# Reliability Prediction for Computer Numerical Control Machine Servo Systems Based on an IPSO-Based RBF Neural Network

Authors: Zheng Jiang; GuangJian Wang; ZuGuang Huang; Ye He; RuiJuan Xue
Journal: Shock and Vibration
 (2022)

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## Abstract

The increasing use of computer numerical control (CNC) machines requires better prediction of the reliability of their servo control systems. A novel reliability prediction model based on radial basis function (RBF) neural network optimized by improved particle swarm optimization (IPSO) was proposed. It can overcome the disadvantages of conventional methods, which are time consuming and resource intensive. The major influences on the reliability of servo system include torque, temperature, current, and complexity. An improved algorithm for predicting the mean time between failure (MTBF) of servo systems based on a particle swarm optimization (PSO) and an RBF neural network algorithm is proposed. Two common problem of the PSO: local minimization and slow convergence were solved by the IPSO. “Zero failure” data preprocessing, data normalization, and small-sample data enhancement were performed on the original data. A homogenized sampling method is proposed to extract training and testing samples. Experimental results show that the improved PSO-based RBF neural network is superior to back propagation (BP) and RBF networks in terms of accuracy in servo system reliability prediction.


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## Body

## 1. Introduction

Generally speaking, computer numerical control (CNC) machine servo systems are integrated mechatronics systems that use a servo motor as the controlling object, a servo driver as the core, and a servo mechanism as the dynamic output. As the field of CNC metal processing develops to the direction of intelligent and unmanned directions, long-period continuous processing, and production lines have been widely applied. These put forward higher requirements for the reliability of the servo systems used in CNC machines. Therefore, reliability prediction methods are attracting increasing attention.Recently, many scholars have applied intelligent algorithms to reliability prediction and have obtained useful conclusions. Yan [1] used an IPSO to identify the inrush type of coal mine water source, solved the problem of premature convergence of general neural network, and improved the prediction accuracy. Mamo [2] established an output prediction method for the water-flooding exploitation of oil reservoirs based on an artificial neural network using the BP neural network. This could successfully predict the yield of petroleum reservoirs. Zhang [3] used neural network method to generate artificial logging curve and carries out experimental verification and application effect analysis. Jia-hui [4] proposed a surface water quality prediction model based on a graph neural network (GNN) and found it to be effective. Zhang and Chen [5] proposed an artificial neural network model of device degradation for the prediction of residual service life. Wang [6] used deep learning method for predicting oil field production at ultrahigh water cut stage from the existing oil field production data, and application effect analysis was carried out. In addition, in recent years, metaheuristic algorithms have been proposed and applied in related engineering problems, such as emperor butterfly optimization (MBO), moth search algorithm (MSA), Harris Eagle optimization (HHO), and the like; these algorithms simulate biological characteristics and apply them. Taking the emperor butterfly optimization (MBO) as an example, by simplifying and idealizing the migration process of emperor butterflies, this article proposes a new natural metaheuristic algorithm, called emperor butterfly optimization algorithm (MBO), which is the first of this algorithm. In MBO, all monarch butterfly individuals are located in two different places, namely. Southern Canada and northern United States (land 1) and Mexico (land 2). Therefore, the location of the monarch butterfly will be updated in two ways. Firstly, the offspring (location update) is generated by the migration operator, and the offspring can be adjusted according to the mobility. Then, adjust the position of other butterflies through the butterfly adjustment operator.However, the above studies mainly predict system reliability based on BP. Generally speaking, whether BP-based predictive models can achieve functional convergence is closely related to the network structure (number of input neurons, number of hidden layers, and transfer function), the algorithm, and the sample capacity. Moreover, these models have the disadvantages of slow convergence and easy limitation within local minima. Hence, BP neural networks have difficulty determining parameters, such as network structures using general laws. Given a small sample size, the modeling is relatively difficult and the prediction errors are relatively high. In contrast, RBF and BP networks can approach any nonlinear function with reasonable accuracy. However, because the transfer function used by RBF in hidden layer is different from BP network, the accuracy of approaching the input function is also different. Poggio and Girosi [7] proved the superiority of RBF networks in approximating continuous functions. Although the symbolic function commonly used in BP network has better generalization ability, its result will affect the weight adjustment of more neurons. In addition, the transfer functions overlap over a large range of input values, so the outputs refer to each other. RBF network improves the training speed using local transfer function, so only a few neurons have nonzero response and become active for each input. Therefore, when using new data, only the weights of a few active neurons need to be modified. To sum up, the training speed of BP network is slower than that of RBF network, and the convergence is difficult. In addition, in training process, RBF network can avoid falling into local minimum.Therefore, this article proposes a reliability prediction model of CNC machine servo systems based on an improved RBF neural network. It is suitable for solving cases of small sample sizes and considers the influences of various stresses. In recent years, scholars have carried out extensive research on reliability in their respective fields. Many models and methods are devised, which is based on RBF and PSO to predict and evaluate different products and systems (see [8–25]). This predictive model considers the mathematical relations among torque, current, temperature, complexity, working time, number of failures, and the mean time between failure (MTBF) of servo systems. With respect to sample selection, data preprocessing of “zero failure” (data without failure during reliability test), small sample data enhancement, and normalization as well as homogenized sampling were applied, which realized sample integrity. Experimental results show that the RBF algorithm based on an improved PSO is superior to ordinary BP and RBF algorithms.The rest of this article is organized as follows. Section1 analyzes the fault detection principle and mechanism of the servo system. Section2 introduces the construction of reliability prediction model based on RBF neural network. Section 3 carries out the simulation and verification of the model. Section 4 compared the model with the general reliability prediction model. Section 5 summarizes the article.


## 2. Servo System Fault Detection Principles and Mechanism Analysis

### 2.1. Relationships between Mechanical Transmission Component Faults with Motor Current and Torque

During the operation of a servo system, any changes in interference load, transmission friction, and working conditions can cause fluctuations in the current (torque) of the motor. Hence, it can be concluded that changes in motor input current reflect the responses to the servo system to the external world.A servo motor overcomes cutting torque and interference torque, which are produced by frictional forces during workpiece processing, mainly by providing driving forces to the servo system. In this study, the CNC servo systems were idling, and frictional force was the major interference. Hence, the following torque equation could be used:(1)Tm=Jdωdt+Tt=Jdωdt+Tf+Ta,where Tm is the output torque of the motor; J is a constant of the rotational inertia of the feed system; ω is the instantaneous angular velocity of the servo motor; Tt is the interference torque; Tf is the torque of frictional force; and Ta is the torque caused by faults in the mechanical transmission components of the system.Without consideration of rotor winding losses, we can obtain(2)Tm=KtIt,(3)It=iab2+iac2+ibc2,where Kt is the constant of motor torque; It is the effective value of three-phase current; and iab, iac, and ibc are the three-phase input currents of the motor.Due toω=v/t=ds/ldtf, s is the displacement of the rail pair of the lead screw and is measured by the grating scale in the servo system of closed-loop control mode. L is the lead of the lead screw.Hence, (1) can be rewritten as(4)Jd2sdt2+Tf+Ta=Tm=KtIt.Equation (4) shows that the positional detection of the servo system has some mathematical relations with the current signal and torque of the servo control system. Therefore, it is possible to use in-built sensors for comprehensive monitoring and assessment of the servo system.When the feed system is moving at a constant speed,Jdω/dt=0. Therefore, equation (4) can be rewritten as(5)Tm=Tf+Ta=KtIt.Given fixed working conditions, the friction torque (Tf) is constant. As a result, the output torque of the motor changes with the fault torque caused by a transmission fault. In other words, the output torque of the motor increases when the mechanical transmission components break down, which is manifested as changes in current. Therefore, the current changes of the motor (output torque) can be used to diagnose faults in the mechanical transmission system.


### 2.2. Relations between Mechanical Transmission Component Faults and Motor Temperature

Based on the above analysis, mechanical transmission system faults of the servo system can be reflected by motor current signals, which are caused by changes in motor output power. Changes in output power and the motor are reflected by changes in motor temperature. For the whole servo system, the output power of the motor is mainly composed of mechanical processing power, frictional loss power, and the power needed to overcome system faults:(6)POUT=Jωdωdt+Tfω+Tαω.When the feed system moves at a constant speed, the mechanical power is 0 because there is no load. Hence, equation (6) is rewritten as(7)POUT=Tfω+Taω.A servo system with good lubrication characteristics has constant frictional losses of power. Hence, the fault-loss power must increase when mechanical transmission components break down, which directly increases the motor output power and thereby increases the motor temperature. Hence, changes in motor temperature can also be used for fault diagnosis of the mechanical transmission components of the servo system.


## 2.1. Relationships between Mechanical Transmission Component Faults with Motor Current and Torque

During the operation of a servo system, any changes in interference load, transmission friction, and working conditions can cause fluctuations in the current (torque) of the motor. Hence, it can be concluded that changes in motor input current reflect the responses to the servo system to the external world.A servo motor overcomes cutting torque and interference torque, which are produced by frictional forces during workpiece processing, mainly by providing driving forces to the servo system. In this study, the CNC servo systems were idling, and frictional force was the major interference. Hence, the following torque equation could be used:(1)Tm=Jdωdt+Tt=Jdωdt+Tf+Ta,where Tm is the output torque of the motor; J is a constant of the rotational inertia of the feed system; ω is the instantaneous angular velocity of the servo motor; Tt is the interference torque; Tf is the torque of frictional force; and Ta is the torque caused by faults in the mechanical transmission components of the system.Without consideration of rotor winding losses, we can obtain(2)Tm=KtIt,(3)It=iab2+iac2+ibc2,where Kt is the constant of motor torque; It is the effective value of three-phase current; and iab, iac, and ibc are the three-phase input currents of the motor.Due toω=v/t=ds/ldtf, s is the displacement of the rail pair of the lead screw and is measured by the grating scale in the servo system of closed-loop control mode. L is the lead of the lead screw.Hence, (1) can be rewritten as(4)Jd2sdt2+Tf+Ta=Tm=KtIt.Equation (4) shows that the positional detection of the servo system has some mathematical relations with the current signal and torque of the servo control system. Therefore, it is possible to use in-built sensors for comprehensive monitoring and assessment of the servo system.When the feed system is moving at a constant speed,Jdω/dt=0. Therefore, equation (4) can be rewritten as(5)Tm=Tf+Ta=KtIt.Given fixed working conditions, the friction torque (Tf) is constant. As a result, the output torque of the motor changes with the fault torque caused by a transmission fault. In other words, the output torque of the motor increases when the mechanical transmission components break down, which is manifested as changes in current. Therefore, the current changes of the motor (output torque) can be used to diagnose faults in the mechanical transmission system.


## 2.2. Relations between Mechanical Transmission Component Faults and Motor Temperature

Based on the above analysis, mechanical transmission system faults of the servo system can be reflected by motor current signals, which are caused by changes in motor output power. Changes in output power and the motor are reflected by changes in motor temperature. For the whole servo system, the output power of the motor is mainly composed of mechanical processing power, frictional loss power, and the power needed to overcome system faults:(6)POUT=Jωdωdt+Tfω+Tαω.When the feed system moves at a constant speed, the mechanical power is 0 because there is no load. Hence, equation (6) is rewritten as(7)POUT=Tfω+Taω.A servo system with good lubrication characteristics has constant frictional losses of power. Hence, the fault-loss power must increase when mechanical transmission components break down, which directly increases the motor output power and thereby increases the motor temperature. Hence, changes in motor temperature can also be used for fault diagnosis of the mechanical transmission components of the servo system.


## 3. Construction of Reliability Prediction Models Based on an RBF Neural Network

### 3.1. RBF Neural Network

In 1988, Broomhead and Lowe introduced the radial basis function (RBF) neural network, which is based on the local response principle of biological neurons. The RBF neural network has been applied to more and more fields because it can realize a consistent approach to nonlinear networking.As shown in Figure1, the RBF neural network is a three-layer feedforward network, including the input layer, the output layer, and the hidden layer. The nodes in adjacent layers are full connected. xi(i = 1, 2, …, n) are inputs, hi(i = 1, 2, …, m) are activation functions of the hidden layer, and wi(i = 1, 2, …, m) are weights of output layer. In RBF neural network, the mapping between the input layer and the hidden layer is constructed by Gaussian radial basis function, and the mapping between the output layer and the hidden layer is constructed by linear function. The activation function of the hidden layer nodes locally responds to input signal. That is to say, its activation is maximal when an input is located at the RBF center, while the activation decreases monotonically when the distance between the RBF center and an input increases.Figure 1
Structure of a generalized RBF neural network.


### 3.2. Improved PSO Algorithm

The particle swarm optimization (PSO) algorithm is a random optimization technology based on particle swarm and has strong global searching ability. Moreover, it has the characteristics of easy operation and few adjustment parameters. Hence, it has been widely applied to various fields.The standard PSO algorithm model can be defined as follows: in the n-dimensional searching space, there is a particle swarmX that is composed of N particles. The position of particle i in the particle swarm is xi = {xi1, xi2,…,xin}T, and its velocity is vi=vi1,vi2,…,vinT. The individual extremum and global extremum of this particle are pi = {pi1, pi2,…, pin} T and pg=pg1,pg2,…,pgnT, respectively. The velocity and position updating formulas of particle xi can be expressed as(8)vidk+1=ωvidk+c1r1pidk−xidk+c2r2pgdk−xidk,xidk+1=xidk+vidk+1,where i = 1,2, …, n (n is the total number of particles); k is the number of iterations in the current stage; vidk is the velocity of the d-dimensional component of the flying velocity vector of particle i during k iterations; xidk is the position of the d-dimensional component of the flying velocity vector of particle i during k iterations; pidk is the d-dimensional component of the individual optimal solution of particle i; pgdk is the d-dimensional component of the best position of the particle swarm; c1 and c2 are acceleration constants; and r1 and r2 are random functions used to generate random numbers within (0, 1). In the implementation process, the PSO algorithm approaches the global optimal solution gradually by continuous updating and iteration of positions, velocity, and target positions. However, it may become trapped during local optimization and be unable to obtain the global optimal solution because particles can easily have inadequate mining capacity in the late implementation stage of the PSO algorithm. To address this problem, some optimization and improvements were made to the velocity and weights of the PSO algorithm.


#### 3.2.1. Optimization of Velocity Updating Formula

For optimization and improvement of the self-cognitive capacity of a single particle in the velocity updating formula, the optimal value of a single particle is replaced by the weighted average optimal value of all particles. The velocity and position updating formulas of the improved PSO algorithm are as follows:(9)vidk+1=ωvidk+c1r1∑i=0npidkfin+c2r2pgdk−xidk,xidk+1=xidk+vidk+1,where fi is the frequency of occurrence of the individual optimal value of particle i; n is the total number of particles; k is the number of current iterations; and d is the dimension of the particles.The decision method of a single particle based on the empirical results of the particle swarm is built by updating and calculating the velocity and position using the weighted average ofpidk. Meanwhile, the overall mining capacity and calculation efficiency are further improved by introducing the concept of the weighted average.


#### 3.2.2. Optimization of Inertia Weight

In the PSO algorithm, the rate of convergence is accelerated when the inertia weight is relatively high, but a more precise convergence outcome is gained when the inertia weight is relatively low. Previous studies indicate that the rate of convergence of a PSO algorithm is highest at inertia weights of 0.8–1.2. Moreover, the rate of convergence within this range is positively related to the inertia weight. When the inertia weight is higher than 1.3, the convergence efficiency decreases significantly. To obtain a high rate of convergence and high precision, this study determined the optimal inertia weight through a self-adaptive method. This implies that the value of the inertia weight decreases with increases in the number of iterations. The formula of inertia weight in this study is(10)ωk=ωmax0.55+e−αk/kmax−0.9,where k is the number of iterations of the current particle swarm; α is a positive coefficient that adjusts the rate of change in inertia weight (determined to be 0.55 in this study according to empirical data); kmax is the maximum number of iterations of the particle swarm; and ωmax is the upper limit of ωk (set to 1.15 in this study).


### 3.3. Construction of the RBF Neural Network

In this study, stress factors that influence system reliability were screened by the self-organizing algorithm with consideration of the performance and computational load of the neural network. A three-layer RBF neural network was built with six types of stress factors that may influence the reliability of the servo system used as input neurons and the MTBF of the servo system as the output neuron. The neurons of the input layer are shown in Figure2.Figure 2
Neural network input neurons.The complexity of the servo system was assessed by industrial experts. The comprehensive judgment is carried out in combination with the performance parameters, communication mode, control mode, and electrical design structure of the servo system. The comprehensive evaluation is carried out by several experts, and the final score is determined according to the pruning average value. The complexity classification principles are shown in Table1.Table 1
Complexity Classification of the servo system.
TypeInfluencing factorsComplicated9–10Relatively complicated7–8Moderate5–6Relatively simple3–4Simple1–2


### 3.4. Improved PSO-Based RBF Neural Network Algorithm

Reliability prediction based on RBF is a nonlinear and complex engineering problem. In order to obtain the neural network model with high prediction accuracy, a large amount of data are required for training. However, if the traditional RBF neural network is directly applied for training and simulation, it is easy to have problems such as long convergence time and falling into local optimization due to the inherent defects of RBF neural network. It will affect the research efficiency and research results. Therefore, combined with the research basis of relevant scholars, this article introduces the improved PSO algorithm to optimize it. The steps are described as follows.(1)
Set the input and output data of the network. The input data are the fault phenomenon, and the output data are the fault causes.(2)
Set the initial parameters of the improved PSO algorithm. The maximum number of iterations (kmax), the scale of the particle swarm, and the c1 and c2 value of the particles are set to 50, 10, 2 and 2, respectively. Meanwhile, the optimal position of a single particle and the global optimal position are initialized, which are denoted as pid and pgd.(3)
The position and velocity of particles are initialized.(4)
Calculate the fitness value of particles.(5)
If the fitness of a particle is better than those at any previous moment, this fitness is used as the individual extremum of the particle. If the fitness of this particle is better than those of all particles at any previous moment, this particle is used as the extremum of the particle swarm.(6)
Update the positions and velocities of particles according to equations (3) and (4). Besides, the weights of the algorithm are adjusted dynamically through a self-adaptive operation in accordance with equation (2).(7)
Judge whether the end condition is reached. If yes, terminate the algorithm and output the optimal results. If no, return to step 4 and continue the optimization process.


### 3.5. Double-Legalization Interpolation Method

At present, the reliability data of servo system for CNC machine tools in China are mainly calculated according to the relevant methods in GJB/899A-2009 reliability appraisal and acceptance standard. Due to the high time and economic cost required by the evaluation method, there are few reliable data that can be used as the basis. In order to ensure the accuracy of reliability prediction using neural network model, in this article, double-legalization interpolation method is used to expand the existing reliability data and apply it to the training of neural network. Double-legalization interpolation method is a special type of approximation function. A basic attribute of the interpolation function is that the sampled data must be consistent with the interpolation section. In other words, iff is a sample function and g is the corresponding interpolation function, then g (xk) = f (xk), k = 1, 2, …, if and only if xk is an interpolation point. In equidistant data and one-dimensional signal, the convolution-based interpolation function g can be written as equation (11).(11)gx=∑k=1kckux−xkh.In equation (12), ck is the coefficient determined by the input data, u is the kernel function, k is the number of given data points, the coefficient ck is some simple values, h represents the sampling increment, and xk is the interpolation node. In this method, the value of function at point (x, y) can be obtained by the weighted average of the nearest 16 sampling points in the rectangular grid. Here, two polynomial interpolation cubic functions need to be used, one in each direction. Bicubic interpolation is calculated by the following formula:(12)∑i=03∑j=03aijxiyi.Compared with the near point interpolation and bilinear interpolation, it consumes more time and memory, but the surface is smoother. Because the main application object of this article is the reliability prediction of servo system, this method is selected to further improve the prediction accuracy.


## 3.1. RBF Neural Network

In 1988, Broomhead and Lowe introduced the radial basis function (RBF) neural network, which is based on the local response principle of biological neurons. The RBF neural network has been applied to more and more fields because it can realize a consistent approach to nonlinear networking.As shown in Figure1, the RBF neural network is a three-layer feedforward network, including the input layer, the output layer, and the hidden layer. The nodes in adjacent layers are full connected. xi(i = 1, 2, …, n) are inputs, hi(i = 1, 2, …, m) are activation functions of the hidden layer, and wi(i = 1, 2, …, m) are weights of output layer. In RBF neural network, the mapping between the input layer and the hidden layer is constructed by Gaussian radial basis function, and the mapping between the output layer and the hidden layer is constructed by linear function. The activation function of the hidden layer nodes locally responds to input signal. That is to say, its activation is maximal when an input is located at the RBF center, while the activation decreases monotonically when the distance between the RBF center and an input increases.Figure 1
Structure of a generalized RBF neural network.


## 3.2. Improved PSO Algorithm

The particle swarm optimization (PSO) algorithm is a random optimization technology based on particle swarm and has strong global searching ability. Moreover, it has the characteristics of easy operation and few adjustment parameters. Hence, it has been widely applied to various fields.The standard PSO algorithm model can be defined as follows: in the n-dimensional searching space, there is a particle swarmX that is composed of N particles. The position of particle i in the particle swarm is xi = {xi1, xi2,…,xin}T, and its velocity is vi=vi1,vi2,…,vinT. The individual extremum and global extremum of this particle are pi = {pi1, pi2,…, pin} T and pg=pg1,pg2,…,pgnT, respectively. The velocity and position updating formulas of particle xi can be expressed as(8)vidk+1=ωvidk+c1r1pidk−xidk+c2r2pgdk−xidk,xidk+1=xidk+vidk+1,where i = 1,2, …, n (n is the total number of particles); k is the number of iterations in the current stage; vidk is the velocity of the d-dimensional component of the flying velocity vector of particle i during k iterations; xidk is the position of the d-dimensional component of the flying velocity vector of particle i during k iterations; pidk is the d-dimensional component of the individual optimal solution of particle i; pgdk is the d-dimensional component of the best position of the particle swarm; c1 and c2 are acceleration constants; and r1 and r2 are random functions used to generate random numbers within (0, 1). In the implementation process, the PSO algorithm approaches the global optimal solution gradually by continuous updating and iteration of positions, velocity, and target positions. However, it may become trapped during local optimization and be unable to obtain the global optimal solution because particles can easily have inadequate mining capacity in the late implementation stage of the PSO algorithm. To address this problem, some optimization and improvements were made to the velocity and weights of the PSO algorithm.


### 3.2.1. Optimization of Velocity Updating Formula

For optimization and improvement of the self-cognitive capacity of a single particle in the velocity updating formula, the optimal value of a single particle is replaced by the weighted average optimal value of all particles. The velocity and position updating formulas of the improved PSO algorithm are as follows:(9)vidk+1=ωvidk+c1r1∑i=0npidkfin+c2r2pgdk−xidk,xidk+1=xidk+vidk+1,where fi is the frequency of occurrence of the individual optimal value of particle i; n is the total number of particles; k is the number of current iterations; and d is the dimension of the particles.The decision method of a single particle based on the empirical results of the particle swarm is built by updating and calculating the velocity and position using the weighted average ofpidk. Meanwhile, the overall mining capacity and calculation efficiency are further improved by introducing the concept of the weighted average.


### 3.2.2. Optimization of Inertia Weight

In the PSO algorithm, the rate of convergence is accelerated when the inertia weight is relatively high, but a more precise convergence outcome is gained when the inertia weight is relatively low. Previous studies indicate that the rate of convergence of a PSO algorithm is highest at inertia weights of 0.8–1.2. Moreover, the rate of convergence within this range is positively related to the inertia weight. When the inertia weight is higher than 1.3, the convergence efficiency decreases significantly. To obtain a high rate of convergence and high precision, this study determined the optimal inertia weight through a self-adaptive method. This implies that the value of the inertia weight decreases with increases in the number of iterations. The formula of inertia weight in this study is(10)ωk=ωmax0.55+e−αk/kmax−0.9,where k is the number of iterations of the current particle swarm; α is a positive coefficient that adjusts the rate of change in inertia weight (determined to be 0.55 in this study according to empirical data); kmax is the maximum number of iterations of the particle swarm; and ωmax is the upper limit of ωk (set to 1.15 in this study).


## 3.2.1. Optimization of Velocity Updating Formula

For optimization and improvement of the self-cognitive capacity of a single particle in the velocity updating formula, the optimal value of a single particle is replaced by the weighted average optimal value of all particles. The velocity and position updating formulas of the improved PSO algorithm are as follows:(9)vidk+1=ωvidk+c1r1∑i=0npidkfin+c2r2pgdk−xidk,xidk+1=xidk+vidk+1,where fi is the frequency of occurrence of the individual optimal value of particle i; n is the total number of particles; k is the number of current iterations; and d is the dimension of the particles.The decision method of a single particle based on the empirical results of the particle swarm is built by updating and calculating the velocity and position using the weighted average ofpidk. Meanwhile, the overall mining capacity and calculation efficiency are further improved by introducing the concept of the weighted average.


## 3.2.2. Optimization of Inertia Weight

In the PSO algorithm, the rate of convergence is accelerated when the inertia weight is relatively high, but a more precise convergence outcome is gained when the inertia weight is relatively low. Previous studies indicate that the rate of convergence of a PSO algorithm is highest at inertia weights of 0.8–1.2. Moreover, the rate of convergence within this range is positively related to the inertia weight. When the inertia weight is higher than 1.3, the convergence efficiency decreases significantly. To obtain a high rate of convergence and high precision, this study determined the optimal inertia weight through a self-adaptive method. This implies that the value of the inertia weight decreases with increases in the number of iterations. The formula of inertia weight in this study is(10)ωk=ωmax0.55+e−αk/kmax−0.9,where k is the number of iterations of the current particle swarm; α is a positive coefficient that adjusts the rate of change in inertia weight (determined to be 0.55 in this study according to empirical data); kmax is the maximum number of iterations of the particle swarm; and ωmax is the upper limit of ωk (set to 1.15 in this study).


## 3.3. Construction of the RBF Neural Network

In this study, stress factors that influence system reliability were screened by the self-organizing algorithm with consideration of the performance and computational load of the neural network. A three-layer RBF neural network was built with six types of stress factors that may influence the reliability of the servo system used as input neurons and the MTBF of the servo system as the output neuron. The neurons of the input layer are shown in Figure2.Figure 2
Neural network input neurons.The complexity of the servo system was assessed by industrial experts. The comprehensive judgment is carried out in combination with the performance parameters, communication mode, control mode, and electrical design structure of the servo system. The comprehensive evaluation is carried out by several experts, and the final score is determined according to the pruning average value. The complexity classification principles are shown in Table1.Table 1
Complexity Classification of the servo system.
TypeInfluencing factorsComplicated9–10Relatively complicated7–8Moderate5–6Relatively simple3–4Simple1–2


## 3.4. Improved PSO-Based RBF Neural Network Algorithm

Reliability prediction based on RBF is a nonlinear and complex engineering problem. In order to obtain the neural network model with high prediction accuracy, a large amount of data are required for training. However, if the traditional RBF neural network is directly applied for training and simulation, it is easy to have problems such as long convergence time and falling into local optimization due to the inherent defects of RBF neural network. It will affect the research efficiency and research results. Therefore, combined with the research basis of relevant scholars, this article introduces the improved PSO algorithm to optimize it. The steps are described as follows.(1)
Set the input and output data of the network. The input data are the fault phenomenon, and the output data are the fault causes.(2)
Set the initial parameters of the improved PSO algorithm. The maximum number of iterations (kmax), the scale of the particle swarm, and the c1 and c2 value of the particles are set to 50, 10, 2 and 2, respectively. Meanwhile, the optimal position of a single particle and the global optimal position are initialized, which are denoted as pid and pgd.(3)
The position and velocity of particles are initialized.(4)
Calculate the fitness value of particles.(5)
If the fitness of a particle is better than those at any previous moment, this fitness is used as the individual extremum of the particle. If the fitness of this particle is better than those of all particles at any previous moment, this particle is used as the extremum of the particle swarm.(6)
Update the positions and velocities of particles according to equations (3) and (4). Besides, the weights of the algorithm are adjusted dynamically through a self-adaptive operation in accordance with equation (2).(7)
Judge whether the end condition is reached. If yes, terminate the algorithm and output the optimal results. If no, return to step 4 and continue the optimization process.


## 3.5. Double-Legalization Interpolation Method

At present, the reliability data of servo system for CNC machine tools in China are mainly calculated according to the relevant methods in GJB/899A-2009 reliability appraisal and acceptance standard. Due to the high time and economic cost required by the evaluation method, there are few reliable data that can be used as the basis. In order to ensure the accuracy of reliability prediction using neural network model, in this article, double-legalization interpolation method is used to expand the existing reliability data and apply it to the training of neural network. Double-legalization interpolation method is a special type of approximation function. A basic attribute of the interpolation function is that the sampled data must be consistent with the interpolation section. In other words, iff is a sample function and g is the corresponding interpolation function, then g (xk) = f (xk), k = 1, 2, …, if and only if xk is an interpolation point. In equidistant data and one-dimensional signal, the convolution-based interpolation function g can be written as equation (11).(11)gx=∑k=1kckux−xkh.In equation (12), ck is the coefficient determined by the input data, u is the kernel function, k is the number of given data points, the coefficient ck is some simple values, h represents the sampling increment, and xk is the interpolation node. In this method, the value of function at point (x, y) can be obtained by the weighted average of the nearest 16 sampling points in the rectangular grid. Here, two polynomial interpolation cubic functions need to be used, one in each direction. Bicubic interpolation is calculated by the following formula:(12)∑i=03∑j=03aijxiyi.Compared with the near point interpolation and bilinear interpolation, it consumes more time and memory, but the surface is smoother. Because the main application object of this article is the reliability prediction of servo system, this method is selected to further improve the prediction accuracy.


## 4. Simulation and Verification of the Predictive Model

### 4.1. Steps Used for Servo System Reliability Prediction

To assure the efficiency and precision of the constructed model, many tests were conducted. The limit of error of the neural network, diffusion factor, and number of neurons were set to 1e−8, 22, and 101, respectively. A predictive model with a relatively perfect structure was gained. In this RBF neural network, the nodes of the hidden layer are uncertain. The function can add new hidden nodes continuously according to the set error target and adjust the center, standard deviation, and weights of nodes until all networks meet the expected error requirements. The steps are shown in Figure 3.Figure 3
Test procedure.


### 4.2. Original Data set Used for Servo System Reliability Prediction

In this study, the proposed reliability prediction model was simulated and verified based on staged reliability test results of 10 products manufactured by five Chinese servo system manufacturers, which were conducted at a national machine tool testing center. The data are listed in Table2.Table 2
Original sample data.
No.Working torque (N·m)Current (A)Temperature change (°C)Working hours (kh)ComplexityNumber of faultsMTBF (kh)11.003.0066.84702.9521.272.00106.84802.9531.272.0086.84802.9542.392.00176.84802.9552.392.00195.50811.36643.00166.21902.68753.00166.21702.68862.00206.01911.49982.00176.21702.68109.552.00206.21702.68The servo system reliability test device is shown in Figure4:Figure 4
Servo system reliability test bed.


### 4.3. Data Preprocessing

#### 4.3.1. “Zero Failure”

In reliability data prediction, “zero failure” data have to be preprocessed because they provide little valid information. The increment function of the reliability coefficient was chosen as the prior distributionπp∝p2 of p, and the posterior distribution of p was calculated based on the Bayesian method.When “zero failure” occurs, the likelihood function ofp is L(0/p) = p2. Through the Bayesian estimation method, the reliability (p˜) was gained under a square-loss condition (equation (13)). The “zero failure” (fi) was transformed to fi˜ (equation (12)).(13)p˜=∫01phpnidp,(14)f˜i=ni1−p˜.


#### 4.3.2. Normalized Data

The torque, current, temperature rise, working hours, complexity, and number of faults in the data set correspond to different units and orders. It is necessary to normalize the relevant data to eliminate the influences of the dimension quantity of the variables and increase the precision of assessment. The normalization method is as follows:(15)x˜i=xi−xmin0.5xmax−xmin−1.


#### 4.3.3. Data Expansion

To use training samples fully and further increase the prediction precision of the model, the data were expanded by 100 items through the double-legalization interpolation method. Using this method, it can effectively avoid overparameterization and improve the prediction accuracy. At the same time, in order to avoid the model learning noise factors and affecting the performance of the model due to the excessive expansion of the number of samples, after repeated testing, it is determined to expand to 100 samples for training. The expanded and normalized data are shown in Table3.Table 3
Expanded and normalized sample data.
No.Working torque (N·m)Current (A)Temperature change (°C)Working hours (kh)ComplexityNumber of faultsMTBF (kh)10.85820.00600.93150.06170.08650.11000.931020.73830.01180.93150.09000.1270.11000.931030.64460.02260.93150.11660.16560.11000.931040.57980.02760.93150.14160.20250.11000.931050.51820.03240.93150.16480.23750.11000.9310……………………970.93580.96590.53170.74990.29080.04490.8296980.93580.97780.52140.75480.19930.06370.8133990.93580.98920.50930.76050.10240.08570.79411000.93581.00000.49540.76710.0000.11000.7722


#### 4.3.4. Homogenizing Sampling Data

In practical application scenarios, there is a data concentration phenomenon, which might make the sample data unable to represent the overall data. Therefore, a homogenizing sampling data acquisition method is proposed. It divides the whole sample set into 10 parts. Nine of them are chosen as the training set in turn, and the other one is used as the testing set (Figure5), thus providing the optimal reliability prediction model. The sample feature deviation caused by polymerization of samples is eliminated.Figure 5
Method of homogenizing sampling data.


#### 4.3.5. Primary Metrics Typology

According to the research results of Botchkarev [26] and other scholars, and the research content of this article, performance metrics absolute error (AE) and squared error (SE) are selected. In order to better illustrate the performance of the function, mean aggregation and maximum aggregation methods are used. Performance metrics is as follows:(16)maxAE=ejj=1,nmax,MSE=1n∑j=1nej2,where ej  = actual values-predicted values; n is the size of the data set.


### 4.4. Construction and Verification of the Reliability Prediction Model Based on the 10-Fold Crossing Method (Divide the Data Set into Ten Parts, 9 of Which Are Used as Training Data and 1 as Test Data)

According to the results of the 10 models built by the 10-fold method, all models provide relatively good predictions. The mean maximum relative error of the 10 models is 0.0017, and the mean square error is 0.00078. Experimental data show that the average MTBF of the servo system is better predicted by applying the model as a whole, and the adaptability of the model is better explained due to the use of ten-folding methods, and the optimal training model is selected. The results are listed in Table4.Table 4
Summary of the results of the reliability prediction model.
ModelTraining iterationsMean square errorMaximum absolute error valueModel 1290.0017390.002454Model 2260.0006330.001151Model 3210.0003280.000328Model 4210.001050.00105Model 5240.0006170.00105Model 6250.0013870.001848Model 7210.0003630.00064Model 8230.000050.0001Model 9220.0003630.00064Model 10310.0000080.000017Model 10 is the best one, showing a mean square error of 0.00008 and a maximum absolute error of 0.00017. Model 1 is the poorest, with a mean square error of 0.001739 and maximum absolute error of 0.002454.The construction process and reliability prediction results of Model 10 are introduced in the following text. Sample data [11–100] were input into the neural network model as training data. After 31 training iterations, the expected target was gained, and the mean square error was 2.5911e−9. The training curve of the RBF neural network is shown in Figure 6.Figure 6
Training curve of RBF neural network.The [1–10] sample data were input into the neural network as testing data and to compare the predicted and measured reliability values (Figure7). The mean square error and maximum relative error of the predicted results were 0.000008 and 0.000017, respectively.Figure 7
Comparison of RBF-predicted and real values.The construction process and reliability prediction results of Model 1 are introduced as follows: Sample data [1–90] were input into the neural network model as training data. After 29 training iterations, the expected target was gained and the mean square error was 4.3814e−7. The training curve of the RBF neural network is shown in Figure 8.Figure 8
Training curve of RBF neural network.The [91–100] sample data were input into the neural network as testing data, and a comparison between the predicted and real reliability values was made (Figure9). The mean square error and maximum relative error of the predicted results were 0.001739 and 0.002454, respectively.Figure 9
Comparison of RBF-predicted and real values.


## 4.1. Steps Used for Servo System Reliability Prediction

To assure the efficiency and precision of the constructed model, many tests were conducted. The limit of error of the neural network, diffusion factor, and number of neurons were set to 1e−8, 22, and 101, respectively. A predictive model with a relatively perfect structure was gained. In this RBF neural network, the nodes of the hidden layer are uncertain. The function can add new hidden nodes continuously according to the set error target and adjust the center, standard deviation, and weights of nodes until all networks meet the expected error requirements. The steps are shown in Figure 3.Figure 3
Test procedure.


## 4.2. Original Data set Used for Servo System Reliability Prediction

In this study, the proposed reliability prediction model was simulated and verified based on staged reliability test results of 10 products manufactured by five Chinese servo system manufacturers, which were conducted at a national machine tool testing center. The data are listed in Table2.Table 2
Original sample data.
No.Working torque (N·m)Current (A)Temperature change (°C)Working hours (kh)ComplexityNumber of faultsMTBF (kh)11.003.0066.84702.9521.272.00106.84802.9531.272.0086.84802.9542.392.00176.84802.9552.392.00195.50811.36643.00166.21902.68753.00166.21702.68862.00206.01911.49982.00176.21702.68109.552.00206.21702.68The servo system reliability test device is shown in Figure4:Figure 4
Servo system reliability test bed.


## 4.3. Data Preprocessing

### 4.3.1. “Zero Failure”

In reliability data prediction, “zero failure” data have to be preprocessed because they provide little valid information. The increment function of the reliability coefficient was chosen as the prior distributionπp∝p2 of p, and the posterior distribution of p was calculated based on the Bayesian method.When “zero failure” occurs, the likelihood function ofp is L(0/p) = p2. Through the Bayesian estimation method, the reliability (p˜) was gained under a square-loss condition (equation (13)). The “zero failure” (fi) was transformed to fi˜ (equation (12)).(13)p˜=∫01phpnidp,(14)f˜i=ni1−p˜.


### 4.3.2. Normalized Data

The torque, current, temperature rise, working hours, complexity, and number of faults in the data set correspond to different units and orders. It is necessary to normalize the relevant data to eliminate the influences of the dimension quantity of the variables and increase the precision of assessment. The normalization method is as follows:(15)x˜i=xi−xmin0.5xmax−xmin−1.


### 4.3.3. Data Expansion

To use training samples fully and further increase the prediction precision of the model, the data were expanded by 100 items through the double-legalization interpolation method. Using this method, it can effectively avoid overparameterization and improve the prediction accuracy. At the same time, in order to avoid the model learning noise factors and affecting the performance of the model due to the excessive expansion of the number of samples, after repeated testing, it is determined to expand to 100 samples for training. The expanded and normalized data are shown in Table3.Table 3
Expanded and normalized sample data.
No.Working torque (N·m)Current (A)Temperature change (°C)Working hours (kh)ComplexityNumber of faultsMTBF (kh)10.85820.00600.93150.06170.08650.11000.931020.73830.01180.93150.09000.1270.11000.931030.64460.02260.93150.11660.16560.11000.931040.57980.02760.93150.14160.20250.11000.931050.51820.03240.93150.16480.23750.11000.9310……………………970.93580.96590.53170.74990.29080.04490.8296980.93580.97780.52140.75480.19930.06370.8133990.93580.98920.50930.76050.10240.08570.79411000.93581.00000.49540.76710.0000.11000.7722


### 4.3.4. Homogenizing Sampling Data

In practical application scenarios, there is a data concentration phenomenon, which might make the sample data unable to represent the overall data. Therefore, a homogenizing sampling data acquisition method is proposed. It divides the whole sample set into 10 parts. Nine of them are chosen as the training set in turn, and the other one is used as the testing set (Figure5), thus providing the optimal reliability prediction model. The sample feature deviation caused by polymerization of samples is eliminated.Figure 5
Method of homogenizing sampling data.


### 4.3.5. Primary Metrics Typology

According to the research results of Botchkarev [26] and other scholars, and the research content of this article, performance metrics absolute error (AE) and squared error (SE) are selected. In order to better illustrate the performance of the function, mean aggregation and maximum aggregation methods are used. Performance metrics is as follows:(16)maxAE=ejj=1,nmax,MSE=1n∑j=1nej2,where ej  = actual values-predicted values; n is the size of the data set.


## 4.3.1. “Zero Failure”

In reliability data prediction, “zero failure” data have to be preprocessed because they provide little valid information. The increment function of the reliability coefficient was chosen as the prior distributionπp∝p2 of p, and the posterior distribution of p was calculated based on the Bayesian method.When “zero failure” occurs, the likelihood function ofp is L(0/p) = p2. Through the Bayesian estimation method, the reliability (p˜) was gained under a square-loss condition (equation (13)). The “zero failure” (fi) was transformed to fi˜ (equation (12)).(13)p˜=∫01phpnidp,(14)f˜i=ni1−p˜.


## 4.3.2. Normalized Data

The torque, current, temperature rise, working hours, complexity, and number of faults in the data set correspond to different units and orders. It is necessary to normalize the relevant data to eliminate the influences of the dimension quantity of the variables and increase the precision of assessment. The normalization method is as follows:(15)x˜i=xi−xmin0.5xmax−xmin−1.


## 4.3.3. Data Expansion

To use training samples fully and further increase the prediction precision of the model, the data were expanded by 100 items through the double-legalization interpolation method. Using this method, it can effectively avoid overparameterization and improve the prediction accuracy. At the same time, in order to avoid the model learning noise factors and affecting the performance of the model due to the excessive expansion of the number of samples, after repeated testing, it is determined to expand to 100 samples for training. The expanded and normalized data are shown in Table3.Table 3
Expanded and normalized sample data.
No.Working torque (N·m)Current (A)Temperature change (°C)Working hours (kh)ComplexityNumber of faultsMTBF (kh)10.85820.00600.93150.06170.08650.11000.931020.73830.01180.93150.09000.1270.11000.931030.64460.02260.93150.11660.16560.11000.931040.57980.02760.93150.14160.20250.11000.931050.51820.03240.93150.16480.23750.11000.9310……………………970.93580.96590.53170.74990.29080.04490.8296980.93580.97780.52140.75480.19930.06370.8133990.93580.98920.50930.76050.10240.08570.79411000.93581.00000.49540.76710.0000.11000.7722


## 4.3.4. Homogenizing Sampling Data

In practical application scenarios, there is a data concentration phenomenon, which might make the sample data unable to represent the overall data. Therefore, a homogenizing sampling data acquisition method is proposed. It divides the whole sample set into 10 parts. Nine of them are chosen as the training set in turn, and the other one is used as the testing set (Figure5), thus providing the optimal reliability prediction model. The sample feature deviation caused by polymerization of samples is eliminated.Figure 5
Method of homogenizing sampling data.


## 4.3.5. Primary Metrics Typology

According to the research results of Botchkarev [26] and other scholars, and the research content of this article, performance metrics absolute error (AE) and squared error (SE) are selected. In order to better illustrate the performance of the function, mean aggregation and maximum aggregation methods are used. Performance metrics is as follows:(16)maxAE=ejj=1,nmax,MSE=1n∑j=1nej2,where ej  = actual values-predicted values; n is the size of the data set.


## 4.4. Construction and Verification of the Reliability Prediction Model Based on the 10-Fold Crossing Method (Divide the Data Set into Ten Parts, 9 of Which Are Used as Training Data and 1 as Test Data)

According to the results of the 10 models built by the 10-fold method, all models provide relatively good predictions. The mean maximum relative error of the 10 models is 0.0017, and the mean square error is 0.00078. Experimental data show that the average MTBF of the servo system is better predicted by applying the model as a whole, and the adaptability of the model is better explained due to the use of ten-folding methods, and the optimal training model is selected. The results are listed in Table4.Table 4
Summary of the results of the reliability prediction model.
ModelTraining iterationsMean square errorMaximum absolute error valueModel 1290.0017390.002454Model 2260.0006330.001151Model 3210.0003280.000328Model 4210.001050.00105Model 5240.0006170.00105Model 6250.0013870.001848Model 7210.0003630.00064Model 8230.000050.0001Model 9220.0003630.00064Model 10310.0000080.000017Model 10 is the best one, showing a mean square error of 0.00008 and a maximum absolute error of 0.00017. Model 1 is the poorest, with a mean square error of 0.001739 and maximum absolute error of 0.002454.The construction process and reliability prediction results of Model 10 are introduced in the following text. Sample data [11–100] were input into the neural network model as training data. After 31 training iterations, the expected target was gained, and the mean square error was 2.5911e−9. The training curve of the RBF neural network is shown in Figure 6.Figure 6
Training curve of RBF neural network.The [1–10] sample data were input into the neural network as testing data and to compare the predicted and measured reliability values (Figure7). The mean square error and maximum relative error of the predicted results were 0.000008 and 0.000017, respectively.Figure 7
Comparison of RBF-predicted and real values.The construction process and reliability prediction results of Model 1 are introduced as follows: Sample data [1–90] were input into the neural network model as training data. After 29 training iterations, the expected target was gained and the mean square error was 4.3814e−7. The training curve of the RBF neural network is shown in Figure 8.Figure 8
Training curve of RBF neural network.The [91–100] sample data were input into the neural network as testing data, and a comparison between the predicted and real reliability values was made (Figure9). The mean square error and maximum relative error of the predicted results were 0.001739 and 0.002454, respectively.Figure 9
Comparison of RBF-predicted and real values.


## 5. Comparison with Ordinary Reliability Prediction Models

To demonstrate the superior efficiency and precision of reliability prediction models based on the improved PSO-based RBF neural network, they were compared with an PSO-RBF and an ordinary BP neural network using the training and testing samples of Model 10.


### 5.1. PSO-RBF Predictive Model

Sample data [11–100] were input into the neural network model as training data. After 141 training iterations, the expected target was gained and the mean square error was6.4869e−6. The training curve of the PSO-BP is shown in Figure 10.Figure 10
Training curve of the BP neural network.The [1–10] sample data were input into the neural network as testing data and a comparison between the PSO-BP neural network’s predicted reliability values and real reliability values was made (Figure11). The mean square error and maximum relative error of the predicted results were 0.000023 and 0.000044, respectively.Figure 11
Comparison of BP-predicted and real values.


### 5.2. BP Neural Network Predictive Model

Sample data [11–100] were input into the neural network model as training data. After 147 training iterations, the expected target was gained and the mean square error was 5.1875e−7. The training curve of the RBF neural network is shown in Figure 12.Figure 12
Training curve of the BP neural network.The [1–10] sample data were input into the neural network as testing data and a comparison between the BP neural network’s predicted reliability values and real reliability values was made (Figure13). The mean square error and maximum relative error of the predicted results were 0.000241 and 0.001296, respectively.Figure 13
Comparison of BP-predicted and real values.By comparison (Table5), the RBF neural network prediction model based on particle swarm optimization algorithm has obvious advantages in prediction accuracy and training efficiency. In addition, through comparison, the RBF neural network optimized by PSO has higher prediction accuracy, but there are still deficiencies compared with the improved PSO. This proves that particle swarm optimization algorithm significantly improves the convergence speed of RBF network and achieves the training goal after only 31 iterations.Table 5
Complexity classification of the servo system.
Model typeTraining iterationsMean square errorMaximum absolute error valueIPSO-RBF (model 10)310.0000080.000017PSO-RBF1410.0000230.000044BP1470.0002410.001296


## 5.1. PSO-RBF Predictive Model

Sample data [11–100] were input into the neural network model as training data. After 141 training iterations, the expected target was gained and the mean square error was6.4869e−6. The training curve of the PSO-BP is shown in Figure 10.Figure 10
Training curve of the BP neural network.The [1–10] sample data were input into the neural network as testing data and a comparison between the PSO-BP neural network’s predicted reliability values and real reliability values was made (Figure11). The mean square error and maximum relative error of the predicted results were 0.000023 and 0.000044, respectively.Figure 11
Comparison of BP-predicted and real values.


## 5.2. BP Neural Network Predictive Model

Sample data [11–100] were input into the neural network model as training data. After 147 training iterations, the expected target was gained and the mean square error was 5.1875e−7. The training curve of the RBF neural network is shown in Figure 12.Figure 12
Training curve of the BP neural network.The [1–10] sample data were input into the neural network as testing data and a comparison between the BP neural network’s predicted reliability values and real reliability values was made (Figure13). The mean square error and maximum relative error of the predicted results were 0.000241 and 0.001296, respectively.Figure 13
Comparison of BP-predicted and real values.By comparison (Table5), the RBF neural network prediction model based on particle swarm optimization algorithm has obvious advantages in prediction accuracy and training efficiency. In addition, through comparison, the RBF neural network optimized by PSO has higher prediction accuracy, but there are still deficiencies compared with the improved PSO. This proves that particle swarm optimization algorithm significantly improves the convergence speed of RBF network and achieves the training goal after only 31 iterations.Table 5
Complexity classification of the servo system.
Model typeTraining iterationsMean square errorMaximum absolute error valueIPSO-RBF (model 10)310.0000080.000017PSO-RBF1410.0000230.000044BP1470.0002410.001296


## 6. Conclusions

This study systematically elaborated the principles of the RBF neural network. The stress factors influencing the reliability of servo systems were analyzed via engineering experience in combination with the reliability prediction characteristics of a CNC machine servo system. These stress factors were used as input neurons. On this basis, based on the evaluation of industry experts, the product complexity factor is added. Meanwhile, the performance parameters of neural networks, including the limit of error, diffusion factor, and number of neurons, were determined. Three-layer RBF neural network models for reliability prediction of CNC machine servo systems were built and then simulated and verified by the 10-fold method. The training data were expanded by introducing the double-legalization interpolation method. The maximum absolute error between the best predicted values and real values was 0.0008%. Meanwhile, the PSO-based RBF neural network model was compared with an RBF neural network model and a BP neural network model. The results prove the advantages of using RBF models for the processing of highly nonlinear data.



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*Source: 2684942-2022-04-21.xml*