Health status evaluation method of CNC machine tools based on grey clustering analysis and fuzzy comprehensive evaluation

Abstract

In order to solve the problem of health evaluation of CNC machine tools, an evaluation method based on grey clustering analysis and fuzzy comprehensive evaluation was proposed. The health status grade of in-service CNC machine tools was divided, and the performance indicator system of CNC machine tools was constructed. On the above basis, the relative importance of each performance and its indicators were combined, and grey clustering analysis and fuzzy comprehensive evaluation was utilized to evaluate the health status of in-service CNC machine tools to determine their health grade. The proposed health status evaluation method was applied to evaluate the health level of an in-service gantry CNC machine that can be used for the machining propellers, and the results shown that the health status of the whole gantry CNC machine tool is healthy. The proposed evaluation method provides useful references for further in-depth research on the health status analysis and optimization of CNC machine tools.

Keywords

CNC machine tools grey clustering fuzzy comprehensive evaluation health evaluation green performance

1 Introduction

Being the “mechanical equipment mother machine”, CNC machine tools are widely used in aerospace, medical equipment, automobiles, ships and other fields. Due to wide applications, the fault analysis, performance testing and safety evaluation of CNC machine tools have always attracted people’s attention. Machinery safety is the ability of machinery to perform its intended function without injury or health hazard to humans during its life cycle, when risks are sufficiently mitigated [1]. Since CNC machine tools are the main production equipment of current manufacturing products, improving their safety, preventing or reducing the occurrence of mechanical injury accidents is a key way to enhance the manufacturing capacity and level. Health status evaluation of in-service CNC machine tools plays an important role in improving the quality and safety assurance of CNC machine tools in service.

At present, some progress has been made in research on the health status evaluation method of CNC machine tools. Shao et al. [2] determined the health status of CNC machine tools based on the relevant parameters of key components of CNC machine tools by combining the combined weighting method and fuzzy grey clustering. In literature [3], an approach of health assessment based on the ADAMS simulation for the feed system of CNC machine tools was proposed, which was used to solve the health status probability for the feed system of the CNC machine tools, and an applied case was used to verify the effectiveness of the approach. Zhao et al. [4] proposed a method to assess the performance degradation and health status of ball screws, and the method combines the Laplace feature dimensionality reduction and the Marxian distance analysis model to establish the nonlinear mapping relationship between the sensor signal sample points in the feature space and the health values under different health status, so as to obtain the quantitative evaluation of the degradation degree of ball screws’ performance. Zhao [5] took key components of CNC machine tools as research objects, and signal monitoring was carried out by feature extraction and selection, and health assessment algorithm models. On above basis, the nonlinear mapping relationship between control flow data or sensor signals and health status was established to realize the preventive maintenance of CNC machine tools. Li [6] analyzed degradation mechanism and health signals of the critical systems and critical components of CNC machine, and the measures are researched of health monitoring of CNC turning center based on multi-sensor information fusion. Deng [7] constructed a HMM health status estimating model based on multi-capability parameter and multiple observation sequences from the perspective of performance degradation, the proposed health estimation model was validated by ball-screw of a Heavy-duty CNC machine tool and the results demonstrated its effectiveness. Liao et al. [8] developed a Plug-and-Prognose (PnP) technology to monitor the health of feed axis of production type machine tools, and reliably identified the normal operation of the machine and diagnose anomalous operating states.

The above research results on the health status evaluation method of CNC machine tools ensure the safe operation of CNC machine tools to a certain extent. However, most researches usually only evaluate the health of key components of CNC machine tools, or evaluate the health of the whole machine based on the experimental data of key components, without a comprehensive analysis of the whole machine situation, so the breadth of application is limited. In order to solve the above problems, on basis of the overall performance of CNC machine tools in this study, the performance and its indicators were analyzed comprehensively, and the health status grade of in-service CNC machine tools was divided. Based on the establishment of the performance indicator system of CNC machine tools, the relative importance of each performance and its indicators were combined, and grey clustering analysis and fuzzy comprehensive evaluation was utilized to evaluate the health status of in-service CNC machine tools to determine their health grade, which provides a scientific basis for the assessment of the production safety of in-service CNC machine tools.

The main contributions in this paper are as followings.

The health evaluation indicator system of CNC machine tools was constructed.

A health status evaluation method of machine tools based on grey clustering analysis and fuzzy comprehensive evaluation was proposed.

The proposed health evaluation method was verified by an in-service gantry CNC machine tool that can be used for machining propellers.

2 Related works

In the recent years, enormous researchers have carried out a lot of research work in the health evaluation and fault diagnosis of CNC machine tools, including the construction of the health assessment model of CNC machine tools, the health status detection system, fault diagnosis and other aspects.

In the construction of the health assessment model of CNC machine tools, Liliya et al. [9] proposed a model of the components of the subsystem for evaluating and monitoring the health of a CNC machine. In literature [10], the Wiener process was used to research the performance degradation modelling and health assessment method, and experiments were carried out on the test platform for feed system of CNC machine tools, so that the correctness and effectiveness of the method was verified. Pan [11] proposed a numerical model and method for the health status of CNC machine tools based on the theory of TOPSIS (The Technique for Order Preference by Similarity to Ideal Solution), which takes the machine feature vector as input and outputs the corresponding health indicator to quantify the health status grade of CNC machine tools. Feng [12] established a health assessment model based on artificial intelligence from the perspective of machine tools maintenance, which is able to predict the future health condition of the machine tools based on the historical data of the machine tools so as to make maintenance appointments.

In terms of the health status detection system, a number of test-beds have been developed by TechSolve to allow extensive application of degradation and faults for evaluation and development of targeted technologies for machine tool health monitoring and characterization [13]. Zhou [14] proposed a system solution, in which introduction domain based on CNC system was combined with cloud service to monitor CNC machine tools condition. Without external sensing equipment, the cloud service platform performs instruction domain analysis, process and display through CNC machine tools run-time data collected by CNC system, thus realizing the goal of CNC machine health condition monitoring. Literature [15] proposed a method to constantly evaluate the health condition of critical components of CNC machine tools by using the monitoring data to perform on-line system diagnostics and prognostics.

In the aspect of fault diagnosis, this literature [16] presented a fault diagnosis system in a cyber-physical manufacturing cloud (CPMC) that allows manufacturers to perform fault diagnosis and maintenance of manufacturing machine tools through remote monitoring and online testing using Machine Tool Communication (MTComm). In literature [17], a smart fault preventive step was taken, a comprehensive fault diagnosis expert system algorithm for CNC machine was designed. Kozlov et al. [18] presented a new fault classification mode and an integrated approach to fault diagnosis; through the intelligent fault diagnosis system to determine the severity, time and location of CNC machine tool fault, and the feasibility of this approach was tested in a simulation environment. In literature [19], vibration signals obtained from the computerized CNC machine drill bit whiledrilling the metal plate, and the vibration data was subjected to feature extraction and then performs fault diagnosis.

3 Health status grading and evaluation system of CNC machine tools

3.1 Health status grading of CNC machine tools

With the increase of working hours, the performance of CNC machine tools will gradually decrease. In engineering practice, the process from performance degradation to the occurrence of faults of most machine tools does not happen overnight, but it presents a clear phase, and it need to go through the break-in period, the stability period and wear period [20]. The degree of performance degradation in each stage is different, which leads to different health status of machine tools. In this study, the health status grade was adopted to describe, and the maximum membership degree principle (MMDP) was used to determine the health status of machine tools.

A system is evaluated, if its risk is general, its safety is also general, so it is more appropriate to divide it into an odd number of grades, such as 5 or 7 grades, of which 5 grades are the best [21]. Based on the above, the relevant evaluation standards and expert experience are combined, the health status of CNC machine tools is divided into five grades: healthy, sub-healthy, qualification, micro-fault and fault, as shown in Table 1.

Table 1
Evaluation grades division of health status of CNC machine tools

Levels Grades Health status description Maintenance strategies

I Healthy All indicators are normal and operating situation is excellent Allow to use and normal maintenance

II Sub-healthy All indicators are qualified and operating situation is good Allow to use and slightly enhanced maintenance

III Qualification All indicators are basically qualified, and operating situation is poor Allow to use and increase maintenance

IV Micro-fault Some indicators are abnormal, and operating situation is slightly abnormal Allow to use and arrange for major repairs as soon as possible

V Fault Most indicators are abnormal and operating situation is abnormal Stop using and arrange repairs

Levels	Grades	Health status description	Maintenance strategies
I	Healthy	All indicators are normal and operating situation is excellent	Allow to use and normal maintenance
II	Sub-healthy	All indicators are qualified and operating situation is good	Allow to use and slightly enhanced maintenance
III	Qualification	All indicators are basically qualified, and operating situation is poor	Allow to use and increase maintenance
IV	Micro-fault	Some indicators are abnormal, and operating situation is slightly abnormal	Allow to use and arrange for major repairs as soon as possible
V	Fault	Most indicators are abnormal and operating situation is abnormal	Stop using and arrange repairs

3.2 Health evaluation indicator system of CNC machine tools

There is a close relationship between the performance and health status of CNC machine tools, the degradation of performance will lead to the generation of faults, while the emergence of faults will also aggravate the degradation of performance. Therefore, the study of the relationship between the performance and faults of CNC machine tools is of great significance to the evaluation of the health status of CNC machine tools.

Pan [22] studied the correlation between multiple performance and multiple faults, and a case was used to verify that machining performance such as positioning accuracy and repeat positioning accuracy can cause feed system faults and directly affect the health status of machine tools. Tool wear of CNC machine tools is one of the important indicators reflecting the health of a machining system [23]. Literature [24] proposed a solution for the diagnosis and control of real-time cutting tool in for edge cutting machining, which can be used to determine and forecast the real-time wear condition of cutting tool. In addition, vibration, cracks, friction, etc. are indicators that reflect the health status of CNC machine tools. In production process, the environmental pollution caused by the waste generated by CNC machine tools cannot be ignored. In order to realize the sustainable development of CNC machine tools, the green performance of CNC machine tools has also become one of the important indicators of the health status of CNC machine tools. In engineering practice, when CNC machine tools needs major repairs or to eliminate safety hazards, there are faults in the operating conditions at this time. Therefore, the evaluation of the health status of the CNC machine tools must consider its maintenance performance. Performance degradation is manifested as the decline of various performance indicators, so the study of performance indicators of CNC machine tools is extremely important for the evaluation of health status of machine tools.

In order to comprehensively and reasonably evaluate the overall health status of CNC machine tools, its scientific, integrity and purpose are considered. The multiple performance indicators are used to evaluate the health status of CNC machine tools in this study, the health evaluation indicator system and evaluation content of CNC machine tools as shown in Table 2 are constructed. As shown in Table 2, the first-level evaluation indicator are the performance of CNC machine tools, such as mechanical performance, machining performance, etc. The second-level evaluation indicator are the specific characteristics (intrinsic attributes) reflecting the performance, such as accuracy, speed, etc. The evaluation content are the parameters or physical phenomena that characterizing the second-level evaluation indicator, which may include both subjective and objective evaluation information.

Table 2
Health evaluation indicator system and evaluation content of CNC machine tools

First-level evaluation indicators Second-level evaluation indicators Evaluation content

Mechanical performance B₁ Quality of key componentsC₁₁ Mainly including strength, static stiffness, local deformation, wear, oxidation, surface deterioration, cracks, porosity, strong stress concentration points and other defects

Dynamic characteristicsC₁₂ Spindle no-load power, feed shaft torque, vibration, friction, etc.

Machining performance B₂ Accuracy C₂₁ Roughness, positioning accuracy, re-orientation accuracy, indexing accuracy, resolution, pulse equivalent, etc.

Coordinate axisC₂₂ Number of controllable axis, number of linked axis

ToolC₂₃ Tool magazine capacity, tool change time, tool deformation, tool wear

Kinematic performance B₃ Speed C₃₁ Machining speed, spindle speed, acceleration, angular speed, feed speed, etc.

Range of motionC₃₂ Coordinate travel, spindle swing angle range

Green performance B₄ Noise pollution C₄₁ Noise ringing intensity, unusual noise, etc.

Air pollution C₄₂ Toxic gas emissions, smoke concentration, dust particles, etc.

Waste pollutionC₄₃ Waste cutting treatment, cutting recycling (solid waste such as leftovers, cutting fluid, etc.)

Oil pollution C₄₄ Oil, suspended matter, toxic compounds, etc.

Green remanufacturing C₄₅ Including the recyclability, reusability and disassembly of CNC machine parts

Resource consumption C₄₆ Material resources, human resources, equipment resources, energy efficiency, etc.

Maintenance performance B₅ Maintenance efficiency C₅₁ Average repair time, median repair time, maximum repair time, etc.

Maintenance costs C₅₂ Maintenance costs

Maintenance degree C₅₃ Removability, versatility of replacement parts, ease of repair, and effectiveness of repair

First-level evaluation indicators	Second-level evaluation indicators	Evaluation content
Mechanical performance B₁	Quality of key componentsC₁₁	Mainly including strength, static stiffness, local deformation, wear, oxidation, surface deterioration, cracks, porosity, strong stress concentration points and other defects
	Dynamic characteristicsC₁₂	Spindle no-load power, feed shaft torque, vibration, friction, etc.
Machining performance B₂	Accuracy C₂₁	Roughness, positioning accuracy, re-orientation accuracy, indexing accuracy, resolution, pulse equivalent, etc.
	Coordinate axisC₂₂	Number of controllable axis, number of linked axis
	ToolC₂₃	Tool magazine capacity, tool change time, tool deformation, tool wear
Kinematic performance B₃	Speed C₃₁	Machining speed, spindle speed, acceleration, angular speed, feed speed, etc.
	Range of motionC₃₂	Coordinate travel, spindle swing angle range
Green performance B₄	Noise pollution C₄₁	Noise ringing intensity, unusual noise, etc.
	Air pollution C₄₂	Toxic gas emissions, smoke concentration, dust particles, etc.
	Waste pollutionC₄₃	Waste cutting treatment, cutting recycling (solid waste such as leftovers, cutting fluid, etc.)
	Oil pollution C₄₄	Oil, suspended matter, toxic compounds, etc.
	Green remanufacturing C₄₅	Including the recyclability, reusability and disassembly of CNC machine parts
	Resource consumption C₄₆	Material resources, human resources, equipment resources, energy efficiency, etc.
Maintenance performance B₅	Maintenance efficiency C₅₁	Average repair time, median repair time, maximum repair time, etc.
	Maintenance costs C₅₂	Maintenance costs
	Maintenance degree C₅₃	Removability, versatility of replacement parts, ease of repair, and effectiveness of repair

4 Health status evaluation method based on grey clustering and fuzzy comprehensive evaluation

In the health evaluation indicator system of CNC machine tools, the importance of each indicator is different, so the weight coefficient is used to express the relative importance of each indicator. The methods of calculating weight coefficients mainly include subjective assignment method and objective assignment method. Subjective assignment method is to determine weight through experts’ knowledge and experience, such as Analytic Hierarchy Process (AHP) [25], Rank Correlation Analysis Method, etc. Objective assignment method is to determine the weight based on the specific data of the indicators, such as Entropy Weight Method [26] and Grey Relational Analysis.

Among the methods mentioned above, in order to more accurately determine the health status of CNC machine tools, based on the widespread use and maturity of Analytic Hierarchy Process and the practicality and effectiveness of the use of the Entropy Weight Method, Analytic Hierarchy Process, Entropy Weight Method and the combination weighting method of combining these two are used to determine the weight coefficients.

Among the current various evaluation methods, grey clustering is an analytical judgment method of grey system theory. Grey clustering can be divided into grey relational clustering and clustering based on whitening weight function according to the clustering object. Grey relational clustering is mainly used to classify similar factors, which makes complex systems simplified. Grey whitening weight function clustering mainly examines whether the observation indicators belong to different categories set in advance, and the uncertainty in the analysis is taken into account, in order to achieve a specific classification of the observation indicators.

The fuzzy comprehensive evaluation method is a fuzzy decision making method using fuzzy mathematical knowledge, which applies the principle of fuzzy relationship set to make a comprehensive evaluation of the object affected by several factors. Since each performance and evaluation indicator of CNC machine tools contains qualitative and quantitative information, the relationship between indicators has uncertainty. The degree of influence of each performance indicator on the health status of CNC machine tools is often a fuzzy number, and the status of the in-service CNC machine tools changes for the randomness and fuzziness. Based on the above mentioned situation, the grey clustering analysis and fuzzy comprehensive evaluation was used to evaluate the health status of in-service CNC machine tools in this study.

In order to solve the problem of implementing health evaluation of in-service CNC machine tools, Entropy Weight Method, Analytic Hierarchy Process, combination weighting method, grey clustering analysis and fuzzy comprehensive evaluation method are introduced to establish a health status evaluation method of in-service CNC machine tools, and the implementation process of the method is shown in Fig. 1.

Fig. 1

Flow chart of health evaluation.

4.1 Calculation method of weight coefficient

4.1.1 Entropy weight method

If there are m evaluation objects and n evaluation indicators, the values of the evaluation objects to the evaluation indicators form an evaluation matrix X = (x_ij) _m×n. $X = [\begin{matrix} x_{11} & x_{12} & \dots & x_{1 n} \\ x_{21} & x_{22} & \dots & x_{2 n} \\ ⋮ & ⋮ & ⋮ \\ x_{m 1} & x_{m 2} & \dots & x_{mn} \end{matrix}]$ (1)

Since each evaluation indicator has its own unit, the data has differences in dimension and order of magnitude, and the evaluation matrix needs to be standardized to obtain the matrix P = (r_ij) _m×n, r_ij is the standard value of the j^th evaluation object on the i^th evaluation indicator, and r_ij ∈ [0, 1]. Among these indicators, to which the bigger the better, the r_ij is as follows. $r_{ij} = \frac{x_{ij} - min_{j} (x_{ij})}{max_{j} (x_{ij}) - min_{j} (x_{ij})}$ (2)

While, the smaller the better, the r_ij is as follows. $r_{ij} = \frac{max_{j} (x_{ij}) - x_{ij}}{max_{j} (x_{ij}) - min_{j} (x_{ij})}$ (3)

According to the definition of entropy, the probability of occurrence of the j^th evaluation object under the i^th indicator is as follows. $u_{ij} = r_{ij} / \sum_{i = 1}^{m} r_{ij}, i = 1, 2, \dots, m; j = 1, 2, \dots, n$ (4)

Where 0 ⩽ u_ij ⩽ 1.

The information entropy of the j^th indicator is as follows. $H_{j} = - k \sum_{i = 1}^{m} (u_{ij} ln u_{ij}), j = 1, 2, \dots, n$ (5)

Where k = 1/lnm, 0 ⩽ H_j ⩽ 1.

The weight value of the j^th indicator is as follows. $S_{j} = \frac{1 - H_{j}}{n - \sum_{j = 1}^{n} H_{j}}, j = 1, 2, \dots, n$ (6)

Where $0 < S_{j} < 1, \sum_{j = 1}^{n} S_{j}$ .

4.1.2 Analytic hierarchy process

Analytical Hierarchy Process (AHP) is a combination of qualitative and quantitative, systematic and hierarchical analysis method [27]. The basic steps are as follows.

(1) Building the hierarchical model structure

On the basis of in-depth analysis of the problem, we can separate various relevant factors into several levels from top to bottom according to their different attributes.

(2) Constructing paired comparison matrix

For any two factors C_i and C_j, a_ij can be used to represent the ratio of the degree of influence C_i and C_j have on the upper layer, the 1∼9 scale method can be applied to measure a_ij, and the scale as shown in Table 3. The scale can be given by experts or engineers based on the nature of the indicators, their relative relationship and practical experience. Therefore, the paired comparison matrix (also called a judgment matrix) A = (a_ij) _n×n can be obtained, and the matrix A = (a_ij) _n×n has the following properties: $a_{ij} > 0, a_{ji} = \frac{1}{a_{ij}}, a_{ii} = 1 (i, j = 1, 2, \cdot \cdot \cdot, n)$ .

Table 3
Meaning of 1∼9 scale

Scale Meaning

1 Factor i and j are equally important

3 Factor i is slightly more important than j

5 Factor i is strongly more important than j

7 Factor i is very strongly more important than j

9 Factor i is extremely more important than j

2, 4, 6, 8 Represents the middle value of the above adjacent judgments

Reciprocal If the ratio of the importance of factor i to factor j is a_ji, then the ratio of the importance of factor i to factor j is $a_{ji} = \frac{1}{a_{ij}}$

Scale	Meaning
1	Factor i and j are equally important
3	Factor i is slightly more important than j
5	Factor i is strongly more important than j
7	Factor i is very strongly more important than j
9	Factor i is extremely more important than j
2, 4, 6, 8	Represents the middle value of the above adjacent judgments
Reciprocal	If the ratio of the importance of factor i to factor j is a_ji, then the ratio of the importance of factor i to factor j is $a_{ji} = \frac{1}{a_{ij}}$

(3) Calculating the relative weight vector

For the general judgment matrix A: $AG = λ_{max} G$ (7)

Where λ_max (λ_max = n) is the largest characteristic root of A, G is the feature vector corresponding to λ_max, and G is the desired weight vector.

(4) Consistency check

In order to ensure the reliability of the calculation results, it is necessary to check the consistency of the judgment matrix, and the steps are as follows:

Calculating the consistency indicator: $CI = \frac{λ_{max} - n}{n - 1}$ (8)

Finding the corresponding random consistency indicator: RI, which is usually given by practical experience, as shown in Table 4.

Calculating consistency ratio:

Table 4

Consistency indicator

n	1	2	3	4	5	6	7	8	9	10	11
RI	0.00	0.00	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49	1.51

CR = \frac{CI}{RI}

(9)

When CR < 0.1, the consistency of the judgment matrix is considered acceptable, which indicates that the weight distribution is reasonable. Otherwise, the corresponding judgment matrix is considered as not reliable and should be modified appropriately.

4.1.3 Combination weighting method

In order to obtain the combination weight with subjective and objective meanings. Assuming that the weight vector determined by AHP is G_i, and the Entropy Weight Method is S_i. Finally, the combination weight vector W_i is obtained by the following equation [28]: $W_{i} = G_{i} S_{i} / (\sum_{i = 1}^{n} G_{i} S_{i})$ (10)

4.2 Grey clustering analysis

The process of grey clustering analysis [29] is as follows: If there are m evaluation objects and n evaluation indicators, the values of the evaluation objects to the evaluation indicators form an evaluation matrix X = (x_ij) _m×n is as follows. $X = [\begin{matrix} x_{11} & x_{12} & \dots & x_{1 n} \\ x_{21} & x_{22} & \dots & x_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{t 1} & x_{t 2} & \dots & x_{tn} \\ ⋮ & ⋮ & \dots & ⋮ \\ x_{m 1} & x_{m 2} & \dots & x_{mn} \end{matrix}]$ (11)

The original data are normalized according to equation (2) or (3) to obtain the normalization matrix Y that is as following. $Y = [\begin{matrix} y_{11} & y_{12} & \dots & y_{1 n} \\ y_{21} & y_{22} & \dots & y_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ y_{t 1} & y_{t 2} & \dots & y_{tn} \\ ⋮ & ⋮ & \dots & ⋮ \\ y_{m 1} & y_{m 2} & \dots & y_{mn} \end{matrix}]$ (12)

Assuming that f (x) is a linear monotonic function of x, x represents the grey scale, and the range of f (x) is between 0 and 1, then f (x) is called a whitening weight function, also called a grey clustering function [30]. The evaluation object is evaluated according to the standard value of the evaluation objects to the evaluation indicators, $f_{j}^{k} (\cdot) (1 ⩽ j ⩽ m, 1 ⩽ k ⩽ s)$ is the k subclass whitening weight function of the evaluation indicator j. Since the health status of CNC machine tools is divided into 5 grades, 5 grey classes are obtained, i.e. s = 5. In summary, the 5 grey class whitening weight functions obtained finally are defined as follows: $f_{j}^{1}, f_{j}^{2}, f_{j}^{3}, f_{j}^{4}$ and $f_{j}^{5}$ respectively indicates that the machine tools status is fault, micro-fault, qualification, sub-health and health. The steps for solving the whitening weight function are as follows:

For the indicator j, its value range is set as [a_j, b_j]. The number of grey class s is divided according to the evaluation requirements, and the turning points $λ_{j}^{1}$ , $λ_{j}^{s}$ of grey class 1 and grey class s and the center points $λ_{j}^{2}, λ_{j}^{3}, \dots λ_{j}^{s - 1}$ of grey class k are determined respectively.

For grey class 1 and grey class s, x is considered as an observation of indicator j, when $x \in [a_{j}, λ_{j}^{2}]$ or $x \in [λ_{j}^{s - 1}, b_{j}]$ , the membership degree $f_{j}^{1} (x)$ and $f_{j}^{s} (x)$ that belong to grey class 1 and grey class s can be calculated by the following two equations. $f_{j}^{1} (x) = {\begin{matrix} 0, & x \notin [a_{j}, λ_{j}^{2}] \\ 1, & x \in [a_{j}, λ_{j}^{1}] \\ \frac{λ_{j}^{2} - x}{λ_{j}^{2} - λ_{j}^{1}}, & x \in [λ_{j}^{1}, λ_{j}^{2}] \end{matrix}$ (13) $f_{j}^{s} (x) = {\begin{matrix} 0, & x \notin [λ_{j}^{s - 1}, b_{j}] \\ \frac{x - λ_{j}^{s - 1}}{λ_{j}^{s} - λ_{j}^{s - 1}}, & x \in [λ_{j}^{s - 1}, λ_{j}^{s}] \\ 1, & x \in [λ_{j}^{s}, b_{j}] \end{matrix}$ (14)

$λ_{j}^{k} (1 ⩽ j ⩽ m, 1 ⩽ k ⩽ s)$ represents the center point of the grey class k, the point $(λ_{j}^{k}, 1)$ , the center point $(λ_{j}^{k - 1}, 0)$ of grey class k-1 and the center point $(λ_{j}^{k + 1}, 0)$ of grey class k + 1 are connected simultaneously, the triangle whitening weight function $f_{j}^{k} (\cdot) (1 ⩽ j ⩽ m, 1 ⩽ k ⩽ s)$ of the indicator j with respect to the grey class k is obtained, as shown in Fig. 2 [29].

Fig. 2

Schematic diagram of the whitening weight function.

For an observation value x of indicator j, the membership degree $f_{j}^{k} (x)$ that belong to grey class k can be calculated by the following equation. $f_{j}^{k} (x) = {\begin{matrix} 0, & x \notin [λ_{j}^{k - 1}, λ_{j}^{k + 1}] \\ \frac{x - λ_{j}^{k - 1}}{λ_{j}^{k} - λ_{j}^{k - 1}}, & x \in [λ_{j}^{k - 1}, λ_{j}^{k}] \\ \frac{λ_{j}^{k + 1} - x}{λ_{j}^{k + 1} - λ_{j}^{k}}, & x \in [λ_{j}^{k}, λ_{j}^{k + 1}] \end{matrix}$ (15)

The combination weight W_j of each indicator obtained by equation (10) is substituted into the following equation.

σ_{i}^{k} = \sum_{j = 1}^{m} f_{j}^{k} (x_{ij}) \cdot W_{j}

(16)

Where $σ_{i}^{k}$ is the clustering coefficient of object i with respect to grey class k, and $f_{j}^{k} (x_{ij})$ is the k subclass whitening weight function of indicator j.

The following equation is used to determine which grey class k the object i belongs to, and the grey scale represented by the corresponding k indicates the current health status of the object.

max_{1 ⩽ k ⩽ s} {σ_{i}^{k}} = max (σ_{i}^{1}, σ_{i}^{2}, \dots σ_{i}^{s})

(17)

4.3 Fuzzy comprehensive evaluation

4.3.1 Single-factor fuzzy comprehensive evaluation

The general steps for single-factor fuzzy comprehensive evaluation are as follows.

Building the factor set: U ={ u₁, u₂, ⋯ , u_n };

Building the evaluation set: V ={ v₁, v₂, ⋯ , v_m }

A single-factor evaluation was conducted to obtain r_i ={ r_i1, r_i2, ⋯ , r_im }, where r_i1 is the membership degree of u_i to v₁.

Building the comprehensive evaluation matrix: $R = [\begin{matrix} r_{11} & r_{12} & \dots & r_{1 n} \\ r_{21} & r_{22} & \dots & r_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ r_{m 1} & r_{m 2} & \dots & r_{mn} \end{matrix}]$ (18)

Comprehensive evaluation:

B = W \circ R = (b_{1}, b_{2}, \dots, b_{m})

(19)

Where W is the weight of each factor, $b_{j} = \sum_{i = 1}^{n} w_{i} r_{ij} (j = 1, 2, \dots, m)$ , evaluation is conducted based on the maximum membership degree principle (MMDP).

4.3.2 Multi- level fuzzy comprehensive evaluation

If there are multiple levels of factor sets, each level factor should be judged, that is, multi-level fuzzy comprehensive evaluation such as second-level fuzzy evaluation and third- level fuzzy evaluation should be used.

The second-level fuzzy comprehensive evaluation is as following. $M = W_{i} \circ B_{i}$ (20)

Where B_i is the single-factor evaluation result of the i^th factor in the first-grade factor set, W_i is the weight of the i^th factor, and M is the fuzzy comprehensive evaluation result among the factors. The corresponding evaluation is obtained according to the maximum membership degree principle (MMDP).

5 Application research on health status evaluation method

CNC machine tools have numerous types and various structures. As large high-grade CNC machine tools, gantry machine tools are widely used in aerospace, ships, metallurgy and other industries. This section took a gantry CNC machine tool that can be used for the machining propellers as the object to conduct engineering application research on the proposed health evaluation method. The gantry CNC machine tool studied in this paper is shown in Fig. 3, whose health status has a great influence on the production safety of the factory.

Fig. 3

Gantry CNC machine tool.

5.1 Data acquisition and processing

Due to the different models and uses of various CNC machine tools, the working environment and maintenance conditions are also inconsistent, and it is relatively difficult to develop a unified rating standard. Therefore, it should be evaluated based on the actual working conditions of each CNC machine tools. In view of the fact that the health evaluation indicators of CNC machine tools include qualitative and quantitative indicators, as shown in Table 2. Each indicator has different dimensions and cannot be calculated uniformly. Therefore, the expert scoring method is the most ideal. For the gantry CNC machine tool shown in Fig. 3, 50 experts with different qualifications and working experience used qualitative evaluation to evaluate second-level evaluation indicators shown in Table 2 to obtain an evaluation result. The results of each evaluation were organized to obtain each evaluation matrix, as shown in Tables 5 to 9.

Table 5
Comprehensive evaluation of mechanical performance

Evaluation indicators Evaluation grades

Poor Slightly poor General Good Excellent

Quality of key components 0.02 0.08 0.06 0.58 0.26

Dynamic characteristics 0.02 0.06 0.08 0.38 0.46

Evaluation indicators	Evaluation grades
Quality of key components	0.02	0.08	0.06	0.58	0.26
Dynamic characteristics	0.02	0.06	0.08	0.38	0.46

Table 6

Comprehensive evaluation of machining performance

Evaluation indicators	Evaluation grades
	Poor	Slightly poor	General	Good	Excellent
Accuracy	0.06	0.04	0.08	0.58	0.24
Coordinate axis	0.04	0.04	0.08	0.22	0.62
Tool	0.04	0.06	0.06	0.34	0.5

Table 7

Comprehensive evaluation of kinematic performance

Evaluation indicators	Evaluation grades
	Poor	Slightly poor	General	Good	Excellent
Speed	0.02	0.06	0.08	0.56	0.28
Range of motion	0.02	0.08	0.1	0.36	0.44

Table 8

Comprehensive evaluation of green performance

Evaluation indicators	Evaluation grades
	Poor	Slightly poor	General	Good	Excellent
Noise pollution	0.06	0.08	0.1	0.52	0.24
Air pollution	0.06	0.08	0.14	0.44	0.28
Waste pollution	0.08	0.06	0.14	0.3	0.42
Oil pollution	0.06	0.08	0.12	0.26	0.48
Green remanufacturing	0.06	0.06	0.1	0.54	0.24
Resource consumption	0.08	0.08	0.12	0.3	0.44

Table 9

Comprehensive evaluation of maintenance performance

Evaluation indicators	Evaluation grades
	Poor	Slightly poor	General	Good	Excellent
Maintenance efficiency	0.02	0.04	0.18	0.3	0.46
Maintenance costs	0.02	0.06	0.16	0.48	0.28
Maintenance degree	0.02	0.06	0.12	0.24	0.56

5.2 Health status evaluation

According to the health evaluation flow chart shown in Fig. 1, the evaluation indicators were evaluated first, and then the health status of the whole gantry CNC machine tool was evaluated on the basis of the health status of the evaluation indicators.

5.2.1 Weight calculation based on entropy weight method

The evaluation data in Table 5 to Table 9 were normalized to obtain the matrices, which are as follows.

$U_{B_{1}} = [\begin{matrix} 0.5 & 0.5714 & 0.4286 & 0.6237 & 0.3611 \\ 0.5 & 0.4286 & 0.5714 & 0.3763 & 0.6389 \end{matrix}]$ $U_{B_{2}} = [\begin{matrix} 0.4286 & 0.2857 & 0.3636 & 0.5088 & 0.1765 \\ 0.2857 & 0.2857 & 0.3636 & 0.193 & 0.4559 \\ 0.2857 & 0.4286 & 0.2727 & 0.2982 & 0.3676 \end{matrix}]$ $U_{B_{3}} = [\begin{matrix} 0.5 & 0.4286 & 0.4444 & 0.6087 & 0.3889 \\ 0.5 & 0.5714 & 0.5556 & 0.3913 & 0.6111 \end{matrix}]$ $U_{B_{4}} = [\begin{matrix} 0.15 & 0.1818 & 0.1389 & 0.2203 & 0.1143 \\ 0.15 & 0.1818 & 0.1944 & 0.1864 & 0.1333 \\ 0.2 & 0.1364 & 0.1944 & 0.1271 & 0.2 \\ 0.15 & 0.1818 & 0.1667 & 0.1102 & 0.2286 \\ 0.15 & 0.1364 & 0.1389 & 0.2288 & 0.1143 \\ 0.2 & 0.1818 & 0.1667 & 0.1271 & 0.2093 \end{matrix}]$ $U_{B_{5}} = [\begin{matrix} 0.3333 & 0.25 & 0.3913 & 0.2941 & 0.3538 \\ 0.3333 & 0.375 & 0.3478 & 0.4706 & 0.2154 \\ 0.3333 & 0.375 & 0.2609 & 0.2353 & 0.4308 \end{matrix}]$

The information entropy can be obtained by equation (9), which are as follows. $H_{1} = {1, 0.9852, 0.9852, 0.9554, 0.9436}$ $H_{2} = {0.9821, 0.9821, 0.9922, 0.9304, 0.9395}$ $H_{3} = {1, 0.9852, 0.9911, 0.9656, 0.9641}$ $H_{4} = {0.9946, 0.9952, 0.9948, 0.9774, 0.9773}$ $H_{5} = {1, 0.9851, 0.9876, 0.9604, 0.9658}$

The weights can be obtained by Equation (10), which are as follows. $S_{1} = {0, 0.1131, 0.1131, 0.3418, 0.4321}$ $S_{2} = {0.1029, 0.1029, 0.0448, 0.4009, 0.3485}$ $S_{3} = {0, 0.1571, 0.0951, 0.3657, 0.3821}$ $S_{4} = {0.0894, 0.0786, 0.0855, 0.3724, 0.3741}$ $S_{5} = {0, 0.1478, 0.1224, 0.3918, 0.3379}$

5.2.2 Calculation of each indicator’s weight based on AHP

AHP was used to calculate the weights of each level of indicators. The judgment matrix was established for the health indicators on the basis of the indicator system in Table 2. The weights of the indicators at each level were calculated. The weight values of the first-level indicators and the first two second-level indicators are respectively shown in Tables 10 and 11.

Table 10
Judgment matrix R (A - B)

B₁ B₂ B₃ B₄ B₅ G

B₁ 1 1/2 2 3 4 0.2848

B₂ 2 1 1 4 3 0.3187

B₃ 1/2 1 1 3 3 0.2258

B₄ 1/3 1/4 1/3 1 1/3 0.0661

B₅ 1/4 1/3 1/3 3 1 0.1046

	B₁	B₂	B₃	B₄	B₅	G
B₁	1	1/2	2	3	4	0.2848
B₂	2	1	1	4	3	0.3187
B₃	1/2	1	1	3	3	0.2258
B₄	1/3	1/4	1/3	1	1/3	0.0661
B₅	1/4	1/3	1/3	3	1	0.1046

Table 11

Judgment matrix R₁ (B₁ - C)

	C₁₁	C₁₂	G₁
C₁₁	1	2	0.6667
C₁₂	1/2	1	0.3333

In the same way, the weights of other indicators can be obtained, which are:

G₁(B₁-C) = (0.6667,0.3333), G₂(B₂-C)=(0.6267, 0.0936,0.2797), G₃(B₃-C) = (0.75,0.25), G₄(B₄-C) = (0.0964,0.0936,0.2440,0.0859,0.3350,0.1451), G₅(B₅-C) = (0.2684,0.1172,0.6144), as shown in Table 12.

Table 12

Evaluation indicator weight values

First-level evaluation indicators	First-level indicator weights	Second-level evaluation indicators	Second -level indicator weights
Mechanical performance	0.2848	Quality of key components	0.6667
		Dynamic characteristics	0.3333
Machining performance	0.3187	Accuracy	0.6267
		Coordinate axis	0.0936
		Tool	0.2797
Kinematic performance	0.2258	Speed	0.75
		Range of motion	0.25
Green performance	0.0661	Noise pollution	0.0964
		Air pollution	0.0936
		Waste pollution	0.244
		Oil pollution	0.0859
		Green remanufacturing	0.335
		Resource consumption	0.1451
Maintenance performance	0.1046	Maintenance efficiency	0.2684
		Maintenance costs	0.1172
		Maintenance degree	0.6144

5.2.3 Grey clustering analysis

After the data was normalized, mechanical performance was taken as an example to obtain a normalization matrix by equation (2), and the obtained normalization matrix Y₁ is as following. $Y_{1} = [\begin{matrix} 0 & 0.1071 & 0.0714 & 1 & 0.4286 \\ 0 & 0.0909 & 0.1364 & 0.8182 & 1 \end{matrix}]$

The whitening weight function can be obtained by the whitening weight function Equations (13), (14) and (15). ${\begin{matrix} f_{j}^{1} (x) = {\begin{matrix} 0, & x \notin [0, 0.334] \\ 1, & x \in [0, 0.167] \\ \frac{0.334 - x}{0.334 - 0.167}, & x \in [0.167, 0.334] \end{matrix} \\ f_{j}^{2} (x) = {\begin{matrix} 0, & x \notin [0.167, 0.5] \\ \frac{x - 0.167}{0.334 - 0.167}, & x \in [0.167, 0.334] \\ \frac{0.5 - x}{0.5 - 0.334}, & x \in [0.334, 0.5] \end{matrix} \\ f_{j}^{3} (x) = {\begin{matrix} 0, & x \notin [0.334, 0.667] \\ \frac{x - 0.334}{0.5 - 0.334}, & x \in [0.334, 0.5] \\ \frac{0.667 - x}{0.667 - 0.5}, & x \in [0.5, 0.667] \end{matrix} \\ f_{j}^{4} (x) = {\begin{matrix} 0, & x \notin [0.5, 0.834] \\ \frac{x - 0.5}{0.667 - 0.5}, & x \in [0.5, 0.667] \\ \frac{0.834 - x}{0.834 - 0.667}, & x \in [0.667, 0.834] \end{matrix} \\ f_{j}^{5} (x) = {\begin{matrix} 0, & x \notin [0.667, 1] \\ \frac{x - 0.667}{0.834 - 0.667}, & x \in [0.667, 0.834] \\ 1, & x \in [0.834, 1] \end{matrix} \end{matrix}$ (21)

Finally, the normalized data in the matrix and the weights calculated by the entropy weight method were brought into Equation (16) to obtain the clustering coefficient. The object i is decided to belong to the grey class k^* by Equation (17). For example, when i = 1, there is: $\begin{matrix} σ_{1}^{1} = \sum_{j = 1}^{5} f_{j}^{1} (x_{1 j}) S_{1} = f_{1}^{1} (0) \times 0 + f_{2}^{1} (0.1071) \\ \times 0.1131 + f_{3}^{1} (0.0714) \times 0.1131 + f_{4}^{1} (1) \\ \times 0.3418 + f_{5}^{1} (0.4286) \times 0.4321 = 0.2262 \end{matrix}$

Similarly, we can obtain: $σ_{1}^{2} = 0.1858, σ_{1}^{3} = 0.2462, σ_{1}^{4} = 0, σ_{1}^{5} = 0.3418$

Therefore, there is: $\begin{matrix} σ_{1} = & (σ_{1}^{1}, σ_{1}^{2}, σ_{1}^{3}, σ_{1}^{4}, σ_{1}^{5}) \\ = (0.2262, 0.1858, 0.2462, 0, 0.3418) \end{matrix}$

Same method was used to calculate:

When i = 2, there is: σ₂ = $(σ_{2}^{1}$ , $σ_{2}^{2}$ , $σ_{2}^{3}$ , $σ_{2}^{4}$ , $σ_{2}^{5}$ ) = (0 . 2262, 0, 0, 0 . 0323, 0 . 7416)

Based on the above results, the grey clustering coefficients of mechanical performance can be obtained as shown in Table 13.

Table 13

Grey clustering results of mechanical performance

	Fault	Micro-fault	Qualification	Sub-healthy	Healthy	Clustering result
Quality of key components	0.2262	0.1858	0.2462	0	0.3418	Healthy
Dynamic characteristics	0.2262	0	0	0.0323	0.7416	Healthy

It can be seen from Table 13 that the clustering degree of key component quality and dynamic characteristics of “healthy” are respectively 0.3418 and 0.7416, while the clustering degree of “fault” of these two indicators is 0.2262. As shown in Fig. 4.

Fig. 4

Clustering degree of mechanical performance indicators.

Figure 4 shows that the clustering degree of dynamic characteristics in “healthy” is significantly greater than that of key components in this status, while the probability of both being in “fault” is the same, both are 0.2262. It indicates that both the dynamic characteristics and the quality of key components have declined with the longer service time, and the decline trend of the key components quality is relatively large.

According to the above method, the grey clustering results of machining performance, kinematic performance, green performance and maintenance performance can be obtained, which are respectively shown in Tables 14 to 17.

Table 14

Grey clustering results of machining performance

	Fault	Micro-fault	Qualification	Sub-healthy	Healthy	Clustering result
Accuracy	0.2506	0.2721	0.0764	0	0.4009	Healthy
Coordinate axis	0.3075	0.344	0	0	0.3485	Healthy
Tool	0.2506	0	0.0355	0.3654	0.3485	Sub-healthy

Table 15

Grey clustering results of kinematic performance

	Fault	Micro-fault	Qualification	Sub-healthy	Healthy	Clustering result
Speed	0.2522	0.0426	0.3395	0	0.3657	Healthy
Range of motion	0.2388	0.0134	0	0.0537	0.6941	Healthy

Table 16

Grey clustering results of green performance

	Fault	Micro-fault	Qualification	Sub-healthy	Healthy	Clustering result
Noise pollution	0.2535	0.245	0.1291	0	0.3724	Healthy
Air pollution	0.2312	0.0223	0.1974	0.1767	0.3724	Healthy
Waste pollution	0.2252	0.0283	0.0007	0.3717	0.3741	Healthy
Oil pollution	0.2535	0.0534	0.319	0	0.3741	Healthy
Green remanufacturing	0.2535	0.2817	0.0924	0	0.3724	Healthy
Resource consumption	0.2535	0	0.1247	0.2477	0.3741	Healthy

Table 17

Grey clustering results of maintenance performance

	Fault	Micro-fault	Qualification	Sub-healthy	Healthy	Clustering result
Maintenance efficiency	0.1478	0.1006	0.0936	0.32	0.3379	Healthy
Maintenance costs	0.1696	0.1006	0.206	0.1319	0.3918	Healthy
Maintenance degree	0.2569	0.2319	0.1722	0	0.3379	Healthy

It can be seen from Table 14 that the status of accuracy and coordinate axis are “healthy” and their clustering degree are respectively 0.4009 and 0.3485. The clustering degree of the coordinate axis in “fault” is 0.3075, which is greater than the degree of accuracy in “fault” of 0.2506. The tool status is “sub-healthy”, and the probability of being in “sub-healthy” and “healthy” is close with clustering degrees are respectively 0.3654 and 0.3485. As shown in Fig. 5.

Fig. 5

Clustering degree of machining performance indicators.

Figure 5 shows that the three indicators of machining performance have declined with the increase of service time. Among them, the degree of coordinate axis in “fault", “micro-fault” and “healthy” status is relatively large proportion, and the three data show that the coordinate axis are prone to fault. The tool indicator is stable with the probability of being in a “fault” status is small. The accuracy indicator is also prone to fault relative to the tool indicator.

As can be seen from Table 15, the clustering degree of “healthy” for speed and range of motion are 0.3657 and 0.6941, and the degree of “fault” between the two indicators are close, which are 0.2522 and 0.2388, as shown in Fig. 6.

Fig. 6

Clustering degree of kinematic performance indicators.

Figure 6 shows that the degree of speed in the “healthy” status is significantly greater than the degree of range of motion. Although both indicators have declined over a long period of service, the trend of the speed indicator to fault is more obvious.

From Table 16, it can be seen that the six indicators of green performance, noise pollution, air pollution, waste pollution, oil pollution, green remanufacturing, and resource consumption, are in “healthy” health status, and their clustering degrees are very close, which are 0.3724, 0.3724, 0.3741, 0.3741, 0.3724 and 0.3741 respectively. The six indicators are also very similar in the degree of “fault", which are 0.2535, 0.2312, 0.2252, 0.2535, 0.2535 and 0.2535 respectively, as shown in Fig. 7.

Fig. 7

Clustering degree of green performance indicators.

Figure 7 clearly shows that the six indicators of green performance are approximately the same degree of “healthy” and close to the degree of “fault", indicating that the six indicators are close to the trend of decrease. Among them, noise pollution and green remanufacturing are prone to fault and the indicator of resource consumption is the least prone to fault.

From Table 17, we can see that the indicators maintenance efficiency, maintenance costs and maintenance degree are in “healthy” status, and their clustering degrees are respectively 0.3379, 0.3918, and 0.3379. The indicator maintenance degree is in “fault” degree is 0.2569, which is greater than the degree of maintenance efficiency and maintenance costs, as shown in Fig. 8.

Fig. 8

Clustering degree of maintenance indicators.

Figure 8 shows that with the in-service time of gantry CNC machine increases, the three indicators of maintenance efficiency, maintenance cost and maintenance degree are decreased. The maintenance degree has decreased by a relatively large extent, and the maintenance efficiency and maintenance costs are relatively stable.

5.2.4 The results and discussion of fuzzy comprehensive evaluation

(1) Determination of factor set U and evaluation set V

Based on the health evaluation indicator system of CNC machine tools, the factor set U is established, that is, U = {u₁, u₂, ⋯ , u_n}= {Mechanical performance, Machining performance, Kinematic performance, Green performance, Maintenance performance}. According to the relevant regulations and standards of the industry, as well as the characteristics of each indicator, the evaluation set is given by the relevant experts for the possible evaluation structure of each factor, with 5 evaluation grades, that is, V = {v₁, v₂, v₃, v₄, v₅}= {Fault, Micro-fault, Qualification, Sub-healthy, Healthy}.

(2) Single-factor fuzzy comprehensive evaluation

The data from Tables 13 to 17 were used to construct single-factor evaluation matrices, which are as follows. $R_{B_{1}} = [\begin{matrix} 0.2262 & 0.1858 & 0.2462 & 0 & 0.3418 \\ 0.2262 & 0 & 0 & 0.0323 & 0.7416 \end{matrix}]$ $R_{B_{2}} = [\begin{matrix} 0.2506 & 0.2721 & 0.0764 & 0 & 0.4009 \\ 0.3075 & 0.344 & 0 & 0 & 0.3485 \\ 0.2506 & 0 & 0.0355 & 0.3654 & 0.3485 \end{matrix}]$ $R_{B_{3}} = [\begin{matrix} 0.2522 & 0.0426 & 0.3395 & 0 & 0.3657 \\ 0.2388 & 0.0134 & 0 & 0.0537 & 0.6941 \end{matrix}]$ $R_{B_{4}} = [\begin{matrix} 0.2535 & 0.245 & 0.1291 & 0 & 0.3724 \\ 0.2312 & 0.0223 & 0.1974 & 0.1767 & 0.3724 \\ 0.2252 & 0.0283 & 0.0007 & 0.3717 & 0.3741 \\ 0.2535 & 0.0534 & 0.319 & 0 & 0.3741 \\ 0.2535 & 0.2817 & 0.0924 & 0 & 0.3724 \\ 0.2535 & 0 & 0.1247 & 0.2477 & 0.3741 \end{matrix}]$ $R_{B_{5}} = [\begin{matrix} 0.1478 & 0.1006 & 0.0936 & 0.32 & 0.3379 \\ 0.1696 & 0.1006 & 0.206 & 0.1319 & 0.3918 \\ 0.2569 & 0.2319 & 0.1722 & 0 & 0.3379 \end{matrix}]$

From 3.2.2 we get G₁(B₁-C) = (0.6667,0.3333), G₂(B₂-C) = (0.6267,0.0936,0.2797), G₃(B₃-C) = (0.75,0.25), G₄(B₄-C) = (0.0964,0.0936,0.2440,0.0859,0.3350,0.1451), G₅(B₅-C) = (0.2684,0.1172,0.6144). Then by Equation (19), the evaluation results of the first-level evaluation indicator are as follows. $\begin{matrix} B_{1} = (0.2262, 0.1239, 0.1641, 0.0108, 0.4751) \\ B_{2} = (0.2559, 0.2027, 0.0578, 0.1022, 0.3813) \\ B_{3} = (0.2488, 0.0353, 0.2546, 0.0134, 0.4478) \\ B_{4} = (0.2445, 0.1316, 0.1075, 0.1432, 0.3732) \\ B_{5} = (0.2174, 0.1813, 0.1551, 0.1013, 0.3442) \end{matrix}$

According to the maximum membership degree principle (MMDP), the health status of each performance indicator is “healthy”.

Based on the above analysis, the fuzzy degree of each performance of the gantry CNC machine tool in each status grade is shown in Fig. 9.

Fig. 9

Fuzzy degree of each performance indicator.

Figure 9 shows that with the increase of in-service time, the performance of the gantry CNC machine tool is degraded. Although the evaluation results show that the performance indicators are “healthy", the fuzzy degree of each performance in “fault” is greater than 0.2 and the fuzzy degree is very close, which indicates that the probability of fault is still relatively large.

(3) Second-level fuzzy comprehensive evaluation

Based on the above evaluation results B₁, B₂, B₃, B₄, B₅, the evaluation matrix R of the whole health evaluation of the gantry CNC machine tool can be constructed, which is as following. $B = [\begin{matrix} 0.2262 & 0.1239 & 0.1641 & 0.0108 & 0.4751 \\ 0.2559 & 0.2027 & 0.0578 & 0.1022 & 0.3813 \\ 0.2488 & 0.0353 & 0.2546 & 0.0134 & 0.4478 \\ 0.2445 & 0.1316 & 0.1075 & 0.1432 & 0.3732 \\ 0.2174 & 0.1813 & 0.1551 & 0.1013 & 0.3442 \end{matrix}]$

According to the obtained matrix B, the weight vector of each performance of the gantry CNC machine tool calculated by the entropy weight method is S = {0.0035, 0.2089, 0.1901, 0.5838, 0.0137}. From section 3.2.2, it can be seen that the performance weight vector of the gantry CNC machine tool calculated by the AHP is G = {0.2848, 0.3187, 0.2258, 0.0661, 0.1046}. Then the combination weight vector of each performance obtained by Equation (10) is W = {0.0066, 0.4423, 0.2852, 0.2564, 0.0095}. Finally, the whole health status of the gantry CNC machine tool calculated by Equation (20) is as shown in Equation (22). $\begin{matrix} M = WR = {0.0066, 0.4423, 0.2852, 0.2564, 0.0095} \\ [\begin{matrix} 0.2262 & 0.1239 & 0.1641 & 0.0108 & 0.4751 \\ 0.2559 & 0.2027 & 0.0578 & 0.1022 & 0.3813 \\ 0.2488 & 0.0353 & 0.2546 & 0.0134 & 0.4478 \\ 0.2445 & 0.1316 & 0.1075 & 0.1432 & 0.3732 \\ 0.2174 & 0.1813 & 0.1551 & 0.1013 & 0.3442 \end{matrix}] \\ = {0.2504, 0.136, 0.1283, 0.0868, 0.3985} \end{matrix}$ (22)

The above results show that the fuzzy degree of health status of the gantry CNC machine tool’s in “healthy” is 0.3985, in the “fault” status is 0.2504. According to the maximum membership degree principle (MMDP), the gantry CNC machine tool’s health status is assessed as “healthy". The results of the whole health status of the gantry CNC machine tool obtained by (22) is as shown in Table 18, and the fuzzy degree of the whole machine in each status grade is shown in Fig. 10.

Table 18

Health status of gantry CNC machine tool

Fault	Micro-fault	Qualification	Sub-healthy	Healthy	Evaluation result
0.2504	0.136	0.1283	0.0868	0.3985	Healthy

Fig. 10

Health fuzzy degree of gantry CNC machine tool.

The results shown in Table 18 indicate that the health status of the whole gantry CNC machine tool is I level (healthy), which means that the indicators of the equipment are normal and operating situation is healthy for production, and it can be used normally with normal maintenance.

Figure 10 shows that although the fuzzy degree of the “healthy” status of the gantry CNC machine tool is the largest, the proportion of the “fault” status also accounts for a quarter, which shows that the whole machine tool’s health status has shown a declining trend with the service time increasing. The main reason is that the whole machine tool’s performance is in a gradually deteriorating trend.

6 Conclusion and future work

In this paper, a health status evaluation method for CNC machine tools based on grey clustering analysis and fuzzy comprehensive evaluation was proposed. Based on the degree of performance degradation of CNC machine tools, the health status grade of in-service CNC machine tools was divided. Starting from 5 major performance indicators of mechanics, machining, kinematics, green and maintenance, and 16 second-level indicators, the health evaluation indicator system and hierarchical relationships of CNC machine tools with the evaluation content of second-level indicators were established. On this basis, the relative importance of each performance and its indicators were combined, and grey clustering analysis and fuzzy comprehensive evaluation was utilized to make a comprehensive evaluation of performance indicators. Finally, the health status grade of each indicator and the health status grade of CNC machine tools were evaluated respectively.

The empirical analysis of health status evaluation of an in-service gantry CNC machine tool shows that the fuzzy degree of the health status of this type of gantry CNC machine tool in “healthy” is 0.3985, and its health status is assessed as I level (healthy). However, from the performance and its evaluation indicators, the health degree of the mechanical performance and kinematic performance of this type of gantry CNC machine tool are relatively high, while the health degree of machining performance, green performance and maintenance performance is relatively low, which are the main indicators affecting the health level of the whole machine. It indicates that the performance of this type of gantry CNC machine tool is gradually deteriorating with the service time increasing, and its health status has shown a declining trend, and the proposed method can comprehensively evaluate the health degree of gantry CNC machine tools.

Although a health status evaluation method for in-service CNC machine tools was constructed in this study, there are still some shortcomings that need to be improved. First of all, during the implementation of the proposed method in this paper, experts evaluate the second-level indicators based on experience to a certain extent, which is subjective. Many experts with different job roles were organized to conduct the evaluation, which consumed a lot of human resources and time. Therefore, how to make more scientific and reliable scoring standards is an urgent problem that need to be solved. Secondly, performance degradation will cause a decline trend in the status of CNC machine tools, but the exact relationship between performance and the health status grade of CNC machine tools requires continuous improvement and optimization of the evaluation model, and more in-depth research in order to obtain more accurate evaluation results. Finally, different types of CNC machine tools have different technical indicators, and the proposed method cannot be used to evaluate the performance indicators of different types of CNC machine tools with a unified evaluation standard.

Future work is to compensate certain drawbacks that this method may have due to the nature of the particular problem. In addition, how to combine different whitening weight functions and fuzzy comprehensive evaluation methods and apply them to other relative fields needs further study in the future.

Footnotes

Acknowledgments

The study was funded by the National Natural Science Foundation of China (Grant No. 51865008), and the Hainan Provincial Natural Science Foundation of China (Grant No. 521RC496).

References

Zhou

and Ju

, A risk assessment method for early warning of mechanical safety risks, China Safety Science Journal 30(8) (2020), 86–92.

Shao

, Luo

, Zhang

and Wang

, Health Evaluation Method of CNC Machine Tools Based on Fuzzy Grey Clustering and Combined Weighting Method, in IEEE 9th Annual International Conference on CYBER Technology in Automation, Control, and Intelligent Systems (CYBER), (2019), 1364–1371.

Chao

, Wu

, Xie

and Bo

, Health status assessment for the feed system of CNC machine tool based on simulation, IEEE 10th Conference on Industrial Electronics and Applications (ICIEA) (2015).

Zhao

, Huang

, Wang

, Liu

and Liang

, Laplacian Eigenmaps and Mahalanobis Distance based Health Assessment Methodology for Ball Screw, Journal of Mechanical Engineering 53(15) (2017), 125–130.

Zhao

, Data-driven based health assessment methodology for key components of CNC machine tools, Shanghai Jiao Tong University (2018).

, The research of health monitoring of CNC turning center based on multi-sensor information fusion, Hebei University of Technology (2017).

Deng

, Sun

, Li

, Wang

and Xiong

, Hidden markov model based on the heavy-duty CNC health state estimate, Computer Integrated Manufacturing Systems (CIMS) 19(3) (2013), 552–558.

Liao

and Pavel

, Machine tool feed axis health monitoring using Plug-and- Prognose technology, PHM: driving efficient operations and maintenance: Society for Machinery Failure Prevention Technology (MFPT) (2012), 395–411.

Martinova

L.I.

, Kozak

N.V.

, Kovalev

I.A.

and Ljubimov

A.B.

, Creation of CNC system’s components for monitoring machine tool health, The International Journal of Advanced Manufacturing Technology (2021), 1–8.

10.

Deng

, Wang

, Jun

, Xia

and Tao

, Performance degradation modelling and health state evaluation for mechanical and electrical equipment, Computer Integrated Manufacturing Systems 24(9) (2018), 2279–2287.

11.

Pan

, Health State Assessment of CNC Machine Tools Based on TOPSIS and Grey Clustering Theory, Huazhong University of Science and Technology (2018).

12.

Feng

, Health state evaluation and prediction of machine tool based on hidden Markov, China Plant Engineering 24 (2018), 126–127.

13.

Pavel

and Iverson

J.C.

, Machine tool health characterization using condition monitoring and prognostic based technologies, Conference of the Society for Machinery Failure Prevention Technology (MFPT) (2012), 395–411.

14.

Zhou

, CNC Machine Tools of Health Condition Monitoring Based on Instruction Domain, Huazhong University of Science and Technology (2016).

15.

Tobon-Mejia

D.A.

, Medjaher

and Zerhouni

, CNC Machine Tool’s wear diagnostic and prognostic by using dynamic bayesian networks, Mechanical Systems & Signal Processing 28 (2012), 167–182.

16.

Sunny

S.M.N.A.

, Liu

and Shahriar

M.R.

, Remote Monitoring and Online Testing of Machine Tools for Fault Diagnosis and Maintenance Using MTComm in a Cyber-Physical Manufacturing Cloud, IEEE 11th International Conference on Cloud Computing (CLOUD) (2018), 532–539.

17.

Al-Jonid

, Kozlov

, N and Baryshev, et al., Design of an expert system for the prediction and comprehensive diagnosis of CNC machining fault, MATEC Web of Conferences 224 (2018).

18.

Kozlov

A.M.

, Al-jonid

K.M.

, Kozlov

A.A.

and Antar

S.D.

, Product quality management based on CNC machine fault prognostics and diagnosis, IOP Conference Series: Materials Science and Engineering 327(2) (2018).

19.

Canbaz

and Polat

, Fault Detection of CNC Machines from Vibration Signals Using Machine Learning Methods, (2020), 365–374.

20.

Lei

, Research on Maintenance Decision of CNC Based on Health Status Estimation, Huazhong University of Science and Technology (2014).

21.

, Chen

and Chen

, Characteristic Quantity of Safety Grade and Its Calculation Method, China Safety Science Journal 6 (1999), 10–16.

22.

Pan.

, Research on Association Matrix between Multi-performance and Multi-faults of CNC Machine Tool, Huazhong University of Science and Technology (2017).

23.

, Shuang

, Tang

and Hao

, A weighted hidden Markov model approach for continuous-state tool wear monitoring and tool life prediction, International Journal of Advanced Manufacturing Technology 91(1–4) (2016), 1–11.

24.

Martinov

G.M.

, Grigoryev

A.S.

and Nikishechkin

P.A.

, Real-Time Diagnosis and Forecasting Algorithms of the Tool Wear in the CNC Systems, Advances in swarm and computational intelligence: 6th International Conference, ICSI, CCI, Beijing, China, 25-28 (2015), 115–126.

25.

Chang

and Jiang

, Research on the Weight of Coefficient through Analytic Hierarchy Process, Journal of Wuhan University of Technology (Information & Management Engineering) 1 (2007), 153–156.

26.

Zhang

, Qiao

, Ma

and Xu

, Study on evaluation of green products in manufacturing industry, Journal of Shenyang University of Technology (Social Science Edition) 1 (2008), 64–70+82.

27.

, Wei

, Li

, Liang

and Liu

, Fuzzy comprehensive performance evaluation method of rolling linear guide based on improved analytic hierarchy process, Journal of Mechanical Science and Technology 34(7) (2020), 2923–2932.

28.

Liu

, Chen

, Li

and Feng

, Generic product quality evaluation method using AHP and entropy weight, Chinese Journal of Construction Machinery 7(4) (2009), 494–498.

29.

Liu

, Grey System Theory and Its Applications, Eighth edition. Beijing: Science Press (2017), 89–90.

30.

Luo

, Research on the status monitoring and health evaluation method of typical CNC machine tools, University of Electronic Science and Technology of China (2020).

Health status evaluation method of CNC machine tools based on grey clustering analysis and fuzzy comprehensive evaluation

Abstract

Keywords

1 Introduction

2 Related works

3 Health status grading and evaluation system of CNC machine tools

3.1 Health status grading of CNC machine tools

4.1.1 Entropy weight method

4.3.1 Single-factor fuzzy comprehensive evaluation

Table 5 Comprehensive evaluation of mechanical performance Evaluation indicators Evaluation grades Poor Slightly poor General Good Excellent Quality of key components 0.02 0.08 0.06 0.58 0.26 Dynamic characteristics 0.02 0.06 0.08 0.38 0.46

5.2.1 Weight calculation based on entropy weight method

5.2.2 Calculation of each indicator’s weight based on AHP

Table 10 Judgment matrix R (A - B) B1 B2 B3 B4 B5 G B1 1 1/2 2 3 4 0.2848 B2 2 1 1 4 3 0.3187 B3 1/2 1 1 3 3 0.2258 B4 1/3 1/4 1/3 1 1/3 0.0661 B5 1/4 1/3 1/3 3 1 0.1046

Footnotes

Acknowledgments

References

Table 5
Comprehensive evaluation of mechanical performance

Evaluation indicators Evaluation grades

Poor Slightly poor General Good Excellent

Quality of key components 0.02 0.08 0.06 0.58 0.26

Dynamic characteristics 0.02 0.06 0.08 0.38 0.46

Table 10
Judgment matrix R (A - B)

B₁ B₂ B₃ B₄ B₅ G

B₁ 1 1/2 2 3 4 0.2848

B₂ 2 1 1 4 3 0.3187

B₃ 1/2 1 1 3 3 0.2258

B₄ 1/3 1/4 1/3 1 1/3 0.0661

B₅ 1/4 1/3 1/3 3 1 0.1046