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Research Papers

Co-Kriging Method for Form Error Estimation Incorporating Condition Variable Measurements

[+] Author and Article Information
Shichang Du

State Key Lab of Mechanical System
and Vibration,
Shanghai Jiaotong University,
Shanghai 200240, China;
Department of Industrial Engineering
and Management,
School of Mechanical Engineering,
Shanghai Jiaotong University,
Shanghai 200240, China

Lan Fei

Department of Industrial Engineering
and Management,
School of Mechanical Engineering,
Shanghai Jiaotong University,
Shanghai 200240, China

1Corresponding author.

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received January 16, 2015; final manuscript received August 24, 2015; published online October 27, 2015. Assoc. Editor: Xiaoping Qian.

J. Manuf. Sci. Eng 138(4), 041003 (Oct 27, 2015) (16 pages) Paper No: MANU-15-1043; doi: 10.1115/1.4031550 History: Received January 16, 2015; Revised August 24, 2015

The form error estimation under various machining conditions is an essential step in the assessment of product surface quality generated in machining processes. Coordinate measuring machines (CMMs) are widely used to measure complicated surface form error. However, considering measurement cost, only a few measurement points are collected offline by a CMM for a part surface. Therefore, spatial statistics is adopted to interpolate more points for more accurate form error estimation. It is of great significance to decrease the deviation between the interpolated height value and the real one. Compared to univariate spatial statistics, only concerning spatial correlation of height value, this paper presents a method based on multivariate spatial statistics, co-Kriging (CK), to estimate surface form error not only concerning spatial correlation but also concerning the influence of machining conditions. This method can reconstruct a more accurate part surface and make the estimation deviation smaller. It characterizes the spatial correlation of machining errors by variogram and cross-variogram, and it is implemented on one of the common features: flatness error. Simulated datasets as well as actual CMM data are applied to demonstrate the improvement achieved by the proposed multivariate spatial statistics method over the univariate method and other interpolation methods.

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Figures

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Fig. 1

Methods of form error estimation: (a) by directly measured points and (b) by interpolated and measured points

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Fig. 2

Influence of machining conditions

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Fig. 3

Comparison of Kriging and CK variables: (a) Kriging and (b) CK

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Fig. 4

Distribution of measured and estimated points

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Fig. 10

Hardware setup for milling and the schematic for vibration data collection

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Fig. 9

Color-coded image of the engine block face with triangular mesh plot

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Fig. 8

Engine cylinder blocks processed by a major domestic car manufacturer

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Fig. 7

Box–whisker plot of flatness error estimation comparison for different cases. (a) Case 1: CK, (b) case 1: OK, (c) case 2: CK, (d) case 2: OK, (e) case 3: CK, and (f) case 3: OK.

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Fig. 6

Theoretical variogram of height and vibration and cross-variogram. (a) Case 1: height value, (b) case 1: vibration, (c) case 1: cross-variogram of height value and vibration, (d) case 2: height value, (e) case 2: vibration, (f) case 2: cross-variogram of height value and vibration, (g) case 3: height value, (h) case 3: vibration, and (i) case 3: cross-variogram of height value and vibration.

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Fig. 13

MSE comparisons between CK and OK estimation

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Fig. 5

Color-coded images of simulated surfaces: (a) Case1, (b) Case 2, and (c) Case 3

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Fig. 15

Box–whisker plot of flatness error estimation comparison by different sample size: (a) CK and (b) OK

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Fig. 16

The workpiece for case study II

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Fig. 17

Curved surface of the workpieces: (a) curved surface with upper and lower plane surfaces and (b) curved surface

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Fig. 11

Theoretical variogram of height in spherical model and exponential model

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Fig. 12

Variogram of vibration and cross-variogram of height and vibration

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Fig. 14

Box–whisker plot of flatness error estimation by different methods

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Fig. 18

Box–whisker plot of MSE comparison by different sample size: (a) OK and (b) CK

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Fig. 19

Diagram of form error estimation for curved surface

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