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Abstract

The computer numerical control machine is important industrial equipment, and its reliability has been one of the most important symbols to measure the modernization of advanced manufacturing and it is critical in the aspects of reliability design improvement, fault monitoring, and fault repair for the computer numerical control machine. The computer numerical control machine’s assembly process is a significant part in its manufacturing process, and assembly operation is a major factor in determining the whole machine’s quality, and assembly process quality analysis is necessary for computer numerical control machines, in which reliability allocation is an essential part of its reliability design. In order to quickly locate the fault of computer numerical control machine tool and accurately judge the fault grade, a method of fault classification decision of computer numerical control machine tool based on motion micro-unit is proposed, which includes the following steps: (1) from the point of the decomposition of system function, the computer numerical control machine tool is decomposed layer by layer into the layer of micro-actions, and the conceptual model of motion unit is given; (2) from the level of action, the types of fault modes of motion units are comprehensively analyzed and summarized; and (3) combining grey clustering theory and rough set theory, a fast and accurate fault classification decision-making method is formed. Finally, the validity of this method is verified by an example analysis of motion micro-units of a computer numerical control machine rack. The contributions of this work can be summarized as (1) the proposed grey fixed weight clustering analysis, (2) the graded fault classification using the decision table approach, (3) the knowledge reduction of decision rules using the rough set theory, and (4) the quicker and accurater decision and effectiveness validated by the given study case.

Keywords

Motion micro-unit fault classification grey clustering decision rough set knowledge reduction

Introduction

As an essential part of system reliability analysis, fault pattern classification analysis is particularly critical in the aspects of reliability design improvement, fault monitoring, and fault repair. Traditional fault mode classification often uses risk priority number (RPN) and hazard matrix (HM). However, for the reason that the subjectivity of the RPN method is high, and the harmfulness matrix method is unsatisfactory in practicability, the result of the analysis is often far from the actual situation. In order to solve these problems as mentioned above, Chen et al.¹ utilized the quantitative parameters in RPN analysis and proposed a new method based on qualitative and quantitative analysis of cost loss and the probability of process fault mode. Wang et al.² combined the data envelope method and RPN and formed a new method of the ranking risk of fault mode. Zhou and Thai³ considered the relative weights of risk factors Severity (S), Occurrence (O), and Detection (D), and improved the traditional RPN analysis. Shi et al.⁴ introduced a fuzzy set theory and fuzzy analytic hierarchy process to analyze fault modes. Du et al.⁵ proposed a new method of fault pattern analysis based on evidence reasoning. Internationally, many useful attempts have been made by Kutlu and Ekmekçioğlu,⁶ Certa et al.,⁷ and Montani et al.⁸ to apply the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method, Dempster–Shafer (D-S) evidence theory, and Bayesian network to fault classification analysis, respectively. However, there are still many problems in industrial applications.

Based on past experience, splitting the complex machine is necessary before researching and analyzing the CNC machine tool. Traditional decomposition of CNC machine tools is based on the hardware structure of the whole machine and is conducted following the idea of “components–module–parts.”^9,10 This method of decomposition is simple and clear, but there are two major problems: First, the lack of consideration of the inter-dependencies between the various components can make the function of the decomposed parts be single and it is difficult to analyze the multi-function and multi-quality features coupling system comprehensively. Second, ignoring the motor function of the device will make the analysis deviate from the actual results. To solve these problems, Zhang et al.¹¹ and Ran et al.¹² tried to analyze the reliability of CNC machine tools from the perspective of CNC machine tool function decomposition and put forward the concept of micro-action. Introducing the motion micro-unit as the main body of the fault analysis of CNC machine tools can refine the granularity of fault analysis, and the main advantages of the following five aspects:

It is easier to set up the mathematical model.

Fault detection is simple and high accuracy.

It is easier to locate the fault and reduces the difficulty of analyzing the reasons for the fault.

Failure mechanism analysis is more straightforward.

Decision-making on fault classification is more objective.

There is a growing trend of the advanced signal processing methods (such as continuous and discrete Fourier transform, Gaber wavelet transform, and statistical analysis method) assisted by artificial intelligence approaches for the applications of machine condition monitoring, fault diagnoses, and fault decision of complex equipment, which include genetic algorithm (GA), support vector machine (SVM), fuzzy logic system (FLS), artificial neural network (ANN), ant colony optimization,¹³ some improved particle swarm optimization,^14,15 collaborative optimization algorithm,¹⁶ entropy approaches,^17–19 clustering theory, and rough set theory. Compared to the conventional signal processing methods, the artificial intelligence–driven techniques are improving the performance and supported by the industrial big data.^20,21

This article attempts to propose a numerical control machine fault classification decision-making method based on motion micro-units which combines grey clustering theory and rough set theory, and it is expected to provide the corresponding basis for the process control of the subsequent maintenance. Compared with the previous research on fault classification, not only this method has the above five advantages, but also its advantage lies in the fact that the grey clustering theory can achieve the goal of accurate decision-making and then simplify the decision-making rules by the rough set theory to make the decision-making more rapid.

Following the “Introduction” section, the remainder of the paper is organized as follows: Section “Micro-action unit” describes the definition of the micro-action unit and related modules; section “Failure analysis of micro-action unit” proposes a quantitative approach to failure analysis of micro-action unit; section “Example analysis and discussion” provides the empirical results and further verifies the design; and section “Conclusion” concludes the paper.

Micro-action unit

FMA structural decomposition

The CNC machine tool is a complex system of mechanical and electrical integration. To facilitate the analysis of quality characteristics, at first, the CNC machine tool should be decomposed into simple basic units to achieve the purpose of easy modeling. Then the quality characteristics of the primary unit will be analyzed. Finally, the quality characteristics of the machine will be analyzed comprehensively. The decomposition method according to the “components–module–parts” is simple and straightforward. However, there are severe problems to be solved in practice, such as the modeling difficulties caused by a large number of parts, the quantitative analysis difficulties caused by the lack of part fault data.²²

Structural decomposition (“Function-Motion-Action” (FMA) decomposition) based on system functionality is to decompose complex systems hierarchically from top to bottom according to the FMA route, and this action which is easy to control and easy to analyze is called micro-action. The primary process of FMA decomposition is shown in Figure 1, in which Figure 1(a) is the ‘part–module–component’ model and Figure 1(b) is the ‘function–movement–action’ model, that is, FMA layout.

Figure 1.

System decomposition process: (a) part–module–component and (b) FMA layer (function–movement–action).

It can be seen from the figure that the central idea of FMA decomposition is to decompose the complicated comprehensive movement of machine tool manufacturing layer by layer until it is decomposed into a series of most essential and non-decomposable action units, which is similar to the process of splitting machine into parts. But the Figure 1(a) takes on the functional movement of the machine tool for the study, while the Figure 2(b) takes the machine parts for the study. Therefore, whether the function of the whole machine is actually functioning correctly depends on whether there is any faulty micro-action.

Figure 2.

The concept model of the micro-action unit.

Conceptual models of the micro-action unit

As shown in Figure 2, the analysis of the mechanism of movement decomposition can give the definition of the micro-action unit: A micro-action unit is a group of parts (including transmission, fastening, supporting, positioning and power input/outputs, and so on) that can perform any basic micro-action. The standard micro-action unit composition should include three units as the power input, the middleware, and the power output.

It can be seen from Figure 2 that the displacement and the angle that the power input of the micro-action unit receives from the output of the upper micro-action unit are transmitted through the middleware and finally, the displacement and the angle are transmitted to the micro-action unit of the next level through the power output. Among them, the power input and power output is an integral component, while the middleware is composed of transmission parts, fasteners, supports, and positioning pieces, which means the middleware may only include some of them for a different micro-action unit. It should be noted that judging which a part belongs to is determined by the role (function or effect) the part plays in the actual work of the micro-action unit to which it belongs.

As for the most basic micro-actions, moving and rotating, the motion units can be divided into moving the micro-action unit and rotating micro-action unit, while CNC machine tools can be considered as the complex electrical–mechanical products assembled with a series of motion micro-units (including moving the micro-action unit and rotating micro-action unit).

Failure analysis of micro-action unit

Fault mode type

Why the CNC machine tool fails is because the micro-action unit itself or the connection between the associated micro-action units have failed, which has an impact on the function or performance of CNC machine tools, and the form of these failures is called fault mode. Taking into account the characteristics of the motion micro-unit itself, the micro-action unit fault mode can be divided into two types: operating type and connecting type, as shown in Figure 3.

Figure 3.

The principle of fault mode division.

“Action” mentioned in the figure refers to the “micro-action,” which causes malfunctioning of the part, eventually resulting in the machine’s performance is affected and even some functions not being able to be completed correctly. “Connection” refers to the structural connection of a micro-action unit itself and the connection between associated micro-action units, including the pipe connection and the cable connection. Combined with related information about relevant enterprise historical data and literature,²³ the article summarizes micro-action unit fault mode, as shown in Table 1.

Table 1.

The fault mode of the micro-action units.

Type	Subtype	Fault mode	Case analysis
Actiontype	Action performancecategory	The action is not flexible	Ball screw rotation is not flexible, stick movement is slow, and so on
		Disoperation	Malfunction of jaw safety clutch, magnetic coupling malfunction, and so on
		Move/rotate is not in place	The guide tube is not in place, the gripper is not in place
		Move/rotate over-travel	Guide tube over-travel, gripper over-travel
		The action is not smooth	The steering tube is not moving smoothly, the lever is not moving smoothly, and so on
		Abnormal sound and noiseduring the action	There are abnormal sound in gear transmission and screw nut movement
		Braking failure	Making heel spindle stop is difficult, dressing wheel idles while it stops
		Action completely invalid,no action	Hydraulic cylinder piston does not move, no tool change for tool magazine
	Action coordinationcategory	Associated micro-movementis not synchronized	Drive motor and grip move, motor, and guide tube
	Action coordinationcategory	Associated micro-actioninterfering	Gear mesh stuck, collision, and so on due to geometric problems
	Action precisioncategory	Associated micro-movement isnot coordinated	The gripper moves and so on when the guide tube is not in place
		Zeroing accuracy outof tolerance	Accuracy is beyond the tolerance when the magazine is back to zero and so on
		Orientation accuracy outof tolerance	Guide tube positioning accuracy out of tolerance, grip movement positioning accuracy out of tolerance, and so on
		Movement/rotation accuracyis not high	Low slotted tube rotation accuracy and rack movement accuracy
Connection type	Connection faultcategory	The connection unitis damaged	Broken screw, gear broken teeth, seal damage, and so on
		Assembly and combination ofconnection units are notin place	The gap between nut and screw assembly is too large, the anti-jamming nut is not tightened, the jaw clutch is installed improperly
		Leakage (including liquid,gas, oil)	Magnetic coupling leakage, worm gear unit oil leakage, and so on
		Clogged (including liquid,gas, oil)	Cylinder blockage and so on

Fault mode classification

As the essential component of the CNC machine tool, the motion unit has many kinds of fault modes, and different fault modes have different effects on the whole machine function. In the fault classification philosophy of identifying the central contradictions, focusing on the key and taking care of general, fault mode of the micro-action unit is divided into the following three levels:

The key fault: Once the key fault occurs, it will lead to the failure of the micro-action unit, so that the system function will be greatly affected, the economic loss will be huge, and the improper handling will threaten the life safety of the relevant personnel.

The main fault: The main fault is difficult to detect and it will lead to damage to the micro-action unit and affect system functions, which will cause greater economic losses.

Secondary fault: Secondary faults have less effect on the system’s functionality, are easy to repair and cost less.

Quickly and accurately determining the level of fault not only helps designers focus on key analysis and design improvements, but also helps maintenance staff to choose maintenance methods to avoid too much cost of dealing with minor problems at a cost and the situation that serious problems are not valued. How to formulate decision rules scientifically and rationally so as to judge the level of fault is a difficult problem to be studied. The decision-making method combined with grey rough combination theory proposed in this article can meet the requirements well.

Fault classification decision based on grey fixed weight clustering

Grey fixed weight clustering analysis is a commonly used multivariate statistical method, which is mainly applied to the problems such as the indefinite correlation analysis of system model, the establishment of the model, the prediction and decision-making under the condition of incomplete information.²⁴

Establishment of the fault information matrix

Suppose that one of the micro-action units of a CNC machine has n-type fault modes, there are a total of m evaluation indicators for these fault modes, and there are s grey decision-making. Then the information matrix with n decision objects, m decision indicator attributes, s decision grey of the decision object i on the quantitative evaluation value of the decision index j is

$X = x_{ij} = {\begin{matrix} x_{11} & x_{12} & \dots & x_{1 m} \\ x_{21} & x_{22} & \dots & x_{2 m} \\ x_{31} & x_{32} & \dots & x_{3 m} \\ \dots & \dots & \dots & \dots \\ x_{n 1} & x_{n 2} & \dots & x_{nm} \end{matrix}}$ (1)

Determination of the specific weighted function $f_{j}^{k} (x_{ij})$ and decision weight $η_{j}$

$f_{j}^{k} (x_{ij}) (j = 1, 2, \dots, m; k = 1, 2, \dots, s)$ is referred to as the specific weighted function of decision index j with respect to k grey and $η_{j} (j = 1, 2, \dots, m)$ is referred to as the decision weight of index j. There exists the following relationship

$\sum_{j = 1}^{m} η_{j} = 1$ (2)

As for grey fixed weight clustering analysis method, the specific weighted function $f_{j}^{k} (x_{ij})$ is usually determined by the experts in related industries. The value of decision weight $η_{j}$ can be determined by experts or determined by the method proposed in Xu,²⁵ which will not be repeated here.

Determination of the grey mode of fault mode

Based on the definite weighted function $f_{j}^{k} (x_{ij}) (j = 1, 2, \dots, m; k = 1, 2, \dots, s)$ obtained from the first step and the second step, decision weight $η_{j} (j = 1, 2, \dots, m)$ and the quantified evaluation value $x_{i j} (i = 1, 2, \dots, n; j = 1, 2, \dots, m)$ of fault i on the decision index j, grey fixed weight clustering coefficient $σ_{i}^{k} (i = 1, 2, \dots, n; k = 1, 2, \dots, s)$ can be calculated as

$σ_{i}^{k} = \sum_{j = 1}^{m} f_{j}^{k} (x_{ij}) η_{j}$ (3)

Determine the grey k in which the fault i belongs to according to the obtained grey fixed weighting clustering coefficient, and the judgment is

$σ_{i}^{k^{*}} = max_{1 \leq k \leq s} {σ_{i}^{k}}$ (4)

Decision table knowledge reduction based on rough set theory

Considering that the grey system itself lacks the strong capability of parallel computing and data reasoning, once the system changes slightly, the subsequent process needs to be recalculated.²⁶ Due to the lack of flexibility in the clustering results, the process of system information needs to be optimized to achieve the goal of improving the grey system theory. The introduction of the rough set theory is a good solution.

As a new mathematical tool for information processing, the rough set theory is mainly used in the situation that the known information is not precise, incomplete, not unified, and so on. The combination of grey system theory and rough set theory scientifically and reasonably can be applied to the field of uncertain and incomplete information processing. Rough set theory focuses on knowledge reduction, using the known decision objects, decision indicators, grey clustering results to form a primitive decision table. Considering the incompleteness of the known data of the micro-action unit fault mode and the grey of clustering result (the inaccurate information), if we directly reduce the knowledge of the original decision table, it is likely to lead to the contradiction between the minimum decision algorithm and the actual meaning of the problem. In order to optimize the decision-making algorithm and make the decision-making rules more flexible, the paper discretized the original decision-making table first and then reduce the knowledge.

Establishment of the original decision table

Establish a decision table with decision-making objects, decision-making indicators, clustering results. Make the decision table as $D T = (U, P \cup D)$ . The subset $P = {a_{j} j = 1, 2, \dots, m}$ is a condition attribute set. The subset $D = {d}$ is the decision attribute set. $U = {x_{1}, x_{2}, \dots, x_{n}}$ is the discourse domain. a_j(x_i) is the value of the decision object on the attribute a_j.

Discretization of the decision table

Combined with the characteristics of grey cluster analysis, this article uses a discretization algorithm based on the importance of attributes that is the weight of decision indexes. The discretization process is as follows:²⁷

Input: the original decision table, among them $U = {x_{1}, x_{2}, \dots, x_{n}}$ , $P = {a_{1}, a_{2}, \dots, a_{m}}$ .

Output: discretization decision table.

Step 1: Initialize the candidate breakpoint set. Suppose that $CUT = {C_{a_{j}} j = 1, 2, \dots, m}$ , $C_{a_{j}}$ is a set of candidate breakpoints for attribute a_j.

Step 2: Sort the condition attributes $a_{j} (j = 1, 2, \dots, m)$ from small to large according to the importance of the attribute (i.e. the decision attribute value of the condition attribute). In the case of the same importance, sort according to the number of conditional attribute candidate breakpoints from more to less.

Step 3: Examine the existence of each candidate breakpoint $C_{a_{j}}$ in the attribute a_j. The specific approach is change the smaller value of two values adjacent to $C_{a_{j}}$ in the original decision table into a larger value. If the decision table at this time will not conflict, so remove $C_{a_{j}}$ from the breakpoint focus; otherwise, restore the modified property value.

Step 4: Finishing the discretized decision table.

A minimal decision algorithm for using recognizable matrix method to get the decision table.

Discriminant matrix C_D is calculated through the discretized decision table. $C_{D} (i, j)$ is used to denote the unit in the jth column of the ith row. The units of the discernable matrix can be calculated by the following formula

$C_{D} (i, j) = {\begin{matrix} {a_{k} | a_{k} \in P \land a_{k} (x_{i}) \neq a_{k} (x_{j})}, d (x_{i}) \neq d (x_{j}) \\ Ø, d (x_{i}) = d (x_{j}) \end{matrix}$ (5)

In the formula $i, j = 1, 2, \dots, n$ .

The Boolean function $Δ$ is called a recognizable function, which is calculated as

$Δ = \underset{(x, y) \in U \times U}{Π} \sum C_{D} (x, y)$ (6)

In formula (6), when $C_{D} (i, j) = {a_{1}, a_{2}, \dots, a_{k}} \neq Ø$ ; otherwise $C_{D} (x, y) = Ø$ , $\sum C_{D} (x, y) = 1$ .

Finally, a minimal decision algorithm is written by recognizable function and discrete decision table.

Example analysis and discussion

Decision table knowledge reduction based on rough set theory

In CNC machines, there are various actuating mechanisms can convert the rotational movement to a translational movement. The efficiency and responsiveness of the actuating mechanism have the greatest influence on the accuracy of the work produced. The actuating mechanisms usually utilized for the slides of CNC machines are the rack and pinion, screw, and so on. As shown in Figure 4, the mechanical transmission system of a CNC machine is in the force and motion transmission paths from the drive motor to the slide, in which the rack and pinion sub-system is given as the study case. The rack and pinion sub-system comprises the elements that convert the rotary motion to linear motion. In this case, the micro-action units of the rack and pinion sub-system (micro-motion unit structure shown in Figure 4, the components shown in Figure 5) are selected as an example, in which the fault classification decision analysis for six kinds of fault modes performed in this micro-motion unit including the fatigue of the tooth surface, the uneven movement of the rack, the abnormal noise of the rack movement, the rack movement over-position, the rack broken teeth, and the oil leakage.

Figure 4.

Diagram of the rack motion micro-action unit of a CNC machine.

Figure 5.

Units of the rack motion micro-action unit.

The above six kinds of fault modes are evaluated according to the four decision indexes of occurrence frequency, the degree of hazard, detection difficulty, and maintenance difficulty. The evaluation method is to invite experts in this field to rate these six fault modes according to four decision indexes (the higher the score, the higher the degree). The obtained quantitative evaluation values are shown in Table 2.

Table 2.

The quantized value of the fault evaluation index of the rack movement micro-action unit.

Fault mode	Occurrence frequency	The level of danger	Detection difficulty	Maintenance difficulty
Rack moving too far	60	93	84	53
Oil spill	88	62	60	45
Rack moving noise	55	75	74	61
Gear tooth surfacefatigue wear	77	86	60	55
Rack broken teeth	86	93	51	69
The rack is notmoving smoothly	67	70	72	49

The information matrix consisting of the quantitative evaluation values $x_{ij} (i = 1, 2, \dots, 6; j = 1, 2, \dots, 4)$ obtained from formula (1) is

$X = x_{6 \times 4} = {\begin{matrix} 60 & 93 & 84 & 53 \\ 88 & 62 & 60 & 45 \\ 55 & 75 & 74 & 61 \\ 77 & 86 & 60 & 55 \\ 86 & 93 & 51 & 69 \\ 67 & 70 & 72 & 49 \end{matrix}}$ (7)

Providing decision rules according to the key fault, the main fault, the secondary fault three greys as clustering decision-making. The definite weighted function of these four decision indexes with respect to the three grey categories determined by the experts that are

$f_{j}^{1} [70, 90, -, -], f_{j}^{2} [55, 70, -, 85], f_{j}^{3} [-, -, 40, 65]$ (8)

In the formula, j = 1, 2, 3, 4.

The decision weight of the four decision indicators including the frequency of occurrence, the degree of harm, the difficulty of testing, and the difficulty of maintenance are determined by the expert group, and the results are $η_{1} = 0.18$ , $η_{2} = 0.28$ , $η_{3} = 0.28$ , and $η_{4} = 0.26$ .

Substitute the information matrix, definite weighted function and decision weight into formula (3) and the grey fixed weighting clustering coefficient matrix can be obtained as given in formula (9)

$\begin{matrix} \sum = (σ_{i}^{k}) = [\begin{matrix} \begin{matrix} σ_{1}^{1} & σ_{1}^{2} & σ_{1}^{3} \\ σ_{2}^{1} & σ_{2}^{2} & σ_{2}^{3} \\ σ_{3}^{1} & σ_{3}^{2} & σ_{3}^{3} \\ σ_{4}^{1} & σ_{4}^{2} & σ_{4}^{3} \\ σ_{5}^{1} & σ_{5}^{2} & σ_{5}^{3} \end{matrix} \\ \begin{matrix} σ_{6}^{1} & σ_{6}^{2} & σ_{6}^{3} \end{matrix} \end{matrix}] \\ = [\begin{matrix} 0.4760 & 0.0787 & 0.1608 \\ 0.1620 & 0.2240 & 0.2976 \\ 0.1260 & 0.4960 & 0.1136 \\ 0.2870 & 0.1893 & 0.1600 \\ 0.4240 & 0.2427 & 0.1568 \\ 0.0280 & 0.6667 & 0.1664 \end{matrix}] \end{matrix}$ (9)

According to formula (4), we can get the conclusion

$\begin{matrix} max_{1 \leq k \leq 3} (σ_{1}^{k}) = 0.4760 = σ_{1}^{1}, max_{1 \leq k \leq 3} (σ_{2}^{k}) = 0.2976 = σ_{2}^{3}, \\ max_{1 \leq k \leq 3} (σ_{3}^{k}) = 0.4960 = σ_{3}^{2} \\ max_{1 \leq k \leq 3} (σ_{4}^{k}) = 0.2870 = σ_{4}^{1}, max_{1 \leq k \leq 3} (σ_{5}^{k}) = 0.4240 = σ_{5}^{1}, \\ max_{1 \leq k \leq 3} (σ_{6}^{k}) = 0.6667 = σ_{6}^{2} \end{matrix}$ (10)

Therefore, it can be seen that the fault mode numbered i = 1, 4, 5 including the rack movement over the bit, the tooth surface fatigue wear and rack broken teeth belong to the key grey, the fault mode numbered i = 3, 6 including the rack moves abnormal sound and unsteady rack movement.

Suppose that x₁, x₂, x₃, x₄, x₅, x₆ represent six fault modes including the rack movement over-position, oil leakage, rack noise, tooth wear, rack teeth, and unstable rack movement; a₁, a₂, a₃, a₄, respectively, represent these four decision indicators of the frequency of occurrence, degree of harm, the difficulty of testing and maintenance; (d, 1)=d₁, (d, 2)=d₂, (d, 3)=d₃ represent the key fault, the main fault, and the secondary fault, respectively. Then the original decision table can be obtained by grey clustering results (Table 3).

Table 3.

Initial decision table.

U	a ₁	a ₂	a ₃	a ₄	d	RPN/10³
x ₁	60	93	84	53	1	468.7
x ₂	88	62	60	45	3	327.3
x ₃	55	75	74	61	2	305.2
x ₄	77	86	60	55	1	397.3
x ₅	86	93	51	69	1	407.8
U	a ₁	a ₂	a ₃	a ₄	d
x ₆	67	70	72	49	2	337.6

According to the original decision table, the set of candidate breakpoints for each indicator is given as follows

$\begin{matrix} C_{a_{1}} = {58, 64, 72, 82, 87}, C_{a_{2}} = {66, 73, 81, 90}, \\ C_{a_{3}} = {56, 66, 73, 79}, C_{a_{4}} = {47, 51, 54, 58, 65} \end{matrix}$ (11)

Due to the decision-making weight of four indicators, $η_{1} < η_{4} < η_{2} = η_{3}$ ; meanwhile, $C_{a_{2}}$ and $C_{a_{3}}$ have the same number of breakpoints; there are two discretization paths in the original decision table: a₁ → a₄ → a₃ → a₂ and a₁ → a₄ → a₂ → a₃, The two final discretized decision tables are shown in Tables 4 and 5.

Table 4.

The decision table after the discretization in the order of a₁ → a₄ → a₃ → a₂.

U	a ₁	a ₂	a ₃	a ₄	d
x ₁	88	93	84	69	1
x ₂	88	62	84	69	3
x ₃	88	75	84	69	2
x ₄	88	93	84	69	1
x ₅	88	93	84	69	1
x ₆	88	75	84	69	2

Table 5.

The decision table after the discretization in the order of a₁ → a₄ → a₂ → a₃.

U	a ₁	a ₂	a ₃	a ₄	d
x ₁	88	93	84	69	1
x ₂	88	75	60	69	3
x ₃	88	75	84	69	2
x ₄	88	93	60	69	1
x ₅	88	93	60	69	1
x ₆	88	75	84	69	2

Use the discernible matrix method to reduce knowledge in Table 4, according to formulas (5) and (6), it can be concluded successively that the discernible matrix and discernable function of Table 4 are

$C_{D} (i, j) = [\begin{matrix} 0 & a_{2} & a_{2} & 0 & 0 & a_{2} \\ 0 & a_{2} & a_{2} & a_{2} & a_{2} \\ 0 & a_{2} & a_{2} & 0 \\ 0 & 0 & a_{2} \\ 0 & a_{2} \\ 0 \end{matrix}], Δ = a_{2}$ (12)

Thus, the minimum decision algorithm of Table 4 can be concluded as

${\begin{matrix} a_{2_{93}} \to d_{1} \\ a_{2_{75}} \to d_{2} \\ a_{2_{62}} \to d_{3} \end{matrix}$ (13)

Considering the discretization process of the original decision table and the minimal decision algorithm finally obtained, the decision rules to get the fault classification can be explained in natural language as when the quantitative value of the evaluation index a₂ is between 86 and 93, the decision object should be classified as the key fault grey; when the quantitative value of evaluation index a₂ is between 70 and 85, the decision object should be divided into the main fault grey; when the quantitative value of evaluation index a₂ is not higher than 69, the decision object should be divided into minor fault grey.

Similarly, matrix, recognizable function, and minimum decision algorithm of Table 5 can be identified as

$\begin{matrix} C_{D} (i, j) = [\begin{matrix} 0 & a_{2} a_{3} & a_{2} & 0 & 0 & a_{2} \\ 0 & a_{3} & a_{2} & a_{2} & a_{3} \\ 0 & a_{2} a_{3} & a_{2} a_{3} & 0 \\ 0 & 0 & a_{2} a_{3} \\ 0 & a_{2} a_{3} \\ 0 \end{matrix}], \\ Δ = a_{2} \land a_{3}, {\begin{matrix} a_{2_{93}} \to d_{1} \\ a_{2_{75}} a_{3_{84}} \to d_{2} \\ a_{2_{75}} a_{3_{60}} \to d_{3} \end{matrix} \end{matrix}$ (14)

Results and discussion

The computer specifications for the simulations are a 2.1 GHz Intel Dual-Core Processor, Windows XP Professional v5.01 Build 2600 service pack 3, a 2.0 GB 800 MHz dual-channel DDR2 SDRAM, and MATLAB R2008a. Considering the discretization process of the original decision table and the minimal decision algorithm finally obtained, the decision rules to get the fault classification can be explained in plain language as:

When the quantitative value of the evaluation index a₂ is between 86 and 93, the decision object should be classified as the key fault grey.

When the quantitative value of the evaluation index a₂ is between 70 and 85, and the quantitative value of the evaluation index a₃ is not less than 72, the decision object should be classified as the main fault grey.

When the quantitative value of evaluation index a₂ is lower than 86, and the quantitative value of evaluation index a₃ is lower than 72, the decision object should be divided into secondary failure grey.

As given in Table 3, the comparison of the proposed method with the RPN method has been demonstrated, in which

Obviously, when the evaluation index a₂ is not higher than 80, the Pareto diagram method using RPN has the risk of discriminating the minor faults of the main fault.

When the evaluation index a₂ is between 86 and 90, the critical fault is possibly identified as the main fault.

Compared with the traditional RPN Pareto diagram method, the quantified value of the evaluation index a₃ can effectively avoid the critical fault as the main fault and the main fault as the two decision risks of the secondary fault. Therefore, the proposed decision algorithm is more precise.

Numerically, the proposed fault classification decision approaches are matrix computation algorithms, which have a high efficiency of convergence and accuracy. Furthermore, the historical data of the enterprise and the discussions of the expert group show that the two indicators of the degree of damage and the detection difficulty of the machine tool have a large proportion when making the classification decision of the fault mode. The indicator of the degree of damage to the failure has a decisive influence on the classification decision of the fault mode. The decision rules obtained above are consistent with the operation of many enterprises in the actual engineering application to determine the fault grade of their CNC machine tools, which means that the qualitative analysis is consistent with the calculation results.

Conclusion

This work proposes a conceptual model of the micro-action unit, introduces its components and two kinds of motion micro-action unit and rotating micro-action unit, and put forward a new type of idea of fault classification decision of CNC machine tools based on the micro-action unit. This article analyzes and summarizes the fault mode types of motion micro-units comprehensively and summarizes the principles of fault classification of the micro-action unit. First, the decision table of graded fault classification of CNC machine tool is established by grey fixed weight clustering analysis method, and the knowledge reduction of decision rules is made by the rough set theory, which makes the decision more quickly and accurately, and then, the effectiveness of this method is proved with examples.

The novelties that highlight this research can be summarized as follows: (1) the graded fault classification using the decision table approach and the grey fixed weight clustering analysis, (2) the knowledge reduction of decision rules using the rough set theory, and (3) the effectiveness validated by the case study.

In future work, we are planned to enhance the CNC design by developing time-variant robustness and reliability indices for capturing the time-varying and nonlinear performance. Furthermore, new artificial intelligence algorithms, such as the swarm dolphin algorithm, the artificial wolf pack algorithm, can be embedded into the computational intelligence assisted design framework for the intelligent design of reliability allocation and design.²⁸

Footnotes

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: The support provided by the National Natural Science Foundation of China (no. 51875090) and the Natural Science Foundation of Guangdong Province (2018A03 0313320).

ORCID iD

Leo Chen

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