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: j-han07@mails.tsinghua.edu.cn

Int J Adv Manuf Technol (2012) 62:205–212DOI 10.1007/s00170-011-3796-2

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

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

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

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

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

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

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