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In-situ prediction of machining errors of thin-walled parts: an engineering knowledge based sparse Bayesian learning approach

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Abstract

Thin-walled parts such as blades are widely used in aerospace field, and their machining quality directly affects the service performance of core components. Due to obvious time-varying nonlinear effect and complex machining system, it is a great challenge to realize accurate and fast prediction of machining errors of such parts. To solve the above problems, an engineering knowledge based sparse Bayesian learning approach is proposed to realize in-situ prediction of machining errors of thin-walled blades. Firstly, an engineering knowledge based strategy is proposed to improve the generalization ability of the model by integrating multi-source engineering knowledge, including machining information, physical information and online monitoring information. Then, principal component analysis method is utilized for the dimensional reduction of features. Sparse Bayesian learning approach is developed for model training, which significantly reduce the complexity of the regression model. Finally, the superiority and effectiveness of the proposed approach have been proven in blade milling experiments. Experimental results show that the average deviation of the proposed in-situ prediction model is about 11 μm, and the model complexity is reduced by 66%.

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Acknowledgements

This research was financially supported by the National Key Research and Development Program of China (Grant No. 2018YFB1701904), the National Natural Science Foundation of China (Grant No. U20A20294) and the Basic Science Center of China (Grant No. 52188102). The authors thanked the support by the Advanced Manufacturing and Technology Experimental Center (School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China).

Author information

Authors and Affiliations

  1. National NC System Engineering Research Center, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Hao Sun, Shengqiang Zhao, Rong Yan, Teng Zhang & Chi Zhang

  2. State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Fangyu Peng

  3. Wuhan Digital Design and Manufacturing Innovation Center Co. Ltd, China, Wuhan, 430074, China

    Lin Zhou

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  1. Hao Sun
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  2. Shengqiang Zhao
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  3. Fangyu Peng
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Correspondence to Fangyu Peng.

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Appendices

Appendix I: Main process for obtaining flexibilities in ABAQUS

Main process for obtaining flexibilities based on finite element simulation is completed in ABAQUS 6.14, as shown in Fig. 13.

  1. (1)

    Establishment of finite element simulation model. The 3D model of thin-walled blade is imported into ABAQUS software. First of all, details such as chamfering are simplified. Then, the material physical parameters of the model are set and meshes are divided. Fixed constraints are imposed on the bottom of the blade.

  2. (2)

    Loading of cutting loads. Python is utilized for the secondary development of ABAQUS. A total of 35 analysis steps are set, and the unit cutting force is applied to nodes 1to nodes 35 in turn.

  3. (3)

    Dynamic removal of materials. The life-and-death element technique is used to simulate the material dynamic removal in machining process.

  4. (4)

    Simulation of machining deformation. Finite element simulation job is created and the submitted to the server for the simulation of machining simulation.

  5. (5)

    Extraction of node displacement. After the simulation process is completed, the displacement of 35 nodes is extracted successively.

  6. (6)

    Acquisition of flexibilities of thin-walled blade. The ratio of displacement to unit cutting force is calculated and the flexibility of each point on the workpiece is obtained.

Fig. 13
figure 13

Schematic diagram of finite element simulation model for thin-walled blade

Full size image

Appendix II: Validation of accuracy and efficiency of AVMD algorithm

Three typical simulation signals are selected to verify the accuracy and efficiency of the proposed AVMD method. \({s}_{1}\left(t\right)=cos\left(4\pi t\right)+cos\left(48\pi t\right)/4+cos\left(800\pi t\right)/16+\eta \), where \(\eta \) represents Gaussian noise, \(\eta \sim N\left(0,\sigma \right)\), \(\sigma =0.005\). \({s}_{2}\left(t\right)=cos\left(200\pi t\right)+cos\left(400\pi t\right)/2+cos\left(800\pi t\right)/3+\eta \), \(\eta \sim N\left(0,\sigma \right)\), \(\sigma =0.05\). \({s}_{3}\left(t\right)=4sin\left(160\pi t\right)+5cos\left(40\pi t\right)+\left[1+0.6sin\left(30\pi t\right)\right]cos\left[300\pi t+1.5sin\left(15\pi t\right)\right]/2\). The decomposition results are listed in Table 5.

Table 5 Decomposition results of three typical simulation signals
Full size table

It is pointed out in reference (Dragomiretskiy, Zosso 2014) that the best decomposition parameters can be determined by observing the time–frequency spectrum diagram. However, this method relies too much on experience when the signal is complex (e.g. \({s}_{3}\left(t\right)\)). In (Liu et al., 2018), the grid search method is adopted to optimize decomposition layer and penalty factor. The decomposition efficiency of this method is significantly lower than that of the proposed AVMD method. The comparison results of the two methods are shown in Table 6 (to ensure the rationality of comparison, parameters in AVMD are set in accordance with those in (Liu et al., 2018)). It can be found that in terms of efficiency and accuracy, the proposed AVMD method shows good decomposition effect and versatility.

Table 6 Comparison of decomposition results of \({s}_{3}\left(t\right)\) under different decomposition approaches
Full size table

Appendix III: PCA weights distribution of cutting forces features

Category

Features and PCA weights

Time domain

\([{Max}_{tx},{Max}_{ty},{Max}_{tz},{Mean}_{tx},{Mean}_{ty},{Mean}_{tz},{Var}_{tx},{Var}_{ty},{Var}_{tz},\)

\({RMS}_{tx},{RMS}_{ty},{RMS}_{tz},{Kur}_{tx},{Kur}_{ty},{Kur}_{tz},{Ske}_{tx},{Ske}_{ty},{Ske}_{tz}]\)

[− 0.101,0.232,0.042,− 0.080,0.047,0.028,− 0.032,0.111,0.005,− 0.045,0.143,0.040,− 0.192,− 0.098,− 0.063,− 0.148,0.214,− 0.018]

Frequency domain

\([{FAE}_{fx},{FAE}_{fy},{FAE}_{fz},{FC}_{fx},{FC}_{fy},{FC}_{fz},{FV}_{fx},{FV}_{fy},{FV}_{fz},\)

\({MSF}_{fx},{MSF}_{fy},{MSF}_{fz}]\)

[− 0.101,0.232,0.042,− 0.080,0.047,0.028,− 0.032,0.111,0.005,− 0.045,0.143,0.040,− 0.192,− 0.098,− 0.063,− 0.148,0.214,− 0.018]

Time–frequency domain (\(K=2\))

\(\left[{E}_{k2x1},{E}_{k2x2},{E}_{k2y1},{E}_{k2y2},{E}_{k2z1},{E}_{k2z2}\right]\)

[− 0.035,− 0.168,− 0.074,− 0.071,0.020,− 0.007]

Time–frequency domain (\(K=3\))

\(\left[{E}_{k3x1},{E}_{k2x2},{E}_{k3x3},{E}_{k3y1},{E}_{k3y2},{E}_{k3y3}{,E}_{k3z1},{E}_{k3z2},{E}_{k3z3}\right]\)

[− 0.212,− 0.064, 0.100,− 0.070,− 0.063,− 0.051,− 0.009,0.045,0.018]

Time–frequency domain (\(K=4\))

\(\left[\begin{array}{c}{E}_{k4x1},{E}_{k4x2},{E}_{k4x3},{E}_{k4x4},{E}_{k4y1},{E}_{k4y2},{E}_{k4y3},{E}_{k4y4},\\ {E}_{k4z1},{E}_{k4z2},{E}_{k4z3},{E}_{k4z4}\end{array}\right]\)

[− 0.295,− 0.011,0.099,− 0.024,− 0.066,− 0.067,− 0.055,− 0.057,− 0.019,0.016,− 0.025,0.088]

Time–frequency domain (\(K=5\))

\(\left[\begin{array}{c}{E}_{k5x1},{E}_{k5x2},{E}_{k5x3},{E}_{k5x4},{{E}_{k5x5},E}_{k5y1},{E}_{k5y2},{E}_{k5y3},{E}_{k5y4},\\ {E}_{k5y5},{E}_{k5z1},{E}_{k5z2},{E}_{k5z3},{E}_{k5z4},{E}_{k5z5}\end{array}\right]\)

[− 0.290,− 0.004,0.007,0.017,− 0.020,− 0.065,− 0.065,− 0.065,− 0.045,− 0.037,− 0.026,− 0.055,0.025,0.010,0.022]

Time–frequency domain (\(K=6\))

\(\left[\begin{array}{c}\begin{array}{c}{E}_{k6x1},{E}_{k6x2},{E}_{k6x3},{E}_{k6x4},{E}_{k6x5},{E}_{k6x6},\\ {E}_{k6y1},{E}_{k6y2},{E}_{k6y3},{E}_{k6y4},{E}_{k6y5},{E}_{k6y6},\end{array}\\ {E}_{k6z1},{E}_{k6z2},{E}_{k6z3},{E}_{k6z4},{E}_{k6z5},{E}_{k6z6}\end{array}\right]\)

[0.127,− 0.026,0.071,0.058,0.192,0.143,− 0.019,0.015,0.018,− 0.001,− 0.007,− 0.006,0.034,− 0.147,0.071,0.147,0.298,0.003]

Time–frequency domain (\(K=7\))

\(\left[\begin{array}{c}\begin{array}{c}{E}_{k7x1},{E}_{k7x2},{E}_{k7x3},{E}_{k7x4},{E}_{k7x5},{E}_{k7x6},{E}_{k7x7},\\ {E}_{k7y1},{E}_{k7y2},{E}_{k7y3},{E}_{k7y4},{E}_{k7y5},{E}_{k7y6},{E}_{k7y7},\end{array}\\ {E}_{k7z1},{E}_{k7z2},{E}_{k7z3},{E}_{k7z4},{E}_{k7z5},{E}_{k7z6},{E}_{k7z7}\end{array}\right]\)

[0.135,− 0.029,0.015,0.107,0.159,0.237,0.081,− 0.020,0.004,0.014,0.008,0.016,− 0.001,− 0.012,0.030,− 0.173,0.168,0.124,0.306,0.051,− 0.048]

Time–frequency domain (\(K=8\))

\(\left[\begin{array}{c}\begin{array}{c}{E}_{k8x1},{E}_{k8x2},{E}_{k8x3},{E}_{k8x4},{E}_{k8x5},{E}_{k8x6},{E}_{k8x7},{E}_{k8x8},\\ {E}_{k8y1},{E}_{k8y2},{E}_{k8y3},{E}_{k8y4},{E}_{k8y5},{E}_{k8y6},{E}_{k8y7},{E}_{k8y8},\end{array}\\ {E}_{k8z1},{E}_{k8z2},{E}_{k8z3},{E}_{k8z4},{E}_{k8z5},{E}_{k8z6},{E}_{k8z7},{E}_{k8z8}\end{array}\right]\)

[0.139,− 0.008,− 0.041,0.133,0.137,0.243,0.143,0.065,− 0.022,0.015,0.028,0.015,− 0.015,0.011,− 0.010,0.009,0.025,− 0.092,0.009,0.167,0.252,0.031,0.264,− 0.102]

Time–frequency domain (\(K=9\))

\(\left[\begin{array}{c}{E}_{k9x1},{E}_{k9x2},{E}_{k9x3},{E}_{k9x4},{E}_{k9x5},{E}_{k9x6},{E}_{k9x7},{E}_{k9x8},,{E}_{k9x9},\\ {E}_{k9y1},{E}_{k9y2},{E}_{k9y3},{E}_{k9y4},{E}_{k9y5},{E}_{k9y6},{E}_{k9y7},{E}_{k9y8},,{E}_{k9y9},\\ {E}_{k9z1},{E}_{k9z2},{E}_{k9z3},{E}_{k9z4},{E}_{k9z5},{E}_{k9z6},{E}_{k9z7},{E}_{k9z8},{E}_{k9z9}\end{array}\right]\)

[0.140,0.028,− 0.016,0.013,0.161,0.124,0.163,0.131,0.024,− 0.018,0.001,0.019,0.016,− 0.006,0.010,0.007,− 0.010,− 0.006,0.022,− 0.075,− 0.031,0.087,0.127,0.283,0.020,0.174,− 0.045]

Time–frequency domain (\(K=10\))

\(\left[\begin{array}{c}{E}_{k10x1},{E}_{k10x2},{E}_{k10x3},{E}_{k10x4},{E}_{k10x5},{E}_{k10x6},{E}_{k10x7},{E}_{k10x8},\\ {E}_{k10x9},{E}_{k10x10},{E}_{k10y1},{E}_{k10y2},{E}_{k10y3},{E}_{k10y4},{E}_{k10y5},{E}_{k10y6},\\ {E}_{k10y7},{E}_{k10y8},{E}_{k10y9},{E}_{k10y10},{E}_{k10z1},{E}_{k10z2},{E}_{k10z3},{E}_{k10z4},\\ {E}_{k10z5},{E}_{k10z6},{E}_{k10z7},{E}_{k10z8},{E}_{k10z9},{E}_{k10z10}\end{array}\right]\)

[0.138,0.048,− 0.036,0.042,0.169,0.023,0.273,0.031,0.064,0.024,− 0.023,0.001,0.031,0.021,0.015,− 0.002,− 0.005,0.008,− 0.007,− 0.004,0.009,− 0.043,− 0.051,0.138,0.213,0.113,0.063,0.106,0.079,− 0.095]

Appendix IV: Final form of the regression function (Taking \(\mathrm{K}=5\) as an example)

Predicting model.

$${{\varvec{y}}}_{{\varvec{P}}}={{\varvec{\Phi}}}_{0}{{\varvec{\omega}}}_{\left(k=5\right)}$$

where

$${\varvec{\Phi}}={\left[\begin{array}{cccccc}\kappa \left({\boldsymbol{\varphi }}_{1},{\boldsymbol{\varphi }}_{1}\right)& \kappa \left({\boldsymbol{\varphi }}_{1},{\boldsymbol{\varphi }}_{2}\right)& \cdots & \kappa \left({\boldsymbol{\varphi }}_{1},{\boldsymbol{\varphi }}_{465}\right)& \kappa \left({\boldsymbol{\varphi }}_{1},{\boldsymbol{\varphi }}_{466}\right)& 1\\ \kappa \left({\boldsymbol{\varphi }}_{2},{\boldsymbol{\varphi }}_{1}\right)& \kappa \left({\boldsymbol{\varphi }}_{2},{\boldsymbol{\varphi }}_{2}\right)& \cdots & \kappa \left({\boldsymbol{\varphi }}_{2},{\boldsymbol{\varphi }}_{465}\right)& \kappa \left({\boldsymbol{\varphi }}_{2},{\boldsymbol{\varphi }}_{466}\right)& 1\\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ \kappa \left({\boldsymbol{\varphi }}_{475},{\boldsymbol{\varphi }}_{1}\right)& \kappa \left({\boldsymbol{\varphi }}_{475},{\boldsymbol{\varphi }}_{2}\right)& \cdots & \kappa \left({\boldsymbol{\varphi }}_{475},{\boldsymbol{\varphi }}_{475}\right)& \kappa \left({\boldsymbol{\varphi }}_{475},{\boldsymbol{\varphi }}_{476}\right)& 1\\ \kappa \left({\boldsymbol{\varphi }}_{476},{\boldsymbol{\varphi }}_{1}\right)& \kappa \left({\boldsymbol{\varphi }}_{476},{\boldsymbol{\varphi }}_{2}\right)& \cdots & \kappa \left({\boldsymbol{\varphi }}_{476},{\boldsymbol{\varphi }}_{475}\right)& \kappa \left({\boldsymbol{\varphi }}_{476},{\boldsymbol{\varphi }}_{476}\right)& 1\end{array}\right]}_{476\times \left(476+1\right)}$$
$${\boldsymbol{\varphi }}_{\left(i\right)}=\left\{\begin{array}{c}{a}_{w\left(i\right)},{\theta }_{L\left(i\right)},{\theta }_{T\left(i\right)},{f}_{\left(i\right)},{V}_{\left(i\right)},{c}_{xx\left(i\right)},{c}_{xy\left(i\right)},{c}_{xz\left(i\right)},{c}_{yy\left(i\right)},{c}_{yz\left(i\right)},{c}_{zz\left(i\right)},{Max}_{tx\left(i\right)},{Max}_{ty\left(i\right)},{Max}_{tz\left(i\right)},\\ {Mean}_{tx\left(i\right)},{Var}_{ty\left(i\right)},{RMS}_{ty\left(i\right)},{{RMS}_{tz\left(i\right)},Kur}_{tx\left(i\right)},{Kur}_{ty\left(i\right)},{Kur}_{tz\left(i\right)},{Ske}_{tx\left(i\right)},{Ske}_{ty\left(i\right)},\\ {FAE}_{fy\left(i\right)},{FC}_{fx\left(i\right)},{FC}_{fy\left(i\right)},{FV}_{fx\left(i\right)},{FV}_{fy\left(i\right)},{MSF}_{fx\left(i\right)},{MSF}_{fy\left(i\right)}\end{array}\right\}$$
$${{\varvec{\Phi}}}_{0}=\left[\begin{array}{cccccc}\kappa \left({\boldsymbol{\varphi }}_{0},{\boldsymbol{\varphi }}_{1}\right)& \kappa \left({\boldsymbol{\varphi }}_{0},{\boldsymbol{\varphi }}_{2}\right)& \cdots & \kappa \left({\boldsymbol{\varphi }}_{0},{\boldsymbol{\varphi }}_{475}\right)& \kappa \left({\boldsymbol{\varphi }}_{0},{\boldsymbol{\varphi }}_{476}\right)& 1\end{array}\right]$$

\({{\varvec{\omega}}}_{\left(k=5\right)}=\) [3.4E + 01;1.8E−17; −8.2E−06; −3.2E−10; −1.5E−09; −3.1E−17; 5.4E  + 01; −3.8E−08; −5.9E−06; −8.9E−09; −1.4E−08; −8.8E−13; −2.1E−06; −2.6 E−09; −6.0 E−14; −3.8 E−07; −9.1 E−10; −3.6 E−08; 7.1E  + 01; 3.7E  + 01;9.8E  + 00; −1.1 E−09; −5.7E−07; 2.5E−21; −6.9E−04; 1.1E  + 00; 5.4E  + 00; 2.3E−01; 3.0E−10; −3.9E−13; −2.3E−11; −2.0E−09; −7.2E−12; −3.1E−11; −1.2E−13; 1.1E−07; −6.6E−11; 9.3E−14; −1.3E  + 02; −1.6E−11; −9.1E  + 00; −6.1E  + 01; 2.4E +  00; −4.1E−11; −2.8E  + 01; 5.8E−14; −1.4E−04; −1.9E  + 02; −5.0E  + 00; 2.9E−08; −1.3E  + 02; 1.1E−02; − − 5.9E  + 01; −1.6E  + 01; 5.9E−19; 1.2E−06; −1.3E−11; 5.9E−13; −2.0E−08; 5.4E−15; −6.1E−14; −1.4E −15; 2.2E−08;4.3E−11; 6.5E  + 01; 9.4E−16; 2.8E−11; 2.2E−03; 1.6E−01; 1.9E−05; 8.0E−10; −8.5E  + 01; −1.7E−07; −2.4E−01; −9.3E−12; −4.9E + 02; −3.5E−01; 1.9E−01; 2.8E−13; −4.8E−14; −3.9E−09; 7.5E−10; 2.4E  + 01; −3.6E  + 02; 2.1E  + 02; 1.9E−09; −3.4E−09; 2.4E−09; −2.4E−09; 1.9E−18; −7.1E−05; 1.7E−15; 4.8E−06; 5.9E−14; −3.9E−15; −3.1E−10; −3.5E−01; 1.4E−11; −1.8E−12; −1.4E−12; 4.3E  + 01; −1.2E−10; 2.2E  + 02; 2.3E−09; 1.6E−03; 2.0E  + 02; 7.9E−15; −4.5E  + 02; 3.3E  + 02; 4.6E−13; 5.0E−07; 2.0E  + 02; 3.8E−01; 2.5E−15;2.5E  + 01; 1.7E−12; 6.0E−16; 2.1E−04; 1.9E−06; 5.6E−02; 8.0E−17; 1.7E  + 01; 1.0E  + 00; 4.8E−18; −5.3E−20; 3.6E−14; 1.1E−01; −3.1E−26; −2.1E−13; −3.8E−14; 1.4E  + 02;1.5E  + 02; 4.6E  + 01; 1.8E  + 01; −1.6E−06; −9.1E−10; 1.2E  + 00; 3.9E−03; −2.3E−07; 3.0E−05; −2.3E−04; −5.2E  + 01; −1.7E−06; −7.8E−17; 2.5E−04; 8.2E−02; −1.1E  + 02; −2.9E−05; −1.4E−08; 2.9E  + 01; −2.2E−38; 1.1E−12; 3.2E−06; −7.1E−02; −7.4E−03; −4.8E−13; −6.3E−12; 8.8E−16; 3.0E  + 01; −3.8E  + 01; −3.8E−02; −2.5E  + 01; −4.3E−06; −1.8E−12; −2.1E−03; −6.1E−02; 4.2E−11; 1.2E  + 02; −3.7E−08; −2.1E−10; 2.5E  + 00; −6.3E−16; 1.3E−09; −3.1E−08; 9.0E−14; −1.2E−09; 1.1E  + 01; 1.4E  + 01; 2.3E  + 00; 1.0E−06; 1.7E  + 00; 7.8E−16; −2.8E  + 00; −9.2E−12; −6.4E−07; −6.8E−06; 2.0E−12; 2.1E−15; −2.3E−02; −1.1E−08; −2.5E−04; −1.0E−13; 2.0E−17; 4.7E−12; −2.6E−08; −1.2E−03; 2.2E−08; −3.0E  + 00; −3.6E  + 01; −5.5E−08; −7.5E−12; 6.9E−09;2.6E  + 02; −6.1E−07; −8.7E−01; −7.6E−03; −1.9E−16; −2.0E−09; 4.0E  + 01; 1.7E−13; −1.6E−14; −6.3E−15; −1.1E−12; −8.9E−11; −4.3E−17; −4.6E−17; 7.6E−01; 2.8E−07; −3.0E−03; −6.0E−04; −2.9E−09; −3.2E−04; 9.4E−09; 1.7E−07; 3.7E−04; −1.0E−01; −9.3E−06; −8.0E−07; 1.8E−12; 1.6E−11; 1.2E−02; −9.4E−05; −4.6E−09; −2.3E−08; 3.8E−02; 1.7E−02; 1.9E−02; 1.0E  + 01; 1.0E−08; −8.2E−17; −1.0E  + 01; 9.8E−30; 3.6E−11; 6.6E−09; 5.9E  + 01; 9.1E−02; 6.2E−19; −3.7E  + 01; 3.0E−11; 8.5E  + 01; −7.8E−15; 1.6E  + 02; −7.2E−13; −6.5E−06; 5.6E−15; −3.5E−10; −1.2E  + 01; −9.7E−02; 5.7E−10; −2.6E−05; −8.9E  + 00; −8.8E−02; −1.1E−10; −8.9E−14; −1.3E−04; −7.5E−01; 6.3E−09; 2.0E−06; 8.0E−04; 6.3E  + 02 ;2.8E  + 01; −5.4E−05; 2.3E−16; 1.0E−06; 1.2E−09; −6.7E−12; −5.6E−04; −7.4E−07; −1.0E−06; −7.2E−06; 7.0E−12; 4.0E−06; −4.9E  + 02; −1.3E−05; 1.8E−13; −4.9E−09; −2.0E−08; 2.1E  + 00; 6.8E−04; −1.7E−01; −4.0E−02; −1.7E−13; −3.4E−02; −1.9E  + 02; −1.1E−06; −5.4E−08; 1.5E  + 02; 2.8E−04; 1.3E−06; 5.6E−10; 3.4E  + 00; 3.9E−05; 5.4E−03; −3.3E  + 01; −1.1E−05; −1.1E−01; −2.7E−01; 5.1E−07; −1.5E  + 02; −6.7E−08; 2.6E−10; 9.5E  + 00; 9.5E  + 01; 4.3E−03; 8.0E−05; 4.9E−03; −4.2E−09; 1.7E−13; 3.2E−09; 4.3E−06; 2.8E−10; 5.0E−04; −4.3E−11; −3.3E−11; 2.2E−01; 2.4E−02; 5.1E  + 01; 1.0E−10; 7.6E−04; 2.2E−10; −1.3E−08; 1.7E−03; 3.1E−02; 3.5E  + 00; 3.5E−07; 4.2E−02; −2.7E−16; −5.4E−16; 8.2E−20; 1.2E−03; 1.3E  + 02; 6.4E−05; 1.8E−01; −9.1E−15; −2.6E−04; 1.1E−04; 1.2E−14; 1.3E  + 02; 2.8E−08; 4.4E−22; −2.2E−06; −5.9E−17; −5.0E−09; −5.0E−12; 3.6E−03; 1.4E  + 01; 1.5E−10; 4.1E  + 00; 2.0E−08; 2.4E−05; 4.4E−04; 1.4E  + 01; 1.4E−08; 1.2E−12; −2.0E−07; 1.9E−05; 1.2E−08; 1.7E−05; 1.2E−04; 3.8E−10; −1.5E−10; −8.9E  + 01; −1.2E−15; 2.7E−04; 7.6E  + 01; 3.5E−10; 9.6E−15; −2.9E−07; 1.3E−05; 8.3E−08; 6.5E−12; 3.2E  + 02; 2.2E  + 03; 1.6E−02; 2.7E−02; −6.7E−03; 1.5E−02; 1.7E−08; 2.5E  + 02; 1.2E−02; −1.8E−05; −2.3E−01; −1.6E  + 03; 5.4E−06; 2.6E−08; 1.7E−05; 2.9E−07; −3.0E−09; −6.2E  + 02; 9.4E−03; −4.2E−07; −1.7E−06; 1.1E−10; −5.3E−13; −6.3E  + 02; 3.4E−10; 2.2E−06; 5.6E−12; 6.5E−13; 4.9E−02; 1.9E  + 00; 1.1E−07; 7.0E−06; −9.7E−08; −2.1E−08; −1.6E−05; −2.3E−05; −2.3E−01; 1.1E  + 03; 5.0E−02; −1.4E−04; 6.8E−09; 2.9E−04; −4.5E−03; 8.5E  + 02; −5.7E  + 02; −2.6E  + 01; −1.4E  + 03; −1.8E−09; −6.5E−04; −2.5E−05; −8.4E−09; −3.9E−11; −1.8E−10; −1.2E  + 02; −9.0E−07; −1.7E−12; 1.4E  + 02; 1.1E−04; −8.7E−13; −2.0E−01; −1.1E−12; −2.6E−05; −4.1E−05; −4.6E−02; 4.3E  + 02; 3.6E  + 01; 2.6E−04; 5.5E−04; 2.4E−05; 3.0E  + 02; 3.0E−02; 5.0E−08; 4.8E  + 01; 6.5E−04; −8.3E  + 02; −1.8E−01; −3.1E−06; −5.3E−12; 2.1E−08; 3.3E−13; 1.3E−11; −1.1E  + 00; −6.0E−04; −4.6E−04; −3.0E−12; 4.5E  + 01; 1.2E−14; −2.0E−02; 1.4E−12; −7.1E−16; −9.5E−12; 5.1E−01; 2.5E−06; 3.6E−13; 4.9E−18; 6.6E−01]477*1.

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Sun, H., Zhao, S., Peng, F. et al. In-situ prediction of machining errors of thin-walled parts: an engineering knowledge based sparse Bayesian learning approach. J Intell Manuf 35, 387–411 (2024). https://doi.org/10.1007/s10845-022-02044-6

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