Bin SHEN，Yi-shun SU，Shu-yu ZHAO

The Sino-German School for Postgraduate Studies，Tongji University，Shanghai 201804，China

The application of CNC machine in modern manufacturing has become widespread．In practical production，if we can make right judgment timely of the CNC machine fault，ascertain the fault part，find out the reason and remedy with lower maintenance costs and in a short period，then it could greatly reduce the maintenance blindness and also improve the production economy and security［1］．In the early 1960s，Bell-Labs in American firstly used the FTA method，and random failures of the militia missile launch control system were successfully predicted．Subsequently，Boeing developed computer program based on the FTA to improve aircraft．The study of FTA used in CNC machine fault diagnosis started in the past few years．The CNC machine fault diagnosis is mainly based on the rules and case-based，but methods have their disadvantages，i．e．，the existed cases are few，and rule-based is always stored in redundant and non-logical，which makes the application of FTA become a trend．Fault tree analysis is a good method to make security and reliability analysis of complex systems，it can obtain good result in running and maintaining CNC machine by using the fault tree analysis to predict and diagnose fault［2］．

1.The key elements of fault tree diagnosis inference

1.1.Modular decomposition of fault tree

CNC machine system is very complex，and the fault tree it generated is very large．Thus，fault tree for making relatively qualitative and quantitative analysis of fault tree seems very difficult［3］．In order to make better analysis of fault tree，it is usually necessary to simplify the fault tree．At present，modular analysis of fault tree is a very effective simplified method for fault tree．The modular decomposition algorithm generally uses DFLM twice．

1.2.Basic structure of fault tree modular

Through the modular analysis of fault tree，one could get three basic modular structures as shown in Figure 1．The basic modularization structure of fault tree is a binary tree，which is composed of two layers．The first layer mainly is“modular Top-event”，and the second layer is random combination of“submodule events(i．e．intermediate events)” and“module end events”．The two layers are connected by logical symbols“and”or“or”．Since there is little redundancy protection of CNC machine，this paper doesn’t put“and”into consideration．

Figure 1．The basic modular structure

1.3.Fuzzy search of fault tree modular library

When traversing the modular library，one should firstly obtain the feature information which is needed to be retrieved．Judging from the instruction form of fault，CNC machine fault includes fault with alarm display and fault without alarm display［4］．Thus，the basic information obtained from fault diagnosis includes not only the alarm display from CNC machine itself，but also the alarm display from pilot lamp．This information has to be described by maintenance staff．Of course，one could not exclude that the fault description from maintenance staff is same to the fault reason of alarm．Therefore，after obtaining the information of these two respects，the inference engine should transfer the alarm into fault description，and also remove duplicate with the fault description from maintenance staff．After removing duplicate of phenomenon description，the inference engine can regard every independent phenomenon as feature to go into successively retrieval phase．The retrieval object is the fault tree modular library．Before retrieval，one will firstly introduce the significance of the introduction of valid modular．

2.The introduction of valid modular and its meaning

Generally，once the basic feature of fault is certained，one could start to locate every feature in fault tree，and find out the corresponding modular name in fault tree．In order to obtain the prime reason of fault(i．e．，the traversal of bottom event)，one just needs to treat every feature modular name as Top-event，and then Downward-searching．However，there exists a problem of this method，i．e．，if there is one feature modular name which is below the sub-fault tree another modular name corresponded to，then it will lead to repeat retrieval when making bottom event traversal of sub-fault tree．Besides，if there are a lot of fault features，then it will lead to a substantial increase in retrieval workload，and that is so-called“NP”problem．For example，as shown in Figure 2，there are three fault features，e1，e3and e4，and e1includes e3，while e4is from an independent sub-fault tree．When one traverses e1and e3，respectively，it will generate repeat traversal in fact．Therefore，the aim of introducing the valid fault tree is to avoid repeat traversal of fault tree．Thus，the design of event library should not only consider the hierarchy relationship of the event in fault tree，but also consider the affiliation between each event(parent-child relationship)．

Figure 2．Multi-Ebene Fehlerkennfeld

3.The process of inference based on module library

3.1.Basic process and principle of feature-retrieval

3.1.1.Basic process of retrieval

1)Firstly，traversing every fault feature in framework library and locating every feature in fault tree(i．e．，corresponding event)．Reading the hierarchy and subordinate frame name that the feature-event corresponded to from event library，and as to fault feature which is subject to a sub-fault tree，treating the highest level of fault features hierarchy as the Top-event of this sub-fault tree．

2)Secondly，traversing every sub-fault tree whose fault feature is treated as Top-event，i．e．，to retrieve the sub-frame and embody the notch(modular bottom event)．After the sub-frame expanding，retrieving other units of the second layer level of this frame．And if it retrieves and discovers that the subframe exists，then it should continue to retrieve the frame this sub-frame corresponded to．Before start to retrieve the next frame，the inference will automatically embody this frame as valid frame and also embody all the notches existing in this frame at the meantime．Continuously downward-searching the sub-frame，until the sub-frame does not appear any more．

3)As to the fault tree system which only have“or”logical connection，the final minimum cut sets of the modular are usually the set of bottom events，i．e．，the bottom event can be regarded as the prerequisites of the deriving rules．

3.1.2.Two prior principles

1)If the fault node exists in the bottom event，then embody it and regard it as the top priority of importance．

2)Through traversal searching，it will generate new bottom events，and their importance could get the sub-priority．

3.2.The composite fault frame based on valid fault modular

Through fuzzy retrieving of fault tree modular library and obtaining the valid modular，one could arrange these modular according to the fault modular synthetic table as shown in Table 1，and then one could get the fault frame figure as shown in Figure 3．Thus，one can just solve the minimal cut sets according to this valid synthetic fault frame．This method greatly simplifies the calculation process，and makes the final minimum cut sets obtained much more effective simultaneously．

Table 1．The composited frame of fault-modular

Figure 3．The fault tree for the composited frame of fault-modular

4.Bayesian calculation of importance based on the whole valid fault modular

4.1.Bayesian theoretical basis and method

The basic point of Bayesian method is that every unknown variable X could be regarded as a random variable，and its unknown can be described by a probability distribution．This probability distribution is a probability description of the existing prior information of variable X before sampling．This probability distribution is called prior distribution or prior in short． Either unknown variable has uncertainty，probability and probability distribution are the best way to describe the uncertainty degree．As a new method to process the uncertainty knowledge，Bayesian method has a certain theoretical basis of probability theory and could well express knowledge structure．Thus，it aroused extensive attention，and it has been adopted in many areas to obtain satisfactory results．Especially in fault diagnosis field，there still has big development room besides the previous study．Therefore，it has a significant study value and good development prospect．At present，Bayesian method has been successfully applied into many areas，such as aerospace fault diagnosis，defense systems，industrial fault diagnosis and pattern recognition．And the most successful application is the application of fault diagnosis area．

Bayesian method is used to elaborate system and solve statistical problem based on Bayesian theorem．By using Bayesian method，the possibility of every cause could be back-inferenced through the phenomena of events［5］．Because of its outstanding ability of dealing with the uncertain information from events with complex causality，Bayesian network is applied into fault diagnosis．

4.2.Bayes’theorem

The substance of total probability formula is to integrate the various causes of events to infer the possibility of events［6］．But in reality，one usually needs to analyze the causes of events if the known events already have taken place．For example，under the above hypothesis of total probability，known event A occurred，then how to get the occurring possibility of event B，Solution of this question leads to the Bayes’theorem．

Bayes’theorem:suppose the sample space of experiment E is S．A is an event of E．B1，B2，B3… are a division of S，and P(A)＞0，P(Bi)＞0，i=1，2，…，n，theni=1，2，…，n．This formular［7］provides a method to calculate the posterior probability through the prior probability．

Figure 4．The composited frame of faults

Figure 5．Event importance distribution

Through CF(A)=CF(A/B)×CF(B)，we can get that:

Thus，the importance sorting of these three bottom events are:SSC1＞SC1＞SC2

5.Importance correction

When making importance calculation，the occurring importance of every event in modular is given by the experts in field who will combine knowledge and experience to make professional assessment and then give the result．Therefore，it has an important reference value．However，the“phenomenon descriptions”on fault diagnosis interface inputs are usually given by operators who intuitively get the suspected failure point based on the actual failure site，and it is the easiest to be excluded by operators，especially when“phenomenon descriptions”are modular bottom events，they will have a high importance．Therefore，the aim of importance correction is to treat the known phenomenon descriptions in inputs as prior investigation objects．

6.Conclusion

This paper build the modular library based on the basic modular structure，and introduced the concept of valid modular and also optimal sorting of importance for minimal cut sets based on Bayesian theory．The introduction of valid modular effectively reduces the workload of traversal search on fault tree，which will greatly improve the solving speed and the efficiency．Besides，optimal sorting of importance on minimal cut sets based on Bayesian theory couls well enhance the credibility and accuracy of troubleshooting，and the diagnostic capabilities and speed are also satisfied．

［1］ Bo-sheng YE，Zeng-shuang HUANG，Bin LI．Study on fault diagnosis system for CNC machine tools based on fault tree analysis［J］．Machinery Design ＆ Manufacture，2006(8):135-137．

［2］ Xun SUN．The study of fault diagnosis algorithm based on FTA［D］．Dalian:dalian university of technology，2009．

［3］ Wen-bin LIU．Research on dynamic fault tree analysis method based on modularization［D］．Nanjing:Nanjing University of Science ＆ Technology，2009．

［4］ Na YANG．Research on no-alarm display to fault diagnosis and repair for CNC machine［J］．Mechanical and E-lectrical Information，2013(12):41-42．

［5］ Jian-chuan LI，Niao-qing HU，Guo-jun QIN，et al．Bayesian network theory and its application in machine fault diagnosis［J］．China mechanical engineering，2003，14(10):896-900．

［6］ Ning FAN．Research on inference algorithm in Bayesian network for fault diagnosis［J］．Information technology，2011(3):158-161．

［7］ Zhou SHENG．Probability theory and mathematical statistics［M］．Beijing:Higher Education Press，2008．