FCE-based health status evaluation and IGM-based failure prediction of tool magazine manipulator for CNC machine tools

Abstract

The reliability of machine tool components, particularly the tool magazine manipulator, significantly affects the overall performance of the machine tool. To address the challenge of accurately evaluating the manipulator’s health status using a single performance indicator, this study proposes a method that combines Fuzzy Comprehensive Evaluation (FCE) and a Combined Weighting Method (CWM). By considering both subjective and objective factors, this method provides a comprehensive evaluation of the manipulator’s health status, enhancing the accuracy and reliability of the assessment. The method utilizes fuzzy distribution to construct membership matrices for different health levels and adopts the CWM that combines the Entropy Weight Method (EWM) and Analytic Hierarchy Process (AHP) to determine the combined weights of the health evaluation indices. This approach improves the accuracy and reliability by considering multiple indicators and objectively weighting them based on their importance. The current health status of the manipulator is evaluated using the fuzzy weighted average operator and the maximum membership principle. Moreover, a fault prediction method based on Particle Swarm Optimization (PSO) and GM(1,1) is proposed to overcome the information gap and small sample problems. The proposed model’s prediction accuracy is verified by comparing it with other models, demonstrating its effectiveness and reliability.

Keywords

Health status evaluation fault prediction fuzzy comprehensive evaluation grey model particle swarm optimization

1 Introduction

Computer Numerical Control (CNC) machine tools are highly integrated manufacturing equipment that combines mechanical, electrical, and hydraulic systems and are widely used in the manufacturing industry. The technological advancement of CNC machine tools represents a significant indicator of a country’s overall manufacturing capability and economic development. At present, China’s high-end CNC machine tools have made significant progress in terms of speed, accuracy, multi-axis linkage and composite machining capabilities. But there is a significant gap between the domestic reliability level and the international advanced level [1]. CNC machine tools are comprised of basic support components and functional units, with the dependability of the latter being crucial for overall machine tool reliability. Thus, the improvement of functional component reliability is key to enhancing CNC machine tool performance. The automatic tool change system (ATC) is among the most failure-prone subsystems in machine tools, with the manipulator being the most commonly malfunctioning constituent within the ATC. Therefore, researches such as condition monitoring, health status evaluation, and failure prediction for tool magazine manipulator, which can predict degenerative failure in advance. It is possible to significantly reduce maintenance costs, unplanned downtime, and improve the operational reliability of machine tool.

Prognostic and Health Management (PHM) is an important technology to achieve predictive maintenance of electromechanical equipment, improve equipment reliability, and reduce equipment maintenance costs. The health status evaluation is to take the extracted features as input and build a health status evaluation model based on statistical methods, in order to achieve real-time evaluation of health status for electromechanical equipment. The establishment of an evaluation indicator system is a pivotal consideration when conducting health status evaluations. In this context, the Analytic Hierarchy Process [1], Entropy Weight Method [2], and VlseKriterijumska Optimizacija I Kompromisno Resenje [4] method can be used to determine the weightings of the evaluation indicators. Kulkarni PG et al. [4]. weighted the selection criteria of mother wavelet based on the AHP, and selected optimal mother wavelet to process vibration signal, in order to evaluate the health of bearings of a lathe machine tool. Che et al. [5]. proposed the development of a vulnerability indicator system for a city’s power grid, utilizing the Analytic Hierarchy Process (AHP) method. The system focuses on four main aspects. Nevertheless, the use of AHP may lead to a subjective evaluation index system, which can significantly affect the outcome of the health assessment. Therefore, Fang et al. [6]. proposed a method based on the AHP, the EWM, fuzzy clustering and information entropy to construct an index system. Liu et al. [7]. proposed a comprehensive weighting algorithm that uses Analytic Hierarchy Process (AHP), Maximum Absolute Weighted Residual (MAWR), and Maximum Entropy Method (MEM) to assess the risk of railway signal equipment. Yang et al. [8]. proposed a method based on fuzzy AHP and VIKOR method to obtain comprehensive weights of risk factors. Health status evaluation can adopt methods such as uncertainty AHP [9], fuzzy AHP [10] and fuzzy comprehensive evaluation (FCE) [11]. For example, in [9], a method was proposed for monitoring the health condition of offshore wind power structures, which utilized genetic algorithm optimization and uncertainty-based Analytic Hierarchy Process (AHP). Zhou et al. [10]. proposed an improved fuzzy AHP for evaluating the health status of power transformers. Yang et al. [11]. proposed a combined FCE method and cuckoo search-support vector regression method for accurate assessment of radar health condition. Currently, supervised machine learning algorithms [12] are widely used for health condition assessment of mechanical and electrical equipment, as observed in existing models. The evaluation indicator system mostly uses a single method for weighting, so the evaluation indicator weight cannot be both subjective and objective. To address this issue, the FCE takes into account the ambiguity among evaluation indicators and has the characteristics of clear results and strong systematicness. Therefore, this paper proposes the Combination Weighting method (CWM)that combines EWM method and AHP to weight evaluation indicators and evaluate the health of manipulator based on the FCE method.

The Fault Prediction relies on the current working condition of equipment as a basis. By analyzing the historical monitoring data of the equipment, fault patterns and future trends can be identified. These predictions help develop practical maintenance strategies to ensure smooth operation of the equipment. The GM(1,1) [15 –17], which refers to grey model of order (1,1), is a univariate differential equation model commonly used for fault prediction [17]. To improve the predictive accuracy of the model, modifications have been made to the GM(1,1) [19, 20]. Modifications have been proposed to improve the predictive accuracy of the GM(1,1) model. For example, Wang et al. [20]. proposed a method to optimize the GM(1,1) model by modifying the initial values. Tabaszewski et al. [21]. used grey system theory and the GM(1,1) prediction model with diversified window sizes to predict fan wear in cogeneration plants. Tangkuman et al. [22]. proposed to improve the predictive accuracy by adding a new time series to the traditional GM(1,1) grey model and obtaining m predicted values and their average value. Liu et al. [23]. proposed an improved GM(1,1)-based method for fault prediction of weak electric thrusters, achieving higher prediction accuracy. Yang et al. [24]. proposed a grey model based on similarity information fusion to predict the remaining useful life of aircraft engines, accurately predicting system failures. In addition, in many fields, the grey model is combined with, the neural network [25], EWM [26], discrete wavelet transform [27] or others to improve the fault prediction accuracy. In the field of industrial rotating machinery, Han et al. [27]. developed a sensor drift detection method based on discrete wavelet transform and GM(1,1) model for diagnosing the faults of temperature sensors. Li et al. [28]. proposed an empirical Bayesian algorithm to upgrade traditional grey models. In the electronic field, Hu et al. [29]. proposed a fault prediction model for circuit orbit using grey theory and expert system. Liu et al. [32]. proposed a hybrid prediction model combining fractional-order stable motion, grey model, and metabolism method to predict the reliability of gearboxes. Xu et al. [30]. proposed a method combining fuzzy theory with GM(1,1) and support vector machines to predict the faults of avionics equipment in manned spacecraft, considering the uncertainty of fault prediction. Due to limited data samples and the reliable nature of tool changer manipulators, traditional statistical modeling methods based on large data sets are not suitable. Gray system theory provides a valuable approach for modeling small sample systems with limited information. It is widely used to extract insights from available data and construct predictive models by exploring system evolution. The GM(1, 1) model, a commonly used gray model, approximates the time response function using an exponential function. However, direct application of the GM(1,1) model for fault prediction may not yield desired results when the sample data does not strictly follow an exponential trend. Moreover, existing fusion methods combining GM(1,1) with other approaches have lower prediction accuracy and inherent limitations. After analyzing the fusion model, we still identified three shortcomings in the gray model: background value construction, least squares estimation of unknown parameters, and initial value selection. To address these issues, we propose an Improved Gray Model (IGM) for fault prediction. The IGM method tackles the initial value selection problem using the correction parameter ɛ, while employing particle swarm optimization (PSO) to enhance the solution process for background value construction and unknown parameter estimation. Our study compares the proposed IGM with GM (1,1) and demonstrates that the IGM achieves higher prediction accuracy, thereby improving the overall validity and accuracy of the model.

This paper takes the tool magazine manipulator as a research object, and carries out the research on health status evaluation and fault prediction. The process is shown in Fig. 1. In Section 2, the basic theories and methods used in this paper are introduced; In Section 3, according to the health evaluation indicators determined by five performance indicators of the manipulator, the CWM is used to determine the weight of health evaluation indicators. In Section 4, aiming at the attention state. A fault prediction method based on an IGM is proposed to solve the problems of small samples and poor information in fault prediction. The Section 5 provide an example case and the conclusions.

Fig. 1

Health status evaluation and failure prediction.

2 Relevant theory and methodology

2.1 Membership Function

The membership function [32] can fully consider the fuzziness between different states of equipment, and describe the evaluated object in the form of a fuzzy set.

1) Relative deterioration degree

According to different types of evaluation indicators, the formula for the deterioration degree is as follows. For intermediate excellent indicators, the formula for relative deterioration degree is: $η_{(x)} = {\begin{matrix} 1, x < x_{min} \\ \frac{x_{0} - x}{x_{0} - x_{min}}, x_{min} ⩽ x ⩽ x_{0} \\ \frac{x - x_{0}}{x_{max} - x_{0}}, x_{0} ⩽ x ⩽ x_{max} \\ 1, x > x_{max} \end{matrix}$ (1)

For the indicator where the bigger the better, the formula for the relative degradation is: $η_{(x)} = {\begin{matrix} 1, x < x_{min} \\ \frac{x_{max} - x}{x_{max} - x_{min}}, x_{min} ⩽ x ⩽ x_{max} \\ 0, x > x_{max} \end{matrix}$ (2)

For the indicator where the smaller the better, the formula for the relative degradation is: $η_{(x)} = {\begin{matrix} 0, x < x_{min} \\ \frac{x - x_{min}}{x_{max} - x_{min}}, x_{min} ⩽ x ⩽ x_{max} \\ 1, x > x_{max} \end{matrix}$ (3)

Where η_(x) is the relative deterioration degree of the evaluation indicator. x_min and x_max respectively represents the minimum and maximum value of the evaluation indicator, and x₀ represents the standard value of the evaluation indicator.

2) Ridge-shaped membership function

When the equipment is in different states such as healthy, good, attention, abnormal and fault, the ridge-shaped membership function is shown in Fig. 2, where η ₁ η ₈ represent the fuzzy number of the membership function. Fuzzy numbers represent the membership relationship corresponding to each interval.

Fig. 2

Membership function.

For positive indicators (the bigger the better),the fuzzy number needs to be adjusted appropriately.

When the relative deterioration degree is η, the membership function of the “healthy” is as follows: $μ (η) = {\begin{matrix} 1, η < η_{1} \\ \frac{1}{2} + \frac{1}{2} sin \frac{π}{η_{2} - η_{1}} (η - \frac{η_{1} + η_{2}}{2}), η_{1} ⩽ η ⩽ η_{2} \\ 0, η > η_{2} \end{matrix}$ (4)

The membership function of the “good” is as follows: $μ (η) = {\begin{matrix} 0, η < η_{1} η > η_{4} \\ \frac{1}{2} + \frac{1}{2} sin \frac{π}{η_{2} - η_{1}} (η - \frac{η_{1} + η_{2}}{2}), η_{1} ⩽ η ⩽ η_{2} \\ 1, η_{2} ⩽ η ⩽ η_{3} \\ \frac{1}{2} + \frac{1}{2} sin \frac{π}{η_{4} - η_{3}} (η - \frac{η_{3} + η_{4}}{2}), η_{3} < η ⩽ η_{4} \end{matrix}$ (5)

The ridge-shaped membership functions of “abnormal” and “attention” are similar to those of the “good”, except that different fuzzy numbers η _i are used. The ridge-shaped membership function of the “fault” is similar to those of the “healthy”, with η ₇ instead of η ₁ and η ₈ instead of η ₂ .

2.2 Entropy weight method (EWM)

The process of Entropy weight method [33] is as follows: Suppose m objects and n evaluation indicators exist, the initial evaluation indicator matrix R_m_×n is normalized to obtain the matrix Q_m_×n = [q_ij] _m _×n. Where q_ij represents the normalized value of the j-th evaluation indicator of the i-th evaluation object. Then, according to the entropy e _j of the j-th evaluation indicator, the evaluation indicator weight ω _j is obtained.

The proportion p_ij of the j-th evaluation indicator value of the i-th evaluation object: $p_{ij} = q_{ij} / - \sum_{i = 1}^{m} q_{ij}$ (6)

The entropy e_j of the j-th evaluation indicator: $e_{j} = - \frac{1}{ln m} \times \sum_{i = 1}^{m} (p_{ij} ln p_{ij})$ (7)

The difference of the j-th indicator H_j: $H_{j} = 1 - e_{j}$ (8)

The j-th evaluation indicator weight ω _j : $ω_{j} = \frac{H_{j}}{\sum_{j = 1}^{n} H_{j}} = \frac{1 - e_{j}}{\sum_{j = 1}^{n} 1 - e_{j}}$ (9)

2.3 Analytic Hierarchy Process (AHP)

The basic steps of hierarchical analysis [34] are as follows: Suppose the evaluation object has m indicators, the indicators are represented by a_i, a_ij represents the importance of the evaluation indicator a_i relative to a_j, and the judgment matrix is A_m_×n = [a_ij] _m _×n. The maximum characteristic root λ _max is obtained by the asymptotic normalization coefficient. $λ_{max} = \sum_{i = 1}^{n} \frac{{(AW)}_{i}}{n ω_{i}}$ (10)

The eigenvector W: $W = {(ω_{1}, ω_{2}, \dots, ω_{n})}^{T}$ (11)

The weight of the i-th evaluation indicator: $ω_{i} = c_{i} / \sum_{i = 1}^{n} c_{i}$ (12)

where $c_{i} = \sum_{j = 1}^{n} (a_{ij} / - \sum_{i = 1}^{n} a_{ij})$ (13)

The largest characteristic root λ _max of matrix A_m_×n must pass the consistency check.

2.4 Grey Model -GM(1,1)

Suppose the initial data sequence X⁽⁰⁾ has n values, i.e. $X^{(0)} = (x^{(0)} (1), x^{(0)} (2), \dots, x^{(0)} (n))$ (14)

the 1-AGO sequence X⁽¹⁾ is obtained by the first order accumulative generation operator(1-AGO), i.e. $X^{(1)} = (x^{(1)} (1), x^{(1)} (2), \dots, x^{(1)} (n))$ (15)

Where $x^{(1)} (k) = \sum_{i = 1}^{k} x^{(0)} (i) i = 1, 2, \dots, n$ (16)

A first-order linear differential equation is established for the new sequence X⁽¹⁾, and the basic form of the GM(1,1) is obtained as follows: $x^{(0)} (k) + {aZ}^{(1)} (k) = b k = 1, 2, \dots, n$ (17)

The estimated values $\hat{a}$ and $\hat{b}$ are obtained by the least squares method, and the prediction model of the initial sequence X⁽⁰⁾ is obtained by solving Equation (17). $\begin{matrix} \overset{\land}{x^{(0)}} (k + 1) = \overset{\land}{x^{(1)}} (k + 1) - \overset{\land}{x^{(1)}} (k) \\ = (1 - e^{\hat{a}}) (x^{(0)} (1) - \frac{\hat{b}}{\hat{a}}) e^{- \hat{ak}} \end{matrix}$ (18)

Before establishing a grey model, it is necessary to perform quasi-smoothness and quasi-exponential tests on the initial data sequence and the new sequence.

3 Health status evaluation based on the fuzzy comprehensive evaluation method

The fuzzy comprehensive judgment method [35] has a clear result and systematic characteristics, it is very suitable for the health status assessment of tool changer manipulator. First of all, combined with the structural characteristics and typical failure modes of manipulator, this paper selects five performance indicators such as manipulator vibration, noise, manipulator angle deviation, manipulator motor current and temperature as the health evaluation indicators. The evaluation set, also known as the health level, is divided into five levels: healthy, good, attention, abnormal and fault. Secondly, in order to accurately quantify the deviation between the evaluation indicator and the health status, the relative deterioration degree is introduced to calculate the membership function and the membership matrix. Then, this paper proposes a method of health evaluation indicator weight based on the combination weighting method by combining EWM and AHP. Finally, according to the membership degree matrix and the combination weight of indicator, the evaluation results of each indicator are integrated by the FCE, and the health status of the manipulator is obtained according to the principle of the maximum membership degree. Figure 1 shows the method for evaluating the health status of the manipulator based on the FCE.

3.1 Determine Health Evaluation Indicators

The selection of evaluation indices directly impacts the authenticity and accuracy of the evaluation results. Based on the structure and working principle of the tool changer and the reliability test data collected, four common faults of the manipulator are identified: tool falling failure, motor failure, abnormal movement failure and jamming of tool failure. Failure analysis is conducted for each type of failure to determine their causes and mechanisms. Specific monitoring indices are then identified for each type of failure to accurately characterize the motion state of the manipulator. The tool falling failure can be attributed to two factors. Firstly, fatigue failure of the sliding spring or wear of the locking pin can lead to a failure in the locking device, causing the knife to drop. Secondly, the wear of non-rigid connecting parts between the manipulator and the shaft can result in a severe offset in the position of the manipulator at the knife’s buckling point, thereby causing the manipulator to drop the knife. To monitor this type of failure, the chosen indicators include manipulator corner deviation, manipulator vibration, and noise. The manipulator motor failure can be attributed to four main reasons: high motor temperature, motor damage, noisy operation caused by damaged or loose parts. Consequently, the selected monitoring indicators for this type of failure are robot motor temperature, robot motor current, and noise. The abnormal movement failure can be classified into two categories: no movement of the manipulator and unsteady movement of the manipulator. The selected monitoring indicators for this type of malfunction are manipulator vibration, manipulator motor current, and noise. The jamming of tool failure is primarily caused by wear and loosening of screws in the positioning key, positional shift of the manipulator due to loosening of the expansion sleeve, and locking device failure due to decreased elasticity of the sliding spring. The chosen monitoring indicators for this type of fault are manipulator vibration, noise, and angular deviation of the manipulator. In summary, five performance indicators, namely vibration, noise, corner deviation, motor current, and temperature, can be considered as evaluation indicators to assess the overall health of the manipulator.

3.2 Determine the health status

In order to accurately evaluate the status of the manipulator, this paper divides the health status evaluation set of the manipulator into five levels: healthy, good, attention, abnormal and fault. “Healthy” means that the manipulator is in excellent condition and is unlikely to fail; “Good” means that the is in good operating condition and is not prone to failure; “Attention” means that the manipulator is in a general operating state and is prone to failure; “Abnormal” means that the manipulator is abnormal operating state and more prone to failure; “Failure” means that the manipulator has malfunctioned and must be repaired immediately.

3.3 Determine the membership function of health status

When the equipment is “abnormal”, the monitored evaluation indicator will deviate from the standard value. In order to accurately quantify the degree of deviation, a relative deterioration degree is introduced to indicate the degree to which the evaluation indicator deviates from the standard value, which is between [0, 1]. The initial data of each evaluation indicator is converted into a relative deterioration degree, which eliminates the influence of physical meaning and unit.

Since the five evaluation indicators of the manipulator are all quantitative indicators, the ridge-shape membership function has the characteristics of wide principal value interval, excessive smoothness, and good indicator resolution. In this paper, it is selected based on the fuzzy distribution method to construct the membership matrix M_ij. Where the i-th row respectively represents manipulator vibration, noise, manipulator angle deviation, manipulator motor current and temperature, the j-th column respectively represents healthy, good, attention, abnormal and failure.

3.4 Weight of health evaluation indicator based on CWM

If the subjective weighting method alone is employed to determine the weights in the health state assessment, it may not fully capture the information from the assessment indicators. Subjective weighting often relies heavily on the decision makers’ personal preferences, neglecting the intrinsic relationships between the indicators, leading to arbitrariness. On the other hand, if expert experience and knowledge embedded in the subjective weights are disregarded, and only objective weights are used for evaluation, it can result in significant discrepancies with the actual situation. This approach may also introduce unreasonable weight coefficients. Therefore, this paper combines subjective and objective weights to achieve comprehensive weighting, ensuring the objective utilization of information from objective data while respecting the subjective guidance of decision makers. It treats subjective and objective weights equally, unifying them in decision-making processes for a more balanced and inclusive approach. In this paper, the weights obtained by EWM and AHP are combined according to their proportions, so that the combined weights are closer to the real weights. The advantages and disadvantages of the EWM and the AHP are combined to determine the weight of the health evaluation indicator using the CWM. The formula is as follows: $ω_{cj} = \frac{a_{j} ω_{aj} + b_{j} ω_{bj}}{\sum_{j = 1}^{n} (a_{j} ω_{aj} + b_{j} ω_{bj})}$ (19)

where ω _cj is the combination weight of the j-th evaluation indicator, ω _aj is the weight of the j-th evaluation indicator determined by the EWM. ω _b _j is the weight of the j-th evaluation indicator determined by the AHP. a_j is the proportion of ω _aj in the weight of the j-th evaluation indicator. b_j is the proportion of ω _bj in the weight of the j-th evaluation indicator.

The formulas for a_j and b_j are as follows: $a_{j} = \frac{ω_{aj}}{ω_{aj} + ω_{bj}}, b_{j} = \frac{ω_{bj}}{ω_{aj} + ω_{bj}}$ (20)

3.5 Health status evaluation based on fuzzy comprehensive evaluation

In this paper, the evaluation results of each indicator are integrated based on the FCE in combination with the membership degree matrix M and the combination weight ω _c . The evaluation object obtained by the principle of maximum membership degree is also called health status. The health level membership matrix H of the evaluation object is as follow: $H = ω_{c} • M$ (21)

where “•” represents the fuzzy weighted average operator, W_c represents a row vector (the number of columns is equal to the number of evaluation indicators j), M represents a matrix with the number of evaluation indicators j as the number of rows and the number of health levels as the number of columns and the health level corresponding to the maximum value of the elements in matrix H is the final health status.

4 Failure prediction based on the improved grey model

4.1 Improved Grey Model (IGM)

The grey model GM(1,1) has problems such as subjectivity of background value construction, inconsistent principles of parameter estimation and model verification, and randomness of initial value selection, resulting in low model prediction accuracy.

Figure 1 shows the IGM-based approach to manipulator failure prediction. This paper aims at the problem that the function curve does not necessarily pass the initial point when solving the differential equation by the Ordinary Least Square, that is, the problem of randomly selecting the initial value. Therefore, the modified parameter ɛ is added to solve this problem. i.e. $\overset{\land}{x^{(1)}} (1) = x^{(0)} (1) + ɛ$ , the model becomes:

${\hat{x}}^{(0)} (k + 1) = (1 - e^{a}) (x^{(0)} (1) + ɛ - \frac{b}{a}) e^{- ak}$ (22)

In order to solve the problem of the subjectivity of background value construction and the inconsistent principles of parameter estimation and model verification, this paper uses the Particle Swarm Optimization (PSO) to optimize the unknown parameters a, b,and ɛ in (17). The process is shown in Fig. 3.

Fig. 3

Particle swarm optimization.

Step 1: Initialize particle’s speed and position, and set relevant parameters.

$F_{i t} = \frac{1}{n} \sum_{k = 1}^{n} \frac{| {\hat{x}}^{(0)} (k) - x^{(0)} (k) |}{| x^{(0)} (k) |}$ (23) ∥Step 2: Determine the fitness function value F_it[i] and select the average relative error as the fitness function.∥Step 3: Compare the fitness function value F_it[i] of each particle with the individual extreme value p_best. If F_it[i] < p_best, let F_it[i] become the new p_best.∥Step 4: Compare the fitness function value F_it[i] of each particle with all extreme values g_best. If F_it[i] < g_best, let F_it[i] become the new g_best.∥Step 5: Update the velocity and position of the particles.∥Step 6: Determine whether the end condition is met, if yes, output the result and end, otherwise go back to step 2.∥

4.2 Model Testing

After solving the IGM, it needs to carry out residual test, correlation test and posterior-variance-test. All the above three tests are passed, which means that the prediction accuracy of the model meets the requirements.∥1) The formula for the residual test is as follows:∥Absolute error: $Δ (k) = | {\hat{x}}^{(0)} (k) - x^{(0)} (k) | k = 1, 2, \dots, n$ (24) ∥Relative error: $Φ (k) = \frac{| {\hat{x}}^{(0)} (k) - x^{(0)} (k) |}{x^{(0)} (k)} \times 100$ (25) ∥2) The formula for the correlation test is as follows: ${\begin{matrix} η (k) = \frac{min_{k} {Δ (k)} + ρ max_{k} {Δ (k)}}{Δ (k) + ρ max_{k} {Δ (k)}} \\ r = \frac{1}{n} \sum_{k = 1}^{n} η (k) k = 1, 2, \dots, n \end{matrix}$ (26) ∥where minΔ(k) and maxΔ(k) respectively represent the minimum and maximum values of absolute error, ρ represents the resolution. It is generally considered that when ρ= 0.5, if r > 0.55, the correlation test is passed.∥3) The formula for the posterior-variance-test is as follows:∥Standard deviation ratio C: $C = S_{2} / - S_{1}$ (27) ∥Small error probability P: $P = P {| Δ (k) - \bar{Δ} | < 0.645 S_{1}}$ (28) ∥where S₁ represents the standard deviation of the initial data sequence. S₂ represents the standard deviation of the absolute error. $\bar{Δ}$ represents the average value of the absolute error.

5 Case analysis

5.1 Health Monitoring System for the ATC

According to the results of fault analysis, five main performance indicators of the manipulator are its vibration, noise, angle deviation, motor current and temperature. This paper selects the installation position and method of each sensor based on the principle of monitoring position selection and the structural characteristics of the manipulator, and configures the corresponding data acquisition hardware and operating software for the sensor, as shown in Fig. 4. the health status monitoring system of the ATC is built to monitor the state of the manipulator.

Fig. 4

Health monitoring system of the ATC.

5.2 Health Status Evaluation of the Tool Magazine Manipulator

To demonstrate the modeling process described in Chapter 3, the health status evaluation of the manipulator will be performed. The tool magazine manipulator has performed 150,000 tool changing experiments.

In this paper, the Health Monitoring System is used to collect the data of five performance indicators of the manipulator, and the health status evaluation system of the manipulator is constructed. And the main modeling procedures are interpreted as follows:

Step 1: Substituting the data of the performance indicators into Equation (1), (2) and (3) to obtain the relative deterioration degree of each health evaluation index, as shown in Table 1.

Table 1
Initial data and relative deterioration degree of evaluation indicators

Evaluation indicator Manipulator Manipulator motor

Vibration(g) Noise(dB) Corner deviation (°) Current(A) Temperature (¡æ)

Measured value 2.74 60.2 0.45 2.40 36.64

Standard value –– –– 0 2.14 38.42

Max 3.62 85 1.05 2.86 55

Min 1.58 45 0 1.42 24.75

relative deterioration degree 0.57 0.38 0.43 0.36 0.13

Evaluation indicator	Manipulator	Manipulator motor
Measured value	2.74	60.2	0.45	2.40	36.64
Standard value	––	––	0	2.14	38.42
Max	3.62	85	1.05	2.86	55
Min	1.58	45	0	1.42	24.75
relative deterioration degree	0.57	0.38	0.43	0.36	0.13

Step 2: According to the characteristics of the manipulator itself, the fuzzy number η is determined as: η= (0.1,0.2,0.35, 0.45,0.55,0.7,0.8,0.95), Substituting it into Equation (4) and (5) to obtain the membership functions of the five health levels.

According to the relative deterioration degree of each evaluation indicators, the membership degree of each evaluation indicators relative to each evaluation set is obtained, so the membership degree matrix M is: $M = [\begin{matrix} 0 & 0 & 0 . 9568 & 0 . 0432 & 0 \\ 0 & 0 . 7939 & 0 . 2061 & 0 & 0 \\ 0 & 0 . 0955 & 0 . 9045 & 0 & 0 \\ 0 & 0 . 9755 & 0 . 9045 & 0 & 0 \\ 0.7939 & 0.2061 & 0 & 0 & 0 \end{matrix}]$

Step 3: The weight of the evaluation indicator determined by the EWM obtained from Equation (6), (7), (8) and (9) can be denoted as:

$ω_{a} = (0.2630, 0.1438, 0.4490, 0.1022, 0.0420)$

the weight of the evaluation indicator determined by the AHP obtained from Equation (10) and (11) can be denoted as:

$ω_{b} = (0.2364, 0.1156, 0.5476, 0.0603, 0.0401)$

Substituting ω _a and ω _b into Equation (14) and (15), the combined weight of each evaluation indicator of the manipulator can be expressed as:

$ω_{c} = (0.2473, 0.1296, 0.4969, 0.0856, 0.0406)$

Step 4: Substituting ωc and M into Equation (21), the health level membership matrix can be described as H = (0.0322,0.2422, 0.7149,0.0107,0)

Step 5: Experiment and Case Analysis

From the matrix M, it can be seen that the evaluation result of the manipulator motor temperature is “healthy", the evaluation results of the manipulator motor current and noise are “good", and the evaluation results of the manipulator vibration and angle deviation are “attention". This fully demonstrates the necessity of selecting multiple evaluation indicators for the health status evaluation.

It can be seen from Table 2 that the weights of each evaluation indicators are sorted in descending order: manipulator angle deviation, manipulator vibration, noise, manipulator motor current and motor temperature. The weight of the manipulator angle deviation is the largest, which is 49.69%. According to the actual situation of the manipulator, this fully reflects the practicability and effectiveness of the CWM.

Table 2

The original data, sequence data and inspection index of the manipulator angle deviation

No.	1	2	3	4	5	6	7	8	9	10
Angle deviation	0.30	0.31	0.33	0.35	0.36	0.38	0.40	0.42	0.45	0.48

According to the membership matrix H, the manipulator is in “attention”, which means that the manipulator’s running state should be observed carefully at this time, and it may change to an “abnormal”. Therefore, this paper uses the tool setting gauge to measure the manipulator angle deviation. it is found that the deviation has reached 0.45°. Although it does not exceed the threshold (1.05°), it has deviated from the standard value (0°). This shows that the evaluation results obtained by the FCE and the CWM are consistent with the actual situation of the manipulator, which proves the effectiveness of the health status evaluation proposed for the manipulator in this paper.

5.3 Fault Prediction of the Tool Magazine Manipulator

The manipulator is currently in “attention”. It is necessary to study the relationship between the number of tool changes and the manipulator angle deviation, in order to predict when the manipulator will fail due to excessive rotation angle deviation and formulate reasonable maintenance strategies before the failure occurs, so the reliability level of the manipulator and the CNC machine tool is improved. Due to the relatively high reliability level of the manipulator, it is difficult to obtain a large amount of data within an effective test time. This paper proposes a fault prediction method based on the IGM for the problems of small samples and poor information.

Step 1: Feasibility Analysis of the Grey Model GM (1,1)

This paper needs to predict the manipulator angle deviation, so 30,000 tool change tests are carried out. the manipulator angle deviation is detected once when the tool is changed 3,000 times so a total of 10 sample data were obtained. Obtain the original data sequence of the manipulator angle deviation X⁽⁰⁾ = (0.30,0.31,0.33,0.35,0.36,0.38,0.40,0.42,0.45,0.48) and generate the1-AGO sequence X⁽¹⁾ = (0.30,0.61, 0.94,1.29,1.65,2.03,2.43,2.85,3.30,3.78). The quasi-smoothness index and the quasi-exponential test index can be denoted as ρ(k) = (1.03,0.54,0.37,0.28,0.23, 0.20,0.17,0.16,0.15) and σ⁽¹⁾(k) = (2.03,1.54,0.37,1.28, 1.23,1.20,1.17,1.16,1.15). In summary, when k > 3, ρ (k) ∈ [0, 0.5) and σ⁽¹⁾ (k) ∈ (1, 1.5) are satisfied, the sequence X⁽⁰⁾ and X⁽¹⁾ pass the quasi-smoothness and the quasi-exponential test. Therefore, it is feasible to apply the model to predict the failure of manipulator.

Step 2: The Improved Grey Model Based on PSO

In this paper, the particle swarm size N = 100, the learning factors c₁ = 1.5, c₂ = 2.5, the inertia weight w = 0.5, the maximum number of iterations M = 1000, and the search space d = 0.3. The minimum value of the fitness function is obtained by MATLAB as 0.0085, and the unknown parameters a = 0.051, b = 0.2868, ɛ= 0.0011. Substitute three parameters into Equation (22), the IGM is obtained as follows: ${\hat{x}}^{(0)} (k + 1) = (1 - e^{- 0.051}) \times (0.3011 + \frac{0.2868}{0.051}) \times e^{0.051 k}$

Step 3: Comparative analysis

The relative error and average relative error between the initial data and the predicted data are calculated, as shown in Table 3. The average relative error is reduced from 4.89% to 0.85%. Then the predicted values were fitted with the initial data. The results are shown in Fig. 5. The predicted value obtained by the GM(1,1) can be denoted as GMPV. The predicted value obtained by the PSO-GM(1,1) can be denoted as PGMPV. The predicted value obtained by the IGM can be denoted as IGMPV.

Fig. 5

Predictive model fitting results.

Table 3

The solution results of two prediction models

Number of	Raw data	Predicted value1	Relative error	Predicted value2	Relative error
times
1	0.30	0.3000	0%	0.3011	0.37%
2	0.31	0.3266	5.36%	0.3100	0%
3	0.33	0.3444	4.37%	0.3262	1.14%
4	0.35	0.3632	3.77%	0.3433	1.91%
5	0.36	0.3830	6.39%	0.3613	0.35%
6	0.38	0.4039	6.29%	0.3802	0.04%
7	0.40	0.4259	6.48%	0.4001	0.02%
8	0.42	0.4492	6.94%	0.4210	0.24%
9	0.45	0.4736	5.25%	0.4430	1.55%
10	0.48	0.4995	4.06%	0.4662	2.87%

The IGMPV, positioned between GMPV and the initial data, exhibits a better fit than PGMPV. By increasing the modified parameter ɛ, IGMPV significantly enhances prediction accuracy and successfully addresses the issues of initial value selection. A comparison of the three models demonstrates that the optimization of solution using the modified parameter ɛ and PSO effectively solves the three problems of the gray prediction model, resulting in a noticeable improvement in prediction accuracy. This confirms the superiority of the model. Figure 6 presents the error analysis between the predicted values of IGMPV, GMPV, and the original data. The error value, denoted as T above the bar, represents the difference between the predicted value and the initial data. A higher T indicates a larger error value. Comparatively, the error value between IGMPV and the original data is lower than that of GMPV, indicating smaller errors and higher prediction accuracy.

Fig. 6

Error analysis.

By adding the modified parameter ɛ and applying the PSO to iteratively solve the model, the influence of the initial value selection problem on the prediction accuracy is reduced, and the background value construction and the inconsistency between the estimation of the unknown parameters and the principle of model test are ignored, so that the prediction accuracy of the IGM is significantly improved. Therefore, this paper finally chooses the IGM to predict the failure of the manipulator.

Step4: Model test

According to Equation (24), (25), (26), (27) and (28), the residual test, correlation test and posterior difference test were carried out. The average relative error of the IGM was 0.85% and less than 10%, indicating that the IGM passed the residual test. The correlation between the predicted data and the initial data is 0.75 and greater than 0.55, indicating that the correlation test is passed. The standard deviations ratio C = 0.0743 and the small error probability P = 1. Since C < 0.35 and P > 0.95, it can be seen that the accuracy of the IGM is level I, and the IGM is considered to pass the posterior difference test. To sum up, the IGM has passed the above three tests. The test results show that it has higher prediction accuracy and is suitable for the fault prediction of manipulator.

6 Conclusion

In order to improve the reliability of the machine tool, the health assessment and failure prediction of the tool magazine manipulator have been carried out. The study also includes a theoretical analysis of the results and practical applications of the established robot maintenance strategies. The paper is summarized as follows:

To address the challenge of accurately evaluating the manipulator’s health status with a single indicator and considering both subjective and objective factors, a method combining FCE and CWM is proposed. This proposed method provides clear results, strong systemics, and considers realistic working conditions.

Aiming at the problems in the process of solving the GM (1,1), a manipulator fault prediction method based on the IGM is proposed. The initial value selection problem is solved by using the modified parameters ɛ and uses the PSO to optimize the solution process of the background value construction and the unknown parameter estimation problem. The IGM demonstrates higher prediction accuracy compared to the grey model and the PSO-optimized grey model, thereby enhancing the overall accuracy and effectiveness of the model.

Through the case analysis, the manipulator’s status is flagged as “attention,” highlighting the importance of closely monitoring its operating state. The proposed method for failure prediction proves to be more accurate and suitable for manipulator failure prediction. By employing this method, the reliability of the manipulator and even the machine tool can be enhanced.

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

This work was supported by Jilin Province Science and Technology Development Plan –Key R&D Program [Grant No. 20220201028GX] and by science and technology research project of Jilin Provincial Department of Education, China (JJKH20220984KJ).

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