only after the
observation of the failure, whereas in the second case they are realized either
systematically or conditionally to the health condition of the system. This
type of maintenance is commonly termed as a Condition Based Maintenance
(CBM). Indeed, the condition of the industrial system is continuously mon-
itored and inspected by a set of sensors. The data recorded by these latter
are then processed in order to extract relevant features that allow to estimate
the current health state and to project this one in the future. The estimated
and projected states are then used to take appropriate maintenance deci-
sions. Diagnostic aims at assessing the component’s current condition and
identifying the cause of its failure, whereas prognostic is used to predict its
future health state in order to anticipate the failure [3–6].
Formally, failure prognostic consists of estimating the time before failure or
the Remaining Useful Life (RUL) and the associated confidence value. It
can be realized by using three main approaches [7, 8], namely: model-based
prognostic, experience-based prognostic and data-driven prognostic. Model-
based prognostic consists of studying each component or sub-system in order
to establish for each one of them a mathematical model of the degradation
phenomenon. The derived model is then used to predict the future evolution
of the degradation and thus the related RUL value. Experience-based prog-
nostic methods use mainly probabilistic or stochastic models of the degra-
dation phenomenon, or of the life cycle of the components, by taking into
account the data and knowledge accumulated by experience during the whole
exploitation period of the industrial system. Data-driven prognostic is based
on the transformation of the monitoring data into relevant behavioral models
permitting to predict the RUL and the associated confidence.
This paper deals with the assessment of the cutting tool’s health condition
of Computer Numerical Control (CNC) machines through the utilization of
Dynamic Bayesian Networks (DBN). The proposed method belongs to the
data-driven prognostic approach and aims at transforming the raw moni-
toring data provided by the sensors into behavioral models that represent
the evolution of the cutting tool’s degradation. The obtained models are

2
then used to continuously estimate the current state of the cutting tool and
calculate its RUL. The choice of this approach dwells in the fact that in
the assessment of the cutting tool’s condition of CNC machines, the main
problem is that deriving a behavioral model in an analytical form that best
fits the dynamic of the tool’s wear is not a trivial task. Furthermore, find-
ing experience data for a long period of time is expensive and not easy in
practice. This is why the utilization of the data provided by the monitor-
ing sensors may be a trade off between the model-based prognostic and the
experience-based prognostic. Thus, the idea behind this contribution is the
transformation of the raw monitoring data into relevant models representing
the wear’s behavior of the cutting tools of CNC machines.
The proposed method relies on two main phases: a learning phase and an as-
sessment phase, as this is done in the framework of data-driven system health
monitoring and prognostic [9, 10]. During the first phase, the raw data are
used to extract reliable features, which are then used to learn behavioral
models representing the dynamic of the degradation in the cutting tool. The
modeling of the degradation is done by using a Mixture of Gaussians Hid-
den Markov Model (MoG-HMM) represented by a DBN. This probabilistic
graphical model allows to use continuous observations and also to speed up
the inference by using the algorithms proposed for DBNs [11]. In the second
phase, the learned models are exploited on line to assess the current health
state of the cutting tool and to estimate the value of the RUL and its asso-
ciated confidence value.
The paper is organized as follows: in section 2 the diagnostic and prognostic
paradigms are presented, where some definitions and the related state of the
art are given, section 3 is dedicated to the proposed diagnostic and prognostic
method and finally, an application example and simulation results are given
in section 4.

2. Diagnostic and prognostic framework

2.1. Definitions
The term prognostic founds its origin in the Greek word “prognostikos”,
which means “to know in advance”. Prognostic is well used in medical do-
main, where doctors try to make predictions about the health of a patient by
taking into account the actual diagnosis of a disease and its evolution com-
pared with other similar observed cases. This reasoning can be transposed

3
into the industrial domain where the patient is a machine, an industrial sys-
tem or a component.
Several definitions have been given in the literature about industrial prognos-
tic [7, 12–14], where three main points are highlighted: the system’s actual
state, the projection (or extrapolation) of this latter, and the estimation of
the remaining time before failure. These definitions are then normalized by
the ISO 13381-1 standard [15] in which prognostic is defined as the estima-
tion of the operating time before failure and the risk of future existence or
appearance of one or several failure modes. This standard defines the out-
lines of prognostic, identifies the data needed to perform prognostic and sets
the alarm thresholds and the limits of system’s reset (total shut-down). The
main steps to perform prognostic, as defined in the standard, are summarized
in Fig. 1.

Monitoring and Projection and Posterior

Diagnostic
Pre-processing prediction actions

Figure 1: Prognostic steps according to ISO 13381-1 [16].

The first step consists of monitoring the system by a set of sensors or in-
spections achieved by operators. The monitored data are then pre-processed
to be used by the diagnostic module. The output of this module is an identi-
fication of the actual operating mode (more details on failure diagnostic can
be found in [3–5]). This mode is then projected in the future, by using ade-
quate mathematical tools, in order to predict the system’s future state. The
intersection point between the value of each projected parameter or feature
and its corresponding alarm threshold permits to estimate the RUL (Fig. 2).
Finally, appropriate maintenance actions can be taken depending on the es-
timated RUL. These actions may aim at eliminating the origin of the failure,
which can lead the system to evolve to any critical failure mode, delaying
the instant of a failure by some maintenance actions or simply stopping the
system if this is judged necessary.
As in any prediction work, a prediction error should be associated to the
estimated value of the RUL (Fig. 3). The sources of the prediction error
may be multiple: modeling hypotheses, non-significant data, used prediction
tools, uncertainty in the thresholds’ values, etc.. In addition, uncertainty is
intrinsic to any prognostic work [17].

4
 RUL
Threshold value of γ

Amplitude of γ

Monitored value of γ
Initial time
Projected value of γ

Time

Figure 2: RUL illustration.

The error associated with a RUL estimation should decrease as the time
of the real failure approaches. This is because the predictions are adjusted
each time new data are acquired. In addition, a confidence degree should be
associated. Indeed, instead of telling an industrial that the machine will fail
in x units of time, it would be more realistic to give an estimated RUL with
a confidence value.
As mentioned previously, the value of the estimated RUL is the output of
some comparison between the projected state of the system and the prede-
termined threshold values (theses values can be determined by using learning
algorithms like those of neuro-fuzzy systems [18]). Note that, at the projec-

RUL
Threshold value of γ
Amplitude of γ

2nd uncertainty
1st uncertainty

Initial time Monitored value of γ

Projected value of γ

Time

Figure 3: Uncertainty inherent to RUL prediction.

5
tion step, what is needed is not necessarily a value of a physical parameter
but can be a desired performance, an achieved function or the availability of
a service, depending on the kind of system on which prognostic is performed.

2.2. Prognostic approaches

Although the novelty of industrial prognostic activity, numerous meth-
ods and tools have been proposed and reported in the literature. By using
as classification criteria the type of data needed to perform failure prognos-
tics, it is possible to group the different techniques in three main categories
[4–6]: model-based prognostic, experience-based prognostic and data-driven
prognostic, as shown in Fig. 4-(a). Note that this classification is not totally
binary as it may exist prognostic methods based on the fusion of more than
one approach [19], see the Fig. 4-(b).

Model-based
Assessment criteria
prognostic
Data used for Applicability
- +
prognostic
Cost
Data + reliability

- +

Sensor data Feedback data Precision

Fusion

Data – driven Experience – based - +

rel

prognostic prognostic
Complexity
-
+
Model – based Data – driven Experience – based
prognostic prognostic prognostic
Prognostic approach

(a) (b)

Figure 4: Prognostic approaches: (a) classification and (b) performance eval-

uation.

In model-based prognostic, the physical component or system and its

degradation phenomenon are represented by a set of mathematical laws. The
obtained behavioral model is then used to predict the future evolution of the
degradation [20, 21]. Whereas, the methods belonging to the experience-
based approach use probabilistic or stochastic models of the degradation
phenomenon, or of the life cycle of the components, by taking into account
the data and the knowledge accumulated by experience [22, 23].
Finally, the data-driven approach aims at transforming the raw monitoring
data into relevant behavior models of the system including its degradation.
Indeed, the degradation model is derived by using only the data provided
by the monitoring system (the sensors mainly), without caring about the

6
system’s analytical model neither on its physical parameters (like material
properties).
Data-driven methods offer an alternative to the model-based and experience-
based ones, especially in cases where obtaining reliable sensor data is easier
than constructing analytical behavior models or waiting to obtain sufficient
experience data. The use of monitoring data to derive a behavioral model
may be considered as a trade off between complexity and precision. Indeed,
in practice most of the degradation phenomena are complex (due to non-
linearities, stochasticity, non-stationarities, etc.) and difficult to model by
using analytical models and experience data.
In the data-driven approach several tools are used, most of them are origi-
nated from artificial intelligence. Among these tools one can cite neuronal
networks, dynamic Bayesian networks, Markovian processes and statistical
regression methods.
In [24] a prognostic method based on the use of recurrent neural networks to
model the crack propagation in a bearing has been proposed. The network
structure used by the authors was able to track the time evolution of the
crack size and to estimate the value of the RUL. In addition to traditional
neural networks, neuro-fuzzy systems were used in failure prognostic. Thus,
Wang et al. [13] have proposed a neuro-fuzzy based prognostic method to
predict the future health state of a pinion. The fuzzy rules were given by ex-
perts whereas the forms of the membership functions were learned by using a
neural network. The authors showed that the results obtained by using neuro-
fuzzy networks were more relevant than those provided by a simple neural
network. The same approach has been applied by Chinnam and Baruah [18]
on a vertical machining center. Moreover, posterior simulations conducted
by Wang [25] have shown that, compared to neural networks, the use of a
feed-forward neuro-fuzzy network can significantly increase the accuracy of
the predictions and the accuracy of the estimated value of the RUL. In [26]
a self organizing map (SOM) has been implemented to perform both failure
diagnostic and prognostic on bearings by exploiting vibration signals along
with a set of historical data. The historical data have been modeled by us-
ing independent neural networks, which estimated separate local RULs. The
global RUL was then calculated by weighting the local RULs, the weightings
were controlled by the degree of representativity of each historical data set.
However, the problem of neural networks based methods is that the definition
of the network’s structure is often complex. Indeed, the generation of the
structure is based on a process of test-error estimation, which necessitates

7
long time calculations and learning.
A second tool, which has been used in data-driven failure prognostic is the
Kalman filter. In [27] a Kalman filter based prognostic has been proposed
in order to track the time evolution of a crack in a tensioned steel band.
The Kalman filter is used to model and to estimate the drift of the modal
frequency of the steel band as a function of the applied vibrations. More
general than Kalman filters, particle filters have also been used to perform
nonlinear projections of features. Orchard et al. [28] have proposed a particle
filter method used to estimate a crack growth in a turbine’s paddle. In their
prognostic model, the current size of the crack and its evolution is recursively
estimated by using the data provided by sensors. Nevertheless, in the case of
Kalman filter and its variants the difficulty to generate a prior model, with
the definition of its parameters, needed for filtering and predictions can limit
their applicability.
In failure diagnostic and prognostic domains, the Hidden Semi-Markov Mod-
els (HSMM), have proved to be a suitable tool as they allow to model the
physical component’s degradation by using continuous observations provided
by the monitoring sensors. They also permit to estimate the stay durations
in each health state leading to the prediction of the RUL value [29]. HSMM
can be used to represent several failure modes by using historical data for
learning. Moreover, the number of observations can be modified depending
on the application and the implementation constraints.
The HSMM permit to do failure prognostic for a long time horizon. Indeed,
once the current health state is identified and assuming that the stay dura-
tions in the states are estimated, the prediction of the RUL is straightforward
[30]. Furthermore, contrary to other tools used in the framework of data-
driven approach, such as the regression models or neural networks where the
structure is not interpretable [31], the states in the HSMM can be interpreted
as the health conditions of the component.
In this same context, Chinam and Baruah [30] have used Hidden Markov
Models (HMM) to assess the degradations of bearings and to estimate the
underlying RUL. In their method the authors considered the degradation as
a stochastic process with several states representing different health states
of the physical component. The degradation levels of each bearing are first
learned by using vibration data (several HMMs corresponding to each state
are obtained during the off-line phase). Then, during the on-line operation
of the bearing the processed data are continuously supplied to each HMM in
order to calculate a likelihood value, which permits to select the model that

8
best represents the current state of the bearing. Finally, knowing the current
state and its corresponding stay duration, it is possible to estimate the value
of the RUL.
More recently, Dynamic Bayesian Networks [11], a tool generalizing the
HMMs and the Kalman filter, have been exploited to perform failure prog-
nostic. Prytzula and Choi [32] proposed an integrated DBNs based diagnos-
tic and prognostic method where the uncertainty related to the operating
conditions is taken into account. Similarly, Muller et al. [12] proposed a
DBNs based procedure integrating both the degradation mechanism and the
maintenance actions in the same model. Medjaher et al. [33] published a
procedure to estimate the RUL of a work station in a manufacturing system,
where maintenance actions on several components were introduced in the
DBN model in order to observe the modifications in the estimated RUL. Fi-
nally, Dong and Yang [34] implemented a particle filtering algorithm applied
to a DBN in order to estimate the RUL of a vertical machining center.

3. Integrated diagnostic and prognostic method

The cutting tool’s wear assessment in CNC machines can be done in
two different ways: by using mathematical or mechanical methods, such as
numerical simulations based on finite elements models, or by exploiting the
monitoring data provided by the sensors installed on the machines. The
first method is a model-based one and can lead to more precise results at a
condition that the derived models represent really the physical phenomena
including the wear process. However, this is practically difficult to reach,
because the wear phenomena are often nonlinear, non-stationary and not
easy to model. To avoid this situation, one can use the monitoring data
to build behavioral models valid for the operating conditions in which the
machine evolves.
This section presents an integrated diagnostic and prognostic method based
on the transformation of the data provided by the monitoring sensors into
models. The method is detailed at the end of the section and its objective
is to assess the health state of the cutting tool in CNC machines and to
estimate its remaining useful life. Before introducing the steps of the method,
a brief introduction of some necessary prerequisites would help the reader
understanding it. These prerequisites concern data clustering, MoG-HMMs,
DBNs and curve fitting.

9
3.1. Data clustering
Cluster analysis is a technique used to identify groups of individuals or
objects that are similar to each other, but different from those in other groups.
Clustering techniques are a simple, but can be considered as a powerful tool
to understand complex phenomena by using statistics. These methods are
principally used by researchers in social sciences and medicine as in [35],
where they are used to identify the factors that cause stress in nurses, or also
in biology as in [36] where a hierarchical cluster analysis is used to identify
interfaces and functional residues in proteins. In the field of clustering two
main methods exist [37]:
1. Hierarchical clustering: this is the most straightforward method of clus-
tering, it can be either agglomerative or divisive. It consists to merge
the data in different clusters using a greedy algorithm, and to repre-
sent their hierarchy using a dendrogram1 . This technique is suitable
for small data sets.
2. Partitional clustering: this method only attempts to directly decom-
pose the data into a set of disjoint clusters. Traditionally, the called
k-means algorithm is used, where k is the exact number of clusters to
partition the data. Contrary to hierarchical clustering in this technique
the quantity of clusters are fixed.
As argued in [38, 39], the tool wear evolution is classified in five wear
stages: initial wear, slight wear (regular stage of wear), moderate wear (micro
breakage stage of wear), severe wear (fast wear stage) and worn-out (or tool
breakage). Thus, the partitional clustering method is suitable to identify in
the learning phase the different wear stages. Using the k-means algorithm
to partition the amount of wear measured after each tool cut may identify
the wear stage at each cut. Then, by knowing the wear stage as a function
of tool life, the features bellowing to a particular wear cluster can be used to
train their respective “wear model”.

3.2. Mixture of Gaussians Hidden Markov Models

The MoG-HMM is primarily a Hidden Markov Model. This latter is a
statistical model used to represent stochastic processes in which the states

1
Tree diagram frequently used to illustrate the arrangement of the clusters produced
by hierarchical clustering.

10
are not directly observed (hidden), but the outputs (observations) dependent
on the hidden state are visible [40]. Each hidden state has a probability
distribution over the possible values of the observations.
A discrete HMM is completely defined by the following parameters:
• N: number of states in the model. The individual states are 1, 2, ..., N ,
and the state at time t is defined as st .
• K: the number of distinct observations for each state. The individual
observation symbols are denoted as V = v1 , v2 , ..., vK .
• A: the state transition probability distribution, A = aij , where aij =
P [st+1 = j |st = i ] , 1 ≤ i, j ≤ N.
• B: the observation probability distribution of a state i, B = bi (k), where
bi (k) = P [vk |st = i ] , 1 ≤ i ≤ N, 1 ≤ k ≤ K.
• π: the initial state distribution π = πi , where πi = P [s1 = i] , 1 ≤ i ≤
N.
For simplicity and clarity of presentation, a compact notation (Θ =
π, A, B) is used for each HMM. In practice, the HMM and its variants are
used to solve three typical problems [40]: identification, decoding and learn-
ing. However, the problem with the discrete HMMs is that the observations
are discrete symbols chosen from a finite alphabet while in most real applica-
tions the observations are continuous signals (or vectors). To overcome this
limitation, one can use a MoG-HMM, where each signal can be expressed as
a combination of a finite number of mixtures [41], each one of the form:
M
X
bj (O) = Cjm ξ(O, µjm , Ujm ), 1 ≤ j ≤ N (1)
m=1
In equation 1, O stands for the observation vector being modeled, Cjm is
the mixture coefficient for the mth mixture in state j and ξ is a Gaussian with
mean vector µjm and covariance matrix Ujm for the mth mixture component
in state j. Similar to an HMM, a MoG-HMM is completely defined by
three parameters: the state transition matrix A, the observation matrix B
and the initial probability distribution π. Moreover, for a MoG-HMM the
observation matrix B is modeled by a Gaussian density with a mean µ, a
standard deviation σ and a mixture matrix M . An illustration of a MoG-
HMM is given in Fig. 5.

11
 S1 S2 S3

M1 M2 M3

O1 O2 O3

Figure 5: A Mixture of Gaussians HMM [11].

3.3. MoG-HMMs and Dynamic Bayesian Networks

In the last decade, a new tool namely the Dynamic Bayesian Networks,
derived from the artificial intelligence domain, became popular thanks to
its modeling, graphical representation and inference capabilities. A DBN is
an extension of the traditional Bayesian Network (BN) [11] used to model
probability distributions over semi-infinite collections of random variables,
Z1 , Z2 , Z3 , . . . Z. For the HMM case, Z is partitioned into Z = (St , Vt )
to represent the hidden and the output variables of a state space model. A
DBN is defined to be a pair, (B1, B ′ ), where B1 is a BN that defines the prior
probability of all variables known as P (Z1 ), and B ′ is a two-slice temporal
Bayes net (2TBN), which defines P (Zt |Zt−1 ) by means of a directed acyclic
graph (DAG) as shown in the following equation:
N
Y   
P (Zt |Zt−1 ) = P Zti |Pa Zti (2)
i=1

Where Zti is the ith node at time t, which could be a component of St ,

Vt (in regular HMMs) and P a(Zti ) are the parents of Zti in the graph. The
nodes in the first slice of a 2TBN do not have any associated parameters,
however the nodes in the second slice have associated conditional probability
distributions (CPD), which define P (Zti = |P a(Zti )) for all t > 1.
A DBN is a powerful tool that allows modeling sequential data [11]. In one
hand, DBNs generalize HMMs by representing the state space in a factored
form, instead of a single random variable. In the other hand, they model
the well known Kalman filters models (KFMs) by using arbitrary probability
distributions instead of the traditional Gaussian models.
The main advantage in modeling HMMs and its variants by DBNs is that
the inference can be done faster. For exemple, the inference in hierarchical

12
HMMs is performed in O(T ) time by using a junction tree algorithm instead
of O(T 3 ) time for the traditional algorithm, where T is the length of the data
sequence [11].
By using a DBN, one can easily represent all the variants of HMMs. The
only requirement is to build the correct DAG and define the correct type of
nodes and CPDs. An example of a MoG-HMM represented by a DBN is
shown in Fig. 6.

S1 St

M1 Mt

V1 Vt

Slice 1 Slice 2

Figure 6: A MoG-HMM represented by a DBN.

In this model, S represents the hidden states, M the mixture coefficients

and V the observation distributions. For an illustration purpose, the corre-
sponding CPDs are presented below.

S1 = π (3)
M1 = Mt = Cjm (4)
V1 = Vt = ξ (O, µjm , Ujm ) (5)
St = aij = P [st+1 = j |st = i ] (6)

Once the parameters of the model are completely defined, the learning
algorithms for DBN [11] can be used to correctly estimate the CPD values.

3.4. Curve fitting

Curve fitting is a numerical process that is used to represent a set of exper-
imentally measured (or estimated) data points by some hypothetic analytical

13
functions [42]. The results of this curve fitting process are the coefficients,
or parameters, that are used to define the fitting function. The points are
constrained to a supposed polynomial with a fixed order. Let us suppose y
the output data and x the input data. By using numerical algorithms, the
input and output data will be fitted to the target function, which can be of
A order, as shown in the following equation:
M
X
M
y(x, β) = β0 + β1 x + . . . + βM x = βα xα (7)
α=0

Where βM stands for the parameters of the supposed polynomial function

to be defined.

3.5. Description of the diagnostic and prognostic method

An integrated diagnostic and prognostic method for estimating the cur-
rent health state of the cutting tool in high-speed CNC machines and pre-
dicting its remaining useful life is proposed in the following of the paper.
The method relies on the utilization of the monitoring data provided by the
sensors installed on the machine to track its tool’s condition represented by
the magnitude of the wear. The main contribution of the method dwells in
the fact that in the generated model the magnitude of wear per cut is learned
by using the features extracted from the monitoring data, instead of doing
it by using a mathematical model of the wear for which material and envi-
ronmental coefficients are not trivial to estimate. Furthermore, the proposed
method may be used to assess the health state of any physical component
(drill bits, milling cutters, tool bits, rolling elements, etc.) at a condition
that the necessary features for learning can be extracted.
The principle of the proposed method relies on two main phases: a learning
phase and an exploitation phase, as shown in Fig. 7. In the first phase,
which is realized off-line, the tool’s wear measured after each cut from the
new state until the failure is used. This latter is fed as input to the k-means
algorithm in order to partition the wear and to identify which information
correspond to a specific health state of the cutting tool. In this way, the mon-
itoring data can be classified in different groups, each one corresponding to a
particular wear level. Then, the appropriate features are extracted from the
raw monitoring signals by using a feature extraction procedure (F.E). In the
feature matrix F , each column vector (of D features at a cut c) corresponds
to a snapshot on the raw signal, and each cell fdc represents the feature d at

14
 Mill cutter On-line
Data Feature
Sensors
Processing extraction

Prognostic
model
Data Feature
Sensors
Processing extraction

Model
learning
Mill cutter
data base
Off-line
1 wear stage
RUL
2 wear stage
Wear stage
identification

N wear stage

Figure 7: The two phases of the proposed method.

a cut c, as shown in the following equation.

F.E
Raw signal −−→ F = (f1c f2c · · · fdc )′
(8)
with 1 6 c 6 C and 1 6 d 6 D
The matrix of extracted features is finally used to estimate the parame-
ters of different MoG-HMMs represented by a DBN. The advantage of using
several features instead of only one is that it can happen that a single feature
may not capture all the information related to the behavior of the physical
component.
The learned models are divided into two groups. The first group contains
the general models per wear level, resuming all the data from all the learning
histories and stored in a data base, called “global model wear base”. The
second group feeds a model data base, which contains one model per wear
level and learning history, named “local wear model base”. In this way if
W are the wear stages and H the learning histories, the global model wear
base contains W models (one per wear stage), whereas the local wear model
base contains W × H models (one model per wear stage and history). This
latter permits to obtain the state sequence and to compute how much wear is
acquired in each model state (hidden states) using its associated wear mea-
sures. Also, by using the Viterbi algorithm [43] it is possible to find the cuts
belonging to a particular state (Fig. 8). Thus, by assuming that the amount

15
of wear acquired in each wear stage is governed by a Gaussian distribution,
it is possible to estimate the mean “µ” and the standard deviation “σ” of the
amount of wear “Wr” and its variation “∆Wr” for each state Si of the models
stored in the local wear model base, as shown in the following equations:

  1 X cl
µ Wrhw (Si ) = wrh (c) (9)
Tc c=sc w
  1 X cl  
µ ∆Wrhw (Si ) = wrhw (c) − wrhw (c − 1) (10)
Tc c=sc+1
v
u
 cl
u1 X 
h
σ Wrw (Si ) = t [wrhw (c) − µ (wrhw (Si ))]
2
(11)
Tc c=sc
v
 u  cl h  i2
u1 X
σ ∆Wrw (Si ) = t
h
(wrhw (c) − wrhw (c − 1)) − µ ∆Wrhw (Si ) (12)
Tc c=sc+1

State Wear

Wear stage 1 Wear stage 2 Wear stage w Wear stage W

Analyzed history
Others histories
3

sc cl
Cut

Figure 8: Viterbi decoding sequence for one history of tool wear evolution.

In the previous equations wrhw stands for the wear associated to the stage
w = 1, . . . , W and the history h = 1, . . . , H, i is the state index and c
the cut index constrained by the limits found in the cluster analysis (sc =
start cut, cl = cut limit and Tc = cl − st + 1, as shown in the Fig. 8).
A compact representation of the learned model, which can then be used to

16
perform health condition diagnostic and prognostic, is given by the following
expression:
 
DBNw (θ), DBNwh (θ), µ(Wrhw (Si )),
λ=  (13)
µ(∆Wrhw (Si )), σ(Wrhw (Si )), σ(∆Wrhw (Si ))

In the above expression, λ represents the model, DBNw (θ) are the parame-
ters of the DBN that capture the behavior of the wear stage w summarizing
all the histories H (“global model wear base”), DBNwh (θ) are the parameters
of the DBN that capture the behavior of the wear stage w from the history
h (“local model wear base”), µ(Wrhw (Si)) and µ(∆(Wrhw (Si)) are respectively
the mean wear and the mean wear variation of the state i in the stage wear w
learned from the history h. In the same way, σ(Wrhw (Si)) and σ(∆(Wrhw (Si))
stand for the standard deviation of the wear and the standard deviation of
the wear variation in the state i for the wear stage w learned from the history
h.
The on-line phase of the proposed method consists of exploiting the learned
models to identify the tool’s current condition (by using DBN inference al-
gorithms) and to estimate the current tool’s wear leading to an estimation of
the value of the corresponding RUL. The processed data and the extracted
features are thus continuously fed to the learned models in order to identify
the actual wear stage. The identification process is based on the evaluation
of the likelihood P (O|λ) of the models (belonging to the global model wear
base) over the observations. By knowing the current wear stage and by us-
ing the wear accumulation and the wear variation learned during the off-line
phase, the cutting tool’s RUL and its associated confidence value can be cal-
culated. The dedicated procedure to estimate the RUL and the associated
confidence value during the on-line phase, based on the use of the DBNs “λ”
are explained through the following steps:
• The first step consists of detecting the appropriate DBN that best
fits and represents the on-line observed sequence of features. Indeed,
the features are continuously fed to the set of learned models in the
global model wear base and a likelihood is calculated in order to select
the appropriate model. The selected model is then used to define the
current wear stage w (Fig. 8).
• The second step concerns the identification of the DBN that best fits
the observations knowing the actual wear stage P (O|λ, w). The DBN

17
 inference algorithm is thus applied on the local wear model base (be-
longing to the known wear stage) to compute the likelihood and to
choose the best DBN. Then, the Viterbi algorithm is applied on the se-
lected model in order to find the hidden state sequence, which permits
to identify the current wear, the wear amount and the wear variation
by choosing the most persistent state in the last observation sequence.
This state number is stored in a global state sequence as Ghw (Si ) in the
cell c, which contains the current and the old states.
 
state sequence = Ghw (si )1 , Ghw (si )2 , ... , Ghw (si )c ,
c = current cut

• The current global state sequence and the wear information µ(Wrhw (Si )),
µ(∆(Wrhw (Si )), σ(Wrhw (Si )) and σ(∆(Wrhw (Si )) are used in the third
step to estimate the wear vector. The idea is to estimate at each
cut c the amount of mean wear and the bounds using the confidence
value. The current state of each cell is compared with the precedent
state (precedent cell). If the states are same the mean wear varia-
tion µ(∆(Wrhw (Si )) is added and the bounds are defined by using the
confidence factor n, otherwise the wear and the bounds are updated,
as shown in the following equations. Three values of wear are esti-
mated: the maximum “Wr d (c)” , the mean “Wr d (c)” and the mini-
u m
d
mum “Wrl (c)” as shown in the following equations.

    


 µ Wrhw (Si ) + n · σ Wrhw (Si ) ,

 h i h i 




 IF (c = 1) OR Ghw (Si ) 6= Ghw (Si )
c c−1
d
Wr (c) =
u

 h    i

 d (c − 1) + µ ∆Wrh (S ) + n.σ ∆Wrh (S ) ,



Wr m w i w i
 h i h i

 IF Gh (S ) = Gh (S )
w i w i
c c−1
(14)

18
   


 µ Whw (Si ) ,

 h i h i 




 IF (c = 1) OR Ghw (Si ) 6= Ghw (Si )
c c−1
d (c) =
Wr (15)
m

  

 d (c − 1) + µ ∆Wrh (S )

 Wr

 hm i h wi i

 IF Gh (S ) = Gh (S )
w i w i
c c−1

    


 µ Wrhw (Si ) − n · σ Wrhw (Si ) ,

 h i h i 




 IF (c = 1) OR Ghw (Si ) 6= Ghw (Si )
c c−1
d
Wr (c) =
l

 h    i

 d (c − 1) + µ ∆Wrh (S ) − n.σ ∆Wrh (S ) ,



Wr m wi i w i
 h i h

 IF Gh (S ) = Gh (S )
w i w i
c c−1
(16)

• Finally, in the fourth step, the estimated wears are used to compute
the RUL. This latter is obtained by using the information
h stored in thei
d d
three vectors obtained in the previous phase Wr = Wru , Wr d , Wrd .
m l
d β) with the same
c (Wr,
Each vector is fitted to a polynomial function W
order than the current wear stage (M = w) to avoid linear variations
and to smooth the data. The RUL is calculated by taking the difference
between the wear limit (Wlimit ) and the mean of the three smoothed
wear estimations, as shown in the following expression:

w
X α
W d β)
c (Wr, = d
βα Wr
α=0
!
d (c), β) + W
c (Wr
W d (c), β) + W
c (Wr d (c), β)
c (Wr
RUL(c) = u m l
Wlimit −
3
(17)

4. Application and simulation results

The CNC machine tools are widely used in industry for achieving pro-
ductivity performance goals. Statistically, 20% of the down-time of these

19
machines is attributed to the cutting tool failure, resulting in productivity
and economic losses [44]. Thus, the prediction of the amount of wear before
reaching the wear limit of the cutters may help improving the reliability,
the availability and the safety of CNC machines, while insuring the surface
roughness requirements and reducing the maintenance costs.
The wear diagnostic and prognostic method presented in the previous sec-
tion was tested on the “prognostic data challenge 2010” data base [45], which
contains several histories of high-speed CNC milling machine 3-flute cutters
used until a significant wear stage (Fig. 9). Each cutter made 315 cuts over
an identical workpiece for a face milling job. The authors of the experiments
recorded the monitoring data from dynamometer, accelerometer and acoustic
sensors during the cut process and measured the amount of wear after each
cut for three experiments (total of 3 sets of 315 cut files). These histories
were named by the authors as: cutter 1, cutter 4 and cutter 6. The details
of each data set are given in the table 1.

Table 1: Test data set description.

Total wear (10−3 mm)

History No of cuts Flute 1 Flute 2 Flute 3
Cutter 1 315 172.6868 164.6378 158.1924
Cutter 4 315 196.1648 210.9193 202.14954
Cutter 6 315 179.4365 233.6742 234.7159

Cutter
Accelerometers
Accelerometer AE Sensor

Workpiece

Dynamometer

Figure 9: CNC milling machine testbed (Röders Tech RFM760).

20
 The experimental records from these tests were obtained under constant
conditions. The cutting parameters were: the spindle speed of the cutter
was 10400 rpm, the feed rate was 1555 mm/min, the Y depth of cut (radial)
was 0.125 mm and the Z depth of cut (axial) was 0.2 mm. The data were
acquired at 50 kHz/channel. For simulation purposes (learning and on-line
RUL estimation), three condition monitoring data were used (two for learn-
ing and one for testing), each cutter was considered as worn at the end of
its associated history. The learning results discussed hereafter are obtained
by choosing the cutters 1 and 4 for learning. However the RUL results of all
possible combinations are presented in Fig. 13.
During the learning phase, five wear stages were defined to set the size of
the global model wear base being consequent by the analysis of the number
of tool degradation regions suggested in [38, 39]. To classify the different
features in the different wear stages, the wear file record of the three flutes
of each cutter was classified by using the k-means algorithm. The figure 10
shows the result of the number of cuts in each cluster for the training sets.
For example, for the cutter 1 the features from the first 32 cuts belong to
the first wear stage, then the next 126 cuts are classified in the wear stage
2, the 59 following cuts are in the wear stage 3, the next 51 are classified in
the wear stage 4 and finally, the last 47 cuts are in the wear stage 5.
For the learning and prognostic phases, different features were extracted
from the raw signals provided by the monitoring sensors by following the
recommendations stated in [46], where the relevant features used in model-
ing machining operations are reported. For the dynamometer signals, the
retained features were the root mean squares (rms), the peak and the stan-
dard deviation (std), for the accelerometer signals the rms and the kurtosis
were computed and finally, for the acoustic emission sensor the mean and
the standard deviation were extracted. Thus, a total of 17 features were
extracted for each cut. These features were chosen because they are more
suitable for tracking the wear evolution [46].
The parameters of the DBNs (see section 3.3) in the global and in the lo-
cal model wear base were first randomly initialized and constrained to be a
three states left-to-right MoG-HMM (representing the degradation phenom-
ena) and then, the extracted features were fed to the learning algorithms
in order to re-estimate the initialized parameters (CPDs). The number of
mixtures in each MoG-HMM represented by a DBN was set to 10 (each coef-
ficient Cjm was uniformly initialized). In order to identify the correct number
of mixtures to be used an analysis of representativeness vs number of mix-

21
 Wear clustering cutter 1
mm)

150
-3
Wear flute 3 (10

Wear stage 1
100
Cuts Wear stage 2
50 Wear stage 3
Wear stage 4
0
180 Wear stage 5
160
140 180
120 160
100 140
120
80 100
80
60 60
40 40
20
Wear flute 2 (10-3 mm) -3
Wear flute 1 (10 mm)

Wear clustering cutter 4

mm)

200
-3

Wear stage 1
Wear flute 3 (10

150

Cuts Wear stage 2

100
Wear stage 3
50 Wear stage 4
Wear stage 5
0
250
200
150 180 200
100 140 160
100 120
50 60 80
0 20 40
-3
Wear flute 2 (10 mm) Wear flute 1 (10-3 mm)

Figure 10: The clustering results.

tures were performed. The clustered data from cutter 1 and 4 were used to
learn a global model for each wear region (1 to 5). Then, the correspond-
ing likelihood was recorded after each learning phase by using the learning
algorithms for DBN [11]. This learning phase was repeated using different
number of mixtures and a plot of likelihood vs the number of mixtures for
each global wear model was obtained. In Fig. 11 it can be observed that the
likelihood of the models representing the different wear regions has tendency
to stabilize after 10 mixtures. In order to ensure good representativeness
and to minimize the learning time, 10 mixtures can be a good choice to allow
a trade-off between precision and computation time (similar to the results
found in [47]).
A total of fifteen DBNs (5 in the global model base and 10 in the local
one) were obtained by using the learning algorithms. The initial and the
re-estimated numerical values of the CPDs parameters related to S1 , St and
Mt associated to the global DBN for the first wear stage are given in the

22
 0.6
Wear region 1
Wear region 2
Wear region 3
Wear region 4
0.4
Wear region 5

0.2
Normalized Log-likelihood

-0.2

-0.4

-0.6

-0.8
1 5 10 15 20
Number of mixtures

Figure 11: Normalized likelihood vs Number of mixtures for the different

wear regions using the cutters 1 and 4 as training sets.

table 2.

Table 2: Initial (I) and final (F) CPDs of the global DBN model for the first
wear stage using the data from cuuters 1 and 4 as training sets.

   
1 1
  
π1 π1I =  0  π1F =  0 

0  
0 
0.5 0.5 0 0.78 0.22 0
  
St StI =  0 0.5 0.5  StF =  0 0.71 0.29 


0 0 1 0 0 1 
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

MtI =  0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 


0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 
Mt
0 0.08 0.18 0.01 0.15 0.18 0.3 0 0.1 0.27

F
Mt =  0 0.1 0 0.03 0.4 0.1 0.1 0.1 0.1 0.07 
0.39 0.14 0.04 0.07 0.07 0.14 0.04 0 0.08 0.04

23
 As defined in the equations 9 to 12, it is also necessary to estimate the
amount of wear obtained and its variation in each state of the models stored
in the local wear model base. To give an example of this estimation, the
corresponding decoding state sequence of the cuts belonging to the first wear
stage (1 to 32 cuts) from the history associated to the cutter 1 is shown in
Fig. 12. Here one can appreciate that the first 6 cuts belongs to the state 1,
the following 6 to the state 2 and the last 30 to the state 3. Then, combining
this information and the equations 9 to 12, the mean wear and the mean
wear variation and their associated standard deviation were estimated, as
presented in table 3.

Table 3: Estimated wear parameters in 10−3 mm.

State
Parameter
 
S 1 S2 S3
1
µ Wr1 30.9867 43.5107 53.9789
 
1
σ Wr1 11.2078 6.5399 4.9714
 
1
µ ∆Wr1 2.9175 1.9883 1.1234
 
σ ∆Wr11 0.7350 0.3365 0.2054

State 1
State 2
State 3

0.8
Probability

0.6

0.4

0.2

0
5 10 15 20 25 30
Cut

Figure 12: The decoding sequence.

In order to simulate an on-line wear monitoring case, the diagnostic and

24
prognostic method presented in the previous section was applied on all data
histories in order to estimate the wear after each cut and compute the cor-
responding RUL. The simulation results for all possible combinations of his-
tories used to predict the mean wear after each cut are shown in Fig. 13,
where the monitoring data used concern 315 cuts and the true wear at each
flute was measured. The results using the data corresponding to the cutter
6 as “test” history are discussed hereafter. For the RUL prediction, it is
supposed that the wear threshold is known in advance. In the benchmark
used in this application no information was available about the surface finish
requirements of the workpiece making impossible the definition of a “logic”
wear limit. However, in order to be able to do prognostic and to evaluate
the error of the proposed method a high threshold limit of 140 × 10−3 mm
was considered, which represents an average tolerance in the surface rough-
ness of the finished part (regular surface quality). In general, the roughness
increases almost constantly with the increase of tool flank wear with a ten-
dency to reach a plateau stage towards the end of tool life. In practice, it is
recommendable to make a study to find the relation between the tool wear
and the surface roughness as the analysis performed in [48] for a hard turning
process of 1117 steel hardened up to 62 ± 1 HRC.
In the results presented in Fig. 13, the confidence bound for the RUL’s es-
timation was fixed to 95%. This value was defined using the information
presented in table 4, where the mean absolute percentage error (MAPE) was
estimated using the confidence levels defined by the three-sigma rule (68%,
95%, 99.7%) [49]. In this table one can appreciate that the MAPE is minimal
when the confidence value is fixed at 95% for the three tests histories.
Table 4: MAPE comparison using the confidence levels from the three-sigma
rule.

MAPE at the confidence bound (n · σ)

History 68% (n = 1) 95% (n = 2) 99.7% (n = 3)
Cutter 1 28.71 24.39 41.67
Cutter 4 20.98 15.71 37.12
Cutter 6 25.47 15.01 40.26

In Fig. 13-(f) one can see that the precision of the estimated RUL is
near to the real value. However, the prediction of the RUL value is slightly

25
over the real value because the confidence bounds were fixed at 95%. This
confidence value gives wide limits leading to a little late prognostic, which
can be corrected by introducing a security factor. This can be useful in a
sense that it suggests a tool replacement before reaching the wear limit in
order to protect the work piece and the CNC machine from damage and
breakdowns.
The RUL predictions presented in Fig. 13 were also used to quantify the
performance of the method using the accuracy metric (equation 18) for failure
prognostics techniques proposed by [4]. This measure is defined to be equal
to 1 for the best performance and 0 for the worst. The table 5 summarizes
the performance results of the three tests. The mean performance accuracy
value is 0.8234 which is near to 1.
C
1 X −
|RULReal (c)−RULEstimated (c)|
Accuracy = e RULReal (c)
(18)
C c=1

Table 5: Accuracy performance measure for the three tests.

History
Measure Cutter 1 Cutter 4 Cutter 6
Accuracy 0.7286 0.8950 0.8467
Mean 0.8234

5. Conclusion
A condition monitoring, diagnostic and prognostic data-driven method
has been proposed in this paper. The main idea of the work relies on the
transformation of the monitoring data into relevant models that capture the
degradation’s behavior of the machining tools. The choice of a data-driven
method instead of a physical model of the degradation dwells in the fact that
obtaining this latter may not a trivial task due to the complexity of the phys-
ical phenomenon of the wear, which is not easy to model. The utilization of
MoG-HMMs allowed thus to model the wear in a stochastic way by taking
into account the different stages of the degradation. The identification of the
current condition of the tool combined with the models learned in the off-line

26
 250 140
Wear flute 1 Real RUL
Wear flute 2 Estimated RUL
Wear flute 3
120
Mean wear
200 Wear limit
Wear prediction
100
Wear (10-3 mm)

RUL (10-3 mm)

150
80

60
100

40

50

20

0 0
50 100 150 200 250 300 50 100 150 200 250
Cut Cut

(a) (b)
250 140
Wear flute 1 Real RUL
Wear flute 2 Estimated RUL
Wear flute 3
120
Mean wear
200 Wear limit
Wear prediction
100
Wear (10-3 mm)

RUL (10-3 mm)

150
80

60
100

40

50

20

0 0
50 100 150 200 250 300 50 100 150 200 250
Cut Cut

(c) (d)
250 140
Wear flute 1 Real RUL
Wear flute 2 Estimated RUL
Wear flute 3
120
Mean wear
200 Wear limit
Wear prediction
100
Wear (10-3 mm)

RUL (10-3 mm)

150
80

60
100

40

50

20

0 0
50 100 150 200 250 300 20 40 60 80 100 120 140 160 180 200
Cut Cut

(e) (f)

Figure 13: Simulation results: (a) wear estimation for the cutter 1 using the
data from cutters 4 and 6 as training sets; (b) RUL prediction for the cutter
1; (c) wear estimation for the cutter 4 using the data from cutters 1 and 6 as
training sets; (d) RUL prediction for the cutter 4; (e) wear estimation for the
cutter 6 using the data from cutters 1 and 4 as training sets; and (f) RUL
prediction for the cutter 6.

27
phase allowed to calculate the RUL and the associated confidence value.
Future works concern the extension of the failure prognostic method by tak-
ing into account the variable operating conditions (load, velocity, tempera-
ture, etc.) and the integration of maintenance actions for RUL prediction.

[1] X. Jiang, Recent development of damage diagnosis and prognosis in

engineering systems, Recent Patents in Engineering 4 (2) (2010) 1–20.

[2] CEN/TC 319 - Maintenance, Maintenance - maintenance terminology.

EN 13306:2010 (2010).

[3] V. Venkatasubramanian, Prognostic and diagnostic monitoring of com-

plex systems for product lifecycle management: Challenges and oppor-
tunities, Computers & Chemical Engineering 29 (6) (2005) 1253 –1263.

[4] G. Vachtsevanos, F. L. Lewis, M. Roemer, A. Hess, B. Wu, Intelligent

fault diagnosis and prognosis for engineering systems, Wiley, 2006.

[5] A. K. Jardine, D. Lin, D. Banjevic, A review on machinery diagnostics

and prognostics implementing condition-based maintenance, Mechanical
Systems and Signal Processing 20 (7) (2006) 1483 – 1510.

[6] A. Heng, S. Zhang, A. C. Tan, J. Mathew, Rotating machinery prognos-

tics: State of the art, challenges and opportunities, Mechanical Systems
and Signal Processing 23 (3) (2009) 724 – 739.

[7] M. Lebold, M. Thurston, Open standards for condition-based mainte-

nance and prognostic systems, in: Maintenance and Reliability Confer-
ence (MARCON), 2001.

[8] D. A. Tobon-Mejia, K. Medjaher, N. Zerhouni, G. Tripot, A mixture

of gaussians hidden markov model for failure diagnostic and prognostic,
in: IEEE Conference on Automation Science and Engineering, CASE’10,
2010.

[9] X. Jiang, S. Mahadevan, An intelligent online damage detection system

for thermal protection panels with active sensors, in: Intelligent Sensors
and Actuators Symposium at Earth & Space 2008 Conference, 2008.

28
[10] X. Jiang, S. Mahadevan, Wavelet energy-based bayesian probabilistic
approach for damage detection of thermal protection system panel,
American Institute of Aeronautics and Astronautics Journal 47 (4)
(2009) 942 – 952.

[11] K. P. Murphy, Dynamic Bayesian Networks: Representation, inference

and learning, Ph.D. thesis, University of California (2002).

[12] A. Muller, M.-C. Suhner, B. Iung, Formalisation of a new prognosis

model for supporting proactive maintenance implementation on indus-
trial system, Reliability Engineering & System Safety 93 (2) (2008) 234
– 253.

[13] W. Q. Wang, M. F. Golnaraghi, F. Ismail, Prognosis of machine health

condition using neuro-fuzzy systems, Mechanical Systems and Signal
Processing 18 (4) (2004) 813 – 831.

[14] C. S. Byington, M. J. Roemer, T. Galie, Prognostic enhancements to

diagnostic systems for improved condition-based maintenance, in: IEEE
Aerospace Conference, Vol. 6, 2002, pp. 2815 – 2824.

[15] D. A. Tobon-Mejia, K. Medjaher, N. Zerhouni, The ISO 13381-1 stan-

dard’s failure prognostics process through an example, in: IEEE - Prog-
nostics & System Health Management Conference, University of Macau,
Macau, China, 2010.

[16] AFNOR, Condition monitoring and diagnostics of machines - prognos-

tics - part 1: General guidelines. NF ISO 13381-1 (2005).

[17] G. Provan, Prognosis and condition-based monitoring: an open systems

architecture, in: IFAC Symposium on Fault Detection, Supervision and
Safety of Technical Processes, no. 5, 2003, pp. 57–62.

[18] R. B. Chinnam, P. Baruah, A neuro-fuzzy approach for estimating mean

residual life in condition-based maintenance systems, International Jour-
nal of Materials and Product Technology 20 (2004) 166 – 179.

[19] D. Tobon-Mejia, K. Medjaher, N. Zerhouni, G. Tripot, Estimation of the

remaining useful life by using wavelet packet decomposition and HMMs,
in: IEEE Aerospace Conference, 2011.

29
[20] J. Luo, M. Namburu, K. Pattipati, L. Qiao, M. Kawamoto, S. Chi-
gusa, Model-based prognostic techniques [maintenance applications], in:
IEEE Systems Readiness Technology Conference (AUTOTESTCON),
2003, pp. 330–340.

[21] D. Chelidze, J. Cusumano, A dynamical systems approach to failure

prognosis, Journal of Vibration and Acoustics 126 (2004) 2 – 8.

[22] A. Keller, U. Perera, A. Kamath, Reliability analysis of CNC machine

tools, Reliability Engineering 3 (6) (1982) 449 – 473.

[23] P. Groer, Analysis of time-to-failure with a weibull model, in: Proceed-

ings of the Maintenance and Reliability Conference, MARCON, 2000.

[24] G. Vachtsevanos, P. Wang, Fault prognosis using dynamic wavelet neural

networks, in: IEEE System Readiness Technology Conference, Autotest-
con Proceedings, 2001, pp. 857 – 870.

[25] W. Wang, An adaptive predictor for dynamic system forecasting, Me-

chanical Systems and Signal Processing 21 (2007) 809 – 823.

[26] R. Huang, L. Xi, X. Li, C. R. Liu, H. Qiu, J. Lee, Residual life predic-
tions for ball bearings based on self-organizing map and back propaga-
tion neural network methods, Mechanical Systems and Signal Processing
21 (1) (2007) 193 – 207.

[27] D. C. Swanson, J. M. Spencer, S. H. Arzoumanian, Prognostic modelling

of crack growth in a tensioned steel band, Mechanical systems and signal
processing 14 (1999) 789–803.

[28] M. Orchard, B. Wu, G. Vachtsevanos, A particle filter framework for fail-

ure prognosis, in: Proceedings of the World Tribology Congress, 2005.

[29] M. Dong, D. He, A segmental hidden semi-markov model (HSMM)-

based diagnostics and prognostics framework and methodology, Mechan-
ical Systems and Signal Processing 21 (2007) 2248–2266.

[30] R. B. Chinnam, P. Baruah, Autonomous diagnostics and prognostics

through competitive learning driven hmm-based clustering, in: Proceed-
ings of the International Joint Conference on Neural Networks, Vol. 4,
2003, pp. 2466–2471.

30
[31] H. Ocak, K. A. Loparo, F. M. Discenzo, Online tracking of bearing
wear using wavelet packet decomposition and probabilistic modeling:
A method for bearing prognostics, Journal of sound and vibration 302
(2007) 951–961.

[32] K. Przytula, A. Choi, An implementation of prognosis with dynamic

bayesian networks, in: Aerospace Conference, 2008, pp. 1 – 8.

[33] K. Medjaher, J.-Y. Moya, N. Zerhouni, Failure prognostic by using dy-

namic bayesian networks, in: 2nd IFAC Workshop on Dependable Con-
trol of Discrete Systems, 2009.

[34] M. Dong, Z.-b. Yang, Dynamic bayesian network based prognosis in

machining processes, Journal of Shanghai Jiaotong University (Science)
13 (2008) 318–322.

[35] J. J. Hillhouse, C. M. Adler, Investigating stress effect patterns in hospi-

tal staff nurses: Results of a cluster analysis, Social Science & Medicine
45 (12) (1997) 1781 – 1788.

[36] R. Landgraf, I. Xenarios, D. Eisenberg, Three-dimensional cluster anal-

ysis identifies interfaces and functional residue clusters in proteins, Jour-
nal of Molecular Biology 307 (5) (2001) 1487 – 1502.

[37] J. Abonyi, B. Feil, Cluster Analysis for Data Mining and System Iden-
tification, Birkhäuser Basel, 2007.

[38] H. M. Ertunc, C. Oysu, Drill wear monitoring using cutting force signals,
Mechatronics 14 (5) (2004) 533 – 548.

[39] T. Liu, K. Anantharaman, Intelligent classification and measurement of

drill wear, Journal of Engineering for Industry 116 (1994) 392–397.

[40] L. R. Rabiner, A tutorial on hidden markov models and selected appli-

cations in speech recognition, in: Proceedings of the IEEE, Vol. 77 (2),
1989, pp. 257–286.

[41] B. H. Juang, Maximum likelihood estimation for mixture multivariate

stochastic observations of marko chains, AT&T Technical Journal 64
(1985) 1235–1249.

31
[42] K. D. Vohnout, Curve fitting and evaluation, in: Mathematical Mod-
eling for System Analysis in Agricultural Research, Elsevier Science,
Amsterdam, 2003, pp. 140 – 178.

[43] A. Viterbi, Error bounds for convolutional codes and an asymptotically

optimal decoding algorithm, IEEE Transaction on Information Theory
13 (1967) 260 – 269.

[44] S. Kurada, C. Bradley, A review of machine vision sensors for tool con-
dition monitoring, Computers in Industry 34 (1997) 55–72.

[45] PHM Society, PHM data chalelnge 2010, in:

https://www.phmsociety.org/competition/phm/10, 2010.

[46] J. V. Abellan-Nebot, F. Romero Subiron, A review of machining mon-

itoring systems based on artificial intelligence process models, Interna-
tional Journal of advanced manufacturing technology 47 (2010) 237–257.

[47] K. Zhu, Y. S. Wong, G. S. Hong, Multi-category micro-milling tool wear

monitoring with continuous hidden markov models, Mechanical Systems
and Signal Processing 23 (2) (2009) 547 – 560.

[48] R. Pavel, I. Marinescu, M. Deis, J. Pillar, Effect of tool wear on surface

finish for a case of continuous and interrupted hard turning, Journal of
Materials Processing Technology 170 (2005) 341–349.

[49] V. Z. Aladjev, V. N. Haritonov, General theory of Statistics, Fultus,

2004.

32