A new thermal error modeling method for CNC machine tools
Transcript of A new thermal error modeling method for CNC machine tools
![Page 1: A new thermal error modeling method for CNC machine tools](https://reader031.fdocuments.in/reader031/viewer/2022020408/575094481a28abbf6bb7a6bb/html5/thumbnails/1.jpg)
ORIGINAL ARTICLE
A new thermal error modeling method for CNC machine tools
Jian Han & Liping Wang & Haitong Wang &
Ningbo Cheng
Received: 21 August 2011 /Accepted: 17 November 2011 /Published online: 3 December 2011# Springer-Verlag London Limited 2011
Abstract A great challenge in a thermal error compensationprocess is to select proper temperature variables and to establishaccurate thermal error models. In this paper, a new approach forbuilding an effective mathematic thermal error for machinetools is presented. Fuzzy clustering analysis is conducted toidentify temperature variables, and then screen the representa-tive variable as an independent variable meanwhile eliminatethe coupling among the variables. Cluster validity is forwardedto measure the reasonability of the clustering and the classifi-cation accuracy for the temperature variables. Furthermore, amathematical model using the robust regression analysis is builtto reveal the relationship between these temperature variablesand thermal deformation. To evaluate the performance of ourproposedmodel, a verification experiment is carried out. Pt-100thermal resistances and Eddy current sensors are used to mon-itor the temperature and thermal shift fluctuation respectively.Fuzzy clustering analysis is utilized to classify 32 temperaturevariables to four clusters. A robust regression thermal errormodel is proposed based on the four key temperature points.The result shows that four representative temperature variablesare precise predictors of the thermal errors of the machine tools.The proposed method is shown to be capable of improving theaccuracy of the machine tools effectively.
Keywords Machine tool . Thermal error . Fuzzy clusteranalysis . Robust regression
1 Introduction
Precision machining has become a critical manufacturingtechnology for industry to improve the machining qualityand meet the fierce international competition [1]. Thermally
induced errors are demonstrated as one of the greatest con-tributors to the accuracy of the machine tool [2–4]. There aregenerally two ways to reduce the thermal errors: erroravoidance and error compensation [5]. In error avoidance,the separation of heat sources, the rearrangement of machinetool structures and materials that have low thermal expan-sion coefficient are used to minimize the thermal error. Eventhough this can also ensure basic machine accuracy, it canresult in a dramatic increase in costs [6]. Error compensationmethod attempts to predict the thermal errors and thencompensate for them with software. The relationship be-tween machine errors and the displacement is modeled,and then the compensation signals are sending to the CNCcontroller, so that the tool and the part have relative motionsin the reverse direction of the predicted machine thermalerrors. Thermal error compensation has been demonstratedas one of the most effective methods to reduce machine toolerrors, enhance machine tool accuracy, and achieve high-quality, cost-effective manufacturing [7–9]. However, thereare two major barriers limit its widespread application: thepoor accuracy of thermal error modeling and the interrela-tion between the temperature variables. The highly interre-lated variables must be screened out; otherwise, theinterrelated property may hide important temperature varia-bles and affect the model accuracy directly. Therefore, a newmodeling approach is needed to overcome this difficulty.
Chen proposed multiple regression analysis and the arti-ficial neural network model for the real-time forecast ofthermal errors with numerous temperature measurements.Both approaches can search automatically for the nonlinearterms among different temperature variables [10]. Lee pre-sented a thermal modeling based on independent componentanalysis to compensate for the thermal errors of an MCH-10machining center [11]. Chih used a modified model adequa-cy criterion based on Mallow’s Cp statistic to optimize theobject function. The optimal model is found with a 0.982 R2
value using four temperature variables selected from all the46 candidates of temperature variables [12].Yang optimized
J. Han (*) : L. Wang :H. Wang :N. ChengInstitute of Manufacturing Engineering, Department of PrecisionInstruments and Mechanology, Tsinghua University,Beijing 100084, People’s Republic of Chinae-mail: [email protected]
Int J Adv Manuf Technol (2012) 62:205–212DOI 10.1007/s00170-011-3796-2
![Page 2: A new thermal error modeling method for CNC machine tools](https://reader031.fdocuments.in/reader031/viewer/2022020408/575094481a28abbf6bb7a6bb/html5/thumbnails/2.jpg)
the thermal sensor’s placement on machine tools based onthe gray correlation model, and decreases the number oftemperature variables [13, 14]. Kun used the k-means theoryto classify the obtained thermal data, which is especiallyuseful in a production environment in which engineers arefacing the urgent requirement for a fast, easily programmedmodeling method [15].
The abovementioned approaches can show satisfactorypredictive accuracy in many occasions, while most are em-pirical and highly dependent on the data collected fromexperiments. Furthermore, proper selection of thermal sen-sors and their locations are vital to the prediction accuracyof models. The most popular approach in choosing propertemperature sensors is to preset a large number of sensorlocations then to select some significant ones in the finalthermal models [16]. A poor selection of the temperaturevariables will degrade the prediction accuracy. The essenceof identifying such temperature variables is to conduct clus-ter analysis among all the temperature variables involved,and then select one variable from each cluster to representthe temperature variable of the same category.
Throughout the machining, the machine tools are alwaysunder the effect of various kinds of the thermal sources [17].Various conditions have made the temperature field of ma-chine tool complicated to express. These error sources alongwith the thermal damping and the thermal inertia of thestructure cause the thermal shift of machine, thus resultingin the decreased accuracy of the machining [18].
The robustness of the thermal error modeling relies onthe distribution of the temperature filed of the machine [13,19]. Screening out the most crucial measuring points is mostimportant. Some temperature variables may correspond ex-actly to other temperature variables in some cases; whilehaving uncertain relationships with some others on otheroccasions. Engineering judgment, correlation analysis, andstepwise regression have been used to select the temperaturevariables for thermal error component models [20]. Therelevance criterion of classification and the standard toselect typical temperature variables also lack a standardquantitative index.
The purpose of this paper is to develop a method competentin determining the best combination of temperature variables,which give sufficient model accuracy of thermal error elements.Since the conception of fuzzy sets was introduced, fuzzy clus-tering has been widely discussed, studied, and applied [21, 22].Among which the c-means clustering is most widely used.In this result, all the temperature variables are first clusteredinto groups using the Fuzzy c-Means method. The clustereddata are then used to identify approximate regions and tocalculate the center of clusters for decision attributes basedon the cluster validity function. Finally, the robust regres-sion is used to establish the relationships between the tem-perature variables and thermal errors. The results show that
the optimal temperature variables can predict the thermalerror of machine tool precision.
2 Basic methods for fuzzy c-means clustering
Data clustering is the process of dividing data elementsinto classes or clusters so that objects in the samecollection are well specified and possess some commonobjects [23].
Fuzzy clustering analysis is a kind of systemic analysisapplying mathematics theory, according to comparabilityamong systemic characteristic parameter series [24]. In theexperimental data processing, fuzzy clustering theory hasthe advantage over traditional statistic theory, namely that itsresults sufficiently embody the inherent properties of asystem with less experimental data and unknown systemicprobability. The aim of this test is to adopt fuzzy clusteringanalysis when experimental data of temperature measuringpoints are analyzed, to seek the relationships among allfactors in the system, and then to find out the key influenc-ing factors of contributing to machine tool thermal errors.
2.1 Fuzzy c-means clustering
One of the most widely used fuzzy clustering algorithms isthe Fuzzy C-Means Algorithm. Clustering of data is done byevaluating nearness of the data. Two basic concepts used forclustering are similarity and cluster center [25].
The extent of the similarity of the sample can beexpressed by the Minikovski:
di;j x; yð Þ ¼Xpk¼1
xik � xjk�� ��q" #1=q
; q > 0 ð1Þ
When q ¼ 1; di;j x; yð Þ is absolute distance:
di;j x; yð Þ ¼Xpk¼1
xik � xjk�� �� ð2Þ
When q ¼ 2; di;j x; yð Þis named Euclidean distance:
di;j x; yð Þ ¼Xpk¼1
xik � xjk� �2" #1=2
ð3Þ
Where x ¼ x1; � � � ; xp� �
and y ¼ y1; � � � ; yp� �
are p-dimensional vectors of Rp.
Most frequently used distance measure is the Euclid-ean distance. A dissimilarity that we use is the squaredEuclidean distance.
D x; yð Þ ¼ d22 x; yð Þ ¼ x� yk k22¼XPj¼1
xj � yj� �2 ð4Þ
206 Int J Adv Manuf Technol (2012) 62:205–212
![Page 3: A new thermal error modeling method for CNC machine tools](https://reader031.fdocuments.in/reader031/viewer/2022020408/575094481a28abbf6bb7a6bb/html5/thumbnails/3.jpg)
A cluster center is used in many algorithms for clustering.It is an element in the space Rp calculated as a function ofelements in X.
An important step in most clustering is to select a dis-tance measure, which will determine how the similarity oftwo elements is calculated. This will influence the shape ofthe clusters, as some elements may be close to one anotheraccording to one distance and farther away according toanother.
Coefficient of similarity is used to weigh the level ofsimilarity. Correlation coefficient is the most popularwhich can be used to estimate the correlation betweenthe variables.
rij ¼Pnk¼1
xik�xið Þ xjk�xjð ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnk¼1
xik�xið Þ2r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn
k¼1
xjk�xjð Þ2r
i; j ¼ 0; � � � ; n
ð5Þ
Where ui ¼ 1n
Pnk¼1
uik ; ui ¼ 1n
Pnk¼1
ujk
2.2 The procedure of the fuzzy c-means algorithm
The fuzzy c-means algorithm attempts to partition a finitecollection of n elements X ¼ x1; � � � ; xnf ginto a collec-tion of c fuzzy clusters with respect to some given criterion,U is the partition matrix. Given a finite set of data, thealgorithm returns a list of c cluster centers C ¼c1; � � � ; cnf g and a partition matrix. In each cluster, a
cluster center vi (i01, 2 . . . c) is determined.Consider the following function having variables X and V.
Jm U ;Vð Þ ¼Xc
i¼1
ðXxk2Xi
dikð Þ2Þ ð6Þ
The value of partition matrix U is rand from the 0∼1,with the constraint condition of normalization:Xc
i¼1
uik ¼ 1 ð7Þ
The function Jm (U, V) is to be minimized for data clustering.In order to introduce the nonlinearity for U, the nonlinear
term (uik)m is introduced into the objective function:
Jfcm U ;Vð Þ ¼Xnk¼1
Xc
i¼1
uikð ÞmXxk2Xi
dikð Þ2 ð8Þ
Where m is a weight index, m 2 1;1ð Þ, if m is too large,the cluster result will worse, which cannot reflect the true-ness condition; while if m is too small, it will not embodythe merit of statistics. Generally, the value is from 1.5 to 2.5.
Construct a new objective function:
J fcm U ;V ; lð Þ
¼ Jfcm U ;Vð Þ þXnj¼1
ljðXc
i¼1
uij � 1Þ
¼Xnk¼1
Xc
i¼1
uikð ÞmXxk2Xi
dikð Þ2 þXnj¼1
ljðXc
i¼1
uij � 1Þ
ð9Þ
lj ¼ 1; 2; � � � ; n, j ¼ 1; 2; � � � ; n is the Lagrange multipli-cation factor.
The minimum is obtained when
@J fcm U ;V ; lð Þ@l
¼ 0 ð10Þ
@J fcm U ;V ; lð Þ@uij
¼ 0 ð11Þ
We can get
vi ¼
Pnj¼1
umij xjPnj¼1
umij
ð12Þ
uij ¼ 1Pck¼1
dijdkj
h i 2m�1ð Þ
ð13Þ
From the result, we can acquire that the fuzzy c-meansalgorithm is an iteration process, the basic algorithm of c-means algorithm is:
Step 1: Generate c initial values for cluster centers vi (i01, 2 . . . c).
Step 2: Find optimal U: Calculate
U ¼ minU
Jfcm U ;V� � ð14Þ
Step 3: Find optimal V: Calculate
V ¼ minV
Jfcm U ;V� � ð15Þ
Step 4: If the clusters are convergent, then stop; otherwisego to step 2.
For a small positive number ε, judge that the solution U isconvergent if
maxk;i
uki � bukij j < " ð16Þ
Where uki is the novel solution and buki is the optimal solutionone step before the last [26].
Int J Adv Manuf Technol (2012) 62:205–212 207
![Page 4: A new thermal error modeling method for CNC machine tools](https://reader031.fdocuments.in/reader031/viewer/2022020408/575094481a28abbf6bb7a6bb/html5/thumbnails/4.jpg)
3 Experiments
3.1 Thermal error measurement
The experiment equipment is composed of a CNC machiningcenter and sensing units, Pt-100 thermal resistances are usedto detect the temperature change in the machine tool with theadvantages of high accuracy and fine calibration. Eddy currentsensors are used to measure the thermal shift of the machinetool. Thirty-two temperature sensors are attached to the spin-dle housing, the column, and the machine body; the locationsof the 32 sensors are listed in Table 1. Figure 1 is themeasurement of the temperature and thermal error.
The temperature sensors at the critical points on themachine tool and the thermal shifts of the spindle inthree directions are measured simultaneously, the signalsof the measured temperatures are sent to a central pro-cessing unit board. The thermal drift of the machinetool is shown in Fig. 2 when the maximum of X-axis is3 μm, the Y-axis is 7 μm, and the Z-axis is 23 μm.
3.2 Application analysis of fuzzy c-means method
In this study, the X-axis thermal shift is much smaller thanthat of Y- and Z-axis, which can be neglected, only the Y- andZ-axis are considered.
First, the correlation coefficient between temperature anddisplacement variables of Z-axis and Y-axis is calculatedusing the function 5, which can be seen as Table 2.
Suppose the temperature variables are clustered to fourclusters. Initial values for cluster centers vi is selected as(0.397, 0.24), (0.724, 0.789), (0.893, 0.714), (0.183, 0.232)through iterative operation with the fuzzy c-means algo-rithm, the cluster result can be seen as Table 3:
Then, one variable from each cluster is chosen accordingto its correlation with the thermal error to represent thetemperature variable of the same category, and they areT1, T17, T25, and T32.
1. Temperature sensor T1 measuring the temperature ofthe leading screw;
2. Temperature sensor T17 measuring the temperature ofthe spindle bearing;
3. Temperature sensor T25 measuring the temperature ofthe headstock;
4. Temperature sensor T32 measuring the temperature ofthe room.
3.3 Cluster analysis validity
In order to check the validity of the cluster we have used, theconception of probability partition coefficient P (U, c) isintroduced, the fuzzy partition coefficient F (U, c) and P (U,c) are combined, a new cluster validity function is defined.
1. Partition coefficient F (U, c)With regard to given cluster center c and membership
Fig. 1 Measurement of the temperature and thermal error
-30
-25
-20
-15
-10
-5
0
5
10
0 1 2 3 4 5
X axis
Y axis
Z axis
The
rmal
shi
ft δ
/ μ m
Fig. 2 Thermal shift of the spindle
Table 1 The location of the temperature sensors
Sensor s number Location of the temperature sensors
1–8 Leading screw
9–12 Column
13–16 Electric motor
17–22 Spindle
23–28 Headstock
29–30 Machine bed
31 Coolant
32 Room
208 Int J Adv Manuf Technol (2012) 62:205–212
![Page 5: A new thermal error modeling method for CNC machine tools](https://reader031.fdocuments.in/reader031/viewer/2022020408/575094481a28abbf6bb7a6bb/html5/thumbnails/5.jpg)
degree U, partition coefficient is defined:
F U ; cð Þ ¼ 1
n
Xc
i¼1
Xnj¼1
μ2ij ð17Þ
If there exist (U*, c*) which satisfied:
F U �; c�ð Þ ¼ maxc
maxΩc
F U ; cð Þ� �
ð18Þ
Then (U*, c*) is the best effective cluster, c* is the bestclassification number.
2. Probability partition coefficient P (U, c)
For every sample xj;Pci¼1
μij ¼ 1;F U ; cð Þ can also be
written as:
F U ; cð Þ ¼ 1
n
Xnj¼1
ðXc
i¼1
μ2ij=
Xc
i¼1
μijÞ ð19Þ
With regard to given cluster center c and mem-bership degree U, probability partition coefficient isdefined:
P U ; cð Þ ¼ 1
c
Xc
j¼1
ðXni¼1
μ2ij=
Xni¼1
μijÞ ð20Þ
Fuzzy cluster validity function is defined:
FP U ; cð Þ ¼ F U ; cð Þ � P U ; cð Þ ð21Þ
If there exist (U*, c*) which satisfied:
FP U�; c�ð Þ ¼ minc
minFP U ; cð ÞΩc
( )ð22Þ
Then (U*, c*) is the most effective cluster.The cluster validity function is calculated. Table 4
shows that when m01.5, FP(U, c) acquire minimum atc04, the second is c03, and the third is c05; when m0
2, FP(U, c) acquire minimum at c03, the second is c04,and the third is c05; when m02.5, FP(U, c) acquireminimum at c04, the second is c03, and the third is c05. From the result, we can seen that the selection of c04is proper, the second probability is c03 or c05.T
able
2Correlatio
ncoefficientbetweentemperature
anddisplacementvariables
Variable
12
34
56
78
910
1112
1314
1516
Z-axis
0.39
70.50
70.32
60.50
70.36
50.48
40.31
70.47
50.22
40.18
90.12
30.15
90.69
80.72
40.67
60.70
4
Y-axis
0.32
40.35
30.28
90.47
60.31
60.28
60.37
40.41
70.20
40.23
20.18
70.23
40.80
30.78
90.77
50.78
6
Variable
1718
1920
2122
2324
2526
2728
2930
3132
Z-axis
0.91
70.89
30.90
20.85
30.87
20.84
00.71
40.73
30.75
20.72
30.70
60.71
70.12
60.18
70.13
80.20
3
Y-axis
0.70
30.71
40.68
90.73
60.65
70.59
80.80
30.82
40.82
70.77
80.76
80.80
60.20
50.17
50.27
60.22
4
Table 3 The cluster result
Class Location of the temperature
1 T1,T2,T3,T4,T5,T6,T7,T8
2 T13,T14,T15,T16,T23,T24,T25,T26, T27,T28
3 T17,T18,T19,T20,T21,T22
4 T9,T10,T11,T12,T29,T30,T31,T32
Int J Adv Manuf Technol (2012) 62:205–212 209
![Page 6: A new thermal error modeling method for CNC machine tools](https://reader031.fdocuments.in/reader031/viewer/2022020408/575094481a28abbf6bb7a6bb/html5/thumbnails/6.jpg)
4 Thermal error modeling and discussion
The machining center can be compensated with the thermalerrors that correlate with the thermal errors to the tempera-ture variables. Various methods have been applied to estab-lish the thermal errors of machining tool. It is difficult to usean accurate model to describe the behavior of the thermaldisplacement of the machining tool. A robust regression isapplied in this study.
The general form of multiple regression analysis modelsis as follows:
d ¼ b0 þ b1t1 þ b2t2 þ � � � þ bmtm þ " ð23Þ
Where:
δ Is the thermal error, which is called the responsevariable
T Is the changes of the temperature variables, which iscalled random variables
β Is called effects, or regression coefficients, which is ap-dimensional parameter vector.
ε Is called the residual, or noise. This variable captures allother factors which influence the dependent variable yi.
The variables can be written in the form of the matrix:
d ¼d1d2� � �dn
26643775; T ¼
1 t t12 � � � t1m1 t21 t22 � � � t2m... ..
. ... . .
. ...
1 tn1 tn2 � � � tnm
2666437775
b ¼b1b2...
bm
2666437775; " ¼
"1"2...
"m
2666437775
Often these m equations are stacked together and writtenin vector form as
d ¼ bT þ " ð24Þ
The goal is to find the unknown parameter bb, the regres-sion model using the least-squares method to minimize thesum of squared residuals J
J ¼ ðd � bbTÞT ðd � bbTÞ ð25Þ
The minimum is obtained when
@J
@bb ¼ 0 ð26Þ
When the Hessian of J is positive semi-definite
@2J
@bb2�����
����� � 0 ð27Þ
Differentiating J will lead to a closed-form expression forthe estimated value of the unknown parameter β. It isconceptually straightforward.
bb ¼ TTT� ��1
TTd ð28ÞAfter the thermal error is established, we should validate
the estimators and detect the outliers. The multiple-linear
regression is to minimize the functionPni¼1
di � tTi b� �2
which
weights all the residuals equally [25]. When there are someextreme points with large residuals in observations, so theresiduals of the potentials outliers will be too small to bedetected.
In the presence of outliers, least squares estimation isinefficient and can be biased. Because the least squarespredictions are dragged towards the outliers, and becausethe variance of the estimates is artificially inflated, the resultis that outliers can be masked.
To overcome the limit of the linear regression model, arobust regression method is introduced. Robust estimation isused when the data contain outliers. Weight is introduced,Pni¼1
di � tTi b� �2
is replaced byPni¼1
wi di � tTi b� �2
, some pop-
ular robust criterion functions are addressed, the robustprocedure dampens the effect of the observations with largeresiduals. Weight functions wi associated with the robustcriterion functions are usually smaller than one when resid-uals get larger.
Huber’s function is used in this report to establish thethermal errors.
wi ¼1:0 rj j � ttrj j rj j > t
�ð29Þ
To minimizePni¼1
wi di � tTi b� �2
, let the first partial deriv-
atives of the function with respect to xj be zero, yielding thesystem of (n+1) equations.
Table 4 The FP(U, c) of the temperature variables
c m01.5 m02 m03
2 0.0183 0.0286 0.0312
3 0.0114 0.0092 0.0118
4 0.0062 0.0103 0.0083
5 0.0144 0.0192 0.0134
6 0.0153 0.0256 0.0277
210 Int J Adv Manuf Technol (2012) 62:205–212
![Page 7: A new thermal error modeling method for CNC machine tools](https://reader031.fdocuments.in/reader031/viewer/2022020408/575094481a28abbf6bb7a6bb/html5/thumbnails/7.jpg)
Iteratively reweighed least squares are used to solvecertain optimization problems. It solves objective functionsof the form:
minXni¼1
wi di � tTi b� �2 ð30Þ
By an iterative method in which each step involvessolving a weighted least squares problem of the form:
b tþ1ð Þ ¼ minPni¼1
wi di � tTi b� �2
¼ TTW ðtÞT� ��1
TTW ðtÞdð31Þ
Where W (t) is the diagonal matrix of weights withelements:
wðtÞi ¼ yi � xTi b
ðtÞ�� ��p�2 ð32ÞIn the case p01, this corresponds to the least absolute
deviation regression. Robust regression method gives resid-uals with different weights. It tends to give large residualswith small weights and leave the residuals associated withextreme points large.
The coefficient of the determination is defined as
R2 ¼ TSS� SSR
TSSð33Þ
Where TSS is the total sum of squares for the dependentvariable and SSR is the residual sum of the squares.R2 willalways be a number between 0 and 1. The coefficient givesthe portion of the variability in Y explained by regression onthe X, with values close to 1 indicating a good degree of fit.
The thermal error model determined by T1, T17, T25,and T32 of the Z-axis by robust regression is:
dZ ¼ 2:56þ 2:18T1 � 4:92T17 þ 4:12T25 � 1:16T32 ð34ÞThe thermal error model determined by T1, T17, T25,
and T32 of the Y-axis by robust regression is:
dY ¼ 0:36þ 1:08T1 þ 3:92T17 � 2:72T25 þ 1:46T32 ð35Þ
Figures 3 and 4 show the measure curve and predictedcurve of the thermal error model, the experiment resultsillustrate that the maximum residual of Z-axis is about1.8 μm and Y-axis is about 0.7 μm. The coefficient of thedetermination is 0.997, which means the ability of predic-tion is 99.7%, four representative temperature variables arepowerful precisely predictors of the thermal errors of themachine tool.
5 Conclusions
This paper proposes a novel thermal error modelingmethod including the fuzzy c-means clustering analysisand robust regression in order to establish the relationshipbetween the temperature ascents and the thermal shifts ofthe machine tool. The following conclusions are drawn bythis study:
1. The fuzzy c-means cluster analysis is able to find theoptimal temperature variables for thermal error compo-nent modeling. The cluster validity can obtain the opti-mal accuracy of classification.
2. The robust regression, which has the advantage ofchecking the outlier, is used to establish the thermalerror model, and assigns larger residuals to smallweights. The maximal residual can be reduced to1.8 μm from 25 μm in Z-axis and to 0.7 μm from7 μm in Y-axis.
3. A verification experiment on the machine tools showsthe representative temperature variables can compensatefor the real time error. This can be conveniently appliedto other CNC machine tools.
Acknowledgments This work is supported by the Chinese KeyNational Science and Technology Specific Projects (grant no.2009ZX04001-112) and National Nature Science Foundation of China(grant no. 50775125).
-30
-25
-20
-15
-10
-5
0
5
0 1 2 3 4 5
error in Z direction
predictor in Z direction
residual
The
rmal
shi
ft δ
/μ m
Measure time t/h
Fig. 3 The measuring curve and predicted curve of Z-direction
-4
-2
0
2
4
6
8
10
error in Y direction
predictor in Y direction
residual
The
rmal
shi
ft δ
/μ m
Measure time t/h
0 1 2 3 4 5
Fig. 4 The measuring curve and predicted curve of Y-direction
Int J Adv Manuf Technol (2012) 62:205–212 211
![Page 8: A new thermal error modeling method for CNC machine tools](https://reader031.fdocuments.in/reader031/viewer/2022020408/575094481a28abbf6bb7a6bb/html5/thumbnails/8.jpg)
References
1. Mehrabi MG, O’Neal G, Min BK, Pasek Z, Koren Y, Szuba P(2002) Improving machining accuracy in precision line boring. JIntel Man 13(5):379–389
2. Bryan J (1990) International status of thermal error research. CIRPAnn-Manuf Tech 39(2):645–656
3. Weck M, McKeown P, Bonse R, Herbst U (1995) Reduction andcompensation of thermal errors in machine tools. CIRP Ann-Manuf Techn 44(2):589–598
4. Wang SM, Liu YL, Kang Y (2002) An efficient error compensa-tion system for CNC multi-axis machines. Int J Mach Tool Man 42(11):1235–1245
5. Ni J (1997) CNC machine accuracy enhancement through real-time error compensation. J Manuf Sci E-TAsme 119(4B):717–725
6. Yuan JX, Ni J (1998) The real-time error compensation techniquefor CNC machining systems. Mechatron 8(4):359–380
7. Tseng PC, Ho JL (2002) A study of high-precision CNC lathethermal errors and compensation. Int J Adv Manuf Technol 19(11):850–858
8. Lei WT, Hsu YY (2003) Accuracy enhancement of five-axis CNCmachines through real-time error compensation. Int J Mach ToolMan 43(9):871–877
9. Wu CH, Kung YT (2006) Thermal analysis and compensation of adouble-column machining centre. P I Mech Eng B-J Eng 220(2):109–117
10. Chen JS (1996) Neural network-based modelling and error com-pensation of thermally-induced spindle errors. Int J Adv ManufTechnol 12(4):303–308
11. Lee DS, Choi JY, Choi DH (2003) ICA based thermal sourceextraction and thermal distortion compensation method for a ma-chine tool. Int J Mach Tool Man 43(6):589–597
12. Lo CH, Yuan JX, Ni J (1999) Optimal temperature variable selec-tion by grouping approach for thermal error modeling and com-pensation. Int J MachinTool Man 39(9):1383–1396
13. Li YX, Yang JG, Gelvis T, Li YY (2008) Optimization of measur-ing points for machine tool thermal error based on grey systemtheory. Int J Adv Manuf Technol 35(7–8):745–750
14. Yan JY, Yang JG (2009) Application of synthetic grey correlationtheory on thermal point optimization for machine tool thermal errorcompensation. Int J Adv Manuf Technol 43(11–12):1124–1132
15. Wang KC, Tseng PC (2010) Thermal error modeling of a machinetool using data mining scheme. J Adv Mech Des Syst 4(2):516–530
16. Guo QJ, Yang JG, Wu H (2010) Application of ACO-BPN tothermal error modeling of NC machine tool. Int J Adv ManufTechnol 50(5–8):667–675
17. Ramesh R, Mannan MA, Poo AN (2000) Error compensation inmachine tools—a review Part I: geometric, cutting-force inducedand fixture-dependent errors. Int J Adv Manuf Technol 40(9):1235–1256
18. Wang YD, Zhang GX, Moon KS, Sutherland JW (1998) Compen-sation for the thermal error of a multi-axis machining center. JMater Process Tech 75(1–3):45–53
19. Guo QJ, Yang JG (2011) Application of projection pursuit regres-sion to thermal error modeling of a CNC machine tool. Int J AdvManuf Technol 55(5–8):623–629
20. Li JW, Zhang WJ, Yang GS, Tu SD, Chen XB (2009) Thermal-error modeling for complex physical systems: the-state-of-artsreview. Int J Adv Manuf Technol 42(1–2):168–179
21. Yang ZY, Sun ML, Li WQ, Liang WY (2011) Modified Elmannetwork for thermal deformation compensation modeling in ma-chine tools. Int J Adv Manuf Technol 54(5–8):669–676
22. Li X (2001) Real-time prediction of workpiece errors for a CNCturning centre, Part 2. Modelling and estimation of thermallyinduced errors. Int J Adv Manuf Technol 17(9):654–658
23. Schwammle V, Jensen ON (2010) A simple and fast method todetermine the parameters for fuzzy c-means cluster analysis. Bio-informatics 26(22):2841–2848
24. Huang KY (2010) Applications of an enhanced cluster validityindex method based on the Fuzzy C-means and rough set theoriesto partition and classification. Eepert Syst Appl 37(12):8757–8769
25. Wen W, Hao ZF, Yang XW (2010) Robust least squares supportvector machine based on recursive outlier elimination. Soft Comput14(11):1241–1251
26. Sadaaki M, Hidetomo I, Katsuhiro H (2008) Algorithms for fuzzyclustering: methods in c-means clustering with applications.Springer, Berlin Heidelberg
212 Int J Adv Manuf Technol (2012) 62:205–212