A Bayesian network approach to root cause diagnosis of process variations

International Journal of Machine Tools & Manufacture 45 (2005) 75–91 www.elsevier.com/locate/ijmactool

A Bayesian network approach to root cause diagnosis of process variations S. Dey, J.A. Stori
Department of Mechanical and Industrial Engineering, 140 Mechanical Engineering Building, 1206 West Green Street, MC-244, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Received 28 October 2003; accepted 29 June 2004

Abstract This paper addresses the challenges inherent in root cause diagnosis of process variations in a production machining environment. We develop and present a process monitoring and diagnosis approach based on a Bayesian belief network for incorporating multiple process metrics from multiple sensor sources in sequential machining operations to identify the root cause of process variations and provide a probabilistic confidence level of the diagnosis. The vast majority of previous work in machining process monitoring has focused on single-operation tool wear monitoring. The focus of the present work is to develop a methodology for diagnosing the root cause of process variations that are often confounded in process monitoring systems, namely workpiece hardness, stock size, and tool wear variations. To achieve this goal, multiple sensor metrics have been identified with statistical correlations to the process faults of interest. Data from multiple sensors on sequential machining operations are then combined through a causal belief network framework to provide a probabilistic diagnosis of the root cause of the process variation. # 2004 Elsevier Ltd. All rights reserved. Keywords: Process monitoring; Process diagnosis; Fault diagnosis; Bayesian belief network

1. Introduction Machining process monitoring has received a great deal of attention in the academic research literature. A wide variety of sensors, modeling, and data analysis techniques have been developed, often focusing on tool condition monitoring. However, the majority of these techniques have yet to experience widespread industrial deployment. Industrial process monitoring systems in use today tend to rely on relatively simple signal thresholds for triggering alarms, and are not generally capable of diagnosing the cause of the alarm. One important challenge in industrial process monitoring is isolating the root cause of a process variation. In automotive transfer line applications, for example, false alarms may be generated in monitoring systems as a result of variations in casting geometry and hardness. Such variations may be difficult to distinguish from increases in the sensor signals resulting from tool wear,
Corresponding author. Tel.: +1-217-244-7762; fax: +1-217-3331942. E-mail address: [email protected] (J.A. Stori).

0890-6955/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2004.06.018

for example. Due to the stochastic nature of manufacturing process variations, a probabilistic analysis technique based on Bayesian inferencing has been developed and will be described in this paper. Before providing an overview of the developed approach, we briefly review some of the relevant prior work in process monitoring and diagnosis. 1.1. Sensors in process monitoring 1.1.1. Tool condition monitoring Tool condition monitoring has received a tremendous amount of attention in the academic literature. Li and Mathew [1] and Dimla [2] provide excellent reviews of the various direct and indirect methods used to predict tool wear and failure in turning. Direct methods include optical techniques as well as radioactivity analysis, workpiece size measurement, and tool/workpiece distance measurement techniques. Indirect methods include cutting force, acoustic emission, sound, vibration, spindle power, cutting temperature and surface roughness measurement. Jantunen [3] recently reviewed prior work in tool condition monitoring for drilling. Below, we

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briefly discuss some of the sensors that have been commonly used in tool condition monitoring applications. Cutting force sensors have proven popular for tool condition monitoring applications. The success of these approaches may be attributed, at least in part, to the prevalence of analytic and mechanistic process models capable of predicting cutting forces as a function of various process parameters. One important practical hindrance to the industrial deployment of cutting force sensors for process monitoring sensors is the high cost and intrusive nature of multi-axis dynamometers in a production environment. A variety of model-based tool condition monitoring techniques using cutting force sensors have been proposed [4–6]. Statistical techniques such as time series and autoregressive models based on cutting force sensors have been developed [7–9] as well as neural network approaches [10–12]. Acoustic emission refers to the transient elastic waves generated from a localized source or sources within a material. These waves are generated at high frequencies, usually above 500 kHz, and are picked up by AE transducers mounted in close proximity to the source of emission. Acoustic emission is sensitive to the microscopic activities related to plastic deformation and friction in the cutting zone. Dornfeld pioneered some of the important early efforts to develop acoustic emission techniques for tool condition monitoring of machining processes [13,14]. A variety of applications of AE techniques to tool condition monitoring by other research groups have been reported in the literature some of which may be found in the following review papers. Jemielniak [15] discusses the different acoustic emission and ultrasonic sensors used for tool condition monitoring. Li [16] presents a compact review of various issues in acoustic emission monitoring for turning, including AE generation, classification and signal correction, signal processing methodologies, and tool wear estimation methods. The majority of AE monitoring applications have relied on the root mean square (rms) value of the acoustic emission signal. Other metrics such as the energy content in various frequency ranges, and time domain statistical parameters such as skew and kurtosis have been explored. Applications of accelerometers or piezoelectric vibration sensors to tool condition monitoring have also been reported. El-Wardany et al. [17] investigated the use of vibration signals for online drill wear monitoring and breakage. Features sensitive to tool wear were identified in both the time domain (ratio of absolute mean and kurtosis metrics) and the frequency domain (power spectra and cepstra ratio). Yao et al. [18] researched the detection and estimation of groove wear at the minor cutting edge of the tool using vibration signatures. Rotberg et al. [19] also described the application of a vibration sensor to tool wear monitoring.

Spindle current sensors have also been applied to tool condition monitoring of drilling and milling processes. The time response of such sensors is inherently limited. For this reason, high frequency components of cutting force are difficult to observe. Feed drive motor sensors have also been used for tool condition monitoring and cutting force estimation. Previous applications of a spindle power sensor to monitor tool wear include [20–23]. Li et al. [24] apply neural network and fuzzy logic analysis techniques to estimate cutting forces and detect the onset of accelerated wear using the feed motor current. 1.1.2. Dimensional and geometric variations Dimensional variations in the incoming raw material will result in unanticipated depth of cut variations, and corresponding increased loads on the cutting tool and spindle. Spindle and cutter run-out will result in varying and unbalanced cutting forces. Seethaler and Yellowly [25] investigated the individual tooth run-out in milling using linear force models. They showed experimentally how the methods could be used to distinguish run-out from edge breakage. Sastry et al. [26] developed a methodology that used variations in spindle speed to compensate for radial run-out. Several researchers have developed techniques for the identification of the depths of cut in milling using cutting force data. Altintas and Yellowly [27,28] used the ratio of two orthogonal cutting force components to identify the axial and radial depth of cut. They assumed a zero helix angle, and required the fitting of polynomial functions for each feedrate value. Choi and Yang [29] proposed a technique for calculating the depths of cut based only on the shape of the cutting force profile. Yang et al. [30,31] has developed a detailed diagnosis methodology for axial and radial depth of cut variations also based on an analysis of the cutting forces and comparison with model predictions. Their approach relies only on the resultant cutting force profile. 1.1.3. Workpiece hardness Workpiece hardness variation does not appear to have received a significant amount of prior attention in the process monitoring and diagnosis literature, yet it remains an important issue challenge for the successful industrial application of process monitoring systems. Production castings may contain significant hardness variations that are difficult to distinguish from other process variations such as tool wear or workpiece geometry variations. We were unable to find any published references addressing sensor-based identification of workpiece hardness variations. 1.2. Inference and diagnostic methods A variety of diagnostic or inferencing techniques have been used in process monitoring, including artificial neural networks, expert systems, fuzzy logic and statistical methods.

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1.2.1. Neural networks Artificial neural networks, also known as connectionist architectures, parallel distributed processing, and neuromorphic systems, are systems where inputs are mapped to known outputs via weights during a training process. The learned weights are used to predict corresponding outputs for given inputs. Their structure is modeled loosely on the neural conduction system existing in the human body. Neural networks have proven successful as pattern recognition engines and robust classifiers, with the ability to generalize in making decisions about imprecise input data. The advantage of artificial neural networks lies in their resilience against distortions in the input data and their capability of learning. Many types of artificial neural networks have been developed with varying structures and connectivity. Dimla et al. [32] have compiled an excellent review of applications of neural networks to tool condition monitoring applications. A number of individual efforts have been referenced previously. Neural networks have also been used successfully to combine inputs from multiple sensors for tool condition monitoring. 1.2.2. Fuzzy logic and hidden Markov models Fuzzy set theory, originally introduced by Lotfi Zadeh in the 1960s, emulates human reasoning in its use of approximate information and uncertainty to generate decisions. Fuzzy set theory implements classes or groupings of data with boundaries that are not sharply defined. Any methodology or theory implementing ‘‘crisp’’ definitions such as classical set theory, arithmetic, and programming, may be ‘‘fuzzified’’ by generalizing the concept of a crisp set to a fuzzy set with blurred boundaries. The benefit of extending crisp theory and analysis methods to fuzzy techniques is the strength in solving real-world problems, which inevitably entail some degree of imprecision and noise in the variables and parameters measured and processed for the application. Accordingly, linguistic variables are a critical aspect of some fuzzy logic applications, where general terms such a ‘‘large’’, ‘‘medium’’, and ‘‘small’’ are each used to capture a range of numerical values. Du et al. [33] demonstrated the effectiveness of using fuzzy set theory in tool condition monitoring. A total of 11 monitoring indices were identified to describe signature characteristics of force, vibration, and spindle power sensor signals. Li et al. [34,35] used both fuzzy set and wavelet transformation techniques to extract features from the spindle and feed servo motor current signals in drilling and establish relationships between these features and the cutting conditions for different tool states. Hidden Markov models are an extension of Markov chains and used to model processes whose characteristics vary with time. A Markov chain is a random pro-

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cess involving a number of states linked by a number of discrete transitions. Each transition has an associated probability and each state has an associated observation output. In hidden Markov models, the transitions and output are stochastic. They have been used extensively and with great effectiveness in speech recognition applications. Ertunc et al. [36] uses hidden Markov models to classify the wear state of the tool in drilling using cutting force signals. Atlas et al. [37] applied hidden Markov models to tool wear monitoring in milling using a vibration sensor. 1.2.3. Statistical techniques Statistical approaches to modeling and monitoring of machining processes most often involve regressionbased and time domain techniques such as time series or moving averages. Liang and Dornfeld [13] used an autoregressive time series model to detect tool wear using acoustic emission signals. Yao and Fang [7] developed a stochastic method based on multivariate autoregressive moving average vector models to quantify the process dynamics embedded in multi-dimensional force and vibration signals to monitor tool wear in finish turning. Kumar et al. [38] used a time series approach to extract features from force and vibration signals and pattern recognition techniques to identify the state of tool wear. Tarn and Tomizuka [23] used an autoregressive model to detect tool breakage using spindle current and cutting force signals. Govekar et al. [39] used spectral and time series analysis methods to develop characteristics for tool condition monitoring using data obtained from multiple sensors including acoustic emission. Yan et al. [40] used an autoregressive model to extract features from acoustic emission and cutting force sensors to detect tool failure in milling. 1.2.4. Bayesian techniques Perhaps the most widely applied technique in probabilistic inferencing is the Bayesian network. Bayesian networks provide a flexible structure for modeling and evaluating uncertainty. The network consists of nodes representing random variables connected by arcs that quantify a causal relationship between the nodes. McCabe [41] provides an overview of Bayesian belief networks and their application. Bayesian networks have proven useful for a variety of monitoring and predictive purposes. Applications have been documented in the medical domain [42–44], image-processing, target recognition [45,46], and pattern matching [47]. Bayesian networks have also been used for monitoring and diagnostic applications in manufacturing. Lewis and Ransing [48] present an interesting study of a semantically constrained Bayesian network for diagnosis in manufacturing. The algorithm they use is based on Bayesian theory but is modified to enable belief

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revision. The presence of uncertainty is also quantified by a relationship. Rigorous constraints have been developed and are used to check the consistency of network values. The network is initialized and then used to diagnose a pressure die casting process. Wolbrecht et al. [49] describe a real time monitoring and diagnosis system that identifies component failures quickly in a multi-stage process. Prateepasen et al. [50] demonstrates the ability to combine signals from acoustic emission and vibration sensors for tool wear monitoring. The AE rms signal and the coherence function between the axial and feed directions for the accelerometer signals were found to be sensitive to tool wear. 1.3. Motivation Despite the tremendous amount of prior work in sensor-based tool condition monitoring, very little attention has been placed on the important task of diagnosing the root cause of process variations when multiple potential sources of variation exist. For example, increases in workpiece hardness, stock size, and tool wear, may be confounded in a production process monitoring application. The intent of the present effort is to explore the possibility of using multiple sensors and data obtained from sequential machining operations to diagnose the root cause of process variations. We have adopted a Bayesian probabilistic framework to explicitly address the uncertainty inherent in the root cause diagnosis. In Section 2 of this paper, we motivate the application of the Bayesian belief network and describe its structure. The procedure for learning from a training database, and rules for the updation and propagation of evidence are described. In Section 3, we describe the experimental procedure and methodology that were used to identify the correlative sensor metrics for process fault diagnosis, as well as the resulting structure of the Bayesian belief network. In Section 4, results are presented demonstrating the effectiveness of the technique for root cause diagnosis of process faults using an acoustic emission and spindle power sensor and data from two subsequent machining operations. 2. Bayesian belief networks Process monitoring and fault diagnosis requires the assimilation of numerous noisy and incomplete sources of evidence in order to infer when a change in process state has occurred and identify the root cause of that change. Such inferences will necessarily be imperfect, and it would be valuable if the conclusions could be augmented with some quantifiable measure of its certainty or uncertainty. Bayesian inferencing techniques are well suited to such problems. In particular, a data structure known as

a Bayesian belief network provides a convenient formalism for representing conditional probabilistic relationships between attributes of interest. 2.1. Bayesian inferencing Given a hypothesis H and an evidence e, Bayes’ theorem may be stated as follows: PðHjeÞ ¼

PðejHÞ  PðHÞ PðeÞ

ð1Þ

where P(H|e) is the conditional probability of hypothesis H being true given the evidence e, also know as the posterior probability or posterior, P(H) is the probability of hypothesis H being true or the prior probability, and P(e|H) is the probability of evidence e occurring given hypothesis H is true. This is also known as the likelihood that evidence e will materialize if hypothesis H is true. As Pearl states: ‘‘The importance of [Bayes’ Theorem] is that it expresses a quantity, P(H|e)—which is often difficult to assess—in terms of quantities that often can be drawn directly from our experiential knowledge’’ [51]. The aim in monitoring and diagnosis is to be able to incorporate previous knowledge of the process with present observed evidence to come up with the most plausible explanation of how the process is behaving. Bayes’ theorem incorporates this type or predictive and diagnostic support. 2.2. The structure of a Bayesian belief network Bayesian networks are directed acyclic graphs in which each node represents a random variable that can take on two or more discrete values. The arcs in the network signify the existence of direct causal influences between the linked variables, and the strengths of these influences are quantified by conditional probabilities. The directionality of the arrows is essential for displaying causality and non-transitive dependencies. The causal relationships expressed through the network structure are based on prior knowledge, experience, or statistically observed correlations. Each node has distinct states or levels that signify the values that the particular node can take. A Bayesian network must be an acyclic graph, although it may be multiply connected. Each node is associated with a conditional probability table that defines probabilities for the different states of the node given the states of the parents. Fig. 1 illustrates the variables and notation to be used in describing a Bayesian network. The notation has been adopted from Ramoni and Sebastiani [52]. In this subset of a Bayesian network, there are three nodes, representing the random variables X1, X2, and X3. The links between nodes X1 and X2 as well as X2 and X3 indicate an explicit conditional dependence. In

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Fig. 1.

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Notation to be used in describing a Bayesian network.

other words, node X3 is conditionally dependent only on the state of nodes X1 and X2. The matrix h3 quantifies this conditional dependence. h3 contains the conditional probabilities of all possible states of node X3 given all possible states of the parents. The individual elements of matrix h3 are denoted h3jk ¼ PðX3 ¼ kjP3 ¼ p3j Þ where p3j indicates the state of the parents of X3. In this particular example, nodes X1 and X2 can each take on two possible states, so P3 can take on four possible values, p3j, j ¼ 1 . . . 4. The vector p3j holds the individual states of the parents. The general notation is as follows: each random variable Xi is discrete and has a finite set of ci states. The state of the node Xi is given by xik. The state of the parents of Xi is stored in the variable Pi. The number of states of Pi is denoted by qi, and a particular state is indicated by pij. Thus, the subscript i indicates the node in question, j stands for the parent state among a total of qi states and k stands for the state of the variable Xi among a total of ci states. Each element of the conditional probability matrix is defined as: hijk ¼ PðXi ¼ xik jPi ¼ pij Þ. The vector pij holds the individual states of the parents of Xi and has cardinality equal to the number of parents. 2.3. Learning conditional probabilities in a Bayesian belief network Before a Bayesian network may be used for process fault diagnosis, the structure of the network must be

identified, and initial values for the conditional probabilities must be obtained. We will discuss the development of an appropriate structure for our application in Section 3 of this paper. We first discuss techniques for estimating the unknown conditional probabilities, hijk, given a prior set of data, or a database, D. The following development follows [52]. A database consists of a series of instances of prior node states. The objective of a learning procedure is to assign values to the conditional probability matrix that are consistent with the database. One common way in which this is formulated is through maximization of the likelihood function, P(D|h). In other words, it is desired to identify conditional probability values that maximize the likelihood of the database having occurred. frequency of (xik|pij) in the dataLet n(xik|pij) be the P base, and let nðpij Þ ¼ k nðxik jpij Þ be the frequency of pij. It is well known that the ML estimate of hijk is the relevant frequency of the cases in the database [53]: hijk ¼ nðxik jpij Þ=nðpij Þ

ð2Þ

A major drawback of this method is that hijk ¼ 0 whenever nðxik jpij Þ ¼ 0, so that the estimate can be too extreme when the database is sparse or incomplete. Thus, the type of database available for learning is what primarily governs the choice of a suitable learning technique. A complete database contains data for all possible combinations of the variables and their states. A database is sparse when data for all the possible combinations of the

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variables and their states are not available. A database is said to be incomplete when in a particular data set, values of some of the variables are missing. Learning for incomplete databases poses the most significant statistical challenges. A summary of a number of common techniques for learning with incomplete databases can be found in [54]. In our process monitoring application, it will typically be possible to observe all sensor metrics for all data instances. Thus, we will not address learning with an incomplete database. Nevertheless, obtaining training data will be expensive, requiring extensive experimentation. For this reason, it is likely that a training database for a process monitoring application will be sparse. The Bayesian approach is to regard hijk as a random variable whose prior distribution represents the observer’s belief about the conditional probability before data are seen. Application of Bayesian learning involves the use of a database of training examples to update the prior belief on the parameters using Bayes’ theorem. We first require an assumption on the prior distribution of the parameters, h. For convenience, a Dirichlet prior distribution is typically chosen for the multinomial parameters, hij. This choice of prior satisfies the desired assumption of mutual independence of the parameters introduced by Spiegelhalter and Lauritzen [53] and results in a conjugate sampling situation, namely that after data are seen, the distribution of parameters remains in the Dirichlet family [55]. Interestingly, Geiger and Heckerman [55] demonstrate that the under these assumptions, a Dirichlet prior distribution is, in fact, the only possible choice. The assumption of complete data, parameter independence, and Dirichlet priors makes it possible to update each vector of parameters, hij, independently [54]. For our application, we also assume the prior distribution of hij to be a Dirichlet distribution with parameters faij1 ; . . . aijci g. The prior density of hij is then given by [52]: P  XC a k aijk hijkijk ð3Þ pðhij Þ ¼ Cða Þ ijk k The parameters aijk encode an observer’s prior belief. In the absence of prior information, we will assume that aijk ¼ a=ðci qi Þ for all i, j and k, so that the prior probability of (xik|pij) is simply 1=ci [55]. As previously discussed, the posterior density of h will remain within the family of Dirichlet distributions, and the posterior expectation of hijk can be readily obtained [52]:  aijk þ nðxik jpij Þ ð4Þ Eðhijk DÞ ¼ aij þ nðpij Þ Eq. (4) will be used to initialize the conditional probabilities for our process monitoring application given a

sparse training database, and the assumption of a uniform prior expectation. 2.4. Bayesian belief updating—propagation of evidence Once a Bayesian network has been constructed and initial conditional probability tables have been generated by training on prior data, the network is ready for the introduction of external evidence based on sensor observations. Upon the introduction of external evidence, those nodes representing sensor observations are forced to take on a specific value while the remaining nodes must adjust their own ‘‘belief’’ (probability of being in each of the potential states) in order to remain consistent with the new evidence. It is these updated beliefs that will be observed and relied upon as the output of the process monitoring and diagnosis system. A variety of techniques for inferencing in a Bayesian network have been developed. We have adopted a message passing methodology for belief updating developed by Pearl [51]. This method offers a number of appealing characteristics relevant for applications in process monitoring and diagnosis. Specifically, the message passing methodology offers a local triggering mechanism and a computationally simple local updating procedure. Finally, the triggering mechanism and updating action is conceptually meaningful. Alternative approaches typically rely on a global supervisor, or monitor that must select the sequence of propagation and computation throughout the network. In this section, we provide our formal implementation of Pearl’s message passing methodology for belief updating in a multiply connected tree (polytree). For a detailed development of the message passing mechanism itself, the reader is directed to Pearl [51], Chapter 4. The notation and implementation of the belief updating rules has been changed significantly so as to maintain consistency with the data structures and notation used previously in this paper. The probabilities of interest will be the belief, or likelihood of each of the states of a particular node, given all evidence received thus far. This evidence includes all training data, as well as the current instantiation of particular nodes as a result of sensor observations. During the belief updation and inferencing, the conditional probability vectors, hij, remain fixed. These updated belief probabilities will be denoted as a vector B(Xi), indicating the probability, or likelihood of each of the possible states of Xi. The message passing procedure occurs as follows: when new evidence is introduced into the network, each node updates its own belief, based on messages received from its parents and children, and correspondingly generates messages to be sent to its parents as well as to its children. Fig. 2 describes the notation to be used in message passing. The set of parents of node

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Fig. 2.

Message passing conventions for belief updating.

Xi will be denoted Xi . The number of parents will equal the cardinality of the set, denoted jXi j. The set of children of node Xi will be denoted Wi , with cardinality jWi j. A message from one of the parents, m 2 Xi , to node Xi will be denoted xim , while a message from a child, n 2 Wi , to node Xi will be denoted win . The steps are as follows: Step 1—Belief updating. When node Xi is activated, it simultaneously inspects the messages communicated by its parents, xmi , and the messages communicated by its children, win . It then updates it is belief according to the following relationships: For all k; Bi ðkÞ ¼ wi ðkÞ  xi ðkÞ;   1 P Bi ¼ Bi k Bi ðkÞ jWi j Y For all k; wi ðkÞ ¼ win ðkÞ; n ¼ Wi ðoÞ o¼1 ( ) jXi j X Y hij ðkÞ xim ðpij ðoÞÞ ; For all k; xi ðkÞ ¼ j

81

o¼1

m ¼ Xi ðoÞ Step 2—Generation of messages to be sent to parents. For each parent, p 2 Xi of node Xi. Create conditional probability matrix N:

For all k, and all j, 8 9 jXj < = Y xmi ðpij ðoÞÞ ; Nkl ¼ Nkl þ hijk : o¼1;XðoÞ6¼p ; l ¼ pij ðojXðoÞ ¼ pÞ Condition N on values of wi : ci X Nkl wi ðkÞ For all l, wip ðlÞ ¼ k¼1

Step 3—Generation of messages to be sent to children. For each child, c, 8 9 ( ) jWi j jXi j < Y =X Y xic ðkÞ ¼ w ðkÞ hijk xmi ðpij ðoÞÞ ; :o¼1;W ðoÞ6¼c in ; j o¼1 i

m ¼ Xi ðoÞ;

n ¼ Wi ðoÞ

The propagation of messages is triggered by instantiation of evidence into the network. Once the evidence enters the network via the evidence nodes, the beliefs of these nodes are updated. The next step is to post messages to all the parents and children of the availability of evidence and set flags on them that indicate that they should be updated. The nodes containing these flags are then required to update their own belief. For each node whose belief is updated, new messages are again gener-

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ated and sent to the children and parents. Flags are also set for the parents and children as the messages are sent. The presence of undirected loops in the network introduces significant challenges to the message propagation and belief updating procedure. Because there may be multiple paths between two nodes, the message passing could propagate indefinitely. The problem, in general, is NP hard. For large networks with many loops, the computational challenges become significant. For small networks with a small number of loops, a threshold approach may be employed [56]. The propagation in such a case would follow all the rules of normal propagation for singly connected networks. It would also compare the change in the belief between successive belief updations of a node. If the difference is not greater than a small threshold, which is predetermined, the propagation is terminated. 2.5. Application of the Bayesian belief network methodology We now provide an overview of the procedure for applying the Bayesian belief network to root cause diagnosis of process variation. In Section 3, we will describe the specific sensor metrics and experimental methodology for training and utilizing the network. Fig. 3 graphically illustrates the procedure for training and applying the network. The first step is the identification of the network structure. For our root

cause diagnosis application, the network is limited to two levels. The top level contains the root nodes encapsulating the state of the process variations. The second level contains evidence nodes representing the states of the various sensor metrics. The selection of the individual process metrics, and the determination of whether a causal link between a particular root node and a particular evidence node is to exist in the network will be described in detail in Section 3. Once the network structure has been determined (initialization), initial values for the conditional probabilities must be learned from a training database. The learning procedure has been described previously in Section 3.2. A training database is provided through an initial experimental design. After the database structure has been identified and the conditional probability tables have been initialized, the network is ready for application. The first step in application of the network is the instantiation of evidence. When new sensor data are available, the corresponding metrics are evaluated, and the evidence nodes are forced into the corresponding state. The next step is propagating the evidence that has been instantiated into the network to the different nodes in the network. Evidence propagation proceeds per the rules described previously in Section 2.4. This is followed by the updating of the belief in light of the evidence received from both parents and children. After the belief of a particular node is updated, its

Fig. 3. Initialization and application of the Bayesian belief network methodology.

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Table 1 Acoustic emission and spindle power metrics investigated Acoustic emission sensor metrics

Spindle power sensor metrics

Standard deviation of the AE data from the mean Mean of the 10% largest amplitude peaks Mean of the absolute value of the AE data Pulse rate with a variety of threshold levels Peak amplitude in the FFT of the data Frequency at which the peak amplitude occurs Power spectral density in 25 kHz frequency bands Power spectral density between 75 and 500 kHz

Mean of the spindle power data Standard deviation of the spindle power data Mean of the 10% largest amplitude peaks Power spectral density at tooth passing frequency (milling) Power spectral density in 10 Hz bands (drilling only) Power spectral density between 10 and 100 Hz (drilling) Mean of data during partial immersion (drilling only)

belief propagated to the surrounding nodes in accordance with the propagation rules. This is an iterative procedure and is carried out until saturation is reached. Saturation is obtained when there are no other nodes to propagate the evidence or its effect to. 3. Development of sensor metrics and network structure A primary objective of this work was to make use of input from multiple sensors on multiple processes in an attempt to more reliably discern the root cause of process faults. The three faults of interest were dimensional workpiece variation, tool wear, and workpiece hardness. These three faults would commonly manifest themselves in similar ways from a process monitoring perspective (i.e. increase in power levels, cutting forces, etc.). We are interested in attempting to discriminate between these commonly confounded root causes. The case study that was developed to validate the methodology involved two sequential machining operations carried out on a single workpiece. The workpiece material was 100 diameter 4140 bar stock with hardness varying from 24 to 32 Rockwell C. A face milling operv ation was conducted using a 45 lead, 3.1500 diameter face mill with four carbide inserts (Kennametal part #SEKT443AEEN7GP) to machine a flat on one end of the bar stock. Next, a blind hole was drilled into this v newly generated flat. The drill was a 118 point angle, v 00 45 helix, 0.375 diameter tin-coated HSS twist drill (Kennametal part #KHSS13324). The process parameters were as recommended by the Machining Data handbook. It was hoped that the cumulative information from the sequential process steps might aid in discriminating between multiple causes. Care was taken to utilize sensors that may be cost effective and practical in a production machining environment. For this reason, direct measurement of cutting forces through a multi-axis dynamometer was avoided. The two sensors employed were an acoustic emission sensor and a spindle power sensor, both provided by Montronix, Inc. Data from each sensor were obtained during each of the two machining operations.

3.1. Development of sensor metrics In order to identify a variety of sensor metrics that had statistically significant correlations with the process faults of interest, a set of preliminary experiments were conducted. A wide range of both time domain and frequency domain metrics were investigated, and ANOVA analysis was used to determine statistical significance. The intent of the metrics is to encapsulate significant features in the data stream with a scalarizing variable that would be suitable as an evidence node in the Bayesian network. The metrics investigated are listed in Table 1. The acoustic emission sensor was sampled at 1 MHz and band pass filtered between 10 and 500 kHz. During face milling, the AE sensor responds in bursts to the engagement of the individual inserts. The sensor metrics were evaluated over these data bursts. 3.1.1. Preliminary experimental study and ANOVA analysis After the candidate set of metrics had been identified, a three variable, two-level full factorial experimental study was performed to identify those metrics with a statistically significant correlation to the process faults of interest. The three variables were the process variations of interest: tool wear, workpiece hardness, and dimensional workpiece variation. After the subset of statistically significant variations was identified, those corresponding metrics were used as nodes in the Bayesian network. A 23 full factorial experimental design was executed with two repetitions. The sequence of experiments was randomized. The average value of the repetitions was used in the ANOVA analysis. The workpiece hardness ranged from 24.0 to 32.0 Rockwell C. The geometric stock size variation was 0.07000 . A full-factorial experimental design was conducted because we are also interested in the interaction effects between the variables. The ANOVA analysis was conducted for each variable using the F-test with a 95% confidence interval. The metrics were chosen based on the following set of rules that were intended to distinguish between significant causal relationships and confounded effects:

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Table 2 Acoustic emission and spindle power metrics identified through ANOVA analysis Process

Sensor

Acoustic emission Face milling

Spindle power

Acoustic emission Drilling Spindle power

Label

Metric

ANOVA sensitivity

AEFM1 AEFM2 AEFM3 AEFM4 AEFM5 AEFM6 PFM1 PFM2 PFM3 AED1 AED2 AED3 AED4 AED5 PD1 PD2 PD3 PD4

Standard deviation Pulse rate, 0.8 threshold PSD (75–475 kHz) Mean of peaks Frequency of peak Mean, absolute value Mean Standard deviation Mean of peaks Mean of peaks Pulse rate, 0.9 threshold Mean, absolute value PSD Standard deviation Difference between means, full and partial engagement Mean during partial engagement Standard deviation PSD (10–100 Hz)

TWM SS, TWM SS, TWM, SS TWM SS SS SS, TWM, SS TWM WH, TWM SS, TWM SS, TWM SS TWD WH TWD TWD TWD TWD WH SS TWD TWD

1 If only the main effects are significant, all the main effects that are significant are included as causative agents for the metric in question. 2 If the main effects and one or more first order interactions are significant, then: – If the interactions are the combinations of variables that show up as significant in the main effects, then include the variables in the interaction as being causative agents for the metric in question. – If the interactions are the combinations of variables that are other than the variables that show up as significant in the main effects, the result is confounded and ignore the result. 3 If none of the main effects are significant and one or more first order interactions are significant then: – If there are multiple interactions, the result is confounded and thus should be ignored. – If there is one interaction then the variables in the interaction are the causative agents for the metric in question. 4 If the second order interaction is significant, then: – If none of the main effects or the first order interactions are significant, then all the variables involved in the second order interaction are taken as significant and causal agents for the metric in question. – If some or all of the main effects are significant with no first order interaction being significant, the results are confounded and thus ignored. – If any of the first order interactions are significant, the results are confounded and ignored. – If all the effects are significant, the results are completely confounded and are ignored. Application of the above rules resulted in the identification of the causally significant sensor metrics listed in Table 2 for the two processes and

two sensors employed. The last column of the table lists the root nodes with which a statistically significant correlation was found. Each of these relationships represents a causal arc that will be added to the Belief network. 3.2. Initialization of the network structure Fig. 4 illustrates the resulting network structure. There are a total of 18 evidence nodes representing individual sensor metrics. There are four root notes, representing the four root causes of concern: stock size variation, workpiece hardness variation, tool wear state of the face mill, and tool wear state of the endmill. Each of these root cause nodes contained two discrete states, high and low. The evidence nodes were discretized with either two or three states. Each of the links represents a causal relationship as enumerated in Table 2. In the event that a statistically significant relationship between a particular metric and an interaction effect exists and was not eliminated by the filtering rules above, a causal link between both of the corresponding root nodes and the sensor metric in question were added to the Bayesian network. 4. Results and conclusions The network shown in Fig. 4 was trained on the 23 full factorial DOE with two repetitions as described in Section 3. The raw training data and the corresponding table of states from which the conditional probabilities were learned is containing Appendix A, Tables A1–A3. Subsequent to training the network, its diagnostic performance was evaluated on 18 new trials. For these trials, experimental training data were obtained, and the metrics were evaluated and introduced as evidence into the network. The belief updation procedure was executed and the beliefs of the root nodes were recorded. Tables 3–5

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Fig. 4. Structure of Bayesian belief network for root cause diagnosis.

summarize the predictive success of the network on these 18 trials. Three different scenarios are reported: belief updation using milling evidence only (Table 3), drilling evidence only (Table 4), and both milling and drilling evidence (Table 5). The percentages reported in the tables represent the belief in the correct state of the trial. For example, consider the first trial (first row of data) in Table 3. In this trial, and as indicated on the first row of the table, the tool wear of the face mill was in the low state, the tool wear of the drill was in the high state, the stock size dimension was in the low state, and the workpiece hardness was in the low state. Because Table 3 represents a situation in which only the milling evidence is utilized, there is no ability to predict the tool wear state of the drill. For this reason, ‘‘N/A’’ appears in the corresponding cell of the table. Belief predictions are available, however, for each of the remaining three diagnoses. In this trial, all three of these beliefs are greater than 80%, indicating a greater than 80% belief in the correct state of the stock size, face mill wear state, and workpiece hardness level. Higher percentages represent greater degrees of certainty, and therefore more successful diagnostic ability.

A percentage in the neighborhood of 50% represents diagnostic ambivalence—the network is unable to discern the state given the training data and current sensor evidence. A percentage below 50% represents an incorrect diagnosis—the belief is stronger in the opposite (incorrect) state. Tables 3–5 have been colorcoded to provide a visual summary of the diagnostic success—lighter shades of gray indicate increasing levels of diagnostic confidence. Table 6 and Fig. 5 summarize the diagnostic performance of the network. As may be seen in Fig. 5, the predictive success increases with the use of data from both processes. This is an important observation. Seldom has data from subsequent machining operations been combined to provide a greater level of confidence in the diagnosis of process faults. Using evidence from both the milling and drilling metrics, the network was able to correctly diagnose the state at a 60% confidence level in all 18 trials for the drilling tool wear, workpiece hardness, and stock size nodes, and provided a correct diagnosis in 17 of 18 cases for the face mill tool wear state. As the confidence threshold is increased to 70% and 80%, the predictive success declines, as would be

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Table 3 Diagnostic beliefs on 18 trials using milling evidence only

Table 4 Diagnostic beliefs on 18 trials using drilling evidence only

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Table 5 Diagnostic beliefs on 18 trials using milling and drilling evidence

Table 6 Diagnostic success on 18 trials at 60%, 70%, and 80% confidence levels Process fault TWD

TWM

Evidence utilized

SS

WH

Diagnosis confidence

60%

70%

80%

60%

70%

80%

60%

70%

80%

60%

70%

80%

Drilling only, both sensors Milling only, both sensors Both processes, both sensors

N/A 16 17

N/A 16 17

N/A 12 13

15 N/A 18

15 N/A 16

8 N/A 10

17 15 18

16 14 16

8 11 12

17 18 18

15 17 18

6 11 13

expected. Nevertheless, using evidence from both processes, the network was capable of achieving an 80% confidence level in between 10 and 13 of the 18 trials. In conclusion, the Bayesian belief network methodology appears to be an effective tool for explicitly addressing uncertainty and utilizing data from multiple sources. In this application, data from two non-intrusive sensors (spindle power and acoustic emission) and two subsequent processes (face milling and drilling) have been combined to successfully diagnose a state of tool wear, workpiece hardness, and a stock size dimensional variation. After training on a data set of 16 trials, the network was able to correctly diagnose the correct state at a 60% confidence level in all but one of the 18 test cases. The belief network was able to correctly diagnose the state of the drill tool wear at the 80% confidence level

in 10 of the 18 trials, and the stock size variation, workpiece hardness, and face mill tool wear state in either 12 or 13 of the 18 trials. Acknowledgements The authors gratefully acknowledge the support of Montronix, Inc., Ford Motor Company, the NIST ATP program, and the National Science Foundation (award DMI-9941214) for support of this work. Dr. W. Kline and R. Sriram of Montronix, Inc. are personally acknowledged for their support and guidance. Appendix A. Tables A.1–A.3.

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Fig. 5. Diagnostic success on 18 trials at 60% and 80% confidence levels.

Table A.1 Face milling metrics from training data Acoustic emission metrics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Spindle power metrics

AEfm1

AEfm4

AEfm6

AEfm2

AEfm5

AEfm3

Pfm1

Pfm2

Pfm3

0.56 0.52 0.61 0.53 0.49 0.51 0.49 0.49 0.56 0.52 0.61 0.53 0.49 0.51 0.49 0.49

1.59 1.54 1.21 1.04 1.48 1.60 1.03 1.07 1.59 1.54 1.21 1.04 1.48 1.60 1.03 1.07

0.42 0.39 1.05 0.88 0.36 0.36 0.80 0.80 0.42 0.39 1.05 0.88 0.36 0.36 0.80 0.80

1519 1328 920 649 1244 1209 416 407 1519 1328 920 649 1244 1209 416 407

140,512 140,343 920 649 136,072 133,606 416 407 140,512 140,343 920 649 136,072 133,606 416 407

0.64 0.57 920.31 649.19 0.53 0.52 416.75 407.99 0.64 0.57 920.31 649.19 0.53 0.52 416.75 407.99

75.84 74.90 80.95 80.04 79.75 78.87 85.27 84.69 76.16 73.86 81.46 79.94 79.83 75.30 85.37 80.63

15.74 15.51 16.79 15.78 13.28 14.21 14.56 15.25 16.18 14.83 16.95 15.66 13.29 13.43 14.32 14.50

16.77 16.60 18.01 16.52 12.88 14.16 14.74 15.59 17.67 15.64 18.42 16.57 12.94 12.17 14.47 13.45

0.18 0.22 0.20 0.20 0.16 0.17 0.15 0.14 0.18 0.20 0.18 0.20 0.18 0.18 0.19 0.15

1.17 1.36 1.47 1.33 1.13 1.12 0.84 1.00 0.99 1.20 1.34 1.33 1.38 1.36 1.27 1.06

AEd1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Trial

SSV

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

TWM

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Process state

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Hardness TWD

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.13 0.15 0.14 0.13 0.12 0.12 0.11 0.10 0.13 0.14 0.12 0.13 0.13 0.12 0.13 0.11

AEd3

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

AEFM1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1

AEFM2 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0

AEFM3

Face milling metric state

Table A.3 Evidence and process states for training data

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

AEd5

Acoustic emission metrics

Table A.2 Drilling metrics from training data

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

AEFM4 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 0

PFM1

247 757 476 543 226 287 51 104 147 513 327 543 561 362 425 244

AEd2

1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 1

PFM2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

AEFM5

0.06 0.10 0.08 0.08 0.05 0.06 0.04 0.04 0.06 0.08 0.06 0.08 0.07 0.06 0.07 0.04

AEd4

1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 1

PFM3

AEFM6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

48.09 61.17 52.21 64.15 49.68 70.05 44.74 43.86 62.81 58.61 39.27 64.15 53.83 52.41 42.78 45.88

Pd1

1 1 0 1 0 1 1 1 1 0 1 1 0 0 1 1

PD1

0 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0

AED1

1 1 0 1 1 0 1 1 1 0 0 1 1 0 0 1

AED2

Drilling metric state

7.66 8.08 7.87 8.26 7.13 7.91 7.12 7.24 8.53 8.04 7.37 8.26 7.57 7.14 7.16 6.97

Pd3

Spindle power metrics

0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1

PD2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

AED3

0 1 1 0 1 0 1 1 0 1 0 0 0 0 0 1

AED4

25.63 29.05 29.20 32.45 25.39 26.19 22.72 21.08 32.23 38.77 28.99 32.45 23.33 21.49 22.63 25.07

Pd4

AED5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 1 0 1 1 0 1 1 1 1 1 1 0 1 1 1

PD3

77.73 74.97 81.40 78.05 73.39 72.43 78.73 81.19 73.84 72.60 80.50 78.05 73.96 73.87 82.04 79.28

Pd2

PD4 1 0 0 0 1 1 1 1 0 1 0 0 1 1 1 1

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