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An integrated knowledge representation scheme and query processing mechanism for fault diagnosis in heterogeneous manufacturing environments
Robotics and Computer Integrated Manufacturing 16 (2000) 133}141
An integrated knowledge representation scheme and query processing mechanism for fault diagnosis in heterogeneous manufacturing environments Jeong P. Son, Jong H. Park*, You Z. Cho Department of Electronic Engineering, Kyungpook National University, Taegu, 702-701 South Korea Received 1 September 1998; received in revised form 20 July 1999; accepted 28 July 1999
Abstract An integrated fault diagnosis system was developed for manufacturing complex products, e.g., automobile, that undergo a variety of production processes. The proposed global knowledge representation scheme is designed to consistently express such heterogeneous diagnostic knowledge in terms of the causal relations among relevant attributes across di!erent processes. This scheme is unifying in that it re#ects the fact that an e!ect for one causal relation may in turn be a cause for another along a chain of subsequent processes. In order to unify independent pieces of causal relation, speci"c attributes of machine-processed products in addition to the usual diagnostic signals such as noise and voltage of the machines. The accompanying probabilistic diagnosis mechanism can then produce an enhanced diagnostic accuracy over other existing schemes as it can simultaneously consider multiple causes of a fault along with their interdependency. The proposed diagnosis mechanism was implemented in a distributed computer environment to identify the functions and procedures required for its actual application. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Fault diagnosis; Causal relation; Integrated diagnosis; Heterogeneous manufacturing
1. Introduction Manufacturing of complex products such as automobile undergoes a number of production processes forming a chain. A fault in some subassembly is often an e!ect propagated from a fault in a preceding process, and sometimes is even caused by a complication of multiple faults. Accordingly, in such heterogeneous environments, the diagnosis of an individual machine cannot correctly locate the cause(s) of faults. However, in contrast to the development of individual machine diagnosis [1,2] diagnostic methods for entire production systems have drawn scant attention [3]. The failure propagation tree (FPT) is a knowledge structure that can represent propagated faults with respect to an entire production system [4]. This structure expresses the functional aspects of processes in terms of the hierarchical process plan. This hierarchical plan
* Corresponding author. Tel.: #82-53-950-6554; fax: #82-53-9505505. E-mail address: [email protected] (J.H. Park)
models heterogeneous processes globally via the input and output of each process by ignoring its functional details. However, an FPT can only indicate those processes that are potentially faulty, as it does not include the analysis capabilities for detailed diagnosis, such as probabilistic inferencing functions. The in#uence diagram [5] models graph-theoretically all diagnosis-related variables and their causal relations. This model is similar to a Bayesian network, belief network and causal network except when considering Decision nodes [6,7]. Once a diagnosis problem is modelled by an in#uence diagram, the probabilistic relationships can be evaluated in terms of a graph. Based on a topological transformation and a greedy algorithm, sophisticated inferencing can then be performed. However, if the number of variables becomes large, a problem can become too complex to use topological transformation as the computational complexity reduces its practicality. Most causal models [1,8] separate the set of causes from the set of symptoms, resulting in inferencing without considering their interdependency. One diagnostic inferencing mechanism [1] does consider such dependency, yet it is con"ned to only two causes. Though the proposed diagnostic scheme also
0736-5845/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 6 - 5 8 4 5 ( 9 9 ) 0 0 0 4 5 - 9
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is based on causal relations, its diagnostic knowledge model utilizes the attributes of the diagnostic objects, e.g., temperature of engine, rather than their states, e.g., increased temperature, as variables associated with causal relations, unlike the existing cause-and-e!ect models [6]. This choice of attributes instead of states paves the way for a close integration of interrelated causal relations. Consequently, more causal relations, which would otherwise remain independent [6}8], can be connected into a causal net via this utilization of their common attributes. As a result, a cause in a causal relation can at the same time be an e!ect in another relation. Modelling a multi-component heterogeneous system via its inputs and outputs [8] requires a function description at the component system level, just short of integrated modelling at a "ner level [9]. To overcome the shortcomings stated above, an integrated fault diagnosis system was developed with the following characteristics: (1) The diagnostic knowledge is represented via a graph based on the causal relations among the diagnostic variables. (2) In consideration of cascading processes, the knowledge representation model re#ects that the sets of causes and e!ects are not static, i.e., the same node can be either a cause or an e!ect depending on the diagnostic query. (3) The probabilistic inferencing mechanism also takes account of the interdependency between the causes. (4) The diagnosis is performed in terms of causalrelation variables rather than process inputs and outputs. (5) The computational complexity of the diagnostic mechanism is kept reasonably low even for a large number of variables. An integrated diagnosis requires a global diagnostic knowledge representation scheme and a global diagnostic mechanism. Various diagnostic signals detected via sensors are primary means for the diagnosis of individual machines [10,11]. However, specifying the myriad of causal relations among di!erent machines in terms only of these signals requires too much expertise to be practical. Therefore, in addition to these signals, the proposed scheme also uses the characteristic values of a product that accumulatively re#ect the state of all the machines along the process chain. These diagnostic attributes are connected according to their causal relations across different processes. These individual causal relations are then meshed via their common attributes into a global diagnostic knowledge structure. Accordingly, this scheme can recognize the integrated reality where the e!ect for one causal relation may also be a cause for another. The utilization of these unifying attributes, therefore, can produce a closely integrated and comprehensive diagnosis [9] across di!erent processes from a factory-wide perspective. In addition, these causal relations can be easily modelled into a (simple) graph, resulting in a low computational complexity.
Based on this knowledge representation scheme, a global fault diagnosis mechanism was then developed. This mechanism quanti"es the causal relations among product attributes based on probabilistic inferencing. This probabilistic diagnosis mechanism can produce an enhanced diagnostic accuracy over existing mechanisms as it can simultaneously consider multiple causes of a fault plus their inter-dependency. In addition, the sets of causes and e!ects are recognized as variable with respect to given symptoms. The proposed diagnosis mechanism was implemented in a distributed computer environment. The main structural feature of the implemented diagnosis system is its ability to accommodate frequent process changes [12].
2. Knowledge representation scheme for diagnostic knowledge As stated above the diagnostic knowledge for the proposed diagnosis is essentially modelled in terms of the causal relations between the diagnostic attributes associated with the manufacturing processes. Each of these attributes corresponds to a property of the assembled product or processed product during one of the processes. If considering a factory producing engines, an attribute associated with a single component could include, e.g., length of valve, whereas another attribute can be associated with the relation between various components, e.g., gap between cylinder head and valve bottom. Plus a general property of a component can equally be treated as di!erent properties if the property value is changed by processing in more than one process. This temporal (besides spatial) distinction of attributes further enhances the diagnostic locality. Conversely, a property associated with a process is only of interest if it is either a newly created property or a variable in the process. These properties are then identi"ed as attributes with respect to the proposed global diagnostic knowledge, and each attribute is assigned a fault probability as estimated by relevant expertise. A property is usually a continuous variable, therefore, such a property is designated a threshold value beyond which the associated attribute is judged faulty. The causal relation is represented in terms of an attribute where a fault, i.e., a cause, can occur and an attribute where a symptom, i.e., a propagated fault, can be observed as an e!ect of a fault. Generally, a fault will involve multiple faulty attributes, and the symptoms of a fault can be observed on multiple attributes. However, the proposed diagnostic knowledge scheme only captures diagnostic knowledge in terms of the causal relations between two attributes [9], which facilitates quantifying the probability associated with each causal relation. Accordingly, using these binary causal relations the proposed diagnosis mechanism can also diagnose faults
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with multiple causes (i.e., complications.) These binary causal relations are meshed via their common attributes into a network. A cascade of (direct) causal relations will then re#ect the transitive (or indirect) causal relations thereby indicating the possibilities where a fault in a particular attribute in the cascade could propagate via various direct causal relations to other attributes further down the cascade. For example, a fault at the valve gap can be caused by anomalies in either the height of the cylinder head or the length of the valve, plus a fault at the valve gap in turn could cause a propagated fault in the power of the engine. As previously mentioned this connection among independent causal relations is possible due to the use of attributes as data items in causal relations. Note that the resulting uniform integration achieves a higher degree of integration of heterogeneous causal relations than the existing causal models [6,8]. This is because attributes themselves are more accommodating than their states, which implies higher probabilities of commonalities among their associated causal relations. Consequently, an attribute corresponding to the e!ect in one causal relation can equally correspond to the cause in another causal relation in a uniform manner. This feature allows a wide range of causal relations to be grafted into an extensively integrated net, which is in contrast to the existing cause-and-e!ect models (i.e., "shbone chart.) Whereas the existing cause-and-e!ect diagram only depicts the causes of each individual symptom, the proposed In#uence Graph (IG) is also concerned with the relations between symptoms and with the relations between causes and other symptoms caused by those same causes. In this sense, the IG can be deemed to be more global than other existing causal graphs. To extract as many pieces of diagnostic knowledge mainly causal relations, an failure mode and e!ect analysis (FMEA) needs to be performed. In the FMEA procedure [15], the steps of the failure modes and causes, and the e!ects are all relevant to the proposed creation of causal knowledge. The probability of a fault causing a propagated fault in each causal relation is determined by experimentation or previous expertise. Note that this probability will vary inversely to the threshold value of the attribute associated with the propagated fault. The task of deciding the probabilities of a fault and its propagation is not as overly unwieldy as it might seem, since the proposed mechanism only requires relative values for these probabilities rather than absolute values. De5nition of in6uence graph. The in#uence graph (IG) is a directed graph. IG"(N, E) where the node set N"+n "n "(d , p ), d is the node id and p , the probG G G G G G ability of fault at n , and the edge set E"+e "e " G GH GH (n , n , c ), where n is the tail node and n the head node, G H GH G H c , the conditional causal probability,. An attribute is GH captured as a node in the IG only if it is a created or changed property in the manufacturing process. An edge
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Fig. 1. Part of typical in#uence graph.
corresponds to the diagnostic causal relation between a pair of attributes, n and n . with a causal probability of G H c . Its tail node n represents an attribute where a fault GH G can occur, whereas its head node n represents an atH tribute where an associated symptom can be observed. For example, if a factory with a cutting, grinding, and drilling process is considered, the IG would have nodes corresponding to the attributes of these processes plus the attributes of their products, as depicted in Fig. 1. The causal relation contained in a piece of diagnostic knowledge, `the length of valve is a function of the height of the cylinder head and the ratio of the rockerarma for instance, would be modelled in the IG by two edges (dotted arrows in Fig. 1): one with a tail node for the height of the cylinder head and a head node for the length of the valve, and the other with similar nodes for the rockerarm ratio and length of the valve. The edges are coupled via their common attributes into a graph, which collectively represents the global diagnostic knowledge in terms of a chain of causal relations. This integrated representation facilitates the consistent modelling of diagnostic knowledge across heterogeneous machines (or processes) using a directed graph. The IG is similar to a FPT [4] in that the processes are modelled in terms of blackboxes, yet is di!erent in the semantics attached to the node, i.e., the diagnostic attributes associated with the processes instead of the pre- and post-conditions of the processes.
3. Global fault diagnostic mechanism The essence of the proposed global diagnostic mechanism is to apply a probabilistic inferencing method to the IG, which represents the causal relations among the attributes associated with the diagnosis. The probabilistic inferencing method proposed is distinctive from other existing methods in several aspects: (1) Unlike other related researches [1,8] where the sets of causes and symptoms are "xed, the proposed inferencing method decides the set of faults dynamically
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according to the given set of symptoms. Thus a variable set of causes is recomputed based on the IG for each diagnosis. (2) Unlike other approaches [1,5,8] which assume no dependency or correlations among causes, the proposed approach considers the correlations among causes as captured in the IG, resulting in a more accurate diagnosis. (3) The proposed diagnostic method can also handle cases where multiple causes are involved. 3.1. The dexnitions and basic concepts for probabilistic inferencing Our global fault diagnostic mechanism requires "ve elements of information, i.e., D, S, C, D and S\. ' D is the set of all disorders (or faults) that can cause the set of symptoms given in the query"+d , d , 2 , d ,, S, L the set of all symptoms that can be caused by D"+s , s ,2, s ,, C-D;S the causal relations exist I ing between two sets, D and S, D -D the set of disorders ' that can account for all the observed symptoms with respect to an individual query, S\-S the set of symptoms among S that actually occurred, that is, the set of symptoms as given in the query. D is obtained by backtracking the IG with respect to S\. D is the subset of disorders in D that can account for ' all the symptoms in S\. S includes, besides the node in S\, all the nodes that can be a!ected by the nodes in D yet which are not elements of D. This inclusion allows our inferencing mechanism to take into account the nonexistence of potential symptoms in addition to the list of symptoms that actually occurred. All the symptoms that actually occurred may not be speci"ed in the query, however, unspeci"ed symptoms can be identi"ed by checking the elements of S associated with D. If only disorder d can cause symptom s , 1d , s 23C. G G G G For each symptom s 3S, the set of its potential causes H can be obtained in a successive manner, CAUSES(s )" H +d "1d , s 23C,6+d "1d , d 23C,6+d "1d , d 23C,6 G G H I I G D D I 22. The set of symptoms that can be caused by each disorder d 3D is EFFECTS(d )"+s "1d , s 2 G G H G H 3C,. EFFECTS(D ) " G ' EFFECTS(d ). Likewise, ' B Z" G CAUSES(S )" H ( CAUSES(s ). Fig. 2 illustrates ( Q Z1 G how D and S can be obtained for an example query that is given with respect to the IG. Example Query: What is the disorder(s) with respect to the symptoms at Nodes 1 and 3? D"+Node2(d ), Node4(d ), Node5(d ), Node6(d ), Node8(d ), S"+Node1(s ), Node3(s ), Node7(s ), Node9(s ), De5nition 1. Causal event: For some d 3D and some ' s 3S, s : d represents an event in which d caused s . H H G G H
Fig. 2. A simple IG for example query.
That is, s : d is true if both d and s occurred, and s was H G G H H caused by d . G Since causality is recognized according to its impacts among attributes, a causal event s : d means a rei"caH G tion of the concept of the edge in the IG. If both d and G s have occurred, but s has not been caused by d , s : d H H G H G is not true. In general, (s d ).(s : d ) and a causal H G H G event is evidently a basic event [13]. De5nition 2. P(s : d "d ) is de"ned the conditional causal H G G probability of s when d is given. H G Since P(s : d "d )"P((s : d )d )/P(d )"P(s : d )/P(d ), H G G H G G G H G G the conditional causal probability represents the average probability that a given d causes s . This probability G H re#ects the causal relation between d and s more accuG H rately than P(s "d ) used in the Bayesian method. Assume H G that the case d cannot cause s but d can cause s . Obviously the causal relation from d to s is 0. If d and d occurred at the same time, however, P(s "d ) would have a value greater than 0. Meanwhile, P(s : d "d )"0 since s : d is always false. In general, P(s : d "d ) H G G 4P(s "d ). H G 3.2. Probabilistic inferencing mechanism The objective of the proposed probabilistic inferencing is to inference the set of disorders D with the highest ' probability by computing the relative likelihood function, ¸(S\, D ), for D with respect to S\ given in the ' ' query. It is assumed that all the basic probability information required for this inferencing is given, i.e., the probability 0(P(d )(1 for all the causal events d 3D G G and the probability 04P(s : d "d )41 for causal events H G G s : d . These probabilities can be obtained from experiH G mental, empirical data, and human experts. Short-hand notations will be used, i.e., p for P(d ) and c for G G G H P(s : d "d ). The probability of D with respect to S\ is H G G '
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obtained by P(S\"D ) * P(D ) ' ' . P(D "S\)" ' P(S\) Assuming, now, that all the disorders occur independently of each other, P(D ) is given by ' P(D )" (p ) G ' BG Z"' P(S\D ) is computed in terms of c . P(s : (d 2 ' G H H d d )) is conveniently expressed by P(s : d 2 P P> H d d ). P P> Theorem 1. Let D "+d , 2 , d ,-D and s 3M, ' P H P(s "D )"P(s : d 2d "d 2d ) H ' H P P "1! (1!c ). G H BG Z"' Proof. (i) When r"1, P(s : d "d )"c "1!(1!c ) H H H (ii) It is assumed that the equation holds for r*1. (iii) It will be shown that the equation holds for r#1 likewise. P(s : d d )"P((s : d ) d )#P((s : d ) d ) H H H !P((s : d ) (s : d )) H H P(s : d 2 d d ) H P P> "P((s : d 2 d ) d ) H P P> #P((s : d ) d 2 d ) H P> P !P((s : d 2 d ) (s : d )) H P H P> Assuming all the disorders occur independently of each other, "P(s : d 2 d "d 2 d ) H G P G P * P(dP> dG 2 dP )#P(sH : dP> "dP> ) * P(d d 2 dP dP> ) !P(s : d 2 d "d 2 d ) H P P * P(d d 2 dP dP> ) * P(sH : dP> "dP> ) Dividing both sides P(d d 2 d d ), P P> P(s : d 2 d d "d 2 d d ) H G P P> G P P> "P(s : d 2 d "d 2 d ) H G P G P #P(s : d "d ) H P> P> !P(s : d 2 d "d 2 d ) H G P G P * P(sH : dP> "dP> )
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P " 1! (1!c ) #c G H P> H G P ! 1! (1!c ) * c G H P> H G P> "1! (1!c ) 䊐 G H G For a query with all s 3S\ given and s 3S!S\ H ' known, one can derive
P(S\"D )" P(s "D ) * P(s "D ) ' H ' G ' QH Z1\ QG Z1\1\
" 1! (1!c ) G H QG Z1\ BG Z"'
(1!c ). G H QG Z1\1\ BG Z"G Since c "0 for 1d s 2 , C, G H G H *
P (S\"D )"! ' QG Z1\
1! (1!c ) G H B' Z"G
(1!c ) * G H BG Z"' QG ZCDDCAR QB' \1\ ¸(S\, D ) is de"ned in terms of P(S\"D ) and P(D ) as ' ' ' derived above. De5nition 3. The relative likelihood function ¸(S\, D ) is ' de"ned by the product of P(S\"D ) and P(D ). ' ' ¸(S\, D )"P(S\"D * P(D ) ' ' '
" 1! (1!c ) G H QH Z1\ BG Z"'
(1!c )* p * G H G BG Z"' QG ZCDDCARQBG \1\ BG Z"' "¸1 * ¸2 * ¸3. It should be noted that the three terms of ¸(S\, D ) ' each have their respective semantics. ¸ quanti"es the probability with which each D causes the symptoms in ' S\. ¸ factors in that in fact some expected symptoms associated with D did not occur. ¸ accounts for the ' probability of D . The resulting relative likelihood func' tion of D allows the ranking of D with respect to a given ' ' set of symptoms. 3.3. Probabilistic inferencing considering correlations among disorders The preceding discussion of the proposed inferencing method assumed that the disorders occurred independently of each other. In general, however, the correlations
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#p(d ) * p(s : d "d ) * p(d ) * (1!p(s : d "d )) #p(d ) * p(d : d "d ) * p(d ) * p(s : d "d ) * (1!p(s :d "d )) !p(d ) * p(s : d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d )) Fig. 3. Considering in#uences among causes of fault.
among the disorders must also be considered for a more accurate diagnosis. The following demonstrates the case where correlations exist between two or more disorders (compare to the existing mechanisms [3].) The ¸(S\, D ) ' can be obtained when S\"+s , and D "+d , d , ' are given, see Figs. 3(a) and (b). Consider "rst a simple case where Node 1 a!ects Node 2 and Node 2 a!ects Node 3 and Node 4, see Fig. 3(a). In order to account for S\"+s , with D "+d , d ,, ' three cases need to be considered: Case 1, a disorder occurs at Node 1 and a disorder occurring at Node 2 without impact from Node 1 (i.e., originated at Node 2), a!ects Node 3. Case 2, a disorder occurs at Node 1, and this disorder a!ects Node 2, and the impact at Node 2 in turn a!ects Node 3. Case 3, the same event as in Case 2 yet with a simultaneous original disorder at Node 2 also a!ecting Node 3. Accordingly, ¸ (S\, D ) * ¸ (S\, D ) * ¸ (S\, D ) ' ' ' "p(d ) * p(d ) * p(sv : d "d ) * (1!p(s : d "d )) #p(d ) * p(d : d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d )) !p(d ) * p(d : d "d ) * p(s : d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d )) "p(d ) * p(d ) * c * (1!c ) #p(d ) * c * p(d ) * c * (1!c ) !p(d ) * c * c * p(d ) * c * (1!c ) "p(d ) * p(d ) * (1!(1!c ) (1!c * c )) * (1!c ) The case illustrated in Fig. 3(b) results from the addition to (a) of a causal relation from Node 1 to Node 3. The ¸(S\, D ) was computed for Fig. 3(b) considering all ' the cases in the same manner as for (a). ¸(S\, D )"p(d ) * p(d ) * p(s : d "d ) ' * (1!p(s : d "d ))
!p(d ) * p(d : d "d ) * p(s : d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d )) !p(d ) * p(s : d "d ) !* p(d : d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d )) #p(d ) * p(s : d "d ) * p(d : d "d )p(s :d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d ))¸ (S\, D' ) * ¸ (S\, D' ) * ¸ (S\, D' ) "p(d ) * p(d ) * (1!(1!c )(1!c ) ;(1!c
c )) (1!c ). * *
De5nition 4. If there is no causal relation between Nodes A and B, yet there are causal relations between Nodes A and C, and between Nodes C and B, we say there is an indirect causal relation between Nodes A and B along the `patha of A!C!B. De5nition 5. The conditional indirect causal probability (CICP) is de"ned as the product of all the conditional causal probabilities for every pair of nodes along a path. Though there is no causal relation between Nodes 1 and 3 in Fig. 3 (a) in contrast to (b), an indirect causal relation can be inferred between Nodes 1 and 3 via Node 2 since Node 2 is in D as a potential disorder. Nodes 1 and ' 2 have an indirect causal relation along the path of 1}2}3. c *c is the conditional indirect causal probability for the path of 1}2}3, denoted by c which is \ distinguished from c i.e., a conditional causal prob ability for a case with a direct causal relation between Nodes 1 and 3 as for (b). As seen above, the previously computed ¸(S\, D ) can ' equally be utilized for cases with dependency between causes as for cases without dependency. When identifying all the paths between each pair of cause and symptom, every cause constituting such a path must be an element of D . The conditional indirect causal probability for ' these paths must be derived in order to consider the dependency between the causes.
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Hence, relative likelihood function for the general case, ¸(S\, D )"P(S\"D ) * P(D ) ' ' '
" 1! (1!c ) (1!c ) G H G HN?RF QH Z1\ BG Z"' N?RFBG Z"' RG Z1\
(1!c ) * p . * G R G \ BG Z"' QG ZCDDCARQBG \1 BG Z"' Note that path (d 3D , s 3S\) means inferencing all G ' H indirect causal relations among every pair of d and s in G H D )c indicates the indirect conditional causal probG G HN?RF ability for each path. The time complexity for computing ¸(S\, D ) is upper-bounded by O(mH2L) where m is the ' number of edges and n is the number of nodes in the IG. With the consideration of the e$ciency, the proposed approach may not be as suitable for diagnostic tasks involving a single cause when compared to other methods. However, the actual time complexity should be much lower than this because 2L corresponds to a completely connected graph. The accurate complexity varies greatly depending on the pattern of linkage among the nodes. Note also that the proposed method is designed for diagnosis in a complex environment where multiple causes can be involved and may be interrelated. 3.4. Application of our probabilistic inferencing mechanism to example query To evaluate the proposed probabilistic inferencing mechanism, it was applied to the previous example query based on the IG in Fig. 2. First all the disorder sets were identi"ed, i.e., the D 's, that can account for all the symp' toms given in the query. D "+d ,, +d , d ,, +d , d ,, ' +d , d ,, +d , d , d ,, +d , d , d ,, +d , d , d ,, +d , d , d ,, +d , d , d ,, +d , d , d ,, +d , d , d , d ,, +d , d , d , d ,, +d , d , d , d ,, +d , d , d , d ,, +d , d , d , d , or +d , d , d , d , d ,. Thereafter the relative likelihood function, the ¸(S\, D ), can be computed. If D " ' ' +d , d , d ,, for instance, ¸ "(1!(1!c ) (1!c ) (1!c )) * (1!(1!c ) (1!c ) (1!c )) "0.8372, ¸ "(1!c ) * (1!c ) * (1!c ) "0.1120, ¸ "p *p *p "0.2 * 0.1 * 0.2 "0.0040, /¸(+d , d , d ,,+s , s ,)"¸ *¸ *¸ "0.0003751.
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If D "+d , d , d , d ,, there are dependencies among ' the disorders. Speci"cally, a disorder at Node 5 can cause disorders at Nodes 2 and 6, while a disorder at Node 6 can cause disorders at Nodes 2 and 8. Hence, paths among the disorders and symptoms must be found in order to compute the conditional indirect causal probabilities by inferencing the indirect causal relations. If D "+d , d , d , d ,, there are four paths such as: ' 5}2}1, 6}2}1, 5}6}2}1, 5}6}8}3, 6}8}3. Hence, their respective conditional indirect causal probabilities can be computed, i.e., c "0.63, c "c *c }} }} "0.72, c c *c "0.648, c } } } } } } "c * "0.5 * 0.9"0.45. "0.9*0.5*0.9"0.405, c }} Also, ¸(S\, D ) can be computed as follows: ' ¸ (S\, D )"(1!(1!c ) (1!c ) ' \\ ;(1!c ) (1!c ) \\\ \\ ;(1!c )) * (1!(1!c ) ;(1!c ) \\\ ;(1!c ) (1!c ))"0.9639 \\ ¸ (S\, D )"(1!c ) *(1!c )"0.56 ' ¸ (S\, D )"0.2 * 0.2 * 0.1 * 0.2"0.0008, ' /¸(S\, D )"0.00043183. ' Table 1 summarizes the "nal result of probabilistic inferencing with respect to the example query. The most probable set of faulty attributes be decided by ranking the disorder sets in the order of the values of their relative likelihood function, ¸(S\, D ). The most likely disorders ' causing the symptom set +s , s ,, as given in the query, is judged to be +d , d , based on Table 1. 4. Implementation An integrated fault diagnosis system was implemented based on the developed diagnostic mechanism. Since the viability of the proposed diagnostic mechanism has already been analytically veri"ed, the main objective of this implementation lies in identifying the functions and corresponding system components that are required and the procedures that need to be followed in order to apply this diagnostic mechanism to a simulated distributed environment. The target manufacturing system was composed of numerous component systems, and in function, they collectively form an interconnected manufacturing line. The individual component systems were equipped with di!ering levels of diagnostic facilities, ranging from primitive sensors to intelligent diagnostic functions. The proposed integrated diagnostic system was simulated in a distributed environment with UNIX-based computers networked via the Internet. Each cell or module consists of a main process plus one or more child processes.
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Table 1 Results of example query by probabilistic inferencing D '
¸ (S\, D)
¸ (S\, D)
¸ (S\, D ) '
¸(S\, D ) '
Ranking
+d , +d , d , +d , d , +d , d , +d , d , d , +d , d , d , +d , d , d , +d , d , d , +d , d , d , +d , d , d , +d , d , d , d , +d , d , d , d , +d , d , d , d , +d , d , d , d , +d , d , d , d , +d , d , d , d , d ,
0.02 0.092 0.81 0.182 0.097 0.0978 0.8372 0.8667 0.8748 0.1901 0.0996 0.9033 0.9292 0.9639 0.1941 0.9671
0.2 0.14 0.56 0.16 0.007 0.035 0.112 0.056 0.56 0.032 0.07 0.0112 0.112 0.56 0.0096 0.112
0.1 0.02 0.04 0.02 0.004 0.002 0.004 0.008 0.004 0.002 0.0004 0.0008 0.0004 0.0008 0.0004 0.00008
0.004 0.0002576 0.018 0.0005824 0.000002716 0.000006846 0.000375 0.000382 0.0021 0.00001217 0.000000278 0.0000081 0.000041628 0.0004318 0.0000007453 0.0000086652
2 8 1 4 14 13 7 6 3 10 16 11 9 5 15 12
The overall architecture of the proposed integrated diagnosis system is shown in Fig. 4. The internal structure consists of local fault diagnosis cells (LFDCs), each of which corresponds to the diagnostic function of each component system, an integrated fault diagnosis cell (IFDC) that performs an integrated diagnosis, the scheduler cell (SCR) that supports an integrated diagnosis, and the communication service cell (CS). These cells function independently of each other and communicate via messages. Meanwhile, the diagnostic function at a manufacturing machine or process was simulated by an independent software module (called a module in distinction from the various cells above), which could sample the required diagnostic signals, and communicate and interpret the various messages. A cell or module was implemented in terms of one or more processes. Representing each module as a cell in the IFDC alleviates the di$culties in modifying the integrated diagnosis system in the case of changes in the manufacturing processes [12]. With this architecture, the e!orts for such a modi"cation can be con"ned to the cell at issue, or the IFDC and SCR [14]. The following is a brief explanation of how each cell functions in the process of acquiring the data required for the proposed integrated diagnosis. The "rst step is to acquire S\. The nodes in S\ may be provided in two ways: by the user or by checking each node in order without user intervention. Once S\ is speci"ed, D for S\ and, in turn, S associated with D are identi"ed from the IG. The next step is for the IFDC to send a message to each LFDC associated with a node in S in order to check if a disorder has actually occurred at the node. With this message, the LFDC requests its corresponding module via the CS to perform an actual checking and report the results back. This procedure iterates as many times as there are elements in S. The probabilistic
Fig. 4. Overall architecture of our integrated diagnosis system.
inferencing diagnosis is then performed with the acquired data as described previously. The presented graph is mainly for visualization purpose. This diagnostic procedure along with the data formats and communication protocols used are described in greater details elsewhere [14].
5. Conclusion An integrated fault diagnosis mechanism has been developed based on the quality characteristic values of products besides multiple diagnostic signals on heterogeneous machines. This approach allows individual monitoring systems and diagnostic functions to be integrated into a global diagnosis system for an entire factory. Initially a knowledge representation model called the in#uence graph was developed that can capture diverse diagnostic expertise, either experimental or empirical, in a consistent manner. This expertise is modelled in terms
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of the causal relations across the heterogeneous machines or processes. Based on this knowledge model a diagnostic mechanism was developed based on probabilistic inferencing. This mechanism can identify the probable sets of disorders, and can compute the values of their relative likelihood function, ¸(D , S\), using the conditional ' causal and conditional indirect causal probabilities of the causal relations among the diagnostic variables. The major features of our approach include an e!ective integration of the causal relations among heterogeneous processes and a full exploitation of this knowledge in fault diagnosis. This diagnostic approach also considers the dependencies and in#uences between disorders in order to develop a more accurate and general diagnostic inferencing method. The proposed diagnostic mechanism was implemented in a UNIX-based distributed environment.
References [1] Tou J. Design of expert systems for integrated production automation. J Manufacturing Systems 1985;4(2):147}56. [2] Rouse W. Model of human problem solving: detection, diagnosis, and compensation for system failures. Automatica 1983;19(6): 613}25. [3] Chung K, Tou J. Knowledge-based approach to fault diagnosis and control in distributed process environments. Appl Artif Intell IX, SPIE 1991;1468:323}32.
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[4] Chang S. Failure propagation trees for diagnosis in manufacturing systems. IEEE Trans Systems Man Cybernet 1991;21(24). [5] Ashutosh R, Alice M. Topological framework for representing and solving probabilistic inference problems in expert systems. IEEE Trans Systems Man Cybernet 1988;18(3):402}14. [6] Luca C, Pietro T. An approach to the compilation of operational knowledge from causal models. IEEE Trans Systems Man Cybernet 1992;22(4):772}89. [7] Kim J, Pearl J. A computational model for causal and diagnostic in inference systems. Appl Artif Intell 1983:190}3. [8] Peng Y, Reggia J. A probabilistic causal model for diagnostic problem solving * Part I and II: integrating symolic causal inference with numerical probabilistic inference. IEEE Trans Systems Man Cybernet 1987;SMC-17(2):146}62. [9] Park J, Su S. An integrated approach to knowledge sharing among heterogeneous rule-based systems. J Appl Intell 1991: 583}92. [10] Yoon J, et al. Detection of faults in tools using double signals. J Korea Soc of Mech Engng 1992;16(4):707}22. [11] Kim H, Ahn J. Development of a multiple monitoring system for intelligent machining centers. J Korea Inst Precision Engrs 1993; 10(4):143}51. [12] Park J, Cha S. An object-oriented database for integrated control of distributed FMS. J KICS 1994;19(10):1935}44. [13] An Y. Knowledge representation scheme and query processing mechanism for heterogeneous fault diagnostic system. Masters Thesis, Kyungpook Nat'l Univ., 1995. [14] Son J. Implementation of a heterogeneous distributed fault diagnosis system based on probabilistic inference and statistical characteristics. Masters Thesis, Kyungpook Nat'l Univ., 1996. [15] Modarres M. Reliability and risk analysis. New York: Marcel Dekker Inc. 1993, p. 127}190.
An integrated knowledge representation scheme and query processing mechanism for fault diagnosis in heterogeneous manufacturing environments Jeong P. Son, Jong H. Park*, You Z. Cho Department of Electronic Engineering, Kyungpook National University, Taegu, 702-701 South Korea Received 1 September 1998; received in revised form 20 July 1999; accepted 28 July 1999
Abstract An integrated fault diagnosis system was developed for manufacturing complex products, e.g., automobile, that undergo a variety of production processes. The proposed global knowledge representation scheme is designed to consistently express such heterogeneous diagnostic knowledge in terms of the causal relations among relevant attributes across di!erent processes. This scheme is unifying in that it re#ects the fact that an e!ect for one causal relation may in turn be a cause for another along a chain of subsequent processes. In order to unify independent pieces of causal relation, speci"c attributes of machine-processed products in addition to the usual diagnostic signals such as noise and voltage of the machines. The accompanying probabilistic diagnosis mechanism can then produce an enhanced diagnostic accuracy over other existing schemes as it can simultaneously consider multiple causes of a fault along with their interdependency. The proposed diagnosis mechanism was implemented in a distributed computer environment to identify the functions and procedures required for its actual application. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Fault diagnosis; Causal relation; Integrated diagnosis; Heterogeneous manufacturing
1. Introduction Manufacturing of complex products such as automobile undergoes a number of production processes forming a chain. A fault in some subassembly is often an e!ect propagated from a fault in a preceding process, and sometimes is even caused by a complication of multiple faults. Accordingly, in such heterogeneous environments, the diagnosis of an individual machine cannot correctly locate the cause(s) of faults. However, in contrast to the development of individual machine diagnosis [1,2] diagnostic methods for entire production systems have drawn scant attention [3]. The failure propagation tree (FPT) is a knowledge structure that can represent propagated faults with respect to an entire production system [4]. This structure expresses the functional aspects of processes in terms of the hierarchical process plan. This hierarchical plan
* Corresponding author. Tel.: #82-53-950-6554; fax: #82-53-9505505. E-mail address: [email protected] (J.H. Park)
models heterogeneous processes globally via the input and output of each process by ignoring its functional details. However, an FPT can only indicate those processes that are potentially faulty, as it does not include the analysis capabilities for detailed diagnosis, such as probabilistic inferencing functions. The in#uence diagram [5] models graph-theoretically all diagnosis-related variables and their causal relations. This model is similar to a Bayesian network, belief network and causal network except when considering Decision nodes [6,7]. Once a diagnosis problem is modelled by an in#uence diagram, the probabilistic relationships can be evaluated in terms of a graph. Based on a topological transformation and a greedy algorithm, sophisticated inferencing can then be performed. However, if the number of variables becomes large, a problem can become too complex to use topological transformation as the computational complexity reduces its practicality. Most causal models [1,8] separate the set of causes from the set of symptoms, resulting in inferencing without considering their interdependency. One diagnostic inferencing mechanism [1] does consider such dependency, yet it is con"ned to only two causes. Though the proposed diagnostic scheme also
0736-5845/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 6 - 5 8 4 5 ( 9 9 ) 0 0 0 4 5 - 9
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is based on causal relations, its diagnostic knowledge model utilizes the attributes of the diagnostic objects, e.g., temperature of engine, rather than their states, e.g., increased temperature, as variables associated with causal relations, unlike the existing cause-and-e!ect models [6]. This choice of attributes instead of states paves the way for a close integration of interrelated causal relations. Consequently, more causal relations, which would otherwise remain independent [6}8], can be connected into a causal net via this utilization of their common attributes. As a result, a cause in a causal relation can at the same time be an e!ect in another relation. Modelling a multi-component heterogeneous system via its inputs and outputs [8] requires a function description at the component system level, just short of integrated modelling at a "ner level [9]. To overcome the shortcomings stated above, an integrated fault diagnosis system was developed with the following characteristics: (1) The diagnostic knowledge is represented via a graph based on the causal relations among the diagnostic variables. (2) In consideration of cascading processes, the knowledge representation model re#ects that the sets of causes and e!ects are not static, i.e., the same node can be either a cause or an e!ect depending on the diagnostic query. (3) The probabilistic inferencing mechanism also takes account of the interdependency between the causes. (4) The diagnosis is performed in terms of causalrelation variables rather than process inputs and outputs. (5) The computational complexity of the diagnostic mechanism is kept reasonably low even for a large number of variables. An integrated diagnosis requires a global diagnostic knowledge representation scheme and a global diagnostic mechanism. Various diagnostic signals detected via sensors are primary means for the diagnosis of individual machines [10,11]. However, specifying the myriad of causal relations among di!erent machines in terms only of these signals requires too much expertise to be practical. Therefore, in addition to these signals, the proposed scheme also uses the characteristic values of a product that accumulatively re#ect the state of all the machines along the process chain. These diagnostic attributes are connected according to their causal relations across different processes. These individual causal relations are then meshed via their common attributes into a global diagnostic knowledge structure. Accordingly, this scheme can recognize the integrated reality where the e!ect for one causal relation may also be a cause for another. The utilization of these unifying attributes, therefore, can produce a closely integrated and comprehensive diagnosis [9] across di!erent processes from a factory-wide perspective. In addition, these causal relations can be easily modelled into a (simple) graph, resulting in a low computational complexity.
Based on this knowledge representation scheme, a global fault diagnosis mechanism was then developed. This mechanism quanti"es the causal relations among product attributes based on probabilistic inferencing. This probabilistic diagnosis mechanism can produce an enhanced diagnostic accuracy over existing mechanisms as it can simultaneously consider multiple causes of a fault plus their inter-dependency. In addition, the sets of causes and e!ects are recognized as variable with respect to given symptoms. The proposed diagnosis mechanism was implemented in a distributed computer environment. The main structural feature of the implemented diagnosis system is its ability to accommodate frequent process changes [12].
2. Knowledge representation scheme for diagnostic knowledge As stated above the diagnostic knowledge for the proposed diagnosis is essentially modelled in terms of the causal relations between the diagnostic attributes associated with the manufacturing processes. Each of these attributes corresponds to a property of the assembled product or processed product during one of the processes. If considering a factory producing engines, an attribute associated with a single component could include, e.g., length of valve, whereas another attribute can be associated with the relation between various components, e.g., gap between cylinder head and valve bottom. Plus a general property of a component can equally be treated as di!erent properties if the property value is changed by processing in more than one process. This temporal (besides spatial) distinction of attributes further enhances the diagnostic locality. Conversely, a property associated with a process is only of interest if it is either a newly created property or a variable in the process. These properties are then identi"ed as attributes with respect to the proposed global diagnostic knowledge, and each attribute is assigned a fault probability as estimated by relevant expertise. A property is usually a continuous variable, therefore, such a property is designated a threshold value beyond which the associated attribute is judged faulty. The causal relation is represented in terms of an attribute where a fault, i.e., a cause, can occur and an attribute where a symptom, i.e., a propagated fault, can be observed as an e!ect of a fault. Generally, a fault will involve multiple faulty attributes, and the symptoms of a fault can be observed on multiple attributes. However, the proposed diagnostic knowledge scheme only captures diagnostic knowledge in terms of the causal relations between two attributes [9], which facilitates quantifying the probability associated with each causal relation. Accordingly, using these binary causal relations the proposed diagnosis mechanism can also diagnose faults
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with multiple causes (i.e., complications.) These binary causal relations are meshed via their common attributes into a network. A cascade of (direct) causal relations will then re#ect the transitive (or indirect) causal relations thereby indicating the possibilities where a fault in a particular attribute in the cascade could propagate via various direct causal relations to other attributes further down the cascade. For example, a fault at the valve gap can be caused by anomalies in either the height of the cylinder head or the length of the valve, plus a fault at the valve gap in turn could cause a propagated fault in the power of the engine. As previously mentioned this connection among independent causal relations is possible due to the use of attributes as data items in causal relations. Note that the resulting uniform integration achieves a higher degree of integration of heterogeneous causal relations than the existing causal models [6,8]. This is because attributes themselves are more accommodating than their states, which implies higher probabilities of commonalities among their associated causal relations. Consequently, an attribute corresponding to the e!ect in one causal relation can equally correspond to the cause in another causal relation in a uniform manner. This feature allows a wide range of causal relations to be grafted into an extensively integrated net, which is in contrast to the existing cause-and-e!ect models (i.e., "shbone chart.) Whereas the existing cause-and-e!ect diagram only depicts the causes of each individual symptom, the proposed In#uence Graph (IG) is also concerned with the relations between symptoms and with the relations between causes and other symptoms caused by those same causes. In this sense, the IG can be deemed to be more global than other existing causal graphs. To extract as many pieces of diagnostic knowledge mainly causal relations, an failure mode and e!ect analysis (FMEA) needs to be performed. In the FMEA procedure [15], the steps of the failure modes and causes, and the e!ects are all relevant to the proposed creation of causal knowledge. The probability of a fault causing a propagated fault in each causal relation is determined by experimentation or previous expertise. Note that this probability will vary inversely to the threshold value of the attribute associated with the propagated fault. The task of deciding the probabilities of a fault and its propagation is not as overly unwieldy as it might seem, since the proposed mechanism only requires relative values for these probabilities rather than absolute values. De5nition of in6uence graph. The in#uence graph (IG) is a directed graph. IG"(N, E) where the node set N"+n "n "(d , p ), d is the node id and p , the probG G G G G G ability of fault at n , and the edge set E"+e "e " G GH GH (n , n , c ), where n is the tail node and n the head node, G H GH G H c , the conditional causal probability,. An attribute is GH captured as a node in the IG only if it is a created or changed property in the manufacturing process. An edge
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Fig. 1. Part of typical in#uence graph.
corresponds to the diagnostic causal relation between a pair of attributes, n and n . with a causal probability of G H c . Its tail node n represents an attribute where a fault GH G can occur, whereas its head node n represents an atH tribute where an associated symptom can be observed. For example, if a factory with a cutting, grinding, and drilling process is considered, the IG would have nodes corresponding to the attributes of these processes plus the attributes of their products, as depicted in Fig. 1. The causal relation contained in a piece of diagnostic knowledge, `the length of valve is a function of the height of the cylinder head and the ratio of the rockerarma for instance, would be modelled in the IG by two edges (dotted arrows in Fig. 1): one with a tail node for the height of the cylinder head and a head node for the length of the valve, and the other with similar nodes for the rockerarm ratio and length of the valve. The edges are coupled via their common attributes into a graph, which collectively represents the global diagnostic knowledge in terms of a chain of causal relations. This integrated representation facilitates the consistent modelling of diagnostic knowledge across heterogeneous machines (or processes) using a directed graph. The IG is similar to a FPT [4] in that the processes are modelled in terms of blackboxes, yet is di!erent in the semantics attached to the node, i.e., the diagnostic attributes associated with the processes instead of the pre- and post-conditions of the processes.
3. Global fault diagnostic mechanism The essence of the proposed global diagnostic mechanism is to apply a probabilistic inferencing method to the IG, which represents the causal relations among the attributes associated with the diagnosis. The probabilistic inferencing method proposed is distinctive from other existing methods in several aspects: (1) Unlike other related researches [1,8] where the sets of causes and symptoms are "xed, the proposed inferencing method decides the set of faults dynamically
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according to the given set of symptoms. Thus a variable set of causes is recomputed based on the IG for each diagnosis. (2) Unlike other approaches [1,5,8] which assume no dependency or correlations among causes, the proposed approach considers the correlations among causes as captured in the IG, resulting in a more accurate diagnosis. (3) The proposed diagnostic method can also handle cases where multiple causes are involved. 3.1. The dexnitions and basic concepts for probabilistic inferencing Our global fault diagnostic mechanism requires "ve elements of information, i.e., D, S, C, D and S\. ' D is the set of all disorders (or faults) that can cause the set of symptoms given in the query"+d , d , 2 , d ,, S, L the set of all symptoms that can be caused by D"+s , s ,2, s ,, C-D;S the causal relations exist I ing between two sets, D and S, D -D the set of disorders ' that can account for all the observed symptoms with respect to an individual query, S\-S the set of symptoms among S that actually occurred, that is, the set of symptoms as given in the query. D is obtained by backtracking the IG with respect to S\. D is the subset of disorders in D that can account for ' all the symptoms in S\. S includes, besides the node in S\, all the nodes that can be a!ected by the nodes in D yet which are not elements of D. This inclusion allows our inferencing mechanism to take into account the nonexistence of potential symptoms in addition to the list of symptoms that actually occurred. All the symptoms that actually occurred may not be speci"ed in the query, however, unspeci"ed symptoms can be identi"ed by checking the elements of S associated with D. If only disorder d can cause symptom s , 1d , s 23C. G G G G For each symptom s 3S, the set of its potential causes H can be obtained in a successive manner, CAUSES(s )" H +d "1d , s 23C,6+d "1d , d 23C,6+d "1d , d 23C,6 G G H I I G D D I 22. The set of symptoms that can be caused by each disorder d 3D is EFFECTS(d )"+s "1d , s 2 G G H G H 3C,. EFFECTS(D ) " G ' EFFECTS(d ). Likewise, ' B Z" G CAUSES(S )" H ( CAUSES(s ). Fig. 2 illustrates ( Q Z1 G how D and S can be obtained for an example query that is given with respect to the IG. Example Query: What is the disorder(s) with respect to the symptoms at Nodes 1 and 3? D"+Node2(d ), Node4(d ), Node5(d ), Node6(d ), Node8(d ), S"+Node1(s ), Node3(s ), Node7(s ), Node9(s ), De5nition 1. Causal event: For some d 3D and some ' s 3S, s : d represents an event in which d caused s . H H G G H
Fig. 2. A simple IG for example query.
That is, s : d is true if both d and s occurred, and s was H G G H H caused by d . G Since causality is recognized according to its impacts among attributes, a causal event s : d means a rei"caH G tion of the concept of the edge in the IG. If both d and G s have occurred, but s has not been caused by d , s : d H H G H G is not true. In general, (s d ).(s : d ) and a causal H G H G event is evidently a basic event [13]. De5nition 2. P(s : d "d ) is de"ned the conditional causal H G G probability of s when d is given. H G Since P(s : d "d )"P((s : d )d )/P(d )"P(s : d )/P(d ), H G G H G G G H G G the conditional causal probability represents the average probability that a given d causes s . This probability G H re#ects the causal relation between d and s more accuG H rately than P(s "d ) used in the Bayesian method. Assume H G that the case d cannot cause s but d can cause s . Obviously the causal relation from d to s is 0. If d and d occurred at the same time, however, P(s "d ) would have a value greater than 0. Meanwhile, P(s : d "d )"0 since s : d is always false. In general, P(s : d "d ) H G G 4P(s "d ). H G 3.2. Probabilistic inferencing mechanism The objective of the proposed probabilistic inferencing is to inference the set of disorders D with the highest ' probability by computing the relative likelihood function, ¸(S\, D ), for D with respect to S\ given in the ' ' query. It is assumed that all the basic probability information required for this inferencing is given, i.e., the probability 0(P(d )(1 for all the causal events d 3D G G and the probability 04P(s : d "d )41 for causal events H G G s : d . These probabilities can be obtained from experiH G mental, empirical data, and human experts. Short-hand notations will be used, i.e., p for P(d ) and c for G G G H P(s : d "d ). The probability of D with respect to S\ is H G G '
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obtained by P(S\"D ) * P(D ) ' ' . P(D "S\)" ' P(S\) Assuming, now, that all the disorders occur independently of each other, P(D ) is given by ' P(D )" (p ) G ' BG Z"' P(S\D ) is computed in terms of c . P(s : (d 2 ' G H H d d )) is conveniently expressed by P(s : d 2 P P> H d d ). P P> Theorem 1. Let D "+d , 2 , d ,-D and s 3M, ' P H P(s "D )"P(s : d 2d "d 2d ) H ' H P P "1! (1!c ). G H BG Z"' Proof. (i) When r"1, P(s : d "d )"c "1!(1!c ) H H H (ii) It is assumed that the equation holds for r*1. (iii) It will be shown that the equation holds for r#1 likewise. P(s : d d )"P((s : d ) d )#P((s : d ) d ) H H H !P((s : d ) (s : d )) H H P(s : d 2 d d ) H P P> "P((s : d 2 d ) d ) H P P> #P((s : d ) d 2 d ) H P> P !P((s : d 2 d ) (s : d )) H P H P> Assuming all the disorders occur independently of each other, "P(s : d 2 d "d 2 d ) H G P G P * P(dP> dG 2 dP )#P(sH : dP> "dP> ) * P(d d 2 dP dP> ) !P(s : d 2 d "d 2 d ) H P P * P(d d 2 dP dP> ) * P(sH : dP> "dP> ) Dividing both sides P(d d 2 d d ), P P> P(s : d 2 d d "d 2 d d ) H G P P> G P P> "P(s : d 2 d "d 2 d ) H G P G P #P(s : d "d ) H P> P> !P(s : d 2 d "d 2 d ) H G P G P * P(sH : dP> "dP> )
137
P " 1! (1!c ) #c G H P> H G P ! 1! (1!c ) * c G H P> H G P> "1! (1!c ) 䊐 G H G For a query with all s 3S\ given and s 3S!S\ H ' known, one can derive
P(S\"D )" P(s "D ) * P(s "D ) ' H ' G ' QH Z1\ QG Z1\1\
" 1! (1!c ) G H QG Z1\ BG Z"'
(1!c ). G H QG Z1\1\ BG Z"G Since c "0 for 1d s 2 , C, G H G H *
P (S\"D )"! ' QG Z1\
1! (1!c ) G H B' Z"G
(1!c ) * G H BG Z"' QG ZCDDCAR QB' \1\ ¸(S\, D ) is de"ned in terms of P(S\"D ) and P(D ) as ' ' ' derived above. De5nition 3. The relative likelihood function ¸(S\, D ) is ' de"ned by the product of P(S\"D ) and P(D ). ' ' ¸(S\, D )"P(S\"D * P(D ) ' ' '
" 1! (1!c ) G H QH Z1\ BG Z"'
(1!c )* p * G H G BG Z"' QG ZCDDCARQBG \1\ BG Z"' "¸1 * ¸2 * ¸3. It should be noted that the three terms of ¸(S\, D ) ' each have their respective semantics. ¸ quanti"es the probability with which each D causes the symptoms in ' S\. ¸ factors in that in fact some expected symptoms associated with D did not occur. ¸ accounts for the ' probability of D . The resulting relative likelihood func' tion of D allows the ranking of D with respect to a given ' ' set of symptoms. 3.3. Probabilistic inferencing considering correlations among disorders The preceding discussion of the proposed inferencing method assumed that the disorders occurred independently of each other. In general, however, the correlations
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#p(d ) * p(s : d "d ) * p(d ) * (1!p(s : d "d )) #p(d ) * p(d : d "d ) * p(d ) * p(s : d "d ) * (1!p(s :d "d )) !p(d ) * p(s : d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d )) Fig. 3. Considering in#uences among causes of fault.
among the disorders must also be considered for a more accurate diagnosis. The following demonstrates the case where correlations exist between two or more disorders (compare to the existing mechanisms [3].) The ¸(S\, D ) ' can be obtained when S\"+s , and D "+d , d , ' are given, see Figs. 3(a) and (b). Consider "rst a simple case where Node 1 a!ects Node 2 and Node 2 a!ects Node 3 and Node 4, see Fig. 3(a). In order to account for S\"+s , with D "+d , d ,, ' three cases need to be considered: Case 1, a disorder occurs at Node 1 and a disorder occurring at Node 2 without impact from Node 1 (i.e., originated at Node 2), a!ects Node 3. Case 2, a disorder occurs at Node 1, and this disorder a!ects Node 2, and the impact at Node 2 in turn a!ects Node 3. Case 3, the same event as in Case 2 yet with a simultaneous original disorder at Node 2 also a!ecting Node 3. Accordingly, ¸ (S\, D ) * ¸ (S\, D ) * ¸ (S\, D ) ' ' ' "p(d ) * p(d ) * p(sv : d "d ) * (1!p(s : d "d )) #p(d ) * p(d : d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d )) !p(d ) * p(d : d "d ) * p(s : d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d )) "p(d ) * p(d ) * c * (1!c ) #p(d ) * c * p(d ) * c * (1!c ) !p(d ) * c * c * p(d ) * c * (1!c ) "p(d ) * p(d ) * (1!(1!c ) (1!c * c )) * (1!c ) The case illustrated in Fig. 3(b) results from the addition to (a) of a causal relation from Node 1 to Node 3. The ¸(S\, D ) was computed for Fig. 3(b) considering all ' the cases in the same manner as for (a). ¸(S\, D )"p(d ) * p(d ) * p(s : d "d ) ' * (1!p(s : d "d ))
!p(d ) * p(d : d "d ) * p(s : d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d )) !p(d ) * p(s : d "d ) !* p(d : d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d )) #p(d ) * p(s : d "d ) * p(d : d "d )p(s :d "d ) * p(d ) * p(s : d "d ) * (1!p(s : d "d ))¸ (S\, D' ) * ¸ (S\, D' ) * ¸ (S\, D' ) "p(d ) * p(d ) * (1!(1!c )(1!c ) ;(1!c
c )) (1!c ). * *
De5nition 4. If there is no causal relation between Nodes A and B, yet there are causal relations between Nodes A and C, and between Nodes C and B, we say there is an indirect causal relation between Nodes A and B along the `patha of A!C!B. De5nition 5. The conditional indirect causal probability (CICP) is de"ned as the product of all the conditional causal probabilities for every pair of nodes along a path. Though there is no causal relation between Nodes 1 and 3 in Fig. 3 (a) in contrast to (b), an indirect causal relation can be inferred between Nodes 1 and 3 via Node 2 since Node 2 is in D as a potential disorder. Nodes 1 and ' 2 have an indirect causal relation along the path of 1}2}3. c *c is the conditional indirect causal probability for the path of 1}2}3, denoted by c which is \ distinguished from c i.e., a conditional causal prob ability for a case with a direct causal relation between Nodes 1 and 3 as for (b). As seen above, the previously computed ¸(S\, D ) can ' equally be utilized for cases with dependency between causes as for cases without dependency. When identifying all the paths between each pair of cause and symptom, every cause constituting such a path must be an element of D . The conditional indirect causal probability for ' these paths must be derived in order to consider the dependency between the causes.
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Hence, relative likelihood function for the general case, ¸(S\, D )"P(S\"D ) * P(D ) ' ' '
" 1! (1!c ) (1!c ) G H G HN?RF QH Z1\ BG Z"' N?RFBG Z"' RG Z1\
(1!c ) * p . * G R G \ BG Z"' QG ZCDDCARQBG \1 BG Z"' Note that path (d 3D , s 3S\) means inferencing all G ' H indirect causal relations among every pair of d and s in G H D )c indicates the indirect conditional causal probG G HN?RF ability for each path. The time complexity for computing ¸(S\, D ) is upper-bounded by O(mH2L) where m is the ' number of edges and n is the number of nodes in the IG. With the consideration of the e$ciency, the proposed approach may not be as suitable for diagnostic tasks involving a single cause when compared to other methods. However, the actual time complexity should be much lower than this because 2L corresponds to a completely connected graph. The accurate complexity varies greatly depending on the pattern of linkage among the nodes. Note also that the proposed method is designed for diagnosis in a complex environment where multiple causes can be involved and may be interrelated. 3.4. Application of our probabilistic inferencing mechanism to example query To evaluate the proposed probabilistic inferencing mechanism, it was applied to the previous example query based on the IG in Fig. 2. First all the disorder sets were identi"ed, i.e., the D 's, that can account for all the symp' toms given in the query. D "+d ,, +d , d ,, +d , d ,, ' +d , d ,, +d , d , d ,, +d , d , d ,, +d , d , d ,, +d , d , d ,, +d , d , d ,, +d , d , d ,, +d , d , d , d ,, +d , d , d , d ,, +d , d , d , d ,, +d , d , d , d ,, +d , d , d , d , or +d , d , d , d , d ,. Thereafter the relative likelihood function, the ¸(S\, D ), can be computed. If D " ' ' +d , d , d ,, for instance, ¸ "(1!(1!c ) (1!c ) (1!c )) * (1!(1!c ) (1!c ) (1!c )) "0.8372, ¸ "(1!c ) * (1!c ) * (1!c ) "0.1120, ¸ "p *p *p "0.2 * 0.1 * 0.2 "0.0040, /¸(+d , d , d ,,+s , s ,)"¸ *¸ *¸ "0.0003751.
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If D "+d , d , d , d ,, there are dependencies among ' the disorders. Speci"cally, a disorder at Node 5 can cause disorders at Nodes 2 and 6, while a disorder at Node 6 can cause disorders at Nodes 2 and 8. Hence, paths among the disorders and symptoms must be found in order to compute the conditional indirect causal probabilities by inferencing the indirect causal relations. If D "+d , d , d , d ,, there are four paths such as: ' 5}2}1, 6}2}1, 5}6}2}1, 5}6}8}3, 6}8}3. Hence, their respective conditional indirect causal probabilities can be computed, i.e., c "0.63, c "c *c }} }} "0.72, c c *c "0.648, c } } } } } } "c * "0.5 * 0.9"0.45. "0.9*0.5*0.9"0.405, c }} Also, ¸(S\, D ) can be computed as follows: ' ¸ (S\, D )"(1!(1!c ) (1!c ) ' \\ ;(1!c ) (1!c ) \\\ \\ ;(1!c )) * (1!(1!c ) ;(1!c ) \\\ ;(1!c ) (1!c ))"0.9639 \\ ¸ (S\, D )"(1!c ) *(1!c )"0.56 ' ¸ (S\, D )"0.2 * 0.2 * 0.1 * 0.2"0.0008, ' /¸(S\, D )"0.00043183. ' Table 1 summarizes the "nal result of probabilistic inferencing with respect to the example query. The most probable set of faulty attributes be decided by ranking the disorder sets in the order of the values of their relative likelihood function, ¸(S\, D ). The most likely disorders ' causing the symptom set +s , s ,, as given in the query, is judged to be +d , d , based on Table 1. 4. Implementation An integrated fault diagnosis system was implemented based on the developed diagnostic mechanism. Since the viability of the proposed diagnostic mechanism has already been analytically veri"ed, the main objective of this implementation lies in identifying the functions and corresponding system components that are required and the procedures that need to be followed in order to apply this diagnostic mechanism to a simulated distributed environment. The target manufacturing system was composed of numerous component systems, and in function, they collectively form an interconnected manufacturing line. The individual component systems were equipped with di!ering levels of diagnostic facilities, ranging from primitive sensors to intelligent diagnostic functions. The proposed integrated diagnostic system was simulated in a distributed environment with UNIX-based computers networked via the Internet. Each cell or module consists of a main process plus one or more child processes.
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Table 1 Results of example query by probabilistic inferencing D '
¸ (S\, D)
¸ (S\, D)
¸ (S\, D ) '
¸(S\, D ) '
Ranking
+d , +d , d , +d , d , +d , d , +d , d , d , +d , d , d , +d , d , d , +d , d , d , +d , d , d , +d , d , d , +d , d , d , d , +d , d , d , d , +d , d , d , d , +d , d , d , d , +d , d , d , d , +d , d , d , d , d ,
0.02 0.092 0.81 0.182 0.097 0.0978 0.8372 0.8667 0.8748 0.1901 0.0996 0.9033 0.9292 0.9639 0.1941 0.9671
0.2 0.14 0.56 0.16 0.007 0.035 0.112 0.056 0.56 0.032 0.07 0.0112 0.112 0.56 0.0096 0.112
0.1 0.02 0.04 0.02 0.004 0.002 0.004 0.008 0.004 0.002 0.0004 0.0008 0.0004 0.0008 0.0004 0.00008
0.004 0.0002576 0.018 0.0005824 0.000002716 0.000006846 0.000375 0.000382 0.0021 0.00001217 0.000000278 0.0000081 0.000041628 0.0004318 0.0000007453 0.0000086652
2 8 1 4 14 13 7 6 3 10 16 11 9 5 15 12
The overall architecture of the proposed integrated diagnosis system is shown in Fig. 4. The internal structure consists of local fault diagnosis cells (LFDCs), each of which corresponds to the diagnostic function of each component system, an integrated fault diagnosis cell (IFDC) that performs an integrated diagnosis, the scheduler cell (SCR) that supports an integrated diagnosis, and the communication service cell (CS). These cells function independently of each other and communicate via messages. Meanwhile, the diagnostic function at a manufacturing machine or process was simulated by an independent software module (called a module in distinction from the various cells above), which could sample the required diagnostic signals, and communicate and interpret the various messages. A cell or module was implemented in terms of one or more processes. Representing each module as a cell in the IFDC alleviates the di$culties in modifying the integrated diagnosis system in the case of changes in the manufacturing processes [12]. With this architecture, the e!orts for such a modi"cation can be con"ned to the cell at issue, or the IFDC and SCR [14]. The following is a brief explanation of how each cell functions in the process of acquiring the data required for the proposed integrated diagnosis. The "rst step is to acquire S\. The nodes in S\ may be provided in two ways: by the user or by checking each node in order without user intervention. Once S\ is speci"ed, D for S\ and, in turn, S associated with D are identi"ed from the IG. The next step is for the IFDC to send a message to each LFDC associated with a node in S in order to check if a disorder has actually occurred at the node. With this message, the LFDC requests its corresponding module via the CS to perform an actual checking and report the results back. This procedure iterates as many times as there are elements in S. The probabilistic
Fig. 4. Overall architecture of our integrated diagnosis system.
inferencing diagnosis is then performed with the acquired data as described previously. The presented graph is mainly for visualization purpose. This diagnostic procedure along with the data formats and communication protocols used are described in greater details elsewhere [14].
5. Conclusion An integrated fault diagnosis mechanism has been developed based on the quality characteristic values of products besides multiple diagnostic signals on heterogeneous machines. This approach allows individual monitoring systems and diagnostic functions to be integrated into a global diagnosis system for an entire factory. Initially a knowledge representation model called the in#uence graph was developed that can capture diverse diagnostic expertise, either experimental or empirical, in a consistent manner. This expertise is modelled in terms
J.P. Son et al. / Robotics and Computer Integrated Manufacturing 16 (2000) 133}141
of the causal relations across the heterogeneous machines or processes. Based on this knowledge model a diagnostic mechanism was developed based on probabilistic inferencing. This mechanism can identify the probable sets of disorders, and can compute the values of their relative likelihood function, ¸(D , S\), using the conditional ' causal and conditional indirect causal probabilities of the causal relations among the diagnostic variables. The major features of our approach include an e!ective integration of the causal relations among heterogeneous processes and a full exploitation of this knowledge in fault diagnosis. This diagnostic approach also considers the dependencies and in#uences between disorders in order to develop a more accurate and general diagnostic inferencing method. The proposed diagnostic mechanism was implemented in a UNIX-based distributed environment.
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