Issue 
Int. J. Metrol. Qual. Eng.
Volume 15, 2024



Article Number  16  
Number of page(s)  11  
DOI  https://doi.org/10.1051/ijmqe/2024011  
Published online  20 August 2024 
Research article
Prediction of machining errors for freeform surfaces based on an improved grey model (1,N)
School of Mechanical and Automotive Engineering, Guangxi University of Science and Technology, Liuzhou 545616, China
^{*} Corresponding author: chenyueping99@126.com
Received:
10
August
2022
Accepted:
14
June
2024
An improved grey model (GM) for predicting freeform surface machining errors was established to address the low measuring efficiencies of coordinate measuring machines (CMMs) and the low prediction accuracy of the GM(1,1) model. A number of points on a freeform surface are measured with a CMM, and machining errors are obtained. Ideas from metabolic methods and the GM(1,N) prediction model were combined. To reduce the impact of random fluctuations of the machining errors, a Markov prediction model was then used to correct the fitted results for the residuals and obtain the predicted results of the machining errors for freeform surfaces, thus improving the prediction accuracy of the model. The predicted results of the metabolic GM(1,1) and metabolic GM(1,N) models were then compared. The experimental results showed that the combination of metabolic theory, a grey model, and Markov theory effectively improved the prediction accuracy.
Key words: GM(1,N) / freeform surfaces / prediction of machining errors / metabolism / Markov theory
© D. Ma and Y. Chen, Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Coordinate measuring technology is widely applied in industrial production [1]. A coordinate measuring machine (CMM) can accurately obtain information about the machining errors at various points on a work piece surface. Freeform parts are generally measured using highprecision CMMs. Measurement accuracy is usually improved by increasing the number of measuring points, which is both time consuming and laborious [2]. A machining error prediction model can predict machining errors at any location on the surface of a freeform work piece from machining error information at a limited number of measurement points to obtain richer machining error information. When the prediction errors are sufficiently small, the prediction model can replace the CMM to obtain information about the machining errors at unmeasured points, thus improving the inspection efficiency.
To make an effective prediction of machining errors in a freeform part [3], a machining error prediction model must first be developed. Thus, it is crucial to seek an efficient, simple, and highly accurate modeling method for machining error prediction. Scholars around the world have conducted considerable research on error prediction, mainly based on grey system theory, artificial neural network theory, and other methods. Accurate prediction of machining errors helps shorten the production cycle time, reduce product scrap rates, and improve the effectiveness of error compensation [4–6].
Some scholars have chosen single prediction models for the prediction of machining errors and optimized the models using other mathematical methods. Qi et al. [7] established a mathematical model for the dynamic distribution of dimensional errors in the bearing grinding process based on the characteristics of mechanical machining errors using grey system theory. Analysis and prediction of the dynamic distribution of current and future machining errors were conducted based on the model. The experimental results showed that the accuracy of the model was high and the predicted values matched well with the measured values. Sun et al. [8] proposed a machining error prediction model for thinwalled parts and obtained data through simulations and experiments. Compared with the mechanism model or datadriven model, the experimental results showed that the proposed model has higher efficiency and accuracy. Wang [9] used a neural network model for the prediction of machining errors. Huang et al. [10] used a simulated annealing algorithm to optimize a back propagation neural network to effectively predict machining errors on freeform surfaces. Wu et al. [11] proposed a gear machining error prediction method based on a parameter significance estimation and probability regression, which was experimentally verified to be superior to other algorithms in predicting gear inspection accuracy. Manikandan et al. [12] developed a geometric error prediction model using process geometry variables, such as the actual feed per revolution, as reference factors. The model enabled more effective prediction and control of geometric errors in unmachined conditions. Yang et al. [13] studied the background values of the grey model (GM) and proposed an optimization method based on threeparameter background values, which could effectively improve the smoothing effect of the background values and weaken the influence of extreme data on the performances of grey models.
Single forecasting models are often inadequate, and many scholars have sought to improve the forecasting accuracy by combining forecasting models. Shen et al. [14] combined a grey model and a neural network model, invoking a particle swarm algorithm to optimize a grey neural network prediction model. Li et al. [15] established a dynamic analysis model by optimizing a grey model through a Taylor approximation and a Markov prediction model and verified the validity of the prediction model through experiments. Zhou et al. [16] proposed a model for predicting machining errors by combining a metabolic grey model and a nonlinear autoregressive neural network for the dynamic and nonlinear characteristics of machine tool spindle machining errors. The results showed that the combined model had high accuracy, speed, and robustness, it outperformed the individual models, and it could be applied for the prediction of other complex machining errors. Zhao et al. [17] implemented a GM(1,1) model for the replacement of old and new information in the prediction process and combined it with Markov chains to correct the prediction results. The method is more suitable for longterm forecasting with a large volume of data and a wide range of fluctuations, which can effectively compensate for the shortcomings of the grey model and improve the feasibility of the forecast values. Yang et al. [18] analyzed the characteristics and shortcomings of the traditional grey model, combined with the advantages of Markov chain theory, which is suitable for dealing with processes with large random fluctuations, and proposed a weighted Markov combined prediction model based on the dimensional prediction of the external turning process. Zou et al. [19] used a GM(1,1) model in grey system theory to model the predicted relative errors of 12 holes drilled and reamed consecutively and used Markov theory to classify the predictions into states and correct the results. The results of this study demonstrated that it is feasible to use a combination of grey models and Markov chain theory for the prediction of processing errors.
Lu et al. [20] analyzed the relevant factors affecting the passenger volume of urban rail transport and proposed a combined forecasting model by integrating the ideas of a GM(1,N) forecasting model, Markov theory, and three metabolic methods. Fung et al. [21] optimized two different prediction models using recursive extended least squares (RELS) and neural network methods, and the results showed that the combined prediction model was more effective. Teng et al. [22] combined metabolic theory, a grey model, and Markov theory to construct a dynamic online public opinion crisis prediction model with a metabolic GM(1,N) Markov model. Wang et al. [23] constructed a grey incremental model and a grey combined prediction model based on the GM(1,1) model principle and used the least squares method to solve for the optimal weight coefficients of the grey combined prediction model.
In summary, although many research results have improved and applied grey models more prominently, the crossfusion application research using metabolic theory, grey models, and Markov theory at the same time has not yet been completed. In particular, the application of these fusion methods for freeform surface machining error prediction needs to be studied in depth. Therefore, this article proposes a method for predicting freeform machining errors based on a metabolic grey Markov model.
2 Development of metabolic GM(1,N) Markov model
2.1 Modeling ideas
Based on the prediction results of the GM(1,N) model, this study combines the ideas of three approaches: grey theory, Markov theory, and metabolic methods. A new freeform surfaces machining error prediction model is proposed called the metabolic GM(1,N) Markov model. The modeling steps are as follows.
Step 1. Obtain the original data series ,…, N. Initialize the raw data and process it nonnegatively.
Step 2. Determine the best predictive dimension N for the selected series.
Step 3. Establish the GM(1,N) model for prediction.
Step 4. Calculate the residuals. Calculate the degree of deviation from the actual values.
Step 5. Perform state partitioning and build a state transfer probability matrix, use a Markov prediction model for residual correction, and then refit the data.
Step 6. Use the new fitted data to optimize the model to obtain new predicted values.
Step 7. Use metabolic ideas for the replacement of old and new values to obtain the final prediction.
The overall modeling process is shown in Figure 1.
Fig. 1 Modeling process for the metabolic GM(1,N) Markov model. 
2.2 GM(1,N) model
Grey theory was proposed by Professor Deng Julong in 1982, and grey prediction models are core elements of grey theory. The grey model GM(1,N) transforms the original data into a series with regular variation, weakening the randomness of the original data, and the constructed firstorder linear differential equation is solved by cumulative reduction to find the predicted value of the original data series. The modelling process of the GM(1,N) model is shown in Figure 2, and the steps are as follows.
The system has N sequences of data containing n elements:
An accumulation on is performed, and the onetime accumulated generating series is
where
The background value Z^{(1)}(k) is defined as follows:
Finally, the GM(1,N) grey differential equation for is constructed:
The GM(1,N) whitening differential equation is
where α,β_{1},β_{2},...,β_{N} are parameters.
The series and the parametric series are defined. Then,
From this it can be determined that
where .
Parameter â is substituted into equation (3), yielding
Where α and β_{i} are parameters.
The solution obtained by equation (9) is the predicted value of the original cumulative series, so the predicted value of the original series under the GM(1,N) model needs to be obtained by cumulative reduction:
is the predicted value of the original series under the GM(1,N) prediction model:
Note that the original residuals generated are ϵ^{0} = (ϵ_{1}, ϵ_{2}, …, ϵ_{n}), where ε^{0} is the sequence of residuals from . The residual series are used to build a GM(1,N) model and find its parameters P = [α_{ϵ}, β_{ϵ}]^{T}, The simulated value of is calculated according to equation (11):
where α_{ε} and β_{ε} are parameters. Then, equation (9) can be corrected to
Fig. 2 Modeling process for the GM(1,N) model. 
2.3 Metabolic GM(1,N) model
When using the GM(1,N) forecasting model, the data source is fixed and the model is unable to provide uptodate data for the data used in the forecasts. Using such a fixed data source for forecasting more distant data is bound to produce large errors in the forecasts, whereas the metabolic idea can incorporate the latest forecasts already produced into the model. Therefore, the idea of metabolism is incorporated in this study by adding the predicted data as current information when predicting the value of the n + 2 location points and then removing the information from the earliest data. The process is as follows. The original data series is added to the predicted data Y(n+1) as a new data source, and then the earliest data is removed. As a result, a new data series can be obtained, which, using this data series, enables the data to be kept up to date. The schematic diagram of the metabolic model is shown in Figure 3.
Fig. 3 Schematic diagram of the metabolic model. 
2.4 Metabolic GM(1,N) Markov model
The GM(1,N) model is generally used for forms where the variation in the original series has an exponential pattern and where the dynamic fluctuations in the surface machining errors can produce large errors in the prediction results. Therefore, based on the prediction situation of the GM(1,N) model and the characteristics of the fluctuations in the freeform surface processing errors, Markov theory was used to classify the prediction results into states, and a Markov model was used to make a quadratic correction to the GM(1,N) prediction results in order to make up for the limitations of the GM(1,N) model and improve the prediction accuracy. The modelling process of the model is shown in Figure 4 and the specific modelling steps are as follows.
(1) Division of state intervals
The Markov model state partitioning method based on the GM(1,N) model is based on the relative error distribution of the prediction results of the GM(1,N) model, resulting in a corresponding scatter plot and a number of lines parallel to the xaxis. The area between each pair of adjacent parallel lines can be regarded as a state, denoted as state E_{i}.
The number of state intervals depends in general on the size of the original data set. As the total number of state transfers is smaller when the volume of raw data is smaller, the number of states should be smaller in order to reflect the transfer of raw data between states more objectively. In contrast, when the volume of the original data is larger, the number of states should be divided into a larger number to extract more information from the state transfer probability matrix to improve the prediction accuracy.
(2) State transfer probability matrix
The number of transfers from state E_{i} to state E_{j} is n_{i→j} and the number of transfers from state E_{j} to another state is N_{i}. The state transfer probability of transferring from state E_{i} to state E_{j} is
According to equation (14), the state transfer probability matrix can be calculated as
where P_{n1} + P_{n2} + P_{n3} + … + P_{nn} = 1, 0 ≤ P_{nn} ≤ 1, i, j = 0, 1,2,…, n.
(3) Calculation of predicted values
After establishing the state transfer probability matrix P, let the object be in state E_{i} at moment n. If the kth row in P satisfies max P_{ij }= P_{kl}, then it is considered that moment n + 1 is most likely to be transferred from state E_{i} to state E_{j}. Thus, the interval of change [E_{1}_{i}, E_{2}_{i}] of the predicted value is also determined, and the median of this interval can be taken as the predicted value at moment n+1:
(4) Error checking
The relative error is calculated as follows:
After completing the modeling of the Markov model, the metabolic model, the GM(1,N) model, and the Markov model were combined to construct the metabolic GM(1,N) Markov dynamic model.
Fig. 4 Prediction process for Markov model. 
3 Determination of the optimal dimensions of the model
From the modeling process of the metabolic model, it was found that the model dimension N was one of the main factors affecting the prediction accuracy of the metabolic model. The determination of the modeling data length (model dimension N) is very important in the metabolic modeling process, and different model dimensions N had different effects on the prediction accuracy. When the model dimension was small, the prediction accuracy of the model was low. As the number of dimensions of the model increased, the prediction accuracy of the model gradually improved. However, further increasing the number of dimensions did not improve the accuracy of the model, but tended to decrease it. There is currently no better way of determining the optimal model dimensionality. In this study, experimental calculations were performed on sample data with lengths between 5 and 15, and the mean absolute errors of the residuals of the prediction model were compared to determine the size of the selected dimension. The variation in the prediction accuracy of the metabolic model for different dimensions N is shown in Figure 5. For this artefact model, when N was 9, the absolute mean error was minimized, and the highest model accuracy prediction precision was achieved.
The prediction accuracy of Metabolic GM(1,N) Markov model was evaluated using three sets of data when different dimensions are taken. In this paper, the mean absolute percentage error was chosen to illustrate the prediction level of the forecasting model. As can be seen in Figure 6, the prediction level of the model is changed with the choice of dimensions. The model has the best accuracy when the optimal dimension is determined to be N = 9, so in the following tests, the model dimension is 9. The variation in the prediction accuracy of Metabolic GM(1,N) Markov model for different dimensions N is shown in Figure 6.
Fig. 5 Variation of prediction accuracy of metabolic GM(1,N) model with different values of dimension N. 
Fig. 6 Variation of prediction accuracy of Metabolic GM(1,N) Markov model with different values of dimension N. 
4 Experimental study of machining error prediction for freeform surfaces
CMMs have an unmatched advantage for dimensional inspection and can provide accurate information on machining errors at various points on a workpiece [24]. Compared with noncontact probes, touch probes have a high level of measurement accuracy. However, when measuring surface machining errors on workpieces, each point along the path has to be measured one by one, which is inefficient. A freeform surface machining error prediction model uses machining information from a limited number of known points to predict machining errors at other locations on the freeform surface. In the event that the predicted error meets the accuracy requirements, it can replace the CMM for predicting the machining error of unknown points, which achieves the purpose of improving efficiency and lowering costs.
4.1 Collecting machining error data
The workpiece used in this study was a freeform surface machined on a computer numerical control (CNC) machine. Figure 7 shows the computeraided design (CAD) model of the freeform surface and the theoretical measurement points. The information from 300 random measurement points on the surface was selected as raw data, and a further 100 points were selected in the same way as a validation sample. The freeform surface was also inspected using a CMM to obtain the machining error of each point on the freeform surface and the corresponding coordinate values. The inspection device was a German Hexagon Leitz Reference HP CMM (PCDMIS software, MPEE = 0.9 + L/400 µm), with a selected ball diameter of 5 mm, positioning and retraction distance of 3 mm, moving speed of 20 mm/s, and touch and retraction speed of 2 mm/s. Figure 8 shows the use of a CMM for freeform machining error detection. The sample data were processed using the MATLAB R2016b platform. The selected X, Y, and Z coordinate values of the theoretical points of the freeform surface were used as influencing factors to obtain a comprehensive prediction of the machining error at each point of the workpiece.
Fig. 7 Freeform surface digital model and theoretical measurement points. 
Fig. 8 Freeform surface part measurement using the coordinate measuring machine. 
4.2 Modeling and prediction of surface machining errors
Based on the machining errors obtained from the CMM inspection, a GM(1,N) model was constructed consisting of the freeform machining errors with coordinate values X, Y, and Z. Since the GM(1,N) model requires the original series to be nonnegative, nonnegative processing of the original data was carried out before the prediction, and the data reduction process was conducted after the prediction results were obtained.
The model accuracy was the highest when the model dimension N was taken as 9. Therefore, the model dimension was taken as 9 for modelling in the modelling process. A metabolic GM(1,N) model was used to fit the experimental data, and the results of the fit are shown in Figure 9. From the fitting results, it can be seen that a better fit was obtained by using the metabolic GM(1,N) prediction model. Thus, the metabolic GM(1,N) prediction model can be used for prediction.
The metabolic GM(1,1) and GM(1,N) prediction models were used to predict the sample data, and the prediction results are shown in Figure 10. It is clear that the metabolic GM(1,N) model had a better prediction accuracy.
Fig. 9 Fitting result of metabolic GM(1,N) model (N = 9). 
Fig. 10 Prediction results of different models. 
4.3 Correction of predicted values by Markov models
The residual values of the prediction results of the metabolic GM(1,N) model were calculated, and then the Markov prediction model was used to correct and optimize the prediction results of the metabolic GM(1,N) model to make up for the shortcomings of the GM(1,N) model and improve the accuracy of the prediction. The absolute values of the residuals before and after the correction are shown in Figure 11.
The prediction results of the metabolic GM(1,1) and the metabolic GM(1,N) prediction models were calculated and analyzed. In this study, the mean absolute error (σ_{MAE}), root mean square error (σ_{RMSE}) and mean absolute percentage error (σ_{MAPE}) were used as evaluation indicators for the analysis. These are defined as follows.
Mean absolute error, denoted by σ_{MAE},
Root mean square error, denoted by σ_{RMSE},
Mean absolute percentage error, denoted by σ_{MAPE},
Here n is the validation sample size, ê_{i} is the predicted value of the validation sample machining error and e_{i} is the actual measured value of the validation sample machining error. The more the above three indicators converge to 0, the better the prediction is. The results of the analysis are shown in Table 1.
The metabolic GM(1,N) Markov model proposed in this study was compared with the metabolic GM(1,1) model and the metabolic GM(1,N) model for freeform surface machining error prediction results, and the σ_{MAE}, σ_{RMSE} and σ_{MAPE} values of each model were calculated separately. As shown in Table 1, when the predictions using the metabolic GM(1,N) model were compared with the predictions using the metabolic GM(1,1) model, the the root mean square error of the metabolic GM(1,N) model decreased by 0.0014 mm, and the mean absolute percentage error decreased by 14.86%. When predictions using the metabolic GM(1,N) Markov model were compared with predictions using the metabolic GM(1,N) model, the mean absolute percentage error decreased by 51.61%. The prediction accuracy of the metabolic GM(1,1) model could be further improved by the metabolic GM(1,N) Markov model. For the prediction of the freeform surface machining errors, the prediction of the metabolic GM(1,N) model can be greatly improved by adapting the metabolic GM(1,1) model, and the prediction accuracy of the model can be improved by adding Markov theory to further optimize the model.
Fig. 11 Absolute values of residuals of predicted results before and after correction. 
Comparison of prediction accuracies of different models.
5 Conclusion
In this study, the prediction of freeform machining errors was studied. In response to the insufficient prediction accuracy of the GM(1,1) model, a GM(1,N) prediction model was established, incorporating the idea of metabolism, which effectively improved the prediction accuracy. On this basis, the prediction results of the metabolic GM(1,N) model were revised using a Markov prediction model, and the experimental results showed that the combination of metabolic theory, a grey model, and Markov theory prediction could further improve the prediction accuracy.
Using Markov's principle to correct the fitting results of the metabolic GM(1,N) model could effectively reduce the impact of random fluctuations of the freeform machining errors on the grey model and enhance the adaptability of the grey prediction model in machining error model prediction. This method has practical implications for extending the use of grey models in mechanical engineering. When measuring machining errors on freeform surfaces, the model can be used to predict machining errors at other locations on the workpiece more accurately with only a small amount of measurement point information, saving time and improving efficiency.
Acknowledgments
We thank LetPub (www.letpub.com) for its linguistic assistance during the preparation of this manuscript.
Funding
This work was financially supported by the National Natural Science Foundation of China (51565006, 52165054), the Natural Science Foundation of Guangxi Province (2018GXNSFAA050085, 2020GXNSFAA1591), the Science Research Innovation Team Project of Guangxi Provincial Education Department, and the Science Research Innovation Team Project of Guangxi University of Science and Technology.
Conflicts of interest
The authors declare no conflict of interest.
Data availability statement
The datasets generated or analyzed during this study are available from the corresponding author on reasonable request.
Author contribution statement
Dayang Ma: Conceptualization, Methodology, Formal Analysis, Writing  Original Draft, Data Curation.
Yueping Chen: Conceptualization, Funding Acquisition, Resources, Project Administration, Supervision, Formal Analysis, Methodology, Writing  Review & Editing.
References
 M. Marek, Selection of the best model of distribution of measurement points in contact coordinate measurements of freeform surfaces of products, Sensors. 19, 5346 (2019) [CrossRef] [Google Scholar]
 D.Z. Zhao, W.H. Wang, J.Z. Zhou et al., Measurement point sampling method for inspection of parts with freeform surfaces, Adv. Mech. Eng. 10, 1–12 (2018) [Google Scholar]
 W. Wolovich, H. Albakri, H. Yalcin, The precise measurement of freeform surfaces, J. Manuf. Sci. Eng. 124, 326–332 (2002) [CrossRef] [Google Scholar]
 S.F. Liu, M.L. Zhou, J.P. Wang, Research on variation model for predicting machining error of milling surface, Manuf. Technol. Machine Tool. 4, 80–84 (2018) [Google Scholar]
 L.X. Gong, J.H. Jiang, A BP neural networks error predictive model of mechanical processing based on LM algorithm, Machine Tool Hydraul. 11, 67–71 (2013) [Google Scholar]
 Q. Zhang, H.X. Wen, Study on the optimization of machining error gray forecast modeling, Machine Tool Hydraul. 36, 61–63 (2008) [Google Scholar]
 S.H. Qi, Y.T. Fei, Prediction of error in manufacturing based on the gray model theory, Trans. Chin. Soc. Agric. Machinery. 32, 89–91+94 (2001) [Google Scholar]
 H. Sun, F.Y. Peng, L. Zhou et al., A hybrid driven approach to integrate surrogate model and Bayesian framework for the prediction of machining errors of thinwalled parts, Int. J. Mech. Sci. 192, 106–111 (2021) [Google Scholar]
 B. Wang, Predicting machining errors by artificial neural network, J. Jiangsu Univ. Sci. Technol. 4, 44–48 (1997) [Google Scholar]
 K.Q. Huang, Y.P. Chen, Y.K. Zhang, Machining errors prediction of freeform surfaces: BP neural network algorithm optimized by simulated annealing algorithm, J. Guangxi Univ. Sci. Technol. 33, 69–73+82 (2022) [Google Scholar]
 D.Y. Wu, P. Yan, Y. Guo et al., A gear machining error prediction method based on adaptive Gaussian mixture regression considering stochastic disturbance, J. Intell. Manuf. 2021, 1–19 (2021) [Google Scholar]
 H. Manikandan, T.C. Bera, Modelling of dimensional and geometric error prediction in turning of thinwalled components, Precis. Eng. 72, 382–396 (2021) [CrossRef] [Google Scholar]
 X.L. Yang, M. Zhou, B. Zeng, A new method for constructing background value of grey prediction model, Statistics & Decision. 19, 14–18 (2018) [Google Scholar]
 M.X. Shen, T. Tao, Research on thermal error compensation of milling machine based on improved grey network, Mach. Design Manuf. 4, 158–161 (2021) [Google Scholar]
 G.D. Li, S. Masuda, D. Yamaguchi et al., A study on the prediction of machining accuracy, Int. J. Adv. Manuf. Technol. 43, 5–6 (2009) [Google Scholar]
 W. Zhou, X. Zhu, J. Wang et al., New error prediction method for machining process based on a combined model, Math. Probl. Eng. 2018, 1–8 (2018) [CrossRef] [Google Scholar]
 Y.Y. Zhao, H.L. Lin, D.H. Zhang et al., Forecasting of grassland carrying capacity of Gannan based on GreyMarkov residual error models, Trans. Chin. Soc. Agric. Eng. 28, 199–204 (2012) [Google Scholar]
 L. Yang, H.F. Ning, J. Gong, Research on turning dimension prediction based on grey weighted Markov combined model, Mach. Design Manuf. 8, 39–42+45 (2016) [Google Scholar]
 L.L. Zou, M. Cong, Y.R. He, Machining error prediction based on GreyMarkov theory, Modular Mach. Tool Automatic Manuf. Technique. 1, 14–16+19 (2014) [Google Scholar]
 C.W. Lu, H. Zhao, S. Jiang et al., A prediction method of rail transit passenger volume based on metabolism − GM(1,N) Markov model, J. Highway Transp. Res. Dev. 35, 141–150 (2018) [Google Scholar]
 E.H.K. Fung, Y.K. Wong, H.F. Ho, Modelling and prediction of machining errors using ARMAX and NARMAX structures, Appl. Math. Modell. 27, 611–627 (2003) [CrossRef] [Google Scholar]
 J. Teng, Z.J. Xia, M.Y. Lou, Dynamic network public opinion crisis prediction based on metabolic GM(1,N) Markov model, Inf. Sci. 38, 88–94 (2020) [Google Scholar]
 A. Wang, Y. Yang, A combined gray forecast model based on least square method and its application, Math. Practice Theory. 48, 291–299 (2018) [Google Scholar]
 Y.P. Chen, N.Q. Shang, Comparison of GA, ACO algorithm, and PSO algorithm for path optimization on freeform surfaces using coordinate measuring machines, Eng. Res. Express. 3, 4 (2021) [Google Scholar]
Cite this article as: Dayang Ma, Yueping Chen, Prediction of machining errors for freeform surfaces based on an improved grey model (1,N), Int. J. Metrol. Qual. Eng. 15, 16 (2024)
All Tables
All Figures
Fig. 1 Modeling process for the metabolic GM(1,N) Markov model. 

In the text 
Fig. 2 Modeling process for the GM(1,N) model. 

In the text 
Fig. 3 Schematic diagram of the metabolic model. 

In the text 
Fig. 4 Prediction process for Markov model. 

In the text 
Fig. 5 Variation of prediction accuracy of metabolic GM(1,N) model with different values of dimension N. 

In the text 
Fig. 6 Variation of prediction accuracy of Metabolic GM(1,N) Markov model with different values of dimension N. 

In the text 
Fig. 7 Freeform surface digital model and theoretical measurement points. 

In the text 
Fig. 8 Freeform surface part measurement using the coordinate measuring machine. 

In the text 
Fig. 9 Fitting result of metabolic GM(1,N) model (N = 9). 

In the text 
Fig. 10 Prediction results of different models. 

In the text 
Fig. 11 Absolute values of residuals of predicted results before and after correction. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.