An Analytical Framework for Error Propagation Effects in Multiprocess Manufacturing

Li, Chunlei; Li, Liang

doi:https://doi.org/10.1155/2023/2769757

Mathematical Problems in Engineering

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 2769757 | https://doi.org/10.1155/2023/2769757

An Analytical Framework for Error Propagation Effects in Multiprocess Manufacturing

Chunlei Li^1,2and Liang Li^1,2

Academic Editor: Ivan Giorgio

Received08 Mar 2023

Revised31 May 2023

Accepted05 Jun 2023

Published27 Jun 2023

Abstract

Analysis of multisource errors accompanying process execution and study of the propagation effects of the corresponding multiprocess manufacturing process are the basic prerequisites for optimizing the generation process and improving the quality of the product. This paper proposes an analytical framework for error propagation effects in multiprocess manufacturing, which completely covers error source analysis, mathematical calculation of error propagation accumulation, and description of error propagation effects, which can provide better support for error traceability, error propagation monitoring, and error dissipation in manufacturing production. First of all, establish the process state model into the features of the manufacturing level, analyze the type of multisource error, and its mode of action when executing the process operation; second, establish the mathematical calculation model of error propagation accumulation based on state space equation is established, so as to realize calculation and analysis of error propagation; and finally, establish the link graph and link matrix of error propagation influence in multiprocess manufacturing process are established to further reveal the scope and path of error propagation influence. In this section of the case study, a plate cover part had been taken as an example, the multisource error in its manufacturing was analyzed, and the propagation and accumulation of these error values are obtained by state space and state transition theory. On this basis, the link graph and its corresponding link matrix were used to explicitly describe the error propagation effects during the multiprocess manufacturing of the example part. The example analysis illustrated the proposed error analysis framework could analyze and calculate the multisource error of multiprocess manufacturing and visually describe the error propagation effects, which showed the effectiveness and feasibility of the proposed method framework.

1. Introduction

Improving product quality is the key factor for enterprises to maintain competitiveness and achieve sustainable development in the face of increasingly fierce market competition. One of the fundamental measures to grasp the quality of mechanical manufacturing is to control the quality of the manufacturing process, especially to control the multisource error generated in mechanical processing and its accompanying influence on the propagation of multiprocess manufacturing process.

The research on the influence of error propagation in mechanical manufacturing mainly covers the following development stages. The early research was mainly based on the quality theory and the concept of selection control chart proposed by Zhang [1], and the error diagnosis and traceability determination in the process articulation transformation was achieved by converting the measured values of quality features into selection control values and combining certain selection control chart quality cause determination strategies. Due to the need of a large number of quality characteristic data to construct a selective control chart and the lack of an accurate mathematical description of the machining process and processing state, the practical application of the control chart-based method is difficult and cannot provide an effective basis for the optimization and improvement of error propagation. Then, some scholars further proposed the concept of a manufacturing system with a stream of variation for the propagation and transformation features of the error form in the manufacturing process. However, the research on propagation and accumulation of manufacturing errors by using the dynamic change features of stream of variation in manufacturing system has been developed until now. For example, by modeling and applying the stream of variation in multiprocess manufacturing process, Ma et al. [2] proposed a chain model of error propagation. The error prediction can be accomplished by substituting the measurement results of error parameters into the model, which can eventually support the process adjustment for error compensation. Ma realized unified modeling and analysis of multierror sources. This improved the problem that many existing error analysis models only consider one error factor, and make the errors are analyzed comprehensively from the global perspective. Liu [3] considered that the stream of variation formed by multisource error propagation, accumulation and coupling in the multiprocess machining process is the key factor affecting the machining quality, and established the SOV function with the multiprocess grinding of rotary body as the research object, and verified the correctness of the SOV function accompanied by multiprocess machining transfer by using the Shewhart control chart. Liu revealed the law of error generation, transmission and accumulation in the multiprocess grinding process of rotary parts, and this made an active contribution to the tracing and control of machining errors of rotary parts. Wang et al. [4] analyzed the production process of remanufactured machine tools, focusing on the deviation transfer of feature change, feature measurement, and adjustment in the assembly process in manufacturing, and accordingly proposed SOV model and SOV correction function to achieve accuracy prediction and quality error correction. Wang et al. described error propagation in the assembly process of a remanufacturing machine tool quantitatively, which solved the problem of uncertainty and randomness of its assembly parts and ensured the consistency and reliability of assembly accuracy. In general, the theoretical basis of the stream of variation method determines that continuous process manufacturing is more applicable, while modern complex part processing often involves multiple process/station transitions and presents the features of discrete manufacturing. In addition, the stream of variation constructed in some studies still lacks a reasonable differentiation and accurate description of multisource error, which reduces the applicability of the method to a certain extent. With the development of network system and modern control theory, it has become a hot topic to study error propagation in multiprocess manufacturing using a network model or state space equation. The method based on the network model is to use many kinds of models such as complex network to model the multiprocess processing system. The manufacturing process of a part involves the flow between multiple processes/stations. If the process/station or errors influencing factors are abstracted as network nodes and the flow relationships are abstracted as edges, then the topology and propagation features of the network can be used to analyze the propagation influence features of machining errors along with the multiprocess machining process. For example, Jia et al. [5] applied a complex network model to model the processing error propagation network, analyzed the quality fluctuation of key control processes, realized the analysis of the fluctuation impact of processing error from the system level, and proposed the control method of error propagation accordingly. Their contribution is that the network model was introduced to analyze the propagation characteristics of errors, which made the fluctuation of processing quality, could be described in more detail by using characteristics of the network, such as dynamic characteristics of the network propagation. Jiang et al. [6] used neural network theory to construct a traditional network model of machining error and realized the detection and monitoring of machining stream of variation by using the inverse solving features of neural networks. Jiang et al. established a machining error propagation model of machining by applying the neural network and other machine learning methods, this not only analyzed the error propagation but also realized the accuracy prediction. Wang et al. [7] established an extended processing error propagation network model with the processing of aircraft landing gear with complex structure as an example, which can quantitatively describe the coupling relationship between various elements related to processing error and analyze the stream of variation and its propagation features by combining the process measurement and sensing data of each processing stage. Zheng et al. [8] constructed an adaptive weighted deviation propagation network based on the mechanism of the generation and propagation of actual machining errors, and proposed an error source diagnosis method for the key machining surfaces, which finally realized the dynamic analysis of the machining process deviation flow propagation. Their study could help technicists effectively identify the weak process and the problematic machining equipment, thus controlling the processing quality better. Zhu et al. [9] constructed a weighted self-adjusting deviation propagation network model and proposed an error tracing method based on the traversal backtracking algorithm. Zhu et al. realized the identification of error propagation paths and key error sources, which provided the basis for grasping the key factors to resolve the error. Due to the difficulty of modeling and analysis and the complexity of application conditions, there are still some challenges in the application of the error propagation network model. The model of error propagation [10–15] based on the state-space equation regards the multiprocess manufacturing process as a discrete dynamic system in the time domain, and establishes the state-space equation that can represent the quality features of the process. Among them, the errors contained in the process execution can be regarded as the inputs that drive the change of the current system state, and the analysis and prediction of the accumulation of error propagation can be realized through the analysis of the state output. Relatively speaking, the state-space equation method is closer to the essence of modern multistep discrete manufacturing process, and easier to realize in mathematical modeling. The disadvantages are mainly reflected in that the process processing state modeling generally only involves the part level and the lack of detailed description of the state evolution of each manufacturing feature on the workpiece accompanied by multiprocess processing. In addition, the state-space model lacks an explicit demonstration of the propagation influence of multisource error during the processing of the accompanying workpiece, which reduces the efficiency of error tracing diagnosis and error compensation digestion. Therefore, it is still not a complete error propagation influence analysis system.

In recent years, in addition to the methods mentioned above, the influence of manufacturing error has been analyzed and modeled by many scholars from various angles, which produces many valuable results. For example, Niu et al. [16] established the spatial error model by using the MBS theory, which could analyze the local influence of geometric error on machining accuracy. Lv et al. [17] summarized and analyzed the main factors affecting the machining quality of high-precision hole, and revised the temperature error model. The revised error model could provide a practical basis for the finishing process scheme of key parts with large size and high precision. Tang et al. [18] analyzed the main geometric errors of the CNC machine tools used to the worm grinding of face gears, and realized to model and compensate the geometric errors by considering the influence of the cutter rotation angle. Liu et al. [19] put forward a digital thread-driven distributed collaboration mechanism between digital twin manufacturing units, and built the graph-based manufacturing task model, which could provide the support to distinguish the error sources and dynamically reconstruct production tasks. Qu et al. [20] extended the relevant concepts of information theory to the field of mechanical manufacturing error analysis, established a reliability analysis model of the machining process, and defined the transfer coefficient based on mutual information to measure the transfer of machining error between two processes. Sun et al. [21] created the machining error transfer network for the diesel engine body, and improved the prediction accuracy of the body processing quality. Wu et al. [22] established the multisource and multiprocess machining error transmission model of a near-net-shaped blade, and qualitatively analyzed the reduction effect of the machining error transmission chain by the adaptive CNC machining process based on the machining error transmission flow model. Most of these methods are based on theoretical research and experimental verification, which provides a good foundation for us to revise and condense the research ideas.

In order to optimize the modern multiprocess complex, manufacturing process and control the quality of the manufacturing process, it is necessary to accurately describe and carefully depict the changes in the quality features of the workpiece accompanied by the production process. A complete analytical framework for error propagation effects is required to provide strong support. Based on this starting point, this paper investigates the propagation effects of multisource error accompanying multiprocess complex manufacturing processes. This paper is dedicated to constructing a complete theoretical framework including error source analysis, error propagation mathematical calculation, and error propagation effects description, so as to provide an effective basis for error traceability, error propagation monitoring, and error compensation and dissipation in multiprocess manufacturing process. First of all, this study established process state model into the features of manufacturing level, analyze the type of multisource error and its mode of action when executing the process operation; second, the mathematical calculation model of error propagation accumulation based on state space equation is established, so as to realize the calculation and analysis of error propagation; and finally, the link graph and link matrix are introduced to analyze the propagation effect of multisource error with multiprocess manufacturing process. This paper adopts the theory of state space to establish the mathematical model of the influence of errors propagation and accumulation in multiprocess manufacturing, and displays the path of error transmission and accumulation visually by applying link graph with low modeling difficulty, which provide a relatively complete framework for analyzing the relationship of propagation and accumulation of processing errors. The proposed framework makes up for the defect that the state space model cannot explicitly and intuitively describe the error propagation path and scope, and provides the basis for the analysis, prediction, traceability, and control of multiprocess machining errors. Of course, it is a relatively mature method to use state-space equation to study the transfer and accumulation of machining errors. However, this paper has some marked differences with existing methods in the setting of model parameters and the construction of model mathematical relations, and strives to achieve a relatively complete analysis framework by using chain diagram. This has a certain contribution and reference to the related research fields, which is also the important embodiment of the innovation of this paper.

2. Propagation Impact Framework for Multiprocess Complex Manufacturing Processes Accompanied by Multisource Machining Errors

Modern mechanical parts usually have complex structures, and the manufacturing process generally involves multiple stations. It is necessary to use a variety of machining equipment in accordance with the process flow in order to complete the processing of modern mechanical parts. In addition, the production process also shows the typical multiprocess manufacturing features of modern mechanical parts. It is precisely because of the features of multiprocess manufacturing that the quality control of the production process is also faced with difficult challenges. In the multiprocess complex manufacturing processes of mechanical parts, an abnormal condition in any one of the processes, such as tool wear, and inaccurate clamping positioning, will increase the risk of quality loss. The complexity of the manufacturing process involves a variety of machining equipment, fixtures, and tools and reduced rigidity of the part due to cut material, resulting in various sources of error along the machining process. At the same time, due to the process organization and connection between processes, multiple-source errors can be accumulated and propagated continuously in the manufacturing process. The complexity of the production process and the nonlinear relationship of multisource error propagation make the quality control, error tracing, and resolution process of complex parts manufacturing very difficult. It is necessary to study the propagation influence framework of multisource errors accompanying the complex manufacturing process of parts in order to better solve this problem, and to further explore the accumulation law of error propagation among multiple processes is the key to the subsequent adoption of targeted error control and dissipation strategies. The propagation influence framework of multisource errors accompanying multiprocess complex manufacturing process proposed in this paper is shown in Figure 1, which contains three specific aspects as follows:

2.1. Multisource Error Analysis

To summarize the potential error types in the multiprocess complex manufacturing processes of mechanical parts, so as to further clarify the error sources. In general, the processing error sources of a process include fixture error, workpiece positioning error, tool error, and deformation error of the process system due to cutting force and rigidity. For multisource error analysis, it is necessary to use testing equipment to measure the actual values of position, size, and geometry and compare them with the theoretical values to form error parameters that can evaluate the quality features of products.

2.2. Analysis of the Propagation and Accumulation of Errors

The manufacturing and processing of modern mechanical parts exhibit the typical features of multiple processes/stations. The error generated in the previous process will be propagated to the next process, showing the propagation and accumulation features of the error. The analysis of error propagation and accumulation requires a clear mathematical model of error propagation between multiple processes so as to accurately calculate the error propagation between processes.

2.3. Analysis of the Propagation Influence of Multisource Error

A complete analytical framework for error propagation effects requires not only the mathematical calculation of multisource error propagation accumulation results but also the description and portrayal of the error propagation effects mechanism between process articulation and conversion. Only by clarifying the propagation range and path of multisource error, and cooperating with the results of error propagation and accumulation analysis, can we comprehensively and deeply understand the whole process of error from generation, propagation to final effect results, which also helps to carry out accuracy prediction and error control based on the analytical framework for error propagation effects.

3. Process State Modeling and Multisource Error Analysis Accompanying Process Execution

The state space theory and state transition model are relatively mature methods for studying the cumulative impact of error propagation in multiprocess manufacturing. On the basis of full reference to the existing state space and its transition theory, this paper analyzed and modeled the processing state and error influence of the process.

3.1. Process State Modeling Based on State Space

Modern machine parts generally have complex structure, and the manufacturing process can be regarded as the formation process of one local characteristic structure on the parts. Moreover, the size and precision of these features are also very important to be guaranteed. Therefore, the geometric features of the part are divided to form manufacturing features [23–25], and the current machining state is described by the orientation vector, position vector, key dimensions, and quality features of the manufacturing features. Assuming that there are a total of n manufacturing features on the design CAD model of a part, the machining state of the part after the completion of k processes is

In equation (1), X_i(k) represents the machining state of the ith manufacturing feature after k processes; DIRE_i(k) = [j_x, j_y, j_z, 1] represents the homogeneous coordinate form of the direction vector of the ith manufacturing feature after k processes. For example, the direction vector of a plane feature is its normal vector, the direction vector of a hole feature is the direction vector of its axis, etc.; LOCA_i(k) = [p_x, p_y, p_z, 1] represents the homogeneous coordinate form of the positioning position vector of the ith manufacturing feature after k processes, which is determined by the position relation of the feature with respect to the reference; SIZE_i(k) = [d₁, d₂, d₃, 1] represents the homogeneous coordinate form of the dimension vector of the ith manufacturing feature after k processes; QUAL_i(k) represents the homogeneous coordinate form of the quality characteristic parameter vector, which is generally composed of form and position errors that need to constrain the manufacturing accuracy of the feature.

3.2. Fixture Error Analysis Based on Transformation Matrix

In the process of machining operation of mechanical parts, the workpiece is first clamped and positioned with a fixture, and then use the tool to sweep off the solid material of the workpiece to complete the established features of processing. If there is a deviation in the mounting position of the fixture, then it will cause a deviation in the clamping position of the workpiece, which will lead to machining errors. This error can be described by the transformation matrix Q as follows:

In equation (2), represents the homogeneous transformation matrix between the theoretical correct position and the actual position of the fixture, represents the rotation matrix between the theoretical correct position and the actual position, represents the translation vector between the theoretical correct position and the actual position of the fixture; represents the homogeneous change matrix between the workpiece’s theoretical coordinate system and the fixture’s theoretical coordinate system, represents the rotation matrix between the workpiece’s theoretical coordinate system and the fixture’s theoretical coordinate system, and represents the translation vector between the workpiece’s theoretical coordinate system and the fixture’s theoretical coordinate system. According to (2), when using Q_f to describe the fixture error, its essence is to first transform the workpiece into the fixture coordinate system, and then introduce the deviation between the theoretical installation position of the fixture and its actual position.

3.3. Positioning Error Analysis Based on Transformation Matrix

In a multiprocess manufacturing process, the workpiece needs to be positioned during each operation. When a workpiece is clamped and positioned with a fixture, positioning errors may occur even if the fixture is mounted in an accurate position. For example, the workpiece datum surface used for positioning may be created by the previous process, and the manufacturing itself may cause form and dimensional errors, or the positioning surface on the fixture may be worn due to long-term use. All of these can lead to positioning errors, which in turn produce machining errors. The positioning error can be expressed by the transformation matrix Q_l as follows:

In equation (3), represents the rotation matrix between the theoretical positioning position and the actual positioning position of the workpiece; represents the translation vector between the theoretical positioning position of the workpiece and its actual positioning position.

3.4. Analysis of Tool Error and Process System Deformation Error Based on Transformation Matrix

The error of the tool will cause the inaccurate cutting position, resulting in the deviation between the actual cutting position and the ideal cutting position. Due to the influence of cutting force, the contact between cutting tool and workpiece will cause a small deformation, which makes the deviation of the original cutting position more complicated. Since the influence of the two on the workpiece processing error is synchronous and superimposed on each other, showing the features of mutual entanglement and superposition, this paper has carried out a unified modeling of the relationship between the two, specifically described by the transformation matrix Q_t:

In equation (4), represents the homogeneous transformation matrix of the process system from the theoretical correct position to the actual position in the machine tool coordinate system, represents the rotation matrix of the process system from the theoretical correct position to the actual position, and represents the translation vector of the process system from the theoretical correct position to the actual position; represents the homogeneous change matrix between the fixture theoretical coordinate system and the machine tool theoretical coordinate system, represents the rotation matrix between the fixture theoretical coordinate system and the machine tool theoretical coordinate system, and represents the translation vector between the fixture theoretical coordinate system and the machine tool theoretical coordinate system.

4. Analysis of Multisource Error Propagation and Accumulation Based on State Transition

The multiprocess manufacturing process can be regarded as a complex system, in which each process is updated and changed after execution. At the same time, the organization and arrangement of the process operations have typical time-domain evolutionary features, which make the whole system exhibit the features of a complex time-varying system. In contrast, the state space model in modern control theory is a proven means of analyzing complex time-varying systems, which can accurately portray the internal connections between the state changes of the system along the time evolution. In summary, it is feasible to use state space theory to analyze the impact of error propagation in multiprocess manufacturing process.

According to Section 3.1, the machining state of the part after k − 1 processes is X(k − 1), and X(k − 1) contains the actual machining results of each manufacturing feature on the part after the previous k − 1 processes have been performed. This result necessarily deviates from the theoretical manufacturing result in the ideal absence of error, further indicating that X(k − 1) contains the total accumulated machining error from the previous k − 1 processes. During the execution of the kth process, the processing state will change from X(k − 1) to X(k) due to the influence of multisource machining errors that may be introduced into the kth process mentioned in Sections 3.2∼3.4. Therefore, the equation of processing error propagation in multiprocess manufacturing process can be established as follows:

In equation (5), X(0) = (X₁(0), X₂(0), …, X_n(0)) represents the state of the workpiece when it has not been machined. Since no error effects have been introduced yet, it represents the ideal state of the workpiece and its corresponding features without error effects; ξ(k) = [ξ₁(k), ξ₂(k), …, ξ_n(k)]^T represents the system model noise, which indicates the random error impact generated when the kth process is executed; C represents the adjustment coefficient of the random error impact in manufacturing, which takes values in the range [0, 1]. The main purpose of this paper is to verify the accuracy of the proposed error propagation accumulation model, so in the subsequent calculations, C is set to 0; and A(k) is the state transition matrix, representing the influence of multisource error brought in during the execution of the kth process. The specific expression is as follows:

After k − 1 processes, a manufacturing feature MF_i on the workpiece will form a machining state of X_i(k − 1). If the k process contains the processing content of the feature, it means that the feature processing state will be updated after the k process. Thus, corresponding to this update is obtained, which represents the transformation of the workpiece from the theoretical workpiece coordinate system to the current actual position in the machine tool coordinate system, and then the reverse transformation back to the theoretical coordinate system of the workpiece under the condition of considering the introduction of multisource machining error in the kth process. If the kth process does not contain the processing content of the feature, it means that the kth process operation will not cause any change to the feature processing state, at which time A_i(k) = E, and the corresponding X_i(k) can be expressed in the following equation:

For the workpiece processing state X(k), the integrated error accumulated from the first k processes can be expressed as X(k). The integrated error of the ith manufacturing feature of X(k) after k processes can be expressed as X_i(k), as shown in the following equation:

According to equation (6), the analysis of error propagation and accumulation after the completion of any process in the multiprocess manufacturing process can be realized according to different values of k. If k represents the serial number of the last process, the quality analysis and error prediction of the final machining results of the parts can be realized by means of equation (8).

5. Analysis of Multisource Error Propagation Effects Based on Link Graph and Link Matrix

The current research on the influence of error propagation in the multiprocess manufacturing process is mostly based on complex network models. The specific method is to use complex networks to study the stream of variation in multiprocess manufacturing, which generally requires analyzing and extracting all factors affecting processing quality and using these factors as network nodes and the coupling relationship between factors as the connecting edge. Finally, the propagation effects of stream of variation are described quantitatively by using mathematical analysis of network topology combined with a large amount of process measurement data. The advantage of this approach is that it can provide a detailed picture of the evolution and development of the stream of variation and facilitate the traceability of the error; however, the disadvantage is that the modeling and analysis are difficult and the application conditions are complicated, which poses a challenge to the final practical application. The use of state space theory to study the accumulation of processing errors can significantly reduce the difficulty of modeling and can also achieve accurate prediction of process errors; while the disadvantage is that the propagation path of errors cannot be visualized, which is not conducive to error traceability and control. Therefore, this paper mainly adopts the state space theory to mathematically model the accumulation effect of error propagation in multiprocess manufacturing and visualizes the accumulation path of error transmission by means of the link graph, which is less difficult to model. It is expected to provide a basis for analysis and prediction of multiprocess processing errors and traceability control without significantly increasing the complexity of the model.

In multiprocess complex manufacturing processes, error propagation is mainly based on two ways: (i) the processing of manufacturing features needs to be realized in multiple processes, in which the errors generated in the previous process will be propagated and accumulated in the next process and (ii) different manufacturing features are related in the process organization, and the processing quality of a feature may affect the processing quality of other features with process-related relationships. The first situation exists in multiprocess machining of a single manufacturing feature. If the processing errors generated in the previous process are not adjusted and eliminated in the next process, such errors will be accumulated and propagated to the next process, which will expand the manufacturing errors of the feature and affect the processing quality; the second situation mainly occurs among manufacturing features with process-related relationships. A feature plane is the positioning basis for processing another feature, and the processing quality of this feature will directly affect the positioning accuracy of the next process, and thus lead to the error in the multiprocess manufacturing process with different features of the workpiece also shows the propagation effect features. Drawing on the idea of studying assembly error propagation in the literature [26], this paper uses the propagation link graph and link matrix of stream of variation to analyze the propagation impact range and path of multisource error accompanying the multiprocess complex manufacturing processes. This study is dedicated to provide a basic basis for error traceability in processing, error propagation monitoring, and error dissipation strategy formulation.

5.1. Link Graph of Error Propagation in Multiprocess Manufacturing with Single Feature

The link graph represents the propagation model of the error propagation and accumulation law between single manufacturing features or multiple manufacturing features with process-related relationships in the multiprocess complex manufacturing processes. The circles represent the multisource machining errors introduced during the process operation, the rectangles represent the feature processing state after the process execution, and the connecting arrows show the impact of the error introduction on the current machining state and the impact of the accumulation of the characteristic machining state formed in this process to the next process. Figure 2 is an example of a link graph of error propagation constructed according to the above principles.

In Figure 2, the unbolded error transformation matrix symbols Q_f, Q_l, and Q_t are used to represent fixture errors, positioning errors, and tool and process system deformation errors introduced during the process execution. These three errors are not necessarily present at the same time in every process execution. For example, if the tool is not worn or the process system is rigid, the impact of Q_t may be very small and can be approximately ignored when analyzing the impact of error propagation. As can be seen in Figure 2, the manufacturing feature MF₁ on a part needs to be processed in at least two processes, Process 1 and Process 2, meanwhile, the manufacturing feature MF₂ also needs to be processed in at least two processes, Process 1 and Process 3. In this case, Process 2 does not include the processing of Feature MF₂. Therefore, according to Section 4 of this paper, it is known that the feature processing state X₂(2) inherits equal to X₂(1), so the propagation and accumulation effect formed by X₂(2) on X₂(3) can be directly simplified to the propagation and accumulation effect of X₂(1) on X₂(3).

In the multiprocess manufacturing process of a feature, the feature processing state formed after the completion of the previous process itself contains manufacturing errors, which are propagated and accumulated to the next process of the feature. At the same time, each process of the feature introduces multiprocess errors, which also have an impact on the feature processing state after the current process is completed. In summary, the total error after the execution of each process is the result of the combined effect of the errors accumulated in the previous process and the newly introduced errors in the current process.

5.2. Link Graph of Error Propagation in Multiprocess Manufacturing with Associated Features

The process correlation of different manufacturing features on the same part is generally transmitted in the form of a process datum. In other words, a machining surface of a manufacturing feature may be a positioning datum for subsequent processing of other features. Once geometric shape and position errors exist in machining surfaces, new positioning errors will inevitably be introduced into other correlation features based on them during machining, thus enabling the link graph of error propagation to reflect the correlation relationships in the manufacturing process of multiprocess with different features, as shown in Figure 3.

As can be seen in Figure 3, the manufacturing feature MF₁ on a part needs to be processed in at least two processes, Process 1 and Process 2, meanwhile, the manufacturing feature MF₃ also needs to be processed in at least two processes, Process 2 and Process 3. Since the processing surface formed by feature MF₁ after the execution of procedure 1 is the positioning datum surface when feature MF₃ executes initial processing procedure 2, the feature processing state X₂(1) will have an impact on the formation of X₃(2), and this influence is mainly introduced into X₃(2) in the form of positioning error. In addition, since the first processing procedure of the part does not include the processing content of the feature MF₃, the feature processing state X₃(1) after the execution of the first processing procedure still remains the ideal unprocessed state X₃(0) of the feature, that is, there is no introduction or propagation of processing errors. Therefore, it is not considered or discussed in the link graph.

5.3. Link Matrix of Error Propagation Effects in Multiprocess Complex Manufacturing

The link graph is a visualization of the propagation impact range and path of multisource machining errors in multiprocess complex manufacturing. Although it is possible to quantify the propagation of errors by combining state space equations, it is not easy to store and calculate, and it also limits the efficiency of the propagation analysis of machining errors. Therefore, it is necessary to convert the link graph into a form that is easy to store and calculate. In this way, with the multiprocess error propagation equation based on state space theory, the propagation impact of multisource error in multiprocess manufacturing can be accurately analyzed and quickly calculated, providing the basis for computer-based precision analysis and error control in manufacturing processes.

In this paper, the link graph is indirectly transformed by the link matrix. Each column of the matrix represents the manufacturing process and is sorted from left to right according to the order of the process arrangement; each row of the matrix represents the manufacturing features and is sorted from top to bottom according to the feature number. The intersecting cells of the rows and columns of the matrix hold the 8-dimensional error propagation accumulation representation vector for the corresponding feature during the execution of the corresponding process. The first 6 dimensions of the vector are mainly used to characterize the types and sources of errors introduced during the execution of the process. The 1st and 2nd dimensions represent fixture errors and their sources, respectively; the 3rd and 4th dimensions represent positioning errors and their sources, respectively; while the 5th and 6th dimensions represent tool errors and process system deformation errors and their sources, respectively. If there is some kind of error, the corresponding vector element of the error type is set to 1; otherwise, it is set to 0. If the error is newly introduced in the process, the corresponding vector element of the error source is set to 1; otherwise, it is set to the feature processing state symbol of the associated feature from which it originates. The 7th dimensional element of the process error propagation accumulation vector represents the processing state symbol of the previous process that should be inherited and updated when the corresponding process is executed, and the 8th dimensional element represents the feature processing state symbol formed after the corresponding process is executed. By constructing the link matrix of multisource error propagation impact, the error propagation relationship in multiprocess complex manufacturing processes can be stored and described precisely in a structured manner. At the same time, combined with the detection results of the error change matrix during process execution and the processing error propagation equation constructed by equation (5), the propagation analysis and accurate prediction of processing error in multiprocess manufacturing process can be realized. Combining the examples of Figures 2 and 3, assuming that a part contains only three manufacturing features and the machining process has only three operations, the link matrix of error propagation effects in the machining of the part can be represented by Table 1.

As can be seen from Table 1, for feature MF₁, fixture error, positioning error, tool error, and process system deformation error will be introduced during the execution of Process 1. At the same time, because it is the first process to process the feature, it will be updated and changed on the basis of inheriting the ideal feature state X₁(0), thus finally forming the feature processing state X₁(1); for feature MF₁, as the processing content of MF₁ is not included in Process 3, the vector elements corresponding to the three main error types and their sources are set to 0. In this case, feature processing state X₁(3) after Process 3 will directly inherit X₁(2); for another example, in the case of the feature MF₃, since the fixture error, positioning error, tool error, and process system deformation error are also present when the Process 2 is performed, the elements of the vector in positions 1, 3, and 5 are all set to 1. However, the positioning error mainly comes from the processing state X₁(1) of the associated feature MF₁ after the completion of the first process, while the fixture error, tool error, and process system deformation error are newly introduced from this process, so the corresponding 2, 4, and 6 elements of the vector are set to 1, X₁(1), and 1, respectively.

6. Case Study

In this paper, a plate cover part processed by an equipment manufacturing enterprise (as shown in Figure 4) is taken as an example to analyze and verify the influence of error propagation in its multiprocess processing. By identifying the manufacturing features of this part and planning the process flow, the results are shown in Figure 5.

As can be seen in Figure 5, the machining process of this part involves five main operations, containing a total of 12 manufacturing features. The five operations are: turning one outer circle and one end face, turning another outer circle and one other end face, turning the center hole, milling the two sides of the plane, and drilling holes that are distributed along the circumference and symmetrical about the two sides of the plane. According to the machining accuracy and dimensional requirements shown in Figure 4, the most common machine tools, fixtures, and tools used in actual machining can meet the needs of case part machining. For example, in the execution of Process 1, a three-jaw chuck can be used to position the rough surface of the outer circle and one rough end face of the plate material, and then a common lathe and an integral high-speed steel turning tool can be used for cutting and machining; and for example, in the execution of Process 2, the workpiece is turned over and positioned with the outer circle and end face processed in Process 1, and the same fixture, tool, and machine tool are used to complete the machining of the other outer circle and the other end face. Considering that the whole manufacturing process does not involve special equipment, fixtures, or tools, and the background and details of the process operation belong to the common sense category in the field of machining, this paper will not elaborate on the above issues but only provide the positioning datum relationship when executing these processes, as shown in Table 2. On this basis, it is defined that the origin of the workpiece coordinate system of the part is on the feature plane MF₂, the z axis coincides with the axis of the cylindrical surface feature MF₁ (the horizontal direction to the right in the front view of Figure 4 is positive), the x axis direction is the horizontal direction in the left view of Figure 4 (the direction to the left is positive), and the y axis direction is the vertical direction in the left view of Figure 4. The workpiece coordinate system for this part is shown in Figure 6.

In the case of only considering the geometric position accuracy of the feature, the ideal state of the workpiece is obtained according to the workpiece coordinate system defined above, as shown in Table 3.

The ideal state model of the workpiece refers to the ideal processing state of the feature without the effect of error. Once a process operation is executed, the influence of multisource error will be introduced, and the processing state of the workpiece will be constantly updated and changed along with the execution of the process, showing the propagation and accumulation effect of errors in the multiprocess manufacturing process. The analysis steps of error propagation and accumulation are as follows: first of all, use a three-coordinate measuring machine to measure the deviation of fixture installation, positioning deviation, tool and process system deformation, and so on in the subsequent process, and construct a multisource error transformation matrix according to equations (2) to (4) by combining the relative position relationship among a machine tool coordinate system, a fixture coordinate system, and a workpiece coordinate system during actual process machining; subsequently, on this basis, introduce the influence of multisource error on the process processing state and establish the state transition matrix of process execution according to equation (6); and finally, use equation (5) to analyze and calculate error propagation and accumulation.

For the plate cover part shown in Figure 4, the manufacturing process involves a variety of different machining methods such as turning, milling, and drilling. The multiprocess manufacturing process requires a variety of different cutting machines, fixtures and tools. Therefore, the multisource error transformation matrix needs to be constructed for each process according to the error measurement results and the correspondence between the machine coordinates, fixture coordinates, and workpiece coordinates, which involves multiple measurement data and coordinate system conversion. Considering that both error (deviation) measurement and coordinate system transformation have very mature methods and are limited by space, they will not be listed one by one. This paper only shows the state transition matrix obtained by substituting the error measurement results and the coordinate system transition relationship when the manufacturing features are processed in the corresponding process, as shown in Table 4.

As can be seen from Table 4, for processing with different features in the same process, the calculated state transformation sub-matrix is completely consistent. Since the same machine tool, fixture, and even tool are used in the same process, and the machining of different features is completed by multiple tooling cuts after one clamping and positioning, the types and sizes of the introduced error sources are almost the same. In addition, the transformation relationship between the workpiece coordinate system, the fixture coordinate system, and the machine tool coordinate system is exactly the same, so the calculated submatrix of state changes with different features but belonging to the same process also remains the same.

The analysis and calculation of error propagation and accumulation can be realized by applying equation (5) in the matrix of state transformation, in which the ideal states of the workpiece and process execution are known. Taking Process 1 as an example, since the first process only involves the processing of manufacturing features MF₁ and MF₂, the state transition submatrix corresponding to Process 1 is as follows:

Given the ideal state X(0) of the workpiece (see Table 3), the processing state X(1) of the workpiece after the execution of Process 1 can be deduced according to equation (5); obviously, only feature processing state X₁(1) = A₁(1)X₁(0), X₂(1) = A₂(1)X₂(0), and the processing states of other features after the execution of Process 1 X_i(1) (i = 2∼12) directly inherits the processing state X_i(0) (i = 2∼12) of the corresponding feature in X(0). By analogy, the feature processing state after the execution of all processes can be predicted through calculation, and the predicted value can be compared with the measured value of the actual processing result, as shown in Table 5. The data in Table 5 can be directly converted into actual error values. For example, taking the relative position errors between MF₇ and MF₈ as an example, the actual and predicted position vectors of MF₇ are [35.0091, 0.0213, and 4.9605] and [35.0082, 0.0192, and 4.9617], respectively, and the actual and predicted position vectors of MF₈ are [−35.0311, −0.0064, and 5.0395] and [−35.0238, −0.0088, and 5.0387]. Then, according to the distance relationship between two points in space, the actual processed distance and predicted distance between MF₇ and MF₈ can be obtained as 70.040 and 70.032, respectively. By comparing Figure 4, it can be seen that the actual machining error and the predicted error are 0.04 and 0.032, respectively, and the predicted value is close to the actual result and within the tolerance of ±0.1. Similarly, if MF₇ and MF₈ are parallel to each other, their direction vectors should be collinear, and the parallelism error can be calculated according to the angle relationship between the direction vectors.

In reality, whether the predicted result meets the requirements is more dependent on the precision and tolerance designed by the given part. For general precision parts within the range of usual size, the processing error is usually required to control between 0.01 mm and 0.1 mm. As shown in Table 5, the actual machining error is very close to the calculation error obtained by the proposed method, which meet the requirement of common accuracy. This also proves the validity of the proposed model of error propagation.

Moreover, based on the process organization relationship of this analysis example and the positioning reference relationship during process processing (see Table 2), the link graph (shown in Figure 7) and the link matrix (shown in Table 6) of the propagation influence of multisource error in multiprocess manufacturing process can be constructed.

The combination of the analysis in Figure 7 and Table 6 and the accumulated data of error propagation calculated in Table 5 enables not only the accurate calculation of the transmission of errors in the manufacture of complex multiprocess parts, but also the full analysis of the propagation of errors between processes and their paths, thus forming a complete analytical framework for error propagation effects. This framework can not only intuitively predict the propagation and accumulation of errors, but also clarify how the prediction errors propagate with the multiprocess manufacturing process. This concept is of positive significance to the accuracy prediction in manufacturing processing, as well as the control, blocking, and elimination of errors, and it can also provide better support for the optimization and quality control of modern multiprocess complex manufacturing processes.

7. Conclusion

This paper analyzes the multisource error existing in the complex multiprocess manufacturing process and clarifies the types of errors that may be introduced by process operations and the corresponding ways of action; subsequently, it further studies the mathematical propagation model of multisource error with process conversion, establishes the error transfer accumulation equation based on state space theory, and realizes the error prediction; and finally, through the introduction of the link graph and the link matrix, this paper realizes the analysis of the propagation and influence of multisource machining errors accompanied by the multiprocess manufacturing process.

It is well known that the structures of modern mechanical parts are relatively complex, involving various elements of the process flow. The analytical framework for error propagation effects proposed in this paper makes it difficult to cover all the elements of error generation and propagation in manufacturing processes. At the same time, the measurement of multiple-source errors and their propagation calculations are not closely coupled with the manufacturing system, which also poses a challenge to realize the manufacturing automation of the whole process. In the future, integrating artificial intelligence algorithms and realizing the deep integration of the proposed framework and manufacturing process system to better adapt to the actual manufacturing situation that will be an important follow-up research direction.

Data Availability

All data generated or analyzed during this study are included in this manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Li C. wrote the manuscript and Li L. revised the manuscript.

Acknowledgments

This study was supported by the Key Research and Development Program of Shaanxi (Program no. 2022GY-254), the Natural Science Basic Research Program of Shaanxi (Program no. 2019JQ-896), the Open Project Program of the State Key Lab of CAD&CG (Program no. A2204), and the Zhejiang University, P.R. China.

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Copyright © 2023 Chunlei Li and Liang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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