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.

References

References
1.
Du
,
S.
,
Liu
,
C.
, and
Xi
,
L.
,
2015
, “
A Selective Multiclass Support Vector Machine Ensemble Classifier for Engineering Surface Classification Using High Definition Metrology
,”
ASME J. Manuf. Sci. Eng.
,
137
(
1
), p.
011003
.
2.
Du
,
S.
,
Liu
,
C.
, and
Huang
,
D.
,
2014
, “
A Shearlet-Based Separation Method of 3D Engineering Surface Using High Definition Metrology
,”
Precis. Eng.
,
40
, pp.
55
73
.
3.
Du
,
S.
,
Huang
,
D.
, and
Wang
,
H.
,
2015
, “
An Adaptive Support Vector Machine-Based Workpiece Surface Classification System Using High Definition Metrology
,”
IEEE Trans. Instrum. Meas.
,
64
(
10
), pp.
2590
2604
.
4.
Wang
,
M.
,
Xi
,
L.
, and
Du
,
S.
,
2014
, “
3D Surface Form Error Evaluation Using High Definition Metrology
,”
Precis. Eng.
,
38
(
1
), pp.
230
236
.
5.
Wang
,
M.
,
Ken
,
T.
,
Du
,
S.
, and
Xi
,
L.
,
2015
, “
Tool Wear Monitoring of Wiper Inserts in Multi-Insert Face Milling Using 3D Surface Form Indicators
,”
ASME J. Manuf. Sci. Eng.
,
137
(
3
), p.
031006
.
6.
Nguyen
,
H.
,
Wang
,
H.
, and
Hu
,
S. J.
,
2013
, “
Characterization of Cutting Force Induced Surface Shape Variation Using High-Definition Metrology
,”
ASME J. Manuf. Sci. Eng.
,
135
(
4
), p.
041014
.
7.
Guo
,
P.
,
Lu
,
Y.
,
Pei
,
P.
, and
Ehmann
,
K. F.
,
2014
, “
Fast Generation of Micro-Channels on Cylindrical Surfaces by Elliptical Vibration Texturing
,”
ASME J. Manuf. Sci. Eng.
,
136
(
4
), p.
041008
.
8.
Rao
,
P.
,
Bukkapatnam
,
S.
,
Beyca.
,
O.
,
Kong
,
Z.
, and
Komanduri
,
R.
,
2014
, “
Real-Time Identification of Incipient Surface Morphology Variations in Ultraprecision Machining Process
,”
ASME J. Manuf. Sci. Eng.
,
136
(
2
), p.
021008
.
9.
Zhu
,
X.
,
Ding
,
H.
, and
Wang
,
M.
,
2004
, “
Form Error Evaluation: An Iterative Reweighted Least Squares Algorithm
,”
ASME J. Manuf. Sci. Eng.
,
126
(
3
), pp.
535
542
.
10.
Raghunandan
,
R.
, and
Rao
,
P. V.
,
2007
, “
Selection of an Optimum Sample Size for Flatness Error Estimation While Using Coordinate Measuring Machine
,”
Int. J. Mach. Tools Manuf.
,
47
(
3–4
), pp.
477
482
.
11.
Badar
,
M. A.
,
Raman
,
S.
, and
Pulat
,
P. S.
,
2005
, “
Experimental Verification of Manufacturing Error Pattern and Its Utilization in Form Tolerance Sampling
,”
Int. J. Mach. Tools Manuf.
,
45
(
1
), pp.
63
73
.
12.
Dowling
,
M.
,
Griffin
,
P.
,
Tusi
,
K.
, and
Zhou
,
C.
,
1997
, “
Statistical Issues in Geometric Feature Inspection Using Coordinate Measuring Machines
,”
Technometrics
,
39
(
1
), pp.
3
17
.
13.
Yan
,
Z.
, and
Menq
,
C.
,
1995
, “
Uncertainty Analysis for Coordinate Estimation Using Discrete Measurement Data
,”
ASME International Mechanical Engineering Congress and Exposition
, Vol.
1
,
San Francisco, CA
, Nov. 12–17, pp.
595
616
.
14.
Yeh
,
K. M.
,
Ni
,
J.
, and
Hu
,
S.
,
1994
, “
Adaptive Sampling and Identification of Feature Deformation for Tolerance Evaluation Using Coordinate Measuring Machines
,” S. M. Wu, Manufacturing Research Laboratory, Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI, Technical Report.
15.
Yang
,
B.
, and
Menq
,
C.
,
1993
, “
Compensation for Form Error of End-Milled Sculptured Surfaces Using Discrete Measurement Data
,”
Int. J. Mach. Tools Manuf.
,
33
(
5
), pp.
725
740
.
16.
Henke
,
R.
,
Summerhays
,
K.
,
Baldwin
,
J.
,
Cassou
,
R.
, and
Brown
,
C.
,
1999
, “
Methods for Evaluation of Systematic Geometric Deviations in Machined Parts and Their Relationships to Process Variables
,”
Precis. Eng.
,
23
(
4
), pp.
273
292
.
17.
Cho
,
N.
, and
Tu
,
J.
,
2001
, “
Roundness Modeling of Machined Parts for Tolerance Analysis
,”
Precis. Eng.
,
25
(
1
), pp.
35
47
.
18.
Desta
,
M. T.
,
Feng
,
H. Y.
, and
Ou Yang
,
D.
,
2003
, “
Characterization of General Systematic Form Errors for Circular Features
,”
Int. J. Mach. Tools Manuf.
,
43
(
11
), pp.
1067
1078
.
19.
Xia
,
H.
,
Ding
,
Y.
, and
Wang
,
J.
,
2008
, “
Gaussian Process Method for Form Error Assessment Using Coordinate Measurements
,”
IIE Trans.
,
40
(
10
), pp.
931
946
.
20.
Walker
,
E.
, and
Wright
,
S. P.
,
2002
, “
Comparing Curves Using Additive Models
,”
J. Qual. Technol.
,
34
(
1
), pp.
118
129
.
21.
Whittle
,
P.
,
1954
, “
On Stationary Processes in the Plane
,”
Biometrika
,
41
(
3/4
), pp.
434
449
.
22.
Sayles
,
R. S.
, and
Thomas
,
T. R.
,
1977
, “
The Spatial Representation of Surface Roughness by Means of the Structure Function: A Practical Alternative to Correlation
,”
Wear
,
42
(
2
), pp.
263
276
.
23.
Colosimo
,
B. M.
, and
Semeraro
,
Q.
,
2008
, “
Statistical Process Control for Geometric Specifications: On the Monitoring of Roundness Profiles
,”
J. Qual. Technol.
,
40
(
1
), pp.
1
18
.
24.
Murthy
,
T. E. R.
, and
Abdin
,
S. Z.
,
1980
, “
Minimum Zone Evaluation of Surfaces
,”
Int. J. Mach. Tools Des. Res.
,
20
(
2
), pp.
123
136
.
25.
Wang
,
Y.
,
1992
, “
Minimum Zone Evaluation of Form Tolerances
,”
Manuf. Rev.
,
5
(
3
), pp.
213
220
.
26.
Kannda
,
T.
, and
Suzuki
,
S.
,
1993
, “
Evaluation of Minimum Zone Flatness by Means of Non-Linear Optimization Technique and Its Verification
,”
Precis. Eng.
,
15
(
2
), pp.
93
99
.
27.
Suriano
,
S.
,
Wang
,
H.
,
Shao
,
C.
,
Hu
,
S. J.
, and
Sekhar
,
P.
,
2015
, “
Progressive Measurement and Monitoring for Multi-Resolution Data in Surface Manufacturing Considering Spatial and Cross Correlations
,”
IIE Trans.
,
47
(
10
), pp.
1033
1052
.
28.
Li
,
J.
, and
Heap
,
A. D.
,
2008
,
A Review of Spatial Interpolation Methods for Environmental Scientists
, Vol.
23
,
Geoscience Australia
, Canberra, Australia, pp.
137
145
.
29.
Chen
,
X.
,
Ankenman
,
B. E.
, and
Nelson
,
B. L.
,
2013
, “
Enhancing Stochastic Kriging Metamodels With Gradient Estimators
,”
Oper. Res.
,
61
(
2
), pp.
512
528
.
30.
Kleijnen
,
J. P. C.
, and
Mehdad
,
E.
,
2014
, “
Multivariate Versus Univariate Kriging Metamodels for Multi-Response Simulation Models
,”
Eur. J. Oper. Res.
,
236
(
2
), pp.
573
582
.
31.
Kleijnen
,
J. P. C.
,
2009
, “
Kriging Metamodeling in Simulation: A Review
,”
Eur. J. Oper. Res.
,
192
(
3
), pp.
707
716
.
32.
Yang
,
T. H.
, and
Jackman
,
J.
,
2000
, “
Form Error Estimation Using Spatial Statistics
,”
ASME J. Manuf. Sci. Eng.
,
122
(
2
), pp.
262
272
.
33.
Morimoto
,
Y.
,
Suzuki
,
N.
,
Kaneko
,
Y.
, and
Isobe
,
M.
,
2014
, “
Vibration Control of Relative Tool-Spindle Displacement for Computer Numerically Controlled Lathe With Pipe Frame Structure
,”
ASME J. Manuf. Sci. Eng.
,
136
(
4
), p.
044502
.
34.
Weckenmann
,
A.
,
Heinrichowski
,
M.
, and
Mordhorst
,
H.
,
1991
, “
Design of Gauges and Multipoint Measuring Systems Using Coordinate Measuring-Machine Data and Computer Simulation
,”
Precis. Eng.
,
13
(
3
), pp.
203
207
.
35.
Yan
,
D.
,
Popplewell
,
N.
,
Balkrishnan
,
S.
, and
Kaye
,
J. E.
,
1996
, “
On-Line Prediction of Surface Roughness in Finish Turning
,”
Eng. Des. Autom.
,
2
(
2
), pp.
115
126
.
36.
Chen
,
L.
,
2008
, “
Study on Prediction of Surface Quality in Machining Process
,”
J. Mater. Process. Technol.
,
205
(
1–3
), pp.
439
450
.
37.
Roth
,
J. T.
,
Mears
,
L.
,
Djurdjanovic
,
D.
,
Yang
,
X.
, and
Kurfess
,
T.
,
2007
, “
Quality and Inspection of Machining Operations: Review of Condition Monitoring and CMM Inspection Techniques
,”
ASME International Conference on Manufacturing Science and Engineering
, Vol.
10
, pp.
15
18
.
38.
Benardo
,
S. P. G.
, and
Vosniakos
,
G. C.
,
2003
, “
Predicting Surface Roughness in Machining a Review
,”
Int. J. Mach. Tools Manuf.
,
43
(
8
), pp.
833
844
.
39.
Dowling
,
M.
,
Griffin
,
P.
,
Tsui
,
K.
, and
Zhou
,
C.
,
1995
, “
A Comparison of the Orthogonal Least Square and Minimum Enclosing Zone Methods for Form Error Estimation
,”
Manuf. Rev.
,
8
, pp.
120
134
.
40.
Matheron
,
G.
,
1973
, “
The Intrinsic Random Functions, and Their Applications
,”
Adv. Appl. Probab.
,
5
(
3
), pp.
439
468
.
41.
Cressie
,
N.
,
1993
,
Statistics for Spatial Data, Revised Edition
,
Wiley
,
New York
.
42.
Zhang
,
R. D.
,
2005
,
Spatial Variogram Theory and Its Application
,
Science and Technology Press
,
Beijing
.
43.
Deutsch
,
C. V.
, and
Journel
,
A. G.
,
1992
,
GSLIB—Geostatistical Software Library and User's Guide
,
Oxford University Press
,
New York
.
44.
Isaaks
,
E. H.
, and
Srivastava
,
R. M.
,
1989
,
An Introduction to Applied Geostatistics.
Oxford University Press
,
New York
.
45.
Goovaerts
,
P.
,
1997
,
Geostatistics for Natural Resources Evaluation
,
Oxford University Press
,
New York
.
46.
Ahmed
,
S.
, and
Marsity
,
G. D.
,
1987
, “
Comparison of Geostatistical Methods for Estimating Transmissivity Using Data on Transmissivity and Specific Capacity
,”
Water Resour. Res.
,
23
(
9
), pp.
1717
1737
.
47.
Gelfand
,
A. E.
,
Diggle
,
P.
,
Guttorp
,
P.
, and
Fuentes
,
M.
, eds.,
2010
,
Handbook of Spatial Statistics
,
CRC Press
,
Boca Raton, FL
.
48.
Kennedy
,
M. C.
, and
O'Hagan
,
A.
,
2000
, “
Predicting the Output From a Complex Computer Code When Fast Approximations are Available
,”
Biometrika
,
87
(
1
), pp.
1
13
.
49.
Zhou
,
Q.
,
Qian
,
P. Z.
, and
Zhou
,
S.
,
2011
, “
A Simple Approach to Emulation for Computer Models With Qualitative and Quantitative Factors
,”
Technometrics
,
53
(
3
), pp.
266
273
.
50.
Qian
,
P. Z.
, and
Wu
,
C. J.
,
2008
, “
Bayesian Hierarchical Modeling for Integrating Low-Accuracy and High-Accuracy Experiments
,”
Technometrics
,
50
(
2
), pp.
192
204
.
51.
Clark
,
I.
,
Basinger
,
K. L.
, and
Harper
,
W. V.
,
1987
, “
A Novel Approach to Co-Kriging
,”
Conference on Geostatistical, Sensitivity, and Uncertainty Methods for Ground-Water Flow and Radionuclide Transport Modeling
,
San Francisco, CA
, pp.
473
493
.
52.
Rehman
,
S. U.
, and
Shapiro
,
A.
,
1996
, “
An Integral Transform Approach to Cross-Variograms Modeling
,”
Comput. Stat. Data Anal.
,
22
(
3
), pp.
213
233
.
53.
Armstrong
,
M.
, and
Diamond
,
P.
,
1984
, “
Testing Variograms for Positive-Definiteness
,”
Math. Geol.
,
16
(
4
), pp.
407
421
.
54.
Myers
,
D. E.
,
1994
, “
Spatial Interpolation: An Overview
,”
Geoderma
,
62
(
1–3
), pp.
17
28
.
55.
Myers
,
D. E.
,
1982
, “
Matrix Formulation of Co-Kriging
,”
Math. Geol.
,
14
(
3
), pp.
249
257
.
56.
Myers
,
D. E.
,
1983
, “
Estimation of Linear Combinations and Co-Kriging
,”
Math. Geol.
,
13
(
5
), pp.
633
637
.
57.
Yang
,
T. H.
, and
Jackman
,
J.
,
1997
, “
A Probabilistic View of Problems in Form Error Estimation
,”
ASME J. Manuf. Sci. Eng.
,
119
(
3
), pp.
375
382
.
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