Research on Machining Quality Prediction Method Based on Machining Error Transfer Network and Grey Neural Network

Abstract

Machining quality prediction is the critical link of quality control in parts machining. With the advent of the Industry 4.0 era, intelligent manufacturing and data-driven technologies bring new ideas for quality control in complex machining processes. Quality control is complicated for multi-process, multi-condition, small-batch, and high-precision parts processing requirements. To solve this problem, this paper proposes a machining quality prediction method based on the machining error transfer network and the grey neural network. Initially, by constructing a processing error transfer network, the error transfer law in part processing is described, and the PageRank algorithm and the influence degree of the nodes are used to determine the critical quality features. Additionally, the problem of low prediction accuracy due to small sample data and multiple coupling relationships is solved using the grey neural network algorithm, and a high accuracy prediction of critical quality features is achieved. Finally, the effectiveness and reliability of the method are verified by the case of medium-speed marine diesel engine fuselage processing. The results indicate that this method not only effectively identifies critical quality features in the machining process of complex parts, but it also maintains a high predictive accuracy for these features, even with small samples and limited data.

1. Introduction

Processing quality prediction is crucial in modern manufacturing, mainly when producing complex parts. By collecting and analyzing various parameters in processing, establishing an effective prediction model has become a meaningful way to achieve intelligent manufacturing. Through accurate quality prediction, not only can we improve the manufacturing quality of parts and reduce the production cost, but we can also effectively reduce the product defective rate and scrap rate [1]. The joint action of multiple processes usually guarantees the final machining quality of complex parts. Such parts have a complex structure and various processes. The machining quality of parts is not only affected by machining machine tools, tools, fixtures, and other machining elements and cutting parameters, but it also may be affected by the output quality errors of the previous process [2]. Therefore, it is necessary to carry out a detailed analysis of each process in the process of parts processing in order to achieve comprehensive quality control.

In recent years, with the rapid development of machine learning methods and extensive data analyses, establishing efficient and accurate quality prediction models has become the mainstream direction of the manufacturing industry [3]. Especially in the era of Industry 4.0, intelligent manufacturing has gradually become a significant development trend in the manufacturing industry. By integrating industrial IoT and big data platforms, the manufacturing process can collect and analyze the processing data of multiple processes in real time to achieve dynamic control and a prediction of processing quality. The wide application of new sensors and the enhancement of computing power have also further promoted progress in this field. Many scholars have proposed various methods to improve the accuracy and efficiency of processing quality prediction. Common quality prediction methods include grey theory [4,5], neural network [6], particle swarm [7,8], and support vector machine [9]. These methods have advantages in different application scenarios and are suitable for different data situations and problem types. Grey theory suits situations with few data and incomplete information, while neural networks deal well with complex nonlinear relationships. Particle swarm optimization searches for the global optimal solution by simulating population intelligence, while support vector machines perform well in dealing with high-dimensional data and minor sample problems. Wang et al. [10] combined grey system theory and artificial neural networks to predict the machining quality of shaft parts more accurately under limited data, thus optimizing the production process. Manikandan et al. [11] solved the problem of short tool life and the poor machinability of Hastealloy C276 by optimizing the EDM parameters through the hybrid grey relational analysis and neural network model. Soepangkat et al. [12] used grey fuzzy analysis and the BPNN-GA method to optimize the cryogenically cooled face milling process, effectively improving the multiple performances regarding surface roughness, cutting force, and the metal removal rate. Jiang et al. [13] used support vector regression based on the error transfer network and particle swarm optimization to solve effectively the problem of insufficient data in aerospace manufacturing. Xu et al. [14] proposed a neural network algorithm based on particle swarm optimization and grey correlation, which significantly improved the accuracy of machining quality prediction. Huang et al. [15] solved the automation problem in grinding machining by using least squares and support vector machines. Lu [16] proposed a radial-based neural network model with adaptive parameter tuning to predict the turning surface quality. Abburi et al. [17] developed an adaptive prediction system for turning surface roughness based on neural networks and fuzzy set theory. This can achieve a more accurate prediction under limited training data. Liu et al. [18] combined a simulated annealing algorithm and BP neural network to achieve the prediction of cutting deformation. Ashtiani et al. [19] developed a neural network model based on a feed-forward propagation learning algorithm for predicting the thermal-deformation behavior of alloys. Lu [20] developed a model to predict surface roughness in the machining process of a part, based on Support Vector Machines (SVMs) and the Artificial Bee Colony (ABC) algorithm. The prediction model significantly improved the prediction accuracy and reduced the parameter adjustment time. However, most studies focus on predicting part quality under a single process. Complex multi-process parts’ processing features inevitably affect the final quality due to the coupling effect of inter-process error transfer. The accuracy of the established quality prediction model may decrease if the role of error transfer in processing between processes is ignored [21].

Many scholars have proposed methods for predicting machining quality features oriented towards complex multi-process parts. Abelian et al. [22] analyzed multi-process parts’ process configuration and machining features. Yin et al. [23] proposed a knowledge fusion model based on a rough set algorithm to achieve a machining quality prediction for the multi-process manufacturing of complex surface parts. Wang et al. [24] proposed a data-driven modeling approach to predict product quality in real time by obtaining the time-varying relationship between the final quality and process through the current process information. Once the state of the machining system changes, the accuracy of the established quality prediction model may also be affected. In addition, when complex parts are primarily produced in small batches, and there are insufficient machining sample data, the prediction accuracy of traditional machine learning models often fails to meet the real-time and accuracy requirements of Industry 4.0.

This paper proposes a processing quality prediction method based on a processing error transfer network and grey neural network. Quality problems in the processing of products are usually the result of the substandard processing of critical quality features in a small number of critical processes. Initially, to identify the critical quality features that have a more significant impact on quality in multi-process processing, a processing error transfer network model is established. The importance of the nodes of this model is ranked by the PageRank algorithm and the node influence degree to obtain the critical quality features of complex parts. Moreover, the number of data samples and the coupling relationship between machining quality features significantly influence the prediction results of machining quality features. The machining quality prediction model constructed by the grey neural network algorithm can produce a high-precision prediction with fewer sample data and multiple coupling relationships. Finally, the proposed method is validated by taking the machining of a medium-speed marine diesel engine’s fuselage as an example.

2. Establishment and Analysis of Machining Error Transfer Network

Complex network theory is an effective method to analyze complex systems’ coupling relationships [25]. It can model complex systems and analyze their important nodes and relationships. In the process of complex parts machining, the machining features are the basic unit of parts machining, and its state change is the change process of quality features. Therefore, using complex network theory to construct a model of error transfer based on the evolution of machining features can effectively reflect the law of error generation, accumulation, and transfer in machining [26]. By analyzing the importance of the nodes in the processing error transfer network, it is possible to identify the critical nodes that have a more significant impact on the processing quality, which provides the basis for the subsequent quality prediction.

2.1. Establishment of Machining Error Transfer Network

The error transfer network model for machining complex parts mainly includes nodes and edges. The network nodes are divided into machining feature nodes, quality feature nodes, and machining element nodes. The quality features depend on the machining features, reflecting the machining level, while the machining element nodes mainly input errors to the machining features. According to the evolution process of machining features in different stages during part machining, the positioning relationship between workpiece datums, the use of different machining elements, and the coupling relationship between machining features and their attached quality features, the connecting edges between the nodes are divided into four categories. These categories are evolution, positioning, machining, and attribute relationships [27]. The specific description of the edge relationships between nodes in the machining error transfer network is shown in Figure 1 below:

Define the complex part machining error transfer network model as follows:

$$\mathrm{G}=<\{\mathrm{W},\mathrm{M},\mathrm{Q}\},\mathrm{E}>$$

(1)

where, $\mathrm{W}=\{{\mathrm{W}}_1,{\mathrm{W}}_2,\cdots ,{\mathrm{W}}_{\mathrm{l}}\}$ represents the complex parts processing process-processing elements node-set. $\mathrm{M}=\{{\mathrm{M}}_1,{\mathrm{M}}_2,\cdots ,{\mathrm{M}}_{\mathrm{m}}\}$ represents the complex parts processing features node-set. $\mathrm{Q}=\{{\mathrm{Q}}_1,{\mathrm{Q}}_2,\cdots ,{\mathrm{Q}}_h\}$ represents the complex parts processing the features’ output quality features node-set. $\mathrm{E}=\{{\mathrm{E}}_1,{\mathrm{E}}_2,\cdots ,{\mathrm{E}}_{\mathrm{q}}\}$ represents the complex parts processing the error transfer network processing quality between processes or between the processing quality and the processing elements of the directed edge-set.

According to the description of each machining quality node and edge relationship in the machining process of the part, the error transfer connection between different machining feature nodes mainly relies on the evolution relationship, positioning relationship, and attribute relationship. The error transfer between machining feature nodes and machining element nodes mainly relies on the machining relationship. The network formed by the nodes with evolutionary (ER), positioning (LR), processing (PR), and attribute relationships (AR) is defined as the machining error transfer network. Figure 2 is a schematic diagram of the part processing error transmission sub-network.

The machining process sub-network is established based on the above. The sub-networks are merged by the machining sequence between the machining qualities to finally form a complete complex part machining error transfer network model.

2.2. Analysis of Critical Nodes in Machining Error Transfer Network

When analyzing objects with complex processing technology and numerous processes, it is difficult to identify the key processes solely through technical analysis, excluding human subjective judgment. Therefore, it is necessary to introduce topological relationship evaluation indexes to analyze the nodes and the topological relationship of the nodes in the processing error transfer network. The commonly used evaluation metrics are the degree of the nodes, average degree, network diameter, aggregation factor, average path length, etc. [28]. Although, such metrics can describe the overall performance of the network and the degree of correlation between the nodes. However, the importance of each node in a complex workpiece machining error transfer network is generally different. To more accurately rank the importance of the nodes in the part machining error transfer network and identify the critical nodes in the machining process, this paper uses the PageRank algorithm to calculate the importance of each node in the network.

The PageRank (PR) algorithm is used to rank web pages, assessing the importance of each web page by analyzing the linking relationships between them [29]. The basic principle of the PR algorithm is that, initially, each node is assumed to have the same weight, and the node weights are calculated through an iterative rule until they converge to a stable value. This stable value is called the node’s value; a higher node value indicates a higher node’s importance in the network.

The iteration rule of the PR algorithm is denoted as follows:

$$\mathrm{A}=\left[\begin{array}{cccc}{\mathrm{a}}_{11}& \cdots & {\mathrm{a}}_{1\mathrm{d}}& \begin{array}{cc}\cdots & {\mathrm{a}}_{1\mathrm{M}}\end{array}\\ ⋮& \ddots & & \begin{array}{cc}& ⋮\end{array}\\ {\mathrm{a}}_{\mathrm{m}1}& & {\mathrm{a}}_{\mathrm{m}\mathrm{d}}& \begin{array}{cc}& {\mathrm{a}}_{\mathrm{m}\mathrm{M}}\end{array}\\ \begin{array}{l}⋮\\ {\mathrm{a}}_{\mathrm{M}1}\end{array}& \begin{array}{l}\\ \cdots \end{array}& \begin{array}{l}\\ {\mathrm{a}}_{\mathrm{M}\mathrm{d}}\end{array}& \begin{array}{l}\begin{array}{cc}\ddots & ⋮\end{array}\\ \begin{array}{cc}\cdots & {\mathrm{a}}_{\mathrm{M}\mathrm{M}}\end{array}\end{array}\end{array}\right]$$

(2)

$${\mathrm{B}}_{\mathrm{i}}=\mathrm{A}·{\mathrm{B}}_{\mathrm{i}-1}$$

(3)

where ${\mathrm{B}}_{\mathrm{i}}$ is the matrix of the PR values of network nodes after the i-th iteration, the initial PR values ${\mathrm{B}}_0$ of the nodes are all set to 1/N, and N is the number of nodes in the network. ${\mathrm{a}}_{\mathrm{m}\mathrm{d}}$ in the matrix A represents the number of edges pointing to the d-th network node from the m-th network node.

The PR algorithm can obtain the weights of each quality feature node in the fuselage machining error transfer network. However, since the quality feature nodes are located at the network’s final output, these weights cannot fully reflect their actual impact. In order to better assess the importance of the quality feature nodes, the influence degree E can be defined as the node importance evaluation index. Firstly, the machining feature nodes with large PR values are filtered out and defined as critical quality control nodes. Then, the corresponding quality features are extracted based on these critical quality control nodes. Finally, the quality feature nodes’ weighted in-degree and weighted out-degree are defined, and their influence degree is solved. The weighted out-degree indicates the influence of the analyzed quality feature node on other quality feature nodes, and the weighted out-degree indicates the degree of influence of other quality feature nodes on the analyzed quality feature node.

$$\mathrm{E}={\mathrm{w}}_{\mathrm{M}}·({\displaystyle \sum _{\mathrm{i}\in {\mathrm{Q}}_{\mathrm{O}\mathrm{i}}}{\mathrm{w}}_{\mathrm{i}}}+{\displaystyle \sum _{\mathrm{j}\in {\mathrm{Q}}_{\mathrm{p}\mathrm{j}}}{\mathrm{w}}_{\mathrm{j}}})$$

(4)

where ${\mathrm{w}}_{\mathrm{M}}$ represents the weight of the machining feature node corresponding to the analyzed quality feature node. ${\mathrm{w}}_{\mathrm{i}}$ represents the weight of other quality features pointed outward by the analyzed quality feature nodes. ${\mathrm{Q}}_{\mathrm{O}\mathrm{i}}$ represents the set of other quality feature nodes that the analyzed quality feature nodes point outward. ${\mathrm{w}}_{\mathrm{j}}$ represents the weight of other quality feature nodes pointing to the analyzed quality feature nodes (including their weight). ${\mathrm{Q}}_{\mathrm{P}\mathrm{j}}$ represents the set of other quality feature nodes pointing to the analyzed quality feature nodes.

By calculating the influence degree E value of the quality feature nodes and ranking them according to the size of E, the one with a higher influence degree ranking is defined as the critical quality feature. Identifying critical quality features provides accurate data and decision support for the subsequent prediction of processing quality for critical quality features.

3. Establishment of Grey Neural Network-Based Machining Quality Prediction Model

The machining quality prediction model is a typical nonlinear multivariate pre-system model. For complex parts with multiple processes, multiple working conditions, small batches, and high accuracy, the input and output of the pre-machining quality prediction model usually show nonlinear features. Therefore, this paper adopts a grey neural network to construct the $\mathrm{G}\mathrm{N}\mathrm{N}\mathrm{M}(1,\mathrm{N})$ prediction model. The algorithm performs well in dealing with nonlinearity and a small sample prediction. The $\mathrm{G}\mathrm{N}\mathrm{N}\mathrm{M}(1,\mathrm{N})$ prediction model has a higher computational accuracy and controllable error than the conventional grey model. Compared with the conventional neural network prediction model, the $\mathrm{G}\mathrm{N}\mathrm{N}\mathrm{M}(1,\mathrm{N})$ prediction model is less computationally intensive and achieves a high prediction accuracy with a small number of samples [30]. With this combination, the accuracy and stability of the prediction model are significantly improved. Figure 3 shows the schematic diagram of the $\mathrm{G}\mathrm{N}\mathrm{N}\mathrm{M}(1,\mathrm{N})$ prediction model.

3.1. Establishment of GM (1, N) Prediction Model

In parts machining, the machining quality is often affected by various factors, so it is suitable to use the grey multivariate prediction $\mathrm{G}\mathrm{M}(1,\mathrm{N})$ model. The $\mathrm{G}\mathrm{M}(1,\mathrm{N})$ model contains N variables, including one feature variable and N−1−related factor variables. The model is used to describe the influence of N variables with correlation to the prediction target, and it takes into account the degree of correlation between the variables and the magnitude of the influence to predict the trend of the feature variables.

Let there be a total of n sets of N sample data consisting of the associated processing quality features X. Define the original feature data sequence as ${\mathrm{X}}_1^{(0)}$, which can be expressed as follows:

$${\mathrm{X}}_1^{(0)}=({\mathrm{x}}_1^{(0)}(1),{\mathrm{x}}_1^{(0)}(2),\cdots ,{\mathrm{x}}_1^{(0)}(\mathrm{n}))$$

(5)

The correlation factor sequence is ${\mathrm{X}}_{\mathrm{i}}^{(0)}$, which can be expressed as follows:

$${\mathrm{X}}_{\mathrm{i}}^{(0)}=({\mathrm{x}}_{\mathrm{i}}^{(0)}(1),{\mathrm{x}}_{\mathrm{i}}^{(0)}(2),\cdots ,{\mathrm{x}}_{\mathrm{i}}^{(0)}(\mathrm{n}))$$

(6)

where $\mathrm{i}=2,3,\cdots ,\mathrm{N}$. For this machining quality prediction system model, a cumulative generation operator on ${\mathrm{X}}_1^{(0)}$ is performed to compute the 1−AG0 sequence ${\mathrm{X}}_{\mathrm{i}}^{(1)}$.

$${\mathrm{X}}_{\mathrm{i}}^{(1)}(\mathrm{k})={\mathrm{x}}_1^{(0)}(\mathrm{k})+{\mathrm{x}}_1^{(0)}(\mathrm{k})+\cdots +{\mathrm{x}}_{\mathrm{n}}^{(0)}(\mathrm{k})$$

(7)

In order to satisfy the necessity of quasi-exponential laws for the $\mathrm{G}\mathrm{M}(1,\mathrm{N})$ model operation, a smoothness test is required for the original sequence ${\mathrm{X}}_1^{(0)}(\mathrm{k})$. If the original sequence is ${\mathrm{X}}_1^{(0)}=({\mathrm{x}}_1^{(0)}(1),{\mathrm{x}}_1^{(0)}(2),\cdots ,{\mathrm{x}}_1^{(0)}(\mathrm{k})),{\mathrm{x}}_1^{(0)}(\mathrm{k})\ge 0,\mathrm{k}=2,3,\cdots ,\mathrm{n}$, then $\mathsf{\rho }(\mathrm{k})={\displaystyle \frac{{\mathrm{x}}_1^{(0)}(\mathrm{k})}{{\mathrm{x}}_1^{(0)}(\mathrm{k}-1)}}$ is said to be the smooth ratio of the sequence ${\mathrm{X}}_1^{(0)}$. If $\mathsf{\rho }(\mathrm{k})\in [0,0.5]$, then $\mathrm{X}$ is said to be a quasi-smooth sequence. That is, it satisfies the quasi-exponential law of model operations.

${\mathrm{Z}}_1^{(1)}(\mathrm{k})$ is the immediate neighboring mean generating sequence of ${\mathrm{X}}_1^{(1)}(\mathrm{k})$, which can be expressed as follows:

$${\mathrm{Z}}_1^{(1)}(\mathrm{k})={\displaystyle \frac{1}{2}}[{\mathrm{X}}_1^{(1)}(\mathrm{k})+{\mathrm{X}}_1^{(1)}(\mathrm{k}-1)]\cdots \mathrm{k}=2,3,\cdots ,\mathrm{n}$$

(8)

Build the $\mathrm{G}\mathrm{M}(1,\mathrm{N})$ prediction model using first-order differential equations.

$${\mathrm{X}}_1^{(1)}(\mathrm{k})+\mathsf{\alpha }{\mathrm{Z}}_1^{(1)}(\mathrm{k})={\displaystyle \sum _{\mathrm{i}=2}^{\mathrm{N}}{\mathrm{b}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}}^{(1)}(\mathrm{k})}$$

(9)

where $\mathsf{\alpha }$ is the development coefficient, ${\mathrm{b}}_{\mathrm{i}}$ is the driving coefficient, ${\mathrm{b}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}}^{(1)}(\mathrm{k})$ is the driving term, and $\widehat{\mathrm{a}}={(\mathsf{\alpha },{\mathrm{b}}_2,\cdots ,{\mathrm{b}}_{\mathrm{n}})}^{\mathrm{T}}$. The parameters of $\widehat{\mathrm{a}}={(\mathsf{\alpha },{\mathrm{b}}_2,\cdots ,{\mathrm{b}}_{\mathrm{n}})}^{\mathrm{T}}$ are estimated by the least squares route and satisfy the following form:

$$\widehat{\mathrm{a}}={({\mathrm{B}}^{\mathrm{T}}\mathrm{B})}^{-1}{\mathrm{B}}^{\mathrm{T}}\mathrm{Y}$$

(10)

$$\mathrm{Y}=\left[\begin{array}{c}{\mathrm{X}}_1^{(0)}(2)\\ {\mathrm{X}}_1^{(0)}(3)\\ ⋮\\ {\mathrm{X}}_1^{(0)}(\mathrm{n})\end{array}\right],\mathrm{B}=\left\{\begin{array}{cccc}-{\mathrm{Z}}_1^{(1)}(2)& {\mathrm{Z}}_2^{(1)}(2)& \cdots & {\mathrm{Z}}_{\mathrm{N}}^{(1)}(2)\\ -{\mathrm{Z}}_1^{(1)}(2)& {\mathrm{Z}}_2^{(1)}(3)& \cdots & {\mathrm{Z}}_{\mathrm{N}}^{(1)}(3)\\ ⋮& ⋮& \ddots & ⋮\\ -{\mathrm{Z}}_1^{(1)}(\mathrm{n})& {\mathrm{Z}}_1^{(1)}(\mathrm{n})& \cdots & {\mathrm{Z}}_{\mathrm{N}}^{(1)}(\mathrm{n})\end{array}\right\}$$

(11)

When the variation of ${\mathrm{X}}_{\mathrm{i}}^{(1)}(\mathrm{i}=1,2,\cdots ,\mathrm{N})$ is small, the approximate time response expression ${\widehat{\mathrm{x}}}_1^{(1)}(\mathrm{k}+1)$ for $\mathrm{G}\mathrm{M}(1,\mathrm{N})$ is as follows:

$${\widehat{\mathrm{x}}}_1^{(1)}(\mathrm{k}+1)=({\mathrm{x}}_1^{(1)}(0)-{\displaystyle \frac{1}{\mathsf{\alpha }}}{\displaystyle \sum _{\mathrm{i}=2}^{\mathrm{N}}{\mathrm{b}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}^{(1)}(\mathrm{k}+1)}){\mathrm{e}}^{-\mathsf{\alpha }\mathrm{k}}+{\displaystyle \frac{1}{\mathsf{\alpha }}}{\displaystyle \sum _{\mathrm{i}=2}^{\mathrm{N}}{\mathrm{b}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}^{(1)}(\mathrm{k}+1)}$$

(12)

According to the principle of information priority, ${\mathrm{X}}_1^{(1)}(0)$ is taken as ${\mathrm{X}}_1^{(0)}(\mathrm{n})$. Through the cumulative reduction of ${\widehat{\mathrm{x}}}_1^{(1)}(\mathrm{k}+1)$, the ${\widehat{\mathrm{x}}}_1^{(0)}(\mathrm{k}+1)$ value of the prediction result sequence can be obtained.

$${\widehat{\mathrm{x}}}_1^{(0)}(\mathrm{k}+1)={\widehat{\mathrm{x}}}_1^{(1)}(\mathrm{k}+1)-{\widehat{\mathrm{x}}}_1^{(1)}(\mathrm{k}),\mathrm{k}=1,2,\cdots ,\mathrm{n}$$

(13)

Although the $\mathrm{G}\mathrm{M}(1,\mathrm{N})$ prediction model can predict the machining quality for small samples and information-poor parts, the prediction results will have errors due to its insufficient sample information and data randomness. Quality prediction accuracy is mainly affected by the residuals between the predicted and actual values. The overall prediction accuracy will be significantly improved if the residuals can be reasonably handled.

3.2. Construction of BP Neural Network Model

After years of development, the BP neural network has become the best and central part of artificial neural networks. In dealing with practical problems, it can transform a set of sample input/-output problems into a nonlinear optimization problem and is a powerful tool for summarizing the laws. The BP neural network has many advantages when fitting sequences. It can imitate various functional forms, such as nonlinear and segmented functions, without assuming the functional relationship between the data and directly using it to establish the correlation. At the same time, the BP neural network has high information utilization and avoids the loss of information due to positive and negative cancellations in data processing [31]. Therefore, the BP neural network is particularly suitable for the residual correction of the $\mathrm{G}\mathrm{M}(1,\mathrm{N})$ model.

Considering that the processing examples and sample data selected in this paper have a small amount of computing power, and referring to the experience of related successful cases, this paper chose the three-layer neural network structure shown in Figure 4. The network consists of the input layer, hidden layer, and output layer. The complete training process includes two stages: forward propagation and backpropagation.

N network training samples are identified, and one of them, sample p, is selected to train the neural network. Under the action of sample p, the input $\mathrm{n}\mathrm{e}{\mathrm{t}}_{\mathrm{i}}^{\mathrm{p}}$ of the i-th neuron in the hidden layer is as follows:

$$\mathrm{n}\mathrm{e}{\mathrm{t}}_{\mathrm{i}}^{\mathrm{p}}={\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{M}}{\mathrm{w}}_{\mathrm{i}\mathrm{j}}{\mathrm{o}}_{\mathrm{j}}^{\mathrm{p}}}-{\theta }_{\mathrm{i}}={\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{M}}{\mathrm{w}}_{\mathrm{i}\mathrm{j}}{\mathrm{x}}_{\mathrm{j}}^{\mathrm{p}}}-{\theta }_{\mathrm{i}}$$

(14)

where $\mathrm{i}=1,2,\cdots ,\mathrm{q}$, $\mathrm{j}=1,2,\cdots ,\mathrm{M}$ and q is the number of nodes in the hidden layer. ${\mathrm{x}}_{\mathrm{j}}^{\mathrm{p}}$ and ${\mathrm{o}}_{\mathrm{j}}^{\mathrm{p}}$ are the input and output of the infuser node j under the action of the sample p, which is the same. ${\mathrm{w}}_{\mathrm{i}\mathrm{j}}$ is the connection weight between the infuser layer neuron j and the hidden layer neuron I. ${\theta }_{\mathrm{i}}$ is the threshold value of the hidden layer neuron i.

The output ${\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}}$ of the i-th neuron of the hidden layer is as follows:

$${\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}}=\mathrm{g}(\mathrm{n}\mathrm{e}{\mathrm{t}}_{\mathrm{i}}^{\mathrm{p}})$$

(15)

where $\mathrm{g}(·)$ is the activation function.

The total input $\mathrm{n}\mathrm{e}{\mathrm{t}}_{\mathrm{k}}^{\mathrm{p}}$ for the kth neuron of the output layer is as follows:

$$\mathrm{n}\mathrm{e}{\mathrm{t}}_{\mathrm{k}}^{\mathrm{p}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{q}}{\mathrm{w}}_{\mathrm{k}\mathrm{i}}{\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}}}-{\theta }_{\mathrm{k}}$$

(16)

where ${\mathrm{w}}_{\mathrm{k}\mathrm{i}}$ is the connection weight between neuron k of the output layer and neuron i of the hidden layer, and $\mathrm{k}=1,2,\cdots ,\mathrm{L}$. ${\theta }_{\mathrm{k}}$ is the threshold value of neuron k of the output layer.

The actual output ${\mathrm{o}}_{\mathrm{k}}^{\mathrm{p}}$ is as follows:

$${\mathrm{o}}_{\mathrm{k}}^{\mathrm{p}}=\mathrm{g}(\mathrm{n}\mathrm{e}{\mathrm{t}}_{\mathrm{k}}^{\mathrm{p}})$$

(17)

Error function ${\mathrm{J}}_{\mathrm{p}}$ is as follows:

$${\mathrm{J}}_{\mathrm{p}}={\displaystyle \frac{1}{2}}{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{L}}{({\mathrm{t}}_{\mathrm{k}}^{\mathrm{p}}-{\mathrm{o}}_{\mathrm{k}}^{\mathrm{p}})}^2}$$

(18)

where ${\mathrm{t}}_{\mathrm{k}}^{\mathrm{p}}$ is the desired output.

The error signal ${\delta }_{\mathrm{k}}^{\mathrm{p}}$ of the output layer node is as follows:

$${\delta }_{\mathrm{k}}^{\mathrm{p}}=-{\displaystyle \frac{\partial {\mathrm{J}}_{\mathrm{p}}}{\partial \mathrm{n}\mathrm{e}{\mathrm{t}}_{\mathrm{k}}^{\mathrm{p}}}}=({\mathrm{t}}_{\mathrm{k}}^{\mathrm{p}}-{\mathrm{o}}_{\mathrm{k}}^{\mathrm{p}}){\mathrm{o}}_{\mathrm{k}}^{\mathrm{p}}(1-{\mathrm{o}}_{\mathrm{k}}^{\mathrm{p}})$$

(19)

The error signal ${\delta }_{\mathrm{i}}^{\mathrm{p}}$ of the hidden layer node is as follows:

$${\delta }_{\mathrm{i}}^{\mathrm{p}}=-{\displaystyle \frac{\partial {\mathrm{J}}_{\mathrm{p}}}{\partial \mathrm{n}\mathrm{e}{\mathrm{t}}_{\mathrm{i}}^{\mathrm{p}}}}=-{\displaystyle \frac{\partial {\mathrm{J}}_{\mathrm{p}}}{\partial {\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}}}}{\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}}(1-{\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}})$$

(20)

The weighting coefficient A4 correction formula for any kth neuron of the output layer is as follows:

$$\Delta {\mathrm{w}}_{\mathrm{k}\mathrm{i}}=\eta {\delta }_{\mathrm{k}}^{\mathrm{p}}{\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}}=\eta {\mathrm{o}}_{\mathrm{k}}^{\mathrm{p}}(1-{\mathrm{o}}_{\mathrm{k}}^{\mathrm{p}})({\mathrm{t}}_{\mathrm{k}}^{\mathrm{p}}-{\mathrm{o}}_{\mathrm{k}}^{\mathrm{p}}){\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}}$$

(21)

The incremental equation is as follows:

$$\Delta {\mathrm{w}}_{\mathrm{k}\mathrm{i}}(\mathrm{k}+1)={\mathrm{w}}_{\mathrm{k}\mathrm{i}}(\mathrm{k})+\eta {\delta }_{\mathrm{k}}^{\mathrm{p}}{\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}}$$

(22)

where $\eta$ is the learning rate and $\eta >0$.

The weighting coefficient correction formula for any i-th neuron of the hidden layer is given by the following:

$$\Delta {\mathrm{w}}_{\mathrm{i}\mathrm{j}}=\eta {\delta }_{\mathrm{i}}^{\mathrm{p}}{\mathrm{o}}_{\mathrm{j}}^{\mathrm{p}}=\eta {\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}}(1-{\mathrm{o}}_{\mathrm{i}}^{\mathrm{p}})({\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{L}}{\delta }_{\mathrm{k}}^{\mathrm{p}}-{\mathrm{w}}_{\mathrm{k}\mathrm{i}}}){\mathrm{o}}_{\mathrm{j}}^{\mathrm{p}}$$

(23)

where A2 is the output of the j-th neuron of the input layer, the incremental formula is as follows:

$$\Delta {\mathrm{w}}_{\mathrm{i}\mathrm{j}}(\mathrm{k}+1)={\mathrm{w}}_{\mathrm{i}\mathrm{j}}(\mathrm{k})+\eta {\delta }_{\mathrm{i}}^{\mathrm{p}}{\mathrm{o}}_{\mathrm{j}}^{\mathrm{p}}$$

(24)

3.3. Construction of the GNNM (1, N) Prediction Model

The grey neural network model is an organic combination of grey prediction and neural network modeling methods, which can solve complex uncertainty problems nonlinearly [32]. This paper uses the BP neural network to correct the residuals of the GM (1, N) prediction model and predict the residuals. The predicted values of the GM (1, N) prediction model are adjusted according to the residuals corrected by the BP neural network to improve the overall prediction accuracy.

Let the difference between the previous mass feature original data sequence ${\mathrm{x}}^{(0)}(\mathrm{t})$ and the GM(1,N) prediction model prediction result sequence ${\widehat{\mathrm{x}}}^{(0)}(\mathrm{t})$ at moment t, called the residual sequence, denoted as ${\mathrm{e}}^{(0)}$, be expressed as follows:

$${\mathrm{e}}^{(0)}={\mathrm{x}}^{(0)}(\mathrm{t})-{\widehat{\mathrm{x}}}^{(0)}(\mathrm{t})$$

(25)

If S is the prediction order, the input sample for BP network training is as follows:

$${\mathrm{e}}^{(0)}(\mathrm{t}-1),\hspace{0.17em}{\mathrm{e}}^{(0)}(\mathrm{t}-2),\cdots ,\hspace{0.17em}{\mathrm{e}}^{(0)}(\mathrm{t}-\mathrm{S})$$

(26)

where $\mathrm{t}=1,2,\cdots ,\mathrm{n}$. ${\mathrm{e}}^{(0)}(\mathrm{t})$ is the expected value of prediction for BP network training. The above BP algorithm is used to train the residual model. The trained network model can be used as an effective tool for residual series prediction.

Let the residual sequence predicted by the network training model be ${\widehat{\mathrm{e}}}^{(0)}(\mathrm{t})$, and construct a new prediction value ${\widehat{\mathrm{X}}}^{(0)}$ based on it.

$${\widehat{\mathrm{X}}}^{(0)}={\widehat{\mathrm{x}}}^{(0)}(\mathrm{t})+{\widehat{\mathrm{e}}}^{(0)}(\mathrm{t})$$

(27)

Then, ${\widehat{\mathrm{X}}}^{(0)}$ is the predicted value of the $\mathrm{G}\mathrm{N}\mathrm{N}\mathrm{M}(1,\mathrm{N})$ prediction model.

3.4. Analysis of GNNM (1, N) Prediction Model Results

There are many ways to validate the accuracy of the GNNM (1, N) prediction model built for critical quality features. In this paper, ${\mathrm{R}}^2$ is chosen to determine the degree of fit of the prediction model. The ${\mathrm{R}}^2$ mathematical model is as follows:

$${\mathrm{R}}^2={\displaystyle \frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{y}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}^{\prime })}^2}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{y}}_{\mathrm{i}}-\overline{{\mathrm{y}}_{\mathrm{i}}})}^2}}}$$

(28)

The Root Mean Square Error (RMSE) is used to measure the deviation of the predicted value of the forecasting model from the actual value. The mathematical model of the RMSE is expressed as follows:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{y}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}^{\prime })}^2}}{\mathrm{n}}}}$$

(29)

In the above two equations, n is the number of validated samples, ${\mathrm{y}}_{\mathrm{i}}$ is the actual value of the validated samples, ${\mathrm{y}}_{\mathrm{i}}^{\prime }$ is the prediction result of the GNNM (1, N) prediction model, and $\overline{{\mathrm{y}}_{\mathrm{i}}}$ is the average value of the validated samples. When the value of ${\mathrm{R}}^2$ is closer to 1, it indicates that the fit of the established prediction model is higher, and when the value of ${\mathrm{R}}^2$ is more significant than 0.9, it indicates that there is a better correlation between the variables in the prediction model. When the value of the RMSE is closer to 0, it indicates that the deviation between the predicted value of the prediction model and the real value is more minor.

4. Instance Analysis

In order to verify the practicality of this method, the machining process of the fuselage of a particular model of a medium-speed marine diesel engine is taken as an example, as shown in Figure 5. The fuselage has a complex structure and many processes involving multiple equipment, multiple stations, and multiple datums. The machining process of the fuselage is affected not only by cutting parameters and machining factors such as machine tools, cutting tools, and fixtures, but also by the output quality error of the previous process. Therefore, it is difficult to objectively and effectively control the quality of critical quality features that affect the quality of fuselage machining in actual machining, resulting in insufficient final quality control. In order to reasonably determine the fuselage machining process and achieve control of the machining quality, it is necessary to predict the critical quality features in machining.

In order to describe more clearly the machining error transfer relationship between the processes and between processes and output quality during fuselage machining, each process of the fuselage is decomposed into multiple machining features. Each processing feature corresponds to different quality features and processing elements. Combined with the machining process of marine diesel engine fuselage and its features, 28 machining features, 62 quality features, and 24 machining elements are finally abstracted and coded. The coding rules are shown in

Table 1. Coding rules of processing technology relationship network nodes.

Node Type	Code Rule	Instance
Machining feature nodes	MF + process ID + machining feature ID	MF020A represents the machining feature A of machining process number 020.
Quality feature nodes	QF + process ID + machining feature ID + quality feature ID	QF020Aa represents the quality feature of machining feature A of machining process number 020.
Machining element nodes	MT + machine model; CT + tool model; FT + fixture model	MT01 represents machine tool 01; CT01 represents tool 01; and FT01 represents fixture 01.

below.

According to the coding rules, the error transfer network model is constructed according to the processing sequence between the fuselage processing qualities. Calculate the PR size of each node in the network according to the PR algorithm. The PR value of each node after the stable convergence of the PR algorithm is shown in

Table 2. PR value of each node feature of the fuselage machining process network.

Node Features	PR Value	Node Features	PR Value	Node Features	PR Value
MF030A	0.00785	MF050B	0.00888	MF140B	0.03068
QF030Aa	0.00612	QF050Ba	0.00642	QF140Ba	0.00599
QF030Ab	0.00745	MF050C	0.00690	QF140Bb	0.00599
MF030B	0.00795	QF050Ca	0.00725	QF140Bc	0.00599
QF030Ba	0.01200	MF050D	0.00690	QF140Bd	0.01112
QF030Bb	0.01182	MF130A	0.02779	MF150A	0.02496
⋮	⋮	⋮	⋮	⋮	⋮
QF040Aa	0.00923	MF130B	0.01812	QF150Ab	0.01355
QF040Ab	0.00650	QF130Ba	0.00826	MF150B	0.03152
MF050A	0.00703	QF130Bb	0.01100	QF150Ac	0.00807
QF050Aa	0.00925	QF130Bc	0.02554	QF150Ad	0.00808

below.

According to the size of the PR value of each node, using Gephi—0.9.7 software, the fuselage error transfer visualization network is established as shown in Figure 6. Where the size of the network node represents the size of the PR value, i.e., the larger the network node, the higher its influence will be.

Among the machining feature nodes, the machining feature node with a more significant value is defined as the critical machining feature of the fuselage. According to the analysis results of the process error transfer network, it can be found that the critical machining features correspond to the numbers MF150B, MF140B, MF130A, MF130C, MF150A, and MF080B. The critical machining feature numbers correspond to the mass formation process corresponding to the cylinder bores, camshafts, and crankshaft bores. The critical machining feature numbers correspond to the mass formation process of cylinder bores, camshafts, and crankshaft bores and the finishing process of hafted surfaces, end surfaces, and cylinder bore surfaces, respectively. The critical machining features for a marine diesel engine body include the dimensional and form accuracy of the top surface, bottom surface, crankshaft bores, camshaft bores, and cylinder bores. These features significantly affect the assembly and use of the diesel engine.

The quality of the machine body processing is mainly affected by machining errors, which are reflected in the quality characteristics output by various processes, such as surface quality, dimensional accuracy, and geometric tolerances. Considering the influence of the position of the quality feature nodes in the machining error transfer network, this paper assesses the criticality of the quality feature nodes based on the influence degree E, as defined in Section 2.2. As shown in Figure 7, the degree of influence of the output quality feature nodes of critical machining features is demonstrated.

According to Figure 7, it can be seen that the influence degree of the quality feature nodes coded as QF130Ac, QF140Bd, QF150Bd, QF150Ba, QF130Cd, QF130Aa, QF140Ba, and QF150Ba is significantly higher than the other nodes. Among them, QF130Ac (fine boring of crankshaft bore size) has the highest influence degree, followed by QF140Bd (fine boring of camshaft bore size), QF150Bd (fine boring of cylinder bore size), QF130Cd (perpendicularity of end faces and crankshaft bores), QF130Aa (coaxial of crankshaft bores), QF140Ba (coaxial camshaft bores), and QF130Aa (verticality of cylinder bores). Therefore, these quality features are defined as critical quality features in the machining process of the diesel engine body.

This paper defines multiple quality features in the machining process of marine diesel engine fuselage as critical quality features, and its specific range can be divided according to the actual situation by the degree of influence. Since the dimensions of the fine-bored crankshaft holes have the highest degree of influence on the quality of the machining process of the body, the dimensions of the crankshaft holes in the body of a certain model of diesel engine were selected as the prediction of the machining quality to be verified.

Table 3. Sample data collection form for finish boring crankshaft bore size.

Serial Number	Input Data							Output Data
	Semi-Finish Crankshaft Bore	Machining Element Error			Process Parameter			Fine Boring Crankshaft Bore
	QF080Aa	${\mathbf{\epsilon }}_{\mathbf{s}}$	${\mathbf{\epsilon }}_{\mathbf{f}}$	${\mathbf{\epsilon }}_{\mathbf{t}}$	$\mathbf{n}$	$\mathbf{f}$	${\mathbf{\alpha }}_{\mathbf{p}}$	QF130Ac
1	223.093	0.3	0.109	0.072	75.398	0.3	0.05	220.014
2	223.082	0.2	0.101	0.088	94.248	0.2	0.05	220.016
3	223.097	0.2	0.114	0.087	117.809	0.3	0.05	220.012
4	223.024	0.1	0.106	0.082	94.248	0.2	0.15	220.017
5	223.044	0.3	0.101	0.088	75.398	0.3	0.05	220.015
6	223.067	0.1	0.118	0.087	94.248	0.2	0.05	220.010
7	223.013	0.1	0.116	0.067	117.809	0.3	0.05	220.016
8	223.059	0.1	0.107	0.065	94.248	0.2	0.1	220.014
9	223.028	0.1	0.109	0.100	94.248	0.2	0.1	220.013
10	223.052	0.1	0.108	0.067	94.248	0.2	0.1	220.011
11	223.013	0.1	0.096	0.061	94.248	0.1	0.1	220.015
12	223.018	0.1	0.099	0.082	94.248	0.2	0.05	220.006
13	223.047	0.1	0.104	0.095	117.809	0.3	0.05	220.013
14	223.018	0.1	0.097	0.087	94.248	0.2	0.1	220.012
15	223.012	0.1	0.115	0.068	94.248	0.1	0.15	220.010
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
27	223.018	0.2	0.103	0.084	75.398	0.1	0.05	220.010
28	223.065	0.1	0.093	0.069	94.248	0.2	0.1	220.006
29	223.021	0.2	0.098	0.075	117.809	0.3	0.15	220.022
30	223.011	0.2	0.102	0.083	75.398	0.2	0.1	220.015
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
47	223.016	0.1	0.105	0.077	75.398	0.2	0.1	220.020
48	223.014	0.2	0.107	0.073	94.248	0.2	0.1	220.015
49	223.092	0.1	0.097	0.066	94.248	0.2	0.1	220.009
50	223.030	0.1	0.104	0.067	94.248	0.2	0.1	220.017

shows the results of the dimensional machining test data collection for the fine-bored crankshaft hole of a diesel engine body. In Table 3 ${\mathsf{\epsilon }}_{\mathrm{s}}$, ${\mathsf{\epsilon }}_{\mathrm{f}}$, and ${\mathsf{\epsilon }}_{\mathrm{t}}$ denote the equipment error, fixture error, and tool error. $\mathrm{n}$, $\mathrm{f}$, and ${\mathsf{\alpha }}_{\mathrm{p}}$ denote the rotational speed $(\mathrm{r}/\min )$, feed rate $(\mathrm{m}\mathrm{m}/\min )$, and depth of cut $(\mathrm{m}\mathrm{m})$.

Based on the above experimental data, Python 3.12.5 software constructed a GNNM (1, N) prediction model for the dimensions of fine-bored crankshaft holes for body machining quality. In order to prove the superiority of GNNM (1, N) in solving the optimization problem, the algorithm is compared with the GM (1, N) prediction model. The GNNM (1, N) parameters are set as follows: the learning rate is chosen to be 0.6, the convergence rate is 0.001, and the mean square error is limited to 0.01. Sample data numbered 1 to 40 in Table 3 are used as the training set, and data numbered 41 to 50 are used as the test set for substitution into the prediction model. The prediction results are shown in Figure 8. The red line in Figure 8 indicates the actual value, the green line indicates the prediction of the GNNM (1, N) model, the black line indicates the prediction of the BPNN(1, N) model, and the blue line indicates the prediction of the GM(1, N) model.

According to Figure 8, the optimized prediction model using neural networks is significantly better than the grey or BP neural network prediction models. Using the test set data for calculating the crankshaft bore size GNNM (1, N) prediction model ${\mathrm{R}}^2=0.9302>0.9$, the $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}$ value is $3.8949\times {10}^{-3}$. Therefore, the model meets the requirements for prediction accuracy and is highly reliable.

5. Conclusions

This paper proposes a machining quality prediction method based on a machining error transfer network and a grey neural network, which is mainly used to predict the machining quality of complex parts with complex structures, numerous processes, multiple varieties, small batches, and high accuracy. Compared with Reference [32], which describes the evaluation of the process reliability of diesel engine blocks, this paper not only comprehensively considers the influence of processing elements and the cutting parameters on the quality of parts, but it also considers the error transfer in the machining process and achieves the identification of crucial quality features consistent with the processing of production enterprises through the establishment of a network model of error transfer in the machining of parts. At the same time, the method successfully solves the prediction problem caused by insufficient quality data in small-lot production by taking advantage of the grey system’s high-precision short-term prediction advantage under the condition of few samples, and the nonlinear mapping, adaptive, and fault-tolerant ability of neural networks. In addition, the method in this paper closely matches the development concept of Industry 4.0, especially in intelligent manufacturing, which can be better applied to the highly automated production environment to achieve real-time prediction and the control of machining quality and improve the consistency and stability of products. The main conclusions are as follows:

(1) A processing error transfer network model is constructed based on the complex network theory, which realizes the visualization of error transfer in the machining process. The model provides a new perspective for understanding and controlling machining errors, which is consistent with the emphasis on transparency and visualization of the production process in Industry 4.0, and it helps to achieve real-time monitoring and the optimization of the manufacturing process.

(2) Using the PageRank algorithm and the degree of influence of the nodes, the critical quality feature nodes of the machining error transfer network are identified. Manufacturing enterprises can focus on the supervision, detection, and control of these features that have the most significant impact on the quality of the final product, which is in line with the requirements of Industry 4.0 for the precise control of crucial quality features, and ti improves the level of intelligence of the production process.

(3) Combining grey theory with the BP neural network and using the BP neural network to optimize the residuals of the grey prediction model, the grey neural network processing quality prediction model GNNM (1, N) was established. Finally, the method is validated through the machining process of the medium-speed marine diesel engine hull, and the results show that the GNNM (1, N) model still has high prediction accuracy with small samples and few data. This provides an effective quality prediction method for small-lot and multi-variety production in an Industry 4.0 environment, which supports adaptive control and decision-making in intelligent manufacturing.