research article thermal-induced errors prediction and ... · measurement and described a method to...

Research ArticleThermal-Induced Errors Prediction and Compensation for aCoordinate Boring Machine Based on Time Series Analysis

Jun Yang Dongsheng Zhang Bin Feng Xuesong Mei and Zhenbang Hu

State Key Laboratory for Manufacturing Systems Engineering School of Mechanical Engineering Xirsquoan Jiaotong UniversityNo28 Xianning West Road Xirsquoan 710049 China

Correspondence should be addressed to Dongsheng Zhang zdsmailxjtueducn

Received 28 April 2014 Revised 22 July 2014 Accepted 5 August 2014 Published 27 August 2014

Academic Editor Qingsong Xu

Copyright copy 2014 Jun Yang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To improve the CNC machine tools precision a thermal error modeling for the motorized spindle was proposed based on timeseries analysis considering the length of cutting tools and thermal declined angles and the real-time error compensation wasimplemented A five-point method was applied to measure radial thermal declinations and axial expansion of the spindle witheddy current sensors solving the problem that the three-point measurement cannot obtain the radial thermal angle errorsThen the stationarity of the thermal error sequences was determined by the Augmented Dickey-Fuller Test Algorithm andthe autocorrelationpartial autocorrelation function was applied to identify the model pattern By combining both Yule-Walkerequations and information criteria the order and parameters of the models were solved effectively which improved the predictionaccuracy and generalization ability The results indicated that the prediction accuracy of the time series model could reach up to90 In addition the axial maximum error decreased from 396 120583m to 7 120583m after error compensation and the machining accuracywas improved by 897 Moreover the 119883119884-direction accuracy can reach up to 774 and 86 respectively which demonstratedthat the proposed methods of measurement modeling and compensation were effective

1 Introduction

The precision CNC coordinate boring machine is a tool forprocessing complex box-type components Thermal errorwill account for a larger proportion of total error as themachine tools become more sophisticated However theaccuracy decreases and becomes far lower than the initialdesign value after the machine is used for a long period oftime This decreased accuracy over time primarily resultsfrom inadequate maintenance and accuracy stability and thethermal error is the main factor for the inadequate accuracyaccounting for 70 of the total number of errors arising fromvarious error sources [1] Donmez et al proposed that chang-ing temperatures produce thermal errors and the thermalerror is a major factor for reducing themachine precision [2]And the motorized spindle has more complicated dynamicnonstationary and speed-dependent thermal characteristicsthan conventional spindles [3] A nonuniform temperaturedistribution causes thermal errors inCNCmachine tools and

this distribution becomes nonlinear and nonstationary andvaries with time Most problems in thermal error researchfield focus on how to exactly measure the thermal character-istics establish thermal-induced error model with accurateprediction and excellent generalization and efficiently com-pensate the thermal deformation to improve the machiningaccuracy

Firstly the method of accurately measuring the temper-ature distribution and thermal error is the most importantissue Vissiere et al measured the spindle geometric errorwith a new method in which the measurement accuracy canreach even the nanometer [4] Vyroubal presented a methodfocused on compensation of machinersquos thermal deformationin spindle axis direction based on decomposition analysiswhich is a cheap and effective strategy [5] Hong and Ibarakistudied thermal characteristics of a rotary axis on the five-axis machine and analyzed effect of thermal error on errorsmotion of the rotary axis [6]Wang et al collected the spindlethermal deformation in three directions with three-point

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 784218 13 pageshttpdxdoiorg1011552014784218

2 Mathematical Problems in Engineering

measurement and described a method to group the dataof thermal sensors [7] Huang et al measured the spindlethermal drifts with five eddy current displacement sensorsand proposed a method combining the back propagation(BP) and genetic algorithm (GA) to model the thermalexpansion in axial direction [8] But the radial declinedangle errors were ignored and the thermal angle errors arekey factors to compensate the spindle terminal processingaccuracy

Secondly the thermal error model must be built accu-rately based on the experiment measurement Yang et al[9] used artificial neural networks (ANNS) to establish arelationship between temperature and the thermal error ofa spindle EL Ouafi et al constructed an artificial neuralnetwork model for spindle thermal errors with the temper-ature drawing on statistical methodology which effectivelyimproves the machining accuracy [10] The support vectormachine is a new machine learning theory which has manyadvantages such as simple algorithm global optimizationversatile and strong generalization ability The scholarsLin et al and Zhao et al established a spindle thermal errormodel based on least square support vector machine theoryand the model has perfect robustness [11 12]

The mentioned researchers have established the axialthermal elongation model but they overlooked the thermalyaw and pitch errors of the spindle

In recent years the finite element method (FEM) isapplied to analyze temperature fields and the thermaldeformation of machine tools Creighton et al used thefinite element method to analyze temperature distributioncharacteristics for a high-speed micromilling spindle andconstructed an exponential model of the axial thermal errorconsidering the spindle speed and running time [13] Haitaoet al proposed amethod for calculating thermal conductivitycoefficient of the spindle surface and simulated and analyzedvariation principles of the temperature field and thermaldeformation of the spindle [14]

However the precision CNC machine tool error is amutual coupling of many complex factors that are affectedby many variables and therefore it is extremely difficult toestablish a theoretical equation based on the perspective ofthermoelasticity and heat transfer

If the measurement and modeling of the thermal errorsare completed the next work could be the compensation Fuet al and Miao et al built the spindle axial thermal errormodel by applying multivariate linear regression method[15 16] Wang and Yang also proposed a prediction modelfor the axial thermal deformation and applied the modelto compensate error on a CNC machine [17] Liu et alcompensated the thermal drift on the milling and boringmachine in the 119885-direction [18] Ouafi et al presented anintegrated comprehensive modeling approach for thermalerrors real-time compensation based on multiple tempera-ture measurements after compensation the spindle thermal-induced errors were reduced from 19 120583m to less than 1 120583m[19] There are other scholars who investigated the spindleaxial thermal error compensation method [20 21] and themachining accuracy was improved effectively Gebhardt etal described a high precision grey-box thermal error model

for compensation on five-axis machines and the thermallyinduced errors of the rotaryswivelling can be reduced upto 85 [22] Pajor and Zapłata presented a set allowing forsupervising the feed screw thermal elongation to reduce ballscrew thermal errors [23] Zhang et al developed a novelcompensation implementation technique for machine toolsbased on the function of themachine external zero point shiftand Ethernet data communication protocol which improvedthe machine precision [24]

The existing literatures were mainly about measurementand modeling of the spindle axial thermal elongation butthe radial thermal angle errors were ignored The spindlethermal deformation of a CNC machine tool is usuallyexpressed as the actual spatial position and gesture may devi-ate from the theoretical value namely the running spindlewould induce the drifts of geometric dimensions and spatialphase thus affecting the precision of machine processingOf course the deviated drifts must include axial elongationand radial thermal pitch and yaw angle errors which reducethe machining accuracy For example the paperrsquos object isa coordinate boring machine and its spindle axial thermalexpansion may affect the bore geometric size Meanwhilethe radial thermal inclination angle errors could influencethe geometric dimensions and surface roughness of the holeso it is extremely necessary to measure axial and radialthermal errors simultaneously In order to realize the errorcompensation the spindle radial thermal angle errors mustbe translated into components of the linear coordinate axisMoreover the thermal error offset components are closelyrelated to both radial thermal inclination angle errors andthe handle length So the radial thermal angle errors cannotbe ignored The three-point method could only measure theabsolute thermal deformation in the single radial directionand it cannot obtain the thermal inclination angle errorTherefore the thermal error compensationmodel deduced bythe method was not accurate enough This indicated that thethree-point method cannot completely reflect the variationof the spindle radial thermal deformation and the radialthermal error compensation model is not accurate based onthis measurement

Based on the above analysis the five-point method isapplied and the measurement contains the following advan-tage (1) five-point method could simultaneously measureaxial and radial thermal drifts of the motorized spindlesystem so the variation of the spindle position and ori-entation could be analyzed solving the challenge that thethree-point method cannot measure the inclined angles ofthe spindle radial thermal errors (2) the integrated thermalerror modeling will provide a more accurate mathematicalequation for the error compensation which includes thespindle elongation radial thermal pitch and yaw angle errorsbased on the five-point method

The time series analysis provides a set of approachesto process dynamic data The method primary meaningis that all types of data are approximately described bymathematical models Through the model analysis the datainternal structure can be mined So we can forecast its trendsand make necessary control on it Wang et al applied timeseries analysis method to establish a spindle thermal error

Mathematical Problems in Engineering 3

ΔDX

D

O Z

P

ΔOx

ΔOzΔOD D0x

O998400

120579x

(a)

X

D

O Z

P

D0x

O998400

DL1

120579x

ΔL3

ΔL1

(b)

Figure 1 The geometric principle of the spindle thermal error compensation

model and compensated errors [25ndash27] and they acquiredbetter results

This paper focuses on the spindle system of a box-typeprecisionCNCcoordinate boringmachineThe thermal errormodeling and compensation equations were proposed Thenthermal balance experiments were performed by using thefive-point method The time series models were establishedfor spindle axial thermal elongation and radial thermal yawand pitch errors based on the experiments After that wetranslated the thermal drifts into coordinate offsets andestablished the final compensation mathematical equationsin three directions Finally the error compensation wascarried out The results show that the machine precision canbe improved efficiently

2 Thermal Error Compensation Principle

The motorized spindle system of CNC machine tools couldproduce plenty of heat during the processing the accumu-lated heat leads to the spindle thermal deformation anddeviation to the relative position between cutting tools andthe workpiece finally it will reduce the accuracy of theterminal processing For the coordinate boring machine inthis paper the thermal deformation of the spindle may affectthe geometric size and surface roughness of the workpieceGenerally thermal drifts occur in three directions In thispaper the axial thermal elongation is defined as 119864 And inthe radial directions the thermal yaw angle in 119883119874119885 plane isdefined as 120579

119909 and the thermal pitch angle in 119884119874119885 plane is

defined as 120579119910

Figure 1 describes the spatial pose of the spindle thermaldrift on119883119874119885 and the point119875 is the deflexion center After thespindle experienced axial elongation 119864 and radial inclination

120579119909 the spindle declined from 997888997888

119875119874 to9978889978889978881198751198741015840 so the offset

component in119883-direction is as follows

Δ119874119909= (1198630119909+ 119863 + Δ119863) sin 120579

119909 (1)

where the offset in 119883-direction is Δ119874119909 1198630119909

is the distancebetween the deflexion center and the spindle nose 119863 is thelength of the cutting tool and Δ119863 is the axial elongation 119864

The compensation offset in 119885-direction is Δ119874119911

Δ119874119911= Δ119863 minus Δ119874

119863= Δ119863 minus (119863

0119909+ 119863 + Δ119863) (1 minus cos 120579

119909)

(2)

Because the axial elongation is less than the length of the toolthat is

Δ119863 ≪ 1198630119909+ 119863 120579

119909997888rarr 0 (3)

so

sin 120579119909997888rarr 120579119909 cos 120579

119909997888rarr 1 (4)

Equations (3)-(4) are substituted into (1)-(2) then thermalerror compensation component in 119883- and 119885-directions canbe obtained respectively

Δ119874119909= (1198630119909+ 119863) 120579

119909 (5)

Δ119874119911= Δ119863 (6)

This indicates that the offset in 119885-direction has no relation-ship with the tool length while 119883-directional compensationoffset is closely related to that

Similarly the thermal error offset Δ119874119910in the 119884-direction

can be obtained

Δ119874119910= (1198630119910+ 119863) 120579

119910 (7)

where 1198630119910

is the distance between the deflexion center andthe spindle nose

3 Time Series Analysis of Thermal Errors

In order to solve the final thermal error compensationcomponents in the axes the comprehensive model of theaxial elongation 119864 and the radial thermal angle errors 120579

119909and

120579119910must be derived Time series analysis could be used to


Table 1 The fitting performance parameters of time series

Output |119890119894|min(120583m10158401015840)

|119890119894|max

(120583m10158401015840)|119890119894|

(120583m10158401015840) RMSE 1198772 120578

()120579119909 000 344 0441 0730 0986 956120579119910 000 162 0242 0416 0996 976119864 000 150 0335 0473 0999 987

accomplish this taskThe basic idea of the time series analysisis that a mathematical model which accurately reflects thesystem dynamic dependency is established through theanalysis of the time sequence samples based on a limitedsample of the observation system and it is applied to predictand monitor the future behavior of the system

31 Thermal Errors Stationarity Judgment and Gauss Stan-dardization Given enactment that the sequence 119883

119905 119905 =

0 1 2 is a discrete stochastic process the AugmentedDickey-Fuller Test (ADF) determines the stationarity ofthe time series Assuming that three time series of thespindle system thermal errors are 119864

119905 120579119909119905 120579119910119905 if the results

exhibit that the thermal error sequences are nonstationarythe sequences are normalized by Gaussian standardizationso that the sequences have a smoothness Of course afterGaussian standardization the stationarity of the new series isstill determined by ADF if the new series are still nonstation-ary the difference method or other function transformationapproaches should be utilized to deal with the time seriesuntil they are stationary

120583 =1

119899

119899

sum

119894=1

119909119894

1205902

=1

119899 minus 1

119899

sum

119894=1

(119909119894minus 120583)2

(8)

In the formula 120583119894and 120590

2

119894are mean and variance of

axial thermal elongation and radial thermal declination angleerrors

The original sequences of the spindle thermal errors aretaking a standardizing processing

119864119905sim 119873(120583

1 1205902

1) 997904rArr 119864

1015840

119905=119864119905minus 1205831

1205901

997904rArr 1198641015840

119905sim 119873 (0 1)

120579119909119905sim 119873(120583

2 1205902

2) 997904rArr 120579

1015840

119909119905=120579119909119905minus 1205832

1205902

997904rArr 1205791015840

119909119905sim 119873 (0 1)

120579119910119905sim 119873(120583

3 1205902

3) 997904rArr 120579

1015840

119910119905=120579119910119905minus 1205833

1205903

997904rArr 1205791015840

119910119905sim 119873 (0 1)

(9)

where 1198641015840119905 1205791015840119909119905 1205791015840119910119905

are new time series of spindle thermalerrors and the new sequences approximate standard normaldistribution

32 Box-Jenkins Model Identification Using the autocorre-lation function (ACF) and partial autocorrelation function

(PACF) to identify the thermal error series pattern if thecalculated results of ACF and PACF have tailing it indicatesthat new time series of the standardized spindle thermalerrors are the autoregressive and moving average hybridmodels ARMA(119901 119902) the model is as follows [28]

119883119905= 1206011119883119905minus1

+ sdot sdot sdot + 120601119901119883119905minus119901

+ 120576119905minus 1205791120576119905minus1

minus sdot sdot sdot minus 120579119902120576119905minus119902

(10)where 120601

119894(119894 = 1 2 119901) is the autoregressive parameter and

120579119895(119895 = 1 2 119902) is the moving average parameter Define

the later operator as 119861119861119883119905= 119883119905minus1 119861

119901

119883119905= 119883119905minus119901 (11)

Thus120579 (119861) = 1 minus 120579

1119861 minus 12057921198612

minus sdot sdot sdot 120579119902119861119902

120601 (119861) = 1 minus 1206011119861 minus 12060121198612


(12)

The ARMAmodel is transformed into120601 (119861)119883

119905= 120579 (119861) 120576

119905 (13)

33 The Parameter Estimation George et al [28] suggestedthat if theARMA(119901 119902)model contains119901 order autoregressiveAR(119901) and 119902 ordermoving averageMA(119902) its autocorrelationfunction is a pattern mixed exponential and attenuationsine wave after 119901-119902 order delay Correspondingly the partialcorrelation function is not exact exponential form but it iscontrolled by amixture of index and decaying sine wave Andthe covariance is120574119905+119896119905

= Cov (119883119905+119896 119883119905) = 119864 [(119883

119905+119896minus 120583119905+119896) (119883119905minus 120583119905)] = 120574

119896

119896 = (0 plusmn1 plusmn2 119899 minus 1)

(14)Autocorrelation function is

120588119896=120574119896

1205740

(15)

According to the statistical theory the covariance func-tion of time series with stationarity and zero mean is esti-mated as follows

120574119896=1

119873

119873

sum

119905=119896+1

119883119905119883119905minus119896 (16)

Thus the autocorrelation function is estimated as follows

120588119896=120574119896

1205740

=sum119873

119905=119896+1119883119905119883119905minus119896

sum119873

119905=11198832119905

(17)

Yule-Walker equations can obtain autoregression coeffi-cients set 119896 = 1 2 119901 so the linear equations are gottennamely

1205881= 1206011+ 12060121205881+ sdot sdot sdot + 120601

119901120588119901minus1

1205882= 12060111205881+ 1206012+ sdot sdot sdot + 120601

119901120588119901minus2

120588119901= 1206011120588119901minus1

+ 1206012120588119901minus2

+ sdot sdot sdot + 120601119901

(18)


Replacing theoretical autocorrelation 120588119896with the estimated

autocorrelation 120588119896 the autoregression coefficients of Yule-

Walker estimation can be obtained Define the vectors asfollows

120601 = (

1206011

1206012

120601119901

) 120588119901= (

1205881

1205882

120588119901

)

119875119901= (

1 1205881

1205882

sdot sdot sdot 120588119901minus1

1205881

1 1205881


120588119901minus1

120588119901minus2

120588119901minus3

sdot sdot sdot 1

)

(19)

The parameter 120601 in (18) can be written as

120601 = 119875minus1

119901120588119901 (20)

Mark 1199081015840119905= 120601(119861)119883

119905 and deal with the ARMA model as a

moving average process as follows

1199081015840

119905= 120579 (119861) 120576

119905 (21)

Covariance 1205741015840119895of 1199081015840119905can be presented by covariance 120574

119895of 119883119905

and 119895 = 0 1 119902

1205741015840

119895=

119901

sum

119894=0

1206012

119894120574119895

+

119901

sum

119894=1

(1206010120601119894+ 1206011120601119894+1

+ sdot sdot sdot + 120601119901minus119894120601119901) (120574119895+119894

+ 120574119895minus119894)

(22)

Convention 1206010= minus1 the covariance function of MA(119902)

process is

1205741015840

0= (1 + 120579

2

1+ sdot sdot sdot + 120579

2

119902) 1205902

120576

1205741015840

119896= (minus120579

119896+ 1205791120579119896+1

+ sdot sdot sdot + 120579119902minus119896120579119902) 1205902

120576

(23)

Estimations of the parameters 1205902120576 120579119902 120579119902minus1

1205791can be

calculated under 119902 order moving average using iteration asfollows

1205902

120576=

1205741015840

0

1 + 12057921+ sdot sdot sdot + 1205792

119902

120579119895= minus(

1205741015840

119895

1205902120576

minus 1205791120579119895+1

minus 1205792120579119895+2

minus sdot sdot sdot minus 120579119902minus119895120579119902)

(24)

And promise 1205790= 0 120579

119902 120579119902minus1

1205791can be made zero also at

the beginning of iterations

4 Thermal Characterization Experiment

In order to establish the comprehensive thermal error modelof the axial elongation and radial thermal angle errors

Figure 2 Experimental setup

S1

S2

S3

S4

S5

XY

Z

Figure 3 Spindle five-point installation diagram

of the spindle system with the application of time seriesanalysis the spindle system of a precision CNC coordinateboring machine was chosen as the research object and thethermal equilibrium experiments were carried on Then thetemperature field and thermal deformation of the spindlesystem were analyzed

41 Experimental Setup The experimental system is shownin Figure 2 which focuses on the spindle of the precisionCNC coordinate boring machine The measuring equipmentand functions are as follows a synchronous acquisitionsystem is used to determine the temperatures and thermaldriftsThis system uses Pt100 precisionmagnetic temperaturesensors to measure the spindle system temperatures High-precision eddy-current sensors are applied to measure thespindle thermal drifts Temperature sensors locations areas follows front bearing (T6 T7) rear bearing (T1) themotor (T8 T11) ambient temperature (T5) spindle base (T2)the cooling fluid inlet (T9) bearing cooling out (T3) frontbearing coolant out (T4) and the motor cooling out (T10)

42 Measurement Principle The spindle thermal drifts aremeasured by using the five-pointmethod [29] the diagram ofdisplacement sensors measurement is shown in Figure 3Thespindle is parallel to 119911-axis and the axial thermal expansioncan be obtained by the displacement sensor 119878

5 The radial

thermal yaw 120579119909partial 119883-direction is measured by the 119878

1


Thermal state

Cold state

D

L03

Li3

Li1L01

120579

ΔL

S1 S3

ΔL3

ΔL1

Figure 4 The spindle thermal inclination sketch

and 1198783 and the radial thermal pitch 120579

119910partial 119884-direction is

measured by the 1198782and 1198784

After the spindle running for a long period the thermalelongation expanded to axial direction and thermal angleinclined to radial direction resulting from the uneven tem-perature gradient distribution which is shown in Figure 4and the thermal yaw angle 120579

119909is

Δ1198713= 119871119894

3minus 1198710

3

Δ1198711= 119871119894

1minus 1198710

1

Δ119871 = Δ1198713minus Δ1198711

tan 120579119909=Δ119871

119863

(25)

where 119894 denotes the number of measurements The thermalyaw angle is too small in this experiment that is 120579

119909rarr 0 so

120579119909sim tan 120579

119909 (26)

As shown in (27) the thermal yaw angle can be obtained byapplying (25)

120579119909=(119871119894

3minus 119871119894

1) minus (119871

0

3minus 1198710

1)

119863 (27)

where 1198710

3and 119871

0

1are the radial displacements between

the sensor probes and the spindle measured by 1198783and

1198781 respectively in the initial state and 119871

119894

3and 119871

119894

1are the

transient displacements during the running operation 119863 isthe distance between 119878

1and 1198783 1198782and 1198784 and119863 = 120mm

Similarly the thermal pitch angle in the 119884-direction canbe obtained

120579119910=(119871119894

4minus 119871119894

2) minus (119871

0

4minus 1198710

2)

119863 (28)

43 Results and Analysis The spindle speed is a majorfactor affecting the thermal characteristics The temperaturegradient and the heat generated by the CNCmachine spindle

0 50 100 150 200 250 300 350 4000

100020003000400050006000700080009000

1000011000

N(r

pm)

t (min)

Figure 5 Step speeds distribution

are different at different speeds thus generating differentthermal drifts So the spindle speed affects the temperaturefield distribution and the magnitude of thermal errorsMeanwhile the thermal error model is closely related tothe intrinsic behavior of thermal drift data Therefore aproper test condition shall be proposed so that the thermalerror model put forward in this paper can be applied to thepractical engineering field and be used to improvemachiningaccuracy In order to create such kind of testing condition thedistribution of spindle speeds in this paper is set to simulatesome common processing conditions of the boring machineThe specific speeds distribution is shown in Figure 5

Thermal characteristics of CNC machine tools cover twomain aspects of the temperature distribution and thermalerror and the nonuniform distribution of the temperaturefield gradient is the direct cause of spindle thermal driftsSo for a more comprehensive study of the spindle thermalproperties a simple time-domain analysis of the temperaturefield is also conducted in the paper The spindle systemtemperature variations are shown in Figure 6 The overalltrends of temperatures on all measuring points increase withtime The rear bearing has the highest temperature reaching304∘C due to large capacity heavy load and severe frictionwhich generates more heat and the following is the motorwhose temperature is 273∘C

Figure 7 presents the spindle thermal drifts The timeuntil equilibrium reached is approximately 385min witha maximum elongation 396 120583m The thermal error on 119909-axis direction is positive and its thermal yaw angle is 120579

119909in

the 119883119874119885 plane the maximum amount of hot offset erroris 35 120583m Thermal error in 119884-direction is negative whichindicates that during operation the spindle is closer to thedisplacement sensors 119878

21198784 and its thermal pitch angle is 120579

119910

in the 119884119874119885 plane the maximum thermal offset is of 202 120583m

5 Thermal ErrorsPrediction and Compensation

After analysis of the experiment the models of the spindleelongation and thermal angle errors could be established bytime series analysis and three thermal errors in the spindle


31

30

29

28

27

26

25

24

23

22

210 50 100 150 200 250 300 350 400 450

Time (min)

Tem

pera

ture

(∘C)

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11

Figure 6 Temperatures of the spindle

0 100 150 200 250 300 350 400 450

0102030405060

Radial near x-axisRadial distal x-axisRadial near y-axis

Z axial directionRadial distal y-axis

Ther

mal

erro

r (120583

m)

minus10

minus20

minus30

minus40

Time (min)50

Figure 7 Thermal drifts of the spindle

can be predicted by the proposed models Subsequently thefinal comprehensive thermal error compensation model isinduced based on the geometric principle and the time seriesmodels

Firstly the 5 thermal error sequences in Figure 7 areconverted into 3 series in Figures 8(a)ndash10(a) based on (27)-(28) The spindle axial elongation time series 119864

119905in the 119885-

direction was directly from the measured data and the radialthermal yaw and pitch angle series 120579

119909119905 120579119910119905

were obtainedby applying (27) and (28) respectively Then the AugmentedDickey-Fuller (ADF) Test Algorithm was applied to identifythe stationarity of the thermal error sequences 119864

119905 120579119909119905 120579119910119905

and the calculation showed that thermal elongation andangles were nonstationary series Subsequently the mean120583119894and variance 120590

2

119894of the axial elongation 119864

119905and radial

thermal declination angle errors 120579119909119905 120579119910119905

were calculated by(8) So the original sequences 119864

119905 120579119909119905 120579119910119905

were standardizedby (9) and the new time series 1198641015840

119905 1205791015840119909119905 1205791015840119910119905

were shown inFigures 8(b) 9(b) and 10(b)

When the original sequences 119864119905 120579119909119905 120579119910119905

of the spindlethermal drifts were translated into new time series 1198641015840

119905 1205791015840119909119905 1205791015840119910119905

through Gaussian standardization the Augmented Dickey-Fuller (ADF) Test Algorithm was carried out to identifythe 1198641015840119905 1205791015840119909119905 1205791015840119910119905

stationarity The calculated results indicatedthat the characteristic roots of new series were within theunit circle demonstrating that the standardized series werestationary and they did not need to be smoothed further

The thermal elongation series is calculated by autocorre-lation function (ACF) and partial autocorrelation function(PACF) ACF and PACF all have tailing and this indicatesthat the new standardized sequence of the spindle thermalelongation is ARMA(119901 119902) model as shown in Figures 11 12and 13 Repeat the above process it exhibits that the radialthermal yaw and pitch angles are also ARMAmodel

51 Parameters Identification and Model Training Set orderrange119901 119902 isin [0 5] then calculate ARMA(119901 119902)model and theoptimal order 119901 and 119902 are determined by Akaika InformationCriterion (AIC) [30]The basic idea is to construct a criterionfunction AIC(119901 119902) the function considers both original datafitting and the number of unknown parameters in the modelWhen the parameters and fitting residual variance 2

120576make

the AIC value reach the minimum orders 119901 and 119902 aredecided

AIC (119901 119902) = ln 2120576+2 (119901 + 119902 + 1)

119899 (29)

where 2120576is the variance estimation of fitting residual errors

119899 is the number of samples and 119901 119902 = 1 2 119871 are highestorders of the model

After calculation the new time series 1198641015840119905of the thermal

elongation is ARMA (5 3)model the thermal yaw angle error1205791015840

119909119905is ARMA (2 5) and the thermal pitch angle error 1205791015840

119910119905is

ARMA (2 1) the corresponding models are

1198641015840

119905= 1583119864

1015840

119905minus1minus 0471119864

1015840

119905minus2+ 0644119864

1015840

119905minus3

minus 12461198641015840

119905minus4+ 0488119864

1015840

119905minus5+ 120576(1)

119905+ 0263120576

(1)

119905minus1

minus 0267120576(1)

119905minus2minus 0959120576

(1)

119905minus3

1205791015840

119909119905= 018120579

1015840

119909119905minus1+ 0779120579

1015840

119909119905minus2+ 120576(2)

119905

+ 1139120576(2)

119905minus1+ 0163120576

(2)

119905minus2minus 0139120576

(2)

119905minus3

+ 0231120576(2)

119905minus4+ 0376120576

(2)

119905minus5

1205791015840

119910119905= 1996120579

1015840

119910119905minus1minus 0997120579

1015840

119910119905minus2+ 120576(3)

119905minus 120576(3)

119905minus1

(30)


50

minus10minus15minus20minus25minus30minus35minus40minus45

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)

The unstandardized data

minus5

(a)

The standardized data

minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)

(b)

Figure 8 Thermal elongation (a) original sequence (b) Gauss standardization

Ther

mal

yaw

(998400998400 )

0

minus10

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

yaw

(998400998400 )

(b)

Figure 9 Radial thermal yaw angle (a) original sequence (b) Gauss standardization

Assume that the vector 120583 includes themeans of three timeseries of the spindle thermal errors and the vector 120590 includesthe standard deviations

120583 = (

1205831

1205832

1205833

) = (

minus24681

minus9845

minus10095

) 120590 = (

1205901

1205902

1205903

) = (

1411

6272

6571

)

(31)

New time series are reversed according to the followingtransformation the final spindle system thermal error modelis

(

119864119905

120579119909119905

120579119910119905

) = (

1205901

1205902

1205903

)(

1198641015840

119905

1205791015840

119909119905

1205791015840

119910119905

)+(

1205831

1205832

1205833

) (32)

52 Thermal-Induced Error Prediction The sample datanumber is 89 And then the time series models are used topredict the spindle thermal drifts The fitted curves and theactual measurements are compared in Figures 14 15 and 16

Now the evaluation criteria of a model fitting are estab-lished Assuming that the absolute value of the residual errorsis |119890119894| set its minimum as |119890

119894|min maximum as |119890

119894|max and

mean value as |119890119894| Root mean square error is RMSE the

determination coefficient is 1198772 and the predictive ability is120578 Consider

RMSE = radic 1

119899

119899

sum

119894=1

(119910119894minus 119910119894)2

1198772

= 1 minussum119899

119894=1(119910119894minus 119910119894)2

sum119899

119894=1(119910119894minus 119910119894)2

120578 = 1 minus(1119899)sum

119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

(1119899)sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

= 1 minussum119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

(33)

where 119910119894is the measurement value 119910

119894is the predicted value

119910119894is the average value of the measurement 119894 = 1 119899

and 119899 is the number of data points The fitting performanceparameters of the time series are shown in Table 1

The absolute mean values of the residual errors are smalland the RMSE is similarly closed to zero the coefficientof determination 119877

2 is close to 1 In addition the modelpredictive ability is more than 90 in the three differentdirections which indicates that the time series model has ahigher prediction accuracy


Ther

mal

pitc

h (998400998400

)

0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)

minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

pitc

h (998400998400

)


(b)

Figure 10 Radial thermal pitch angle (a) original sequence (b) Gauss standardization

0 5 10 15 20Lag

Sample autocorrelation function

minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag

Sample partial autocorrelation function

Sam

ple p

artia

l aut

ocor

relat

ions

minus1

minus05

0

05

1

(b)

Figure 11 The ACF and PACF map of thermal elongation sequence

0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus05

0

05

1

(b)

Figure 12 The ACF and PACF map of thermal yaw angle sequence

0 5 10 15 20

0

05

1

Lag

Sam

ple a

utoc

orre

latio

n Sample autocorrelation function

minus05

(a)

0 5 10 15 20LagSa

mpl

e par

tial a

utoc

orre

latio

ns


0

05

1

minus05

(b)

Figure 13 The ACF and PACF map of thermal pitch angle sequence


MeasurementTime series

0

minus10

minus20

minus30

minus40

Elon

gatio

n (120583

m)

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Resid

ual (120583

m)

2

1

0

minus1

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)

Figure 14 Axial thermal elongation (a) the prediction and measurement (b) residual error

0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


Ther

mal

yaw

(998400998400 )

(a)

4

3

2

1

0

minus1

minus2

minus3

Resid

ual (

998400998400)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)

Figure 15 Radial thermal yaw angle (a) the prediction and measurement (b) residual error

53 Thermal Error Compensation Implementation Figure 17is a schematic diagram of the spindle thermal error com-pensation and the CNC system is the Siemens 840D Thetemperature module acquires signal from PT100 and sendsit to the CNC system by RS-232 A thermal error compen-sation module is embedded into CNC based on secondarydevelopment of 840D and it can receive error compensationparameters and passes them to PLC Finally the thermalerror offsets are calculated and sent to the CNC to achievecompensation by PLC While the thermal yaw and pitcherrors are translated into the components of coordinateaxis three components are compensated by the principledescribed by this compensation system

Assuming that the distances between the deflexion centerand the spindle nose are 119863

0119909and 119863

0119910in 119883119884-direction

respectively as is shown in Figure 1 there is

1198630119909=

Δ1198711

tan 120579119909

minus 1198631198711= 548659mm

1198630119910=

Δ1198712

tan 120579119910

minus 1198631198712= 508706mm

(34)

The distances between displacement sensors 1198781 1198782and the

spindle nose are 1198631198711 1198631198712 respectively and 119863

1198711= 1198631198712

=

243158mmThe thermal components of the coordinate can be

obtained by applying (32) and (5)ndash(7) as follows

Δ119874119909= (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

Δ119874119910= (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

Δ119874119911= 12059011198641015840

119905+ 1205831

(35)

Set the coordinate of the original point119882 on the workpieceas (119875119909 119875119910 119875119911) then the new coordinate for thermal error

compensation is1198821015840(1198751015840119909 1198751015840

119910 1198751015840

119911)

1198751015840

119909= 119875119909+ Δ119874119909= 119875119909+ (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

1198751015840

119910= 119875119910+ Δ119874119910= 119875119910+ (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

1198751015840

119911= 119875119911+ Δ119874119911= 119875119911+ 12059011198641015840

119905+ 1205831

(36)

After the thermal compensation errors were reducedsignificantly which are shown in Figures 18ndash20 The spin-dle thermal drift measuring results before and after error


Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)

Figure 16 Radial thermal pitch angle (a) the prediction and measurement (b) residual error

Sensor

Servo controllerEncoder

HMI interface

Compensation value

Display Axis-selection

Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller

Filter Amplification

AD

times

Machine tool

RS232

Part program

Temperaturesdisplacement

Theoretical value

FeedbackPC

OperationInitial parameters

Compensation model

USB

Acquisition thermal drifts

Figure 17 Thermal error compensation control

Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500

PostcompensationZ measurement

30

15

0

minus15

minus30

minus45

Time (min)

Figure 18 Axial thermal elongation compensation

compensation in the three directions are shown in Figures18 and 20 the red curves are the original measurements

before the compensation of the spindle system while the bluecurves are the measurements after the spindle compensationThe existing literatures mainly compensated the axial elonga-tion which was similar to the results in Figure 18 and did notpropose a model or compensate for the radial thermal errorsIf there was no thermal error compensation in radial direc-tions namely unconsidering the radial thermal tilt angleerrors after the completion of the axial error compensationthe radial thermal errors are still the red curves in Figures 19-20 In this paper we not only compensate the axial elongationbut also establish thermal error compensation equationsin three directions of the spindle taking into account theradial thermal tilt angle errors and the length of cuttingtools the equations are (36) The newly measured thermaldrifts are the blue curves shown in Figures 19-20 after theerror compensation in radial119883119884-directions Comparing thetwo color curves it is obvious that the errors of the bluecurves which have considered the thermal tilt angles are


S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)

Figure 19 Radial thermal error compensation in119883-direction

S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)

Figure 20 Radial thermal error compensation in 119884-direction

much smaller than the errors of the red curves which havenot considered the thermal tilt angles The maximum errordecreased from 396 120583m to 7 120583m in axial direction and theaverage error reduced from 246120583m to 28 120583m namely theaverage offset is about 22 120583m Axial accuracy is improvedby 897 which demonstrates the method of the proposedmeasurement and modeling effectively The absolute averagethermal error value 119878

3in the radial119883-direction reduced from

146 120583m to 33 120583m and the accuracy is improved by 774Meanwhile the thermal error absolute maximum value 119878

4in

the radial 119884-direction declined 121120583m into 37 120583m and theaccuracy is advanced by 86

6 Conclusions

The spindle thermal error modeling containing axial elon-gation and radial thermal angle errors is more suitablefor actual conditions because it could exactly describe thespace-pose of the thermal deformation and consequentlybe utilized to compensate spindle thermal drifts improvingthe machining accuracy But radial thermal-induced angleerrors were ignored in current literatures To solve thisproblem the five-point method was applied to measure thespindle thermal drifts and the thermal errormodel includingaxial elongation and radial yaw and pitch angle errors wasproposed based on the time series analysis The time series

thermal error model can fully exploit the inherent dynamiccharacteristics of the spindle system thermal deformationso the prediction of the presented modeling could reachup to 90 with excellent generalization and robustnessIn addition considering the length of the cutting toolsand the radial thermal angle errors the final mathematicalcompensation equation of the spindle thermal drifts waspresented and the real-time compensationwas implementedThe result indicated that the axial machining accuracy wasimproved by 897 the 119883119884-direction accuracy can reachup to 774 and 86 respectively which demonstrated thatthe proposed methodology of measurement modeling andcompensation was effective

Conflict of Interests

The authors declare that they have no financial and personalrelationships with other people or organizations that caninappropriately influence their work there is no professionalor other personal interest of any nature or kind in anyproduct or company that could be construed as influencingthe position presented in or the review of the paper

Acknowledgment

This research is supported by the National High-TechRampD Program of China (863 Program) under Grant no2012AA040701

References

[1] J B Bryan ldquoInternational status of thermal error researchrdquoCIRP AnnalsmdashManufacturing Technology vol 39 no 2 pp645ndash656 1990

[2] M A Donmez M H Hahn and J A Soons ldquoA novel coolingsystem to reduce thermally-induced errors of machine toolsrdquoCIRPAnnalsmdashManufacturing Technology vol 56 no 1 pp 521ndash524 2007

[3] J-S Chen and W-Y Hsu ldquoCharacterizations and modelsfor the thermal growth of a motorized high speed spindlerdquoInternational Journal ofMachine Tools andManufacture vol 43no 11 pp 1163ndash1170 2003

[4] A Vissiere H Nouira M Damak O Gibaru and J-M DavidldquoA newly conceived cylinder measuring machine and methodsthat eliminate the spindle errorsrdquo Measurement Science andTechnology vol 23 no 9 Article ID 094015 11 pages 2012

[5] J Vyroubal ldquoCompensation of machine tool thermal deforma-tion in spindle axis direction based on decomposition methodrdquoPrecision Engineering vol 36 no 1 pp 121ndash127 2012

[6] C Hong and S Ibaraki ldquoObservation of thermal influence onerrormotions of rotary axes on a five-axismachine tool by staticR-testrdquo International Journal of Automation Technology vol 6no 2 pp 196ndash204 2012

[7] H Wang L Wang T Li and J Han ldquoThermal sensor selectionfor the thermal error modeling of machine tool based on thefuzzy clustering methodrdquo International Journal of AdvancedManufacturing Technology vol 69 no 1ndash4 pp 121ndash126 2013

[8] Y Huang J Zhang X Li and L Tian ldquoThermal error modelingby integrating GA and BP algorithms for the high-speed


spindlerdquo International Journal of AdvancedManufacturing Tech-nology vol 71 pp 1669ndash1675 2014

[9] S Yang J Yuan and J Ni ldquoThe improvement of thermalerror modeling and compensation on machine tools by CMACneural networkrdquo International Journal of Machine Tools andManufacture vol 36 no 4 pp 527ndash537 1996

[10] A El Ouafi M Guillot and N Barka ldquoAn integrated modelingapproach for ANN-based real-time thermal error compensa-tion on a CNC turning centerrdquo Advanced Materials ResearchEnvironmental andMaterials Engineering vol 664 pp 907ndash9152013

[11] W Q Lin Y Z Xu J Z Fu and Z Chen ldquoThermal errormodeling and compensation of spindles based on LS-SVMrdquoin Proceeding of the International Technology and InnovationConference (ITIC 06) pp 841ndash846 chn January 2006

[12] C Zhao Y Wang and X Guan ldquoThe thermal error predictionof NCmachine tool based on LS-SVM and grey theoryrdquoAppliedMechanics and Materials vol 16ndash19 pp 410ndash414 2009

[13] E Creighton A Honegger A Tulsian and D MukhopadhyayldquoAnalysis of thermal errors in a high-speed micro-milling spin-dlerdquo International Journal of Machine Tools and Manufacturevol 50 no 4 pp 386ndash393 2010

[14] Z Haitao Y Jianguo and S Jinhua ldquoSimulation of thermalbehavior of a CNCmachine tool spindlerdquo International Journalof Machine Tools and Manufacture vol 47 no 6 pp 1003ndash10102007

[15] Y Q Fu W Guo Gao J Yu Yang Q Zhang and D Wei ZhangldquoThermal error measurement modeling and compensation formotorized spindle and the research on compensation effectvalidationrdquo in Advanced Materials Research vol 889-890 pp1003ndash1008 2014

[16] E Miao Y Gong P Niu C Ji and H Chen ldquoRobustness ofthermal error compensationmodelingmodels of CNCmachinetoolsrdquo International Journal of Advanced Manufacturing Tech-nology vol 69 no 9ndash12 pp 2593ndash2603 2013

[17] W Wang and J G Yang ldquoA combined error model for thermalerror compensation of machine toolsrdquo Advanced MaterialsResearchMetallicMaterials andManufacturing Technology vol820 pp 147ndash150 2013

[18] Y Liu Y Lu D Gao and Z Hao ldquoThermally induced volumet-ric error modeling based on thermal drift and its compensationin Z-axisrdquo International Journal of Advanced ManufacturingTechnology vol 69 no 9ndash12 pp 2735ndash2745 2013

[19] A E Ouafi M Guillot and N Barka ldquoAn integrated modelingapproach for ANN-based real-time thermal error compensa-tion on a CNC turning centerrdquo Environmental and MaterialsEngineering vol 664 pp 907ndash915 2013

[20] C Wu C Tang C Chang and Y Shiao ldquoThermal error com-pensation method for machine centerrdquo International Journal ofAdvanced Manufacturing Technology vol 59 no 5ndash8 pp 681ndash689 2012

[21] Y Li and W Zhao ldquoAxial thermal error compensation methodfor the spindle of a precision horizontal machining centerrdquo inProceedings of the IEEE International Conference on Mechatron-ics andAutomation (ICMA rsquo12) pp 2319ndash2323 ChengduChinaAugust 2012

[22] M Gebhardt J Mayr N Furrer T Widmer S Weikert andW Knapp ldquoHigh precision grey-box model for compensa-tion of thermal errors on five-axis machinesrdquo CIRP AnnalsmdashManufacturing Technology vol 63 no 1 pp 509ndash512 2014

[23] M Pajor and J Zapłata ldquoSupervising and compensation ofthermal error of CNC feed ball screwrdquo Diagnostyka vol 14 no2 pp 37ndash42 2013

[24] Y Zhang J Yang S Xiang and H Xiao ldquoVolumetric errormodeling and compensation considering thermal effect on five-axis machine toolsrdquo Proceedings of the Institution of MechanicalEngineers C Journal of Mechanical Engineering Science vol 227no 5 pp 1102ndash1115 2013

[25] W Wang and J G Yang ldquoA combined error model for thermalerror compensation of machine toolsrdquo Advanced MaterialsResearch-Metallic Materials andManufacturing Technology vol820 pp 147ndash150 2013

[26] EMiao Y Yan andY Fei ldquoApplication of time series to thermalerror compensation of machine toolsrdquo in Proceeding of the 4thInternational Seminar on Modern Cutting and MeasurementEngineering Beijing China December 2010

[27] H Wang S X Tan G Liao R Quintanilla and A GuptaldquoFull-chip runtime error-tolerant thermal estimation and pre-diction for practical thermalmanagementrdquo in Proceedings of theIEEEACM International Conference on Computer-Aided Design(ICCAD rsquo11) Digest of Technical Papers pp 716ndash723November2011

[28] E P B George M J Gwilym and C R Gregory Time SeriesAnalysis Forecasting and Control China Machine Press 4thedition 2011

[29] ISO ldquoTest code for machine tools part 3 determination ofthermal effectsrdquo ISO 230-3 ISO Copyright Office ZurichSwitzerland 2007

[30] H T Akaike ldquoNew Look at the Statistical Model IdentificationrdquoIEEE Transactions on Automatic Control vol 19 pp 716ndash7231974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


measurement and described a method to group the dataof thermal sensors [7] Huang et al measured the spindlethermal drifts with five eddy current displacement sensorsand proposed a method combining the back propagation(BP) and genetic algorithm (GA) to model the thermalexpansion in axial direction [8] But the radial declinedangle errors were ignored and the thermal angle errors arekey factors to compensate the spindle terminal processingaccuracy

Secondly the thermal error model must be built accu-rately based on the experiment measurement Yang et al[9] used artificial neural networks (ANNS) to establish arelationship between temperature and the thermal error ofa spindle EL Ouafi et al constructed an artificial neuralnetwork model for spindle thermal errors with the temper-ature drawing on statistical methodology which effectivelyimproves the machining accuracy [10] The support vectormachine is a new machine learning theory which has manyadvantages such as simple algorithm global optimizationversatile and strong generalization ability The scholarsLin et al and Zhao et al established a spindle thermal errormodel based on least square support vector machine theoryand the model has perfect robustness [11 12]

The mentioned researchers have established the axialthermal elongation model but they overlooked the thermalyaw and pitch errors of the spindle

In recent years the finite element method (FEM) isapplied to analyze temperature fields and the thermaldeformation of machine tools Creighton et al used thefinite element method to analyze temperature distributioncharacteristics for a high-speed micromilling spindle andconstructed an exponential model of the axial thermal errorconsidering the spindle speed and running time [13] Haitaoet al proposed amethod for calculating thermal conductivitycoefficient of the spindle surface and simulated and analyzedvariation principles of the temperature field and thermaldeformation of the spindle [14]

However the precision CNC machine tool error is amutual coupling of many complex factors that are affectedby many variables and therefore it is extremely difficult toestablish a theoretical equation based on the perspective ofthermoelasticity and heat transfer

If the measurement and modeling of the thermal errorsare completed the next work could be the compensation Fuet al and Miao et al built the spindle axial thermal errormodel by applying multivariate linear regression method[15 16] Wang and Yang also proposed a prediction modelfor the axial thermal deformation and applied the modelto compensate error on a CNC machine [17] Liu et alcompensated the thermal drift on the milling and boringmachine in the 119885-direction [18] Ouafi et al presented anintegrated comprehensive modeling approach for thermalerrors real-time compensation based on multiple tempera-ture measurements after compensation the spindle thermal-induced errors were reduced from 19 120583m to less than 1 120583m[19] There are other scholars who investigated the spindleaxial thermal error compensation method [20 21] and themachining accuracy was improved effectively Gebhardt etal described a high precision grey-box thermal error model

for compensation on five-axis machines and the thermallyinduced errors of the rotaryswivelling can be reduced upto 85 [22] Pajor and Zapłata presented a set allowing forsupervising the feed screw thermal elongation to reduce ballscrew thermal errors [23] Zhang et al developed a novelcompensation implementation technique for machine toolsbased on the function of themachine external zero point shiftand Ethernet data communication protocol which improvedthe machine precision [24]

The existing literatures were mainly about measurementand modeling of the spindle axial thermal elongation butthe radial thermal angle errors were ignored The spindlethermal deformation of a CNC machine tool is usuallyexpressed as the actual spatial position and gesture may devi-ate from the theoretical value namely the running spindlewould induce the drifts of geometric dimensions and spatialphase thus affecting the precision of machine processingOf course the deviated drifts must include axial elongationand radial thermal pitch and yaw angle errors which reducethe machining accuracy For example the paperrsquos object isa coordinate boring machine and its spindle axial thermalexpansion may affect the bore geometric size Meanwhilethe radial thermal inclination angle errors could influencethe geometric dimensions and surface roughness of the holeso it is extremely necessary to measure axial and radialthermal errors simultaneously In order to realize the errorcompensation the spindle radial thermal angle errors mustbe translated into components of the linear coordinate axisMoreover the thermal error offset components are closelyrelated to both radial thermal inclination angle errors andthe handle length So the radial thermal angle errors cannotbe ignored The three-point method could only measure theabsolute thermal deformation in the single radial directionand it cannot obtain the thermal inclination angle errorTherefore the thermal error compensationmodel deduced bythe method was not accurate enough This indicated that thethree-point method cannot completely reflect the variationof the spindle radial thermal deformation and the radialthermal error compensation model is not accurate based onthis measurement

Based on the above analysis the five-point method isapplied and the measurement contains the following advan-tage (1) five-point method could simultaneously measureaxial and radial thermal drifts of the motorized spindlesystem so the variation of the spindle position and ori-entation could be analyzed solving the challenge that thethree-point method cannot measure the inclined angles ofthe spindle radial thermal errors (2) the integrated thermalerror modeling will provide a more accurate mathematicalequation for the error compensation which includes thespindle elongation radial thermal pitch and yaw angle errorsbased on the five-point method

The time series analysis provides a set of approachesto process dynamic data The method primary meaningis that all types of data are approximately described bymathematical models Through the model analysis the datainternal structure can be mined So we can forecast its trendsand make necessary control on it Wang et al applied timeseries analysis method to establish a spindle thermal error


ΔDX

D

O Z

P

ΔOx

ΔOzΔOD D0x

O998400

120579x

(a)

X

D

O Z

P

D0x

O998400

DL1

120579x

ΔL3

ΔL1

(b)







defined as 120579119910



119875119874 to9978889978889978881198751198741015840 so the offset


Δ119874119909= (1198630119909+ 119863 + Δ119863) sin 120579

119909 (1)




Δ119874119911= Δ119863 minus Δ119874

119863= Δ119863 minus (119863

0119909+ 119863 + Δ119863) (1 minus cos 120579

119909)

(2)


Δ119863 ≪ 1198630119909+ 119863 120579

119909997888rarr 0 (3)

so

sin 120579119909997888rarr 120579119909 cos 120579

119909997888rarr 1 (4)


Δ119874119909= (1198630119909+ 119863) 120579

119909 (5)

Δ119874119911= Δ119863 (6)



can be obtained

Δ119874119910= (1198630119910+ 119863) 120579

119910 (7)

where 1198630119910




119909and




Output |119890119894|min(120583m10158401015840)

|119890119894|max

(120583m10158401015840)|119890119894|

(120583m10158401015840) RMSE 1198772 120578

()120579119909 000 344 0441 0730 0986 956120579119910 000 162 0242 0416 0996 976119864 000 150 0335 0473 0999 987



119905 119905 =


119905 120579119909119905 120579119910119905 if the results


120583 =1

119899

119899

sum

119894=1

119909119894

1205902

=1

119899 minus 1

119899

sum

119894=1

(119909119894minus 120583)2

(8)


2




119864119905sim 119873(120583

1 1205902

1) 997904rArr 119864

1015840

119905=119864119905minus 1205831

1205901

997904rArr 1198641015840

119905sim 119873 (0 1)

120579119909119905sim 119873(120583

2 1205902

2) 997904rArr 120579

1015840

119909119905=120579119909119905minus 1205832

1205902

997904rArr 1205791015840

119909119905sim 119873 (0 1)

120579119910119905sim 119873(120583

3 1205902

3) 997904rArr 120579

1015840

119910119905=120579119910119905minus 1205833

1205903

997904rArr 1205791015840

119910119905sim 119873 (0 1)

(9)

where 1198641015840119905 1205791015840119909119905 1205791015840119910119905




119883119905= 1206011119883119905minus1


+ 120576119905minus 1205791120576119905minus1


(10)where 120601




119901

119883119905= 119883119905minus119901 (11)

Thus120579 (119861) = 1 minus 120579

1119861 minus 12057921198612


120601 (119861) = 1 minus 1206011119861 minus 12060121198612


(12)


119905= 120579 (119861) 120576

119905 (13)


= Cov (119883119905+119896 119883119905) = 119864 [(119883

119905+119896minus 120583119905+119896) (119883119905minus 120583119905)] = 120574

119896



120588119896=120574119896

1205740

(15)


120574119896=1

119873

119873

sum

119905=119896+1

119883119905119883119905minus119896 (16)


120588119896=120574119896

1205740

=sum119873

119905=119896+1119883119905119883119905minus119896

sum119873

119905=11198832119905

(17)


1205881= 1206011+ 12060121205881+ sdot sdot sdot + 120601

119901120588119901minus1

1205882= 12060111205881+ 1206012+ sdot sdot sdot + 120601

119901120588119901minus2

120588119901= 1206011120588119901minus1

+ 1206012120588119901minus2

+ sdot sdot sdot + 120601119901

(18)





120601 = (

1206011

1206012

120601119901

) 120588119901= (

1205881

1205882

120588119901

)

119875119901= (

1 1205881

1205882


1205881

1 1205881


120588119901minus1

120588119901minus2

120588119901minus3

sdot sdot sdot 1

)

(19)


120601 = 119875minus1

119901120588119901 (20)

Mark 1199081015840119905= 120601(119861)119883



1199081015840

119905= 120579 (119861) 120576

119905 (21)


119895of 119883119905

and 119895 = 0 1 119902

1205741015840

119895=

119901

sum

119894=0

1206012

119894120574119895

+

119901

sum

119894=1

(1206010120601119894+ 1206011120601119894+1


+ 120574119895minus119894)

(22)


process is

1205741015840

0= (1 + 120579

2


2

119902) 1205902

120576

1205741015840

119896= (minus120579

119896+ 1205791120579119896+1


120576

(23)


1205791can be


1205902

120576=

1205741015840

0

1 + 12057921+ sdot sdot sdot + 1205792

119902

120579119895= minus(

1205741015840

119895

1205902120576

minus 1205791120579119895+1

minus 1205792120579119895+2


(24)

And promise 1205790= 0 120579

119902 120579119902minus1






S1

S2

S3

S4

S5

XY

Z





5 The radial


1


Thermal state

Cold state

D

L03

Li3

Li1L01

120579

ΔL

S1 S3

ΔL3

ΔL1






119909is

Δ1198713= 119871119894

3minus 1198710

3

Δ1198711= 119871119894

1minus 1198710

1

Δ119871 = Δ1198713minus Δ1198711

tan 120579119909=Δ119871

119863

(25)


119909rarr 0 so

120579119909sim tan 120579

119909 (26)


120579119909=(119871119894

3minus 119871119894

1) minus (119871

0

3minus 1198710

1)

119863 (27)

where 1198710

3and 119871

0




119894

3and 119871

119894

1are the


1and 1198783 1198782and 1198784 and119863 = 120mm


120579119910=(119871119894

4minus 119871119894

2) minus (119871

0

4minus 1198710

2)

119863 (28)


0 50 100 150 200 250 300 350 4000

100020003000400050006000700080009000

1000011000

N(r

pm)

t (min)





119909in



119910





31

30

29

28

27

26

25

24

23

22

210 50 100 150 200 250 300 350 400 450

Time (min)

Tem

pera

ture

(∘C)

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11


0 100 150 200 250 300 350 400 450

0102030405060



Ther

mal

erro

r (120583

m)

minus10

minus20

minus30

minus40

Time (min)50




119905in the 119885-


119909119905 120579119910119905


119905 120579119909119905 120579119910119905


2


119905and radial



119905 120579119909119905 120579119910119905


119905 1205791015840119909119905 1205791015840119910119905




119905 1205791015840119909119905 1205791015840119910119905





120576make


AIC (119901 119902) = ln 2120576+2 (119901 + 119902 + 1)

119899 (29)






119910119905is


1198641015840

119905= 1583119864

1015840

119905minus1minus 0471119864

1015840

119905minus2+ 0644119864

1015840

119905minus3

minus 12461198641015840

119905minus4+ 0488119864

1015840

119905minus5+ 120576(1)

119905+ 0263120576

(1)

119905minus1

minus 0267120576(1)

119905minus2minus 0959120576

(1)

119905minus3

1205791015840

119909119905= 018120579

1015840

119909119905minus1+ 0779120579

1015840

119909119905minus2+ 120576(2)

119905

+ 1139120576(2)

119905minus1+ 0163120576

(2)

119905minus2minus 0139120576

(2)

119905minus3

+ 0231120576(2)

119905minus4+ 0376120576

(2)

119905minus5

1205791015840

119910119905= 1996120579

1015840

119910119905minus1minus 0997120579

1015840

119910119905minus2+ 120576(3)

119905minus 120576(3)

119905minus1

(30)


50


minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)


minus5

(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)

(b)


Ther

mal

yaw

(998400998400 )

0

minus10

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

yaw

(998400998400 )

(b)



120583 = (

1205831

1205832

1205833

) = (

minus24681

minus9845

minus10095

) 120590 = (

1205901

1205902

1205903

) = (

1411

6272

6571

)

(31)


(

119864119905

120579119909119905

120579119910119905

) = (

1205901

1205902

1205903

)(

1198641015840

119905

1205791015840

119909119905

1205791015840

119910119905

)+(

1205831

1205832

1205833

) (32)




119894|max and



RMSE = radic 1

119899

119899

sum

119894=1

(119910119894minus 119910119894)2

1198772

= 1 minussum119899

119894=1(119910119894minus 119910119894)2

sum119899

119894=1(119910119894minus 119910119894)2

120578 = 1 minus(1119899)sum

119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

(1119899)sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

= 1 minussum119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

(33)








Ther

mal

pitc

h (998400998400

)

0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)

minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

pitc

h (998400998400

)


(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus1

minus05

0

05

1

(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus05

0

05

1

(b)


0 5 10 15 20

0

05

1

Lag

Sam

ple a

utoc

orre

latio


minus05

(a)

0 5 10 15 20LagSa

mpl

e par

tial a

utoc

orre

latio

ns


0

05

1

minus05

(b)




0

minus10

minus20

minus30

minus40

Elon

gatio

n (120583

m)

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Resid

ual (120583

m)

2

1

0

minus1

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


Ther

mal

yaw

(998400998400 )

(a)

4

3

2

1

0

minus1

minus2

minus3

Resid

ual (

998400998400)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)




0119909and 119863

0119910in 119883119884-direction


1198630119909=

Δ1198711

tan 120579119909

minus 1198631198711= 548659mm

1198630119910=

Δ1198712

tan 120579119910

minus 1198631198712= 508706mm

(34)



1198711= 1198631198712

=



Δ119874119909= (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

Δ119874119910= (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

Δ119874119911= 12059011198641015840

119905+ 1205831

(35)


compensation is1198821015840(1198751015840119909 1198751015840

119910 1198751015840

119911)

1198751015840

119909= 119875119909+ Δ119874119909= 119875119909+ (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

1198751015840

119910= 119875119910+ Δ119874119910= 119875119910+ (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

1198751015840

119911= 119875119911+ Δ119874119911= 119875119911+ 12059011198641015840

119905+ 1205831

(36)



Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)


Sensor


HMI interface

Compensation value


Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller


AD

times

Machine tool

RS232

Part program


Theoretical value

FeedbackPC


Compensation model

USB



Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500


30

15

0

minus15

minus30

minus45

Time (min)





S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ΔDX

D

O Z

P

ΔOx

ΔOzΔOD D0x

O998400

120579x

(a)

X

D

O Z

P

D0x

O998400

DL1

120579x

ΔL3

ΔL1

(b)







defined as 120579119910



119875119874 to9978889978889978881198751198741015840 so the offset


Δ119874119909= (1198630119909+ 119863 + Δ119863) sin 120579

119909 (1)




Δ119874119911= Δ119863 minus Δ119874

119863= Δ119863 minus (119863

0119909+ 119863 + Δ119863) (1 minus cos 120579

119909)

(2)


Δ119863 ≪ 1198630119909+ 119863 120579

119909997888rarr 0 (3)

so

sin 120579119909997888rarr 120579119909 cos 120579

119909997888rarr 1 (4)


Δ119874119909= (1198630119909+ 119863) 120579

119909 (5)

Δ119874119911= Δ119863 (6)



can be obtained

Δ119874119910= (1198630119910+ 119863) 120579

119910 (7)

where 1198630119910




119909and




Output |119890119894|min(120583m10158401015840)

|119890119894|max

(120583m10158401015840)|119890119894|

(120583m10158401015840) RMSE 1198772 120578

()120579119909 000 344 0441 0730 0986 956120579119910 000 162 0242 0416 0996 976119864 000 150 0335 0473 0999 987



119905 119905 =


119905 120579119909119905 120579119910119905 if the results


120583 =1

119899

119899

sum

119894=1

119909119894

1205902

=1

119899 minus 1

119899

sum

119894=1

(119909119894minus 120583)2

(8)


2




119864119905sim 119873(120583

1 1205902

1) 997904rArr 119864

1015840

119905=119864119905minus 1205831

1205901

997904rArr 1198641015840

119905sim 119873 (0 1)

120579119909119905sim 119873(120583

2 1205902

2) 997904rArr 120579

1015840

119909119905=120579119909119905minus 1205832

1205902

997904rArr 1205791015840

119909119905sim 119873 (0 1)

120579119910119905sim 119873(120583

3 1205902

3) 997904rArr 120579

1015840

119910119905=120579119910119905minus 1205833

1205903

997904rArr 1205791015840

119910119905sim 119873 (0 1)

(9)

where 1198641015840119905 1205791015840119909119905 1205791015840119910119905




119883119905= 1206011119883119905minus1


+ 120576119905minus 1205791120576119905minus1


(10)where 120601




119901

119883119905= 119883119905minus119901 (11)

Thus120579 (119861) = 1 minus 120579

1119861 minus 12057921198612


120601 (119861) = 1 minus 1206011119861 minus 12060121198612


(12)


119905= 120579 (119861) 120576

119905 (13)


= Cov (119883119905+119896 119883119905) = 119864 [(119883

119905+119896minus 120583119905+119896) (119883119905minus 120583119905)] = 120574

119896



120588119896=120574119896

1205740

(15)


120574119896=1

119873

119873

sum

119905=119896+1

119883119905119883119905minus119896 (16)


120588119896=120574119896

1205740

=sum119873

119905=119896+1119883119905119883119905minus119896

sum119873

119905=11198832119905

(17)


1205881= 1206011+ 12060121205881+ sdot sdot sdot + 120601

119901120588119901minus1

1205882= 12060111205881+ 1206012+ sdot sdot sdot + 120601

119901120588119901minus2

120588119901= 1206011120588119901minus1

+ 1206012120588119901minus2

+ sdot sdot sdot + 120601119901

(18)





120601 = (

1206011

1206012

120601119901

) 120588119901= (

1205881

1205882

120588119901

)

119875119901= (

1 1205881

1205882


1205881

1 1205881


120588119901minus1

120588119901minus2

120588119901minus3

sdot sdot sdot 1

)

(19)


120601 = 119875minus1

119901120588119901 (20)

Mark 1199081015840119905= 120601(119861)119883



1199081015840

119905= 120579 (119861) 120576

119905 (21)


119895of 119883119905

and 119895 = 0 1 119902

1205741015840

119895=

119901

sum

119894=0

1206012

119894120574119895

+

119901

sum

119894=1

(1206010120601119894+ 1206011120601119894+1


+ 120574119895minus119894)

(22)


process is

1205741015840

0= (1 + 120579

2


2

119902) 1205902

120576

1205741015840

119896= (minus120579

119896+ 1205791120579119896+1


120576

(23)


1205791can be


1205902

120576=

1205741015840

0

1 + 12057921+ sdot sdot sdot + 1205792

119902

120579119895= minus(

1205741015840

119895

1205902120576

minus 1205791120579119895+1

minus 1205792120579119895+2


(24)

And promise 1205790= 0 120579

119902 120579119902minus1






S1

S2

S3

S4

S5

XY

Z





5 The radial


1


Thermal state

Cold state

D

L03

Li3

Li1L01

120579

ΔL

S1 S3

ΔL3

ΔL1






119909is

Δ1198713= 119871119894

3minus 1198710

3

Δ1198711= 119871119894

1minus 1198710

1

Δ119871 = Δ1198713minus Δ1198711

tan 120579119909=Δ119871

119863

(25)


119909rarr 0 so

120579119909sim tan 120579

119909 (26)


120579119909=(119871119894

3minus 119871119894

1) minus (119871

0

3minus 1198710

1)

119863 (27)

where 1198710

3and 119871

0




119894

3and 119871

119894

1are the


1and 1198783 1198782and 1198784 and119863 = 120mm


120579119910=(119871119894

4minus 119871119894

2) minus (119871

0

4minus 1198710

2)

119863 (28)


0 50 100 150 200 250 300 350 4000

100020003000400050006000700080009000

1000011000

N(r

pm)

t (min)





119909in



119910





31

30

29

28

27

26

25

24

23

22

210 50 100 150 200 250 300 350 400 450

Time (min)

Tem

pera

ture

(∘C)

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11


0 100 150 200 250 300 350 400 450

0102030405060



Ther

mal

erro

r (120583

m)

minus10

minus20

minus30

minus40

Time (min)50




119905in the 119885-


119909119905 120579119910119905


119905 120579119909119905 120579119910119905


2


119905and radial



119905 120579119909119905 120579119910119905


119905 1205791015840119909119905 1205791015840119910119905




119905 1205791015840119909119905 1205791015840119910119905





120576make


AIC (119901 119902) = ln 2120576+2 (119901 + 119902 + 1)

119899 (29)






119910119905is


1198641015840

119905= 1583119864

1015840

119905minus1minus 0471119864

1015840

119905minus2+ 0644119864

1015840

119905minus3

minus 12461198641015840

119905minus4+ 0488119864

1015840

119905minus5+ 120576(1)

119905+ 0263120576

(1)

119905minus1

minus 0267120576(1)

119905minus2minus 0959120576

(1)

119905minus3

1205791015840

119909119905= 018120579

1015840

119909119905minus1+ 0779120579

1015840

119909119905minus2+ 120576(2)

119905

+ 1139120576(2)

119905minus1+ 0163120576

(2)

119905minus2minus 0139120576

(2)

119905minus3

+ 0231120576(2)

119905minus4+ 0376120576

(2)

119905minus5

1205791015840

119910119905= 1996120579

1015840

119910119905minus1minus 0997120579

1015840

119910119905minus2+ 120576(3)

119905minus 120576(3)

119905minus1

(30)


50


minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)


minus5

(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)

(b)


Ther

mal

yaw

(998400998400 )

0

minus10

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

yaw

(998400998400 )

(b)



120583 = (

1205831

1205832

1205833

) = (

minus24681

minus9845

minus10095

) 120590 = (

1205901

1205902

1205903

) = (

1411

6272

6571

)

(31)


(

119864119905

120579119909119905

120579119910119905

) = (

1205901

1205902

1205903

)(

1198641015840

119905

1205791015840

119909119905

1205791015840

119910119905

)+(

1205831

1205832

1205833

) (32)




119894|max and



RMSE = radic 1

119899

119899

sum

119894=1

(119910119894minus 119910119894)2

1198772

= 1 minussum119899

119894=1(119910119894minus 119910119894)2

sum119899

119894=1(119910119894minus 119910119894)2

120578 = 1 minus(1119899)sum

119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

(1119899)sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

= 1 minussum119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

(33)








Ther

mal

pitc

h (998400998400

)

0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)

minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

pitc

h (998400998400

)


(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus1

minus05

0

05

1

(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus05

0

05

1

(b)


0 5 10 15 20

0

05

1

Lag

Sam

ple a

utoc

orre

latio


minus05

(a)

0 5 10 15 20LagSa

mpl

e par

tial a

utoc

orre

latio

ns


0

05

1

minus05

(b)




0

minus10

minus20

minus30

minus40

Elon

gatio

n (120583

m)

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Resid

ual (120583

m)

2

1

0

minus1

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


Ther

mal

yaw

(998400998400 )

(a)

4

3

2

1

0

minus1

minus2

minus3

Resid

ual (

998400998400)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)




0119909and 119863

0119910in 119883119884-direction


1198630119909=

Δ1198711

tan 120579119909

minus 1198631198711= 548659mm

1198630119910=

Δ1198712

tan 120579119910

minus 1198631198712= 508706mm

(34)



1198711= 1198631198712

=



Δ119874119909= (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

Δ119874119910= (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

Δ119874119911= 12059011198641015840

119905+ 1205831

(35)


compensation is1198821015840(1198751015840119909 1198751015840

119910 1198751015840

119911)

1198751015840

119909= 119875119909+ Δ119874119909= 119875119909+ (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

1198751015840

119910= 119875119910+ Δ119874119910= 119875119910+ (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

1198751015840

119911= 119875119911+ Δ119874119911= 119875119911+ 12059011198641015840

119905+ 1205831

(36)



Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)


Sensor


HMI interface

Compensation value


Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller


AD

times

Machine tool

RS232

Part program


Theoretical value

FeedbackPC


Compensation model

USB



Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500


30

15

0

minus15

minus30

minus45

Time (min)





S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Output |119890119894|min(120583m10158401015840)

|119890119894|max

(120583m10158401015840)|119890119894|

(120583m10158401015840) RMSE 1198772 120578

()120579119909 000 344 0441 0730 0986 956120579119910 000 162 0242 0416 0996 976119864 000 150 0335 0473 0999 987



119905 119905 =


119905 120579119909119905 120579119910119905 if the results


120583 =1

119899

119899

sum

119894=1

119909119894

1205902

=1

119899 minus 1

119899

sum

119894=1

(119909119894minus 120583)2

(8)


2




119864119905sim 119873(120583

1 1205902

1) 997904rArr 119864

1015840

119905=119864119905minus 1205831

1205901

997904rArr 1198641015840

119905sim 119873 (0 1)

120579119909119905sim 119873(120583

2 1205902

2) 997904rArr 120579

1015840

119909119905=120579119909119905minus 1205832

1205902

997904rArr 1205791015840

119909119905sim 119873 (0 1)

120579119910119905sim 119873(120583

3 1205902

3) 997904rArr 120579

1015840

119910119905=120579119910119905minus 1205833

1205903

997904rArr 1205791015840

119910119905sim 119873 (0 1)

(9)

where 1198641015840119905 1205791015840119909119905 1205791015840119910119905




119883119905= 1206011119883119905minus1


+ 120576119905minus 1205791120576119905minus1


(10)where 120601




119901

119883119905= 119883119905minus119901 (11)

Thus120579 (119861) = 1 minus 120579

1119861 minus 12057921198612


120601 (119861) = 1 minus 1206011119861 minus 12060121198612


(12)


119905= 120579 (119861) 120576

119905 (13)


= Cov (119883119905+119896 119883119905) = 119864 [(119883

119905+119896minus 120583119905+119896) (119883119905minus 120583119905)] = 120574

119896



120588119896=120574119896

1205740

(15)


120574119896=1

119873

119873

sum

119905=119896+1

119883119905119883119905minus119896 (16)


120588119896=120574119896

1205740

=sum119873

119905=119896+1119883119905119883119905minus119896

sum119873

119905=11198832119905

(17)


1205881= 1206011+ 12060121205881+ sdot sdot sdot + 120601

119901120588119901minus1

1205882= 12060111205881+ 1206012+ sdot sdot sdot + 120601

119901120588119901minus2

120588119901= 1206011120588119901minus1

+ 1206012120588119901minus2

+ sdot sdot sdot + 120601119901

(18)





120601 = (

1206011

1206012

120601119901

) 120588119901= (

1205881

1205882

120588119901

)

119875119901= (

1 1205881

1205882


1205881

1 1205881


120588119901minus1

120588119901minus2

120588119901minus3

sdot sdot sdot 1

)

(19)


120601 = 119875minus1

119901120588119901 (20)

Mark 1199081015840119905= 120601(119861)119883



1199081015840

119905= 120579 (119861) 120576

119905 (21)


119895of 119883119905

and 119895 = 0 1 119902

1205741015840

119895=

119901

sum

119894=0

1206012

119894120574119895

+

119901

sum

119894=1

(1206010120601119894+ 1206011120601119894+1


+ 120574119895minus119894)

(22)


process is

1205741015840

0= (1 + 120579

2


2

119902) 1205902

120576

1205741015840

119896= (minus120579

119896+ 1205791120579119896+1


120576

(23)


1205791can be


1205902

120576=

1205741015840

0

1 + 12057921+ sdot sdot sdot + 1205792

119902

120579119895= minus(

1205741015840

119895

1205902120576

minus 1205791120579119895+1

minus 1205792120579119895+2


(24)

And promise 1205790= 0 120579

119902 120579119902minus1






S1

S2

S3

S4

S5

XY

Z





5 The radial


1


Thermal state

Cold state

D

L03

Li3

Li1L01

120579

ΔL

S1 S3

ΔL3

ΔL1






119909is

Δ1198713= 119871119894

3minus 1198710

3

Δ1198711= 119871119894

1minus 1198710

1

Δ119871 = Δ1198713minus Δ1198711

tan 120579119909=Δ119871

119863

(25)


119909rarr 0 so

120579119909sim tan 120579

119909 (26)


120579119909=(119871119894

3minus 119871119894

1) minus (119871

0

3minus 1198710

1)

119863 (27)

where 1198710

3and 119871

0




119894

3and 119871

119894

1are the


1and 1198783 1198782and 1198784 and119863 = 120mm


120579119910=(119871119894

4minus 119871119894

2) minus (119871

0

4minus 1198710

2)

119863 (28)


0 50 100 150 200 250 300 350 4000

100020003000400050006000700080009000

1000011000

N(r

pm)

t (min)





119909in



119910





31

30

29

28

27

26

25

24

23

22

210 50 100 150 200 250 300 350 400 450

Time (min)

Tem

pera

ture

(∘C)

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11


0 100 150 200 250 300 350 400 450

0102030405060



Ther

mal

erro

r (120583

m)

minus10

minus20

minus30

minus40

Time (min)50




119905in the 119885-


119909119905 120579119910119905


119905 120579119909119905 120579119910119905


2


119905and radial



119905 120579119909119905 120579119910119905


119905 1205791015840119909119905 1205791015840119910119905




119905 1205791015840119909119905 1205791015840119910119905





120576make


AIC (119901 119902) = ln 2120576+2 (119901 + 119902 + 1)

119899 (29)






119910119905is


1198641015840

119905= 1583119864

1015840

119905minus1minus 0471119864

1015840

119905minus2+ 0644119864

1015840

119905minus3

minus 12461198641015840

119905minus4+ 0488119864

1015840

119905minus5+ 120576(1)

119905+ 0263120576

(1)

119905minus1

minus 0267120576(1)

119905minus2minus 0959120576

(1)

119905minus3

1205791015840

119909119905= 018120579

1015840

119909119905minus1+ 0779120579

1015840

119909119905minus2+ 120576(2)

119905

+ 1139120576(2)

119905minus1+ 0163120576

(2)

119905minus2minus 0139120576

(2)

119905minus3

+ 0231120576(2)

119905minus4+ 0376120576

(2)

119905minus5

1205791015840

119910119905= 1996120579

1015840

119910119905minus1minus 0997120579

1015840

119910119905minus2+ 120576(3)

119905minus 120576(3)

119905minus1

(30)


50


minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)


minus5

(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)

(b)


Ther

mal

yaw

(998400998400 )

0

minus10

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

yaw

(998400998400 )

(b)



120583 = (

1205831

1205832

1205833

) = (

minus24681

minus9845

minus10095

) 120590 = (

1205901

1205902

1205903

) = (

1411

6272

6571

)

(31)


(

119864119905

120579119909119905

120579119910119905

) = (

1205901

1205902

1205903

)(

1198641015840

119905

1205791015840

119909119905

1205791015840

119910119905

)+(

1205831

1205832

1205833

) (32)




119894|max and



RMSE = radic 1

119899

119899

sum

119894=1

(119910119894minus 119910119894)2

1198772

= 1 minussum119899

119894=1(119910119894minus 119910119894)2

sum119899

119894=1(119910119894minus 119910119894)2

120578 = 1 minus(1119899)sum

119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

(1119899)sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

= 1 minussum119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

(33)








Ther

mal

pitc

h (998400998400

)

0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)

minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

pitc

h (998400998400

)


(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus1

minus05

0

05

1

(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus05

0

05

1

(b)


0 5 10 15 20

0

05

1

Lag

Sam

ple a

utoc

orre

latio


minus05

(a)

0 5 10 15 20LagSa

mpl

e par

tial a

utoc

orre

latio

ns


0

05

1

minus05

(b)




0

minus10

minus20

minus30

minus40

Elon

gatio

n (120583

m)

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Resid

ual (120583

m)

2

1

0

minus1

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


Ther

mal

yaw

(998400998400 )

(a)

4

3

2

1

0

minus1

minus2

minus3

Resid

ual (

998400998400)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)




0119909and 119863

0119910in 119883119884-direction


1198630119909=

Δ1198711

tan 120579119909

minus 1198631198711= 548659mm

1198630119910=

Δ1198712

tan 120579119910

minus 1198631198712= 508706mm

(34)



1198711= 1198631198712

=



Δ119874119909= (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

Δ119874119910= (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

Δ119874119911= 12059011198641015840

119905+ 1205831

(35)


compensation is1198821015840(1198751015840119909 1198751015840

119910 1198751015840

119911)

1198751015840

119909= 119875119909+ Δ119874119909= 119875119909+ (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

1198751015840

119910= 119875119910+ Δ119874119910= 119875119910+ (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

1198751015840

119911= 119875119911+ Δ119874119911= 119875119911+ 12059011198641015840

119905+ 1205831

(36)



Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)


Sensor


HMI interface

Compensation value


Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller


AD

times

Machine tool

RS232

Part program


Theoretical value

FeedbackPC


Compensation model

USB



Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500


30

15

0

minus15

minus30

minus45

Time (min)





S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120601 = (

1206011

1206012

120601119901

) 120588119901= (

1205881

1205882

120588119901

)

119875119901= (

1 1205881

1205882


1205881

1 1205881


120588119901minus1

120588119901minus2

120588119901minus3

sdot sdot sdot 1

)

(19)


120601 = 119875minus1

119901120588119901 (20)

Mark 1199081015840119905= 120601(119861)119883



1199081015840

119905= 120579 (119861) 120576

119905 (21)


119895of 119883119905

and 119895 = 0 1 119902

1205741015840

119895=

119901

sum

119894=0

1206012

119894120574119895

+

119901

sum

119894=1

(1206010120601119894+ 1206011120601119894+1


+ 120574119895minus119894)

(22)


process is

1205741015840

0= (1 + 120579

2


2

119902) 1205902

120576

1205741015840

119896= (minus120579

119896+ 1205791120579119896+1


120576

(23)


1205791can be


1205902

120576=

1205741015840

0

1 + 12057921+ sdot sdot sdot + 1205792

119902

120579119895= minus(

1205741015840

119895

1205902120576

minus 1205791120579119895+1

minus 1205792120579119895+2


(24)

And promise 1205790= 0 120579

119902 120579119902minus1






S1

S2

S3

S4

S5

XY

Z





5 The radial


1


Thermal state

Cold state

D

L03

Li3

Li1L01

120579

ΔL

S1 S3

ΔL3

ΔL1






119909is

Δ1198713= 119871119894

3minus 1198710

3

Δ1198711= 119871119894

1minus 1198710

1

Δ119871 = Δ1198713minus Δ1198711

tan 120579119909=Δ119871

119863

(25)


119909rarr 0 so

120579119909sim tan 120579

119909 (26)


120579119909=(119871119894

3minus 119871119894

1) minus (119871

0

3minus 1198710

1)

119863 (27)

where 1198710

3and 119871

0




119894

3and 119871

119894

1are the


1and 1198783 1198782and 1198784 and119863 = 120mm


120579119910=(119871119894

4minus 119871119894

2) minus (119871

0

4minus 1198710

2)

119863 (28)


0 50 100 150 200 250 300 350 4000

100020003000400050006000700080009000

1000011000

N(r

pm)

t (min)





119909in



119910





31

30

29

28

27

26

25

24

23

22

210 50 100 150 200 250 300 350 400 450

Time (min)

Tem

pera

ture

(∘C)

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11


0 100 150 200 250 300 350 400 450

0102030405060



Ther

mal

erro

r (120583

m)

minus10

minus20

minus30

minus40

Time (min)50




119905in the 119885-


119909119905 120579119910119905


119905 120579119909119905 120579119910119905


2


119905and radial



119905 120579119909119905 120579119910119905


119905 1205791015840119909119905 1205791015840119910119905




119905 1205791015840119909119905 1205791015840119910119905





120576make


AIC (119901 119902) = ln 2120576+2 (119901 + 119902 + 1)

119899 (29)






119910119905is


1198641015840

119905= 1583119864

1015840

119905minus1minus 0471119864

1015840

119905minus2+ 0644119864

1015840

119905minus3

minus 12461198641015840

119905minus4+ 0488119864

1015840

119905minus5+ 120576(1)

119905+ 0263120576

(1)

119905minus1

minus 0267120576(1)

119905minus2minus 0959120576

(1)

119905minus3

1205791015840

119909119905= 018120579

1015840

119909119905minus1+ 0779120579

1015840

119909119905minus2+ 120576(2)

119905

+ 1139120576(2)

119905minus1+ 0163120576

(2)

119905minus2minus 0139120576

(2)

119905minus3

+ 0231120576(2)

119905minus4+ 0376120576

(2)

119905minus5

1205791015840

119910119905= 1996120579

1015840

119910119905minus1minus 0997120579

1015840

119910119905minus2+ 120576(3)

119905minus 120576(3)

119905minus1

(30)


50


minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)


minus5

(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)

(b)


Ther

mal

yaw

(998400998400 )

0

minus10

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

yaw

(998400998400 )

(b)



120583 = (

1205831

1205832

1205833

) = (

minus24681

minus9845

minus10095

) 120590 = (

1205901

1205902

1205903

) = (

1411

6272

6571

)

(31)


(

119864119905

120579119909119905

120579119910119905

) = (

1205901

1205902

1205903

)(

1198641015840

119905

1205791015840

119909119905

1205791015840

119910119905

)+(

1205831

1205832

1205833

) (32)




119894|max and



RMSE = radic 1

119899

119899

sum

119894=1

(119910119894minus 119910119894)2

1198772

= 1 minussum119899

119894=1(119910119894minus 119910119894)2

sum119899

119894=1(119910119894minus 119910119894)2

120578 = 1 minus(1119899)sum

119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

(1119899)sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

= 1 minussum119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

(33)








Ther

mal

pitc

h (998400998400

)

0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)

minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

pitc

h (998400998400

)


(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus1

minus05

0

05

1

(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus05

0

05

1

(b)


0 5 10 15 20

0

05

1

Lag

Sam

ple a

utoc

orre

latio


minus05

(a)

0 5 10 15 20LagSa

mpl

e par

tial a

utoc

orre

latio

ns


0

05

1

minus05

(b)




0

minus10

minus20

minus30

minus40

Elon

gatio

n (120583

m)

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Resid

ual (120583

m)

2

1

0

minus1

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


Ther

mal

yaw

(998400998400 )

(a)

4

3

2

1

0

minus1

minus2

minus3

Resid

ual (

998400998400)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)




0119909and 119863

0119910in 119883119884-direction


1198630119909=

Δ1198711

tan 120579119909

minus 1198631198711= 548659mm

1198630119910=

Δ1198712

tan 120579119910

minus 1198631198712= 508706mm

(34)



1198711= 1198631198712

=



Δ119874119909= (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

Δ119874119910= (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

Δ119874119911= 12059011198641015840

119905+ 1205831

(35)


compensation is1198821015840(1198751015840119909 1198751015840

119910 1198751015840

119911)

1198751015840

119909= 119875119909+ Δ119874119909= 119875119909+ (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

1198751015840

119910= 119875119910+ Δ119874119910= 119875119910+ (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

1198751015840

119911= 119875119911+ Δ119874119911= 119875119911+ 12059011198641015840

119905+ 1205831

(36)



Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)


Sensor


HMI interface

Compensation value


Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller


AD

times

Machine tool

RS232

Part program


Theoretical value

FeedbackPC


Compensation model

USB



Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500


30

15

0

minus15

minus30

minus45

Time (min)





S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Thermal state

Cold state

D

L03

Li3

Li1L01

120579

ΔL

S1 S3

ΔL3

ΔL1






119909is

Δ1198713= 119871119894

3minus 1198710

3

Δ1198711= 119871119894

1minus 1198710

1

Δ119871 = Δ1198713minus Δ1198711

tan 120579119909=Δ119871

119863

(25)


119909rarr 0 so

120579119909sim tan 120579

119909 (26)


120579119909=(119871119894

3minus 119871119894

1) minus (119871

0

3minus 1198710

1)

119863 (27)

where 1198710

3and 119871

0




119894

3and 119871

119894

1are the


1and 1198783 1198782and 1198784 and119863 = 120mm


120579119910=(119871119894

4minus 119871119894

2) minus (119871

0

4minus 1198710

2)

119863 (28)


0 50 100 150 200 250 300 350 4000

100020003000400050006000700080009000

1000011000

N(r

pm)

t (min)





119909in



119910





31

30

29

28

27

26

25

24

23

22

210 50 100 150 200 250 300 350 400 450

Time (min)

Tem

pera

ture

(∘C)

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11


0 100 150 200 250 300 350 400 450

0102030405060



Ther

mal

erro

r (120583

m)

minus10

minus20

minus30

minus40

Time (min)50




119905in the 119885-


119909119905 120579119910119905


119905 120579119909119905 120579119910119905


2


119905and radial



119905 120579119909119905 120579119910119905


119905 1205791015840119909119905 1205791015840119910119905




119905 1205791015840119909119905 1205791015840119910119905





120576make


AIC (119901 119902) = ln 2120576+2 (119901 + 119902 + 1)

119899 (29)






119910119905is


1198641015840

119905= 1583119864

1015840

119905minus1minus 0471119864

1015840

119905minus2+ 0644119864

1015840

119905minus3

minus 12461198641015840

119905minus4+ 0488119864

1015840

119905minus5+ 120576(1)

119905+ 0263120576

(1)

119905minus1

minus 0267120576(1)

119905minus2minus 0959120576

(1)

119905minus3

1205791015840

119909119905= 018120579

1015840

119909119905minus1+ 0779120579

1015840

119909119905minus2+ 120576(2)

119905

+ 1139120576(2)

119905minus1+ 0163120576

(2)

119905minus2minus 0139120576

(2)

119905minus3

+ 0231120576(2)

119905minus4+ 0376120576

(2)

119905minus5

1205791015840

119910119905= 1996120579

1015840

119910119905minus1minus 0997120579

1015840

119910119905minus2+ 120576(3)

119905minus 120576(3)

119905minus1

(30)


50


minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)


minus5

(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)

(b)


Ther

mal

yaw

(998400998400 )

0

minus10

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

yaw

(998400998400 )

(b)



120583 = (

1205831

1205832

1205833

) = (

minus24681

minus9845

minus10095

) 120590 = (

1205901

1205902

1205903

) = (

1411

6272

6571

)

(31)


(

119864119905

120579119909119905

120579119910119905

) = (

1205901

1205902

1205903

)(

1198641015840

119905

1205791015840

119909119905

1205791015840

119910119905

)+(

1205831

1205832

1205833

) (32)




119894|max and



RMSE = radic 1

119899

119899

sum

119894=1

(119910119894minus 119910119894)2

1198772

= 1 minussum119899

119894=1(119910119894minus 119910119894)2

sum119899

119894=1(119910119894minus 119910119894)2

120578 = 1 minus(1119899)sum

119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

(1119899)sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

= 1 minussum119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

(33)








Ther

mal

pitc

h (998400998400

)

0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)

minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

pitc

h (998400998400

)


(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus1

minus05

0

05

1

(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus05

0

05

1

(b)


0 5 10 15 20

0

05

1

Lag

Sam

ple a

utoc

orre

latio


minus05

(a)

0 5 10 15 20LagSa

mpl

e par

tial a

utoc

orre

latio

ns


0

05

1

minus05

(b)




0

minus10

minus20

minus30

minus40

Elon

gatio

n (120583

m)

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Resid

ual (120583

m)

2

1

0

minus1

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


Ther

mal

yaw

(998400998400 )

(a)

4

3

2

1

0

minus1

minus2

minus3

Resid

ual (

998400998400)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)




0119909and 119863

0119910in 119883119884-direction


1198630119909=

Δ1198711

tan 120579119909

minus 1198631198711= 548659mm

1198630119910=

Δ1198712

tan 120579119910

minus 1198631198712= 508706mm

(34)



1198711= 1198631198712

=



Δ119874119909= (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

Δ119874119910= (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

Δ119874119911= 12059011198641015840

119905+ 1205831

(35)


compensation is1198821015840(1198751015840119909 1198751015840

119910 1198751015840

119911)

1198751015840

119909= 119875119909+ Δ119874119909= 119875119909+ (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

1198751015840

119910= 119875119910+ Δ119874119910= 119875119910+ (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

1198751015840

119911= 119875119911+ Δ119874119911= 119875119911+ 12059011198641015840

119905+ 1205831

(36)



Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)


Sensor


HMI interface

Compensation value


Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller


AD

times

Machine tool

RS232

Part program


Theoretical value

FeedbackPC


Compensation model

USB



Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500


30

15

0

minus15

minus30

minus45

Time (min)





S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










31

30

29

28

27

26

25

24

23

22

210 50 100 150 200 250 300 350 400 450

Time (min)

Tem

pera

ture

(∘C)

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11


0 100 150 200 250 300 350 400 450

0102030405060



Ther

mal

erro

r (120583

m)

minus10

minus20

minus30

minus40

Time (min)50




119905in the 119885-


119909119905 120579119910119905


119905 120579119909119905 120579119910119905


2


119905and radial



119905 120579119909119905 120579119910119905


119905 1205791015840119909119905 1205791015840119910119905




119905 1205791015840119909119905 1205791015840119910119905





120576make


AIC (119901 119902) = ln 2120576+2 (119901 + 119902 + 1)

119899 (29)






119910119905is


1198641015840

119905= 1583119864

1015840

119905minus1minus 0471119864

1015840

119905minus2+ 0644119864

1015840

119905minus3

minus 12461198641015840

119905minus4+ 0488119864

1015840

119905minus5+ 120576(1)

119905+ 0263120576

(1)

119905minus1

minus 0267120576(1)

119905minus2minus 0959120576

(1)

119905minus3

1205791015840

119909119905= 018120579

1015840

119909119905minus1+ 0779120579

1015840

119909119905minus2+ 120576(2)

119905

+ 1139120576(2)

119905minus1+ 0163120576

(2)

119905minus2minus 0139120576

(2)

119905minus3

+ 0231120576(2)

119905minus4+ 0376120576

(2)

119905minus5

1205791015840

119910119905= 1996120579

1015840

119910119905minus1minus 0997120579

1015840

119910119905minus2+ 120576(3)

119905minus 120576(3)

119905minus1

(30)


50


minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)


minus5

(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)

(b)


Ther

mal

yaw

(998400998400 )

0

minus10

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

yaw

(998400998400 )

(b)



120583 = (

1205831

1205832

1205833

) = (

minus24681

minus9845

minus10095

) 120590 = (

1205901

1205902

1205903

) = (

1411

6272

6571

)

(31)


(

119864119905

120579119909119905

120579119910119905

) = (

1205901

1205902

1205903

)(

1198641015840

119905

1205791015840

119909119905

1205791015840

119910119905

)+(

1205831

1205832

1205833

) (32)




119894|max and



RMSE = radic 1

119899

119899

sum

119894=1

(119910119894minus 119910119894)2

1198772

= 1 minussum119899

119894=1(119910119894minus 119910119894)2

sum119899

119894=1(119910119894minus 119910119894)2

120578 = 1 minus(1119899)sum

119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

(1119899)sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

= 1 minussum119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

(33)








Ther

mal

pitc

h (998400998400

)

0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)

minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

pitc

h (998400998400

)


(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus1

minus05

0

05

1

(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus05

0

05

1

(b)


0 5 10 15 20

0

05

1

Lag

Sam

ple a

utoc

orre

latio


minus05

(a)

0 5 10 15 20LagSa

mpl

e par

tial a

utoc

orre

latio

ns


0

05

1

minus05

(b)




0

minus10

minus20

minus30

minus40

Elon

gatio

n (120583

m)

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Resid

ual (120583

m)

2

1

0

minus1

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


Ther

mal

yaw

(998400998400 )

(a)

4

3

2

1

0

minus1

minus2

minus3

Resid

ual (

998400998400)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)




0119909and 119863

0119910in 119883119884-direction


1198630119909=

Δ1198711

tan 120579119909

minus 1198631198711= 548659mm

1198630119910=

Δ1198712

tan 120579119910

minus 1198631198712= 508706mm

(34)



1198711= 1198631198712

=



Δ119874119909= (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

Δ119874119910= (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

Δ119874119911= 12059011198641015840

119905+ 1205831

(35)


compensation is1198821015840(1198751015840119909 1198751015840

119910 1198751015840

119911)

1198751015840

119909= 119875119909+ Δ119874119909= 119875119909+ (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

1198751015840

119910= 119875119910+ Δ119874119910= 119875119910+ (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

1198751015840

119911= 119875119911+ Δ119874119911= 119875119911+ 12059011198641015840

119905+ 1205831

(36)



Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)


Sensor


HMI interface

Compensation value


Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller


AD

times

Machine tool

RS232

Part program


Theoretical value

FeedbackPC


Compensation model

USB



Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500


30

15

0

minus15

minus30

minus45

Time (min)





S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50


minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)


minus5

(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Elon

gatio

n (120583

m)

(b)


Ther

mal

yaw

(998400998400 )

0

minus10

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)


minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

yaw

(998400998400 )

(b)



120583 = (

1205831

1205832

1205833

) = (

minus24681

minus9845

minus10095

) 120590 = (

1205901

1205902

1205903

) = (

1411

6272

6571

)

(31)


(

119864119905

120579119909119905

120579119910119905

) = (

1205901

1205902

1205903

)(

1198641015840

119905

1205791015840

119909119905

1205791015840

119910119905

)+(

1205831

1205832

1205833

) (32)




119894|max and



RMSE = radic 1

119899

119899

sum

119894=1

(119910119894minus 119910119894)2

1198772

= 1 minussum119899

119894=1(119910119894minus 119910119894)2

sum119899

119894=1(119910119894minus 119910119894)2

120578 = 1 minus(1119899)sum

119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

(1119899)sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

= 1 minussum119899

119894=1

1003816100381610038161003816119910119894 minus 1199101198941003816100381610038161003816

sum119899

119894=1

10038161003816100381610038161199101198941003816100381610038161003816

(33)








Ther

mal

pitc

h (998400998400

)

0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)

minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

pitc

h (998400998400

)


(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus1

minus05

0

05

1

(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus05

0

05

1

(b)


0 5 10 15 20

0

05

1

Lag

Sam

ple a

utoc

orre

latio


minus05

(a)

0 5 10 15 20LagSa

mpl

e par

tial a

utoc

orre

latio

ns


0

05

1

minus05

(b)




0

minus10

minus20

minus30

minus40

Elon

gatio

n (120583

m)

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Resid

ual (120583

m)

2

1

0

minus1

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


Ther

mal

yaw

(998400998400 )

(a)

4

3

2

1

0

minus1

minus2

minus3

Resid

ual (

998400998400)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)




0119909and 119863

0119910in 119883119884-direction


1198630119909=

Δ1198711

tan 120579119909

minus 1198631198711= 548659mm

1198630119910=

Δ1198712

tan 120579119910

minus 1198631198712= 508706mm

(34)



1198711= 1198631198712

=



Δ119874119909= (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

Δ119874119910= (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

Δ119874119911= 12059011198641015840

119905+ 1205831

(35)


compensation is1198821015840(1198751015840119909 1198751015840

119910 1198751015840

119911)

1198751015840

119909= 119875119909+ Δ119874119909= 119875119909+ (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

1198751015840

119910= 119875119910+ Δ119874119910= 119875119910+ (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

1198751015840

119911= 119875119911+ Δ119874119911= 119875119911+ 12059011198641015840

119905+ 1205831

(36)



Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)


Sensor


HMI interface

Compensation value


Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller


AD

times

Machine tool

RS232

Part program


Theoretical value

FeedbackPC


Compensation model

USB



Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500


30

15

0

minus15

minus30

minus45

Time (min)





S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ther

mal

pitc

h (998400998400

)

0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


(a)

minus15

minus10

minus05

00

05

10

15

20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

Ther

mal

pitc

h (998400998400

)


(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus1

minus05

0

05

1

(b)


0 5 10 15 20Lag


minus05

0

05

1

Sam

ple a

utoc

orre

latio

n

(a)

0 5 10 15 20Lag


Sam

ple p

artia

l aut

ocor

relat

ions

minus05

0

05

1

(b)


0 5 10 15 20

0

05

1

Lag

Sam

ple a

utoc

orre

latio


minus05

(a)

0 5 10 15 20LagSa

mpl

e par

tial a

utoc

orre

latio

ns


0

05

1

minus05

(b)




0

minus10

minus20

minus30

minus40

Elon

gatio

n (120583

m)

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Resid

ual (120583

m)

2

1

0

minus1

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


Ther

mal

yaw

(998400998400 )

(a)

4

3

2

1

0

minus1

minus2

minus3

Resid

ual (

998400998400)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)




0119909and 119863

0119910in 119883119884-direction


1198630119909=

Δ1198711

tan 120579119909

minus 1198631198711= 548659mm

1198630119910=

Δ1198712

tan 120579119910

minus 1198631198712= 508706mm

(34)



1198711= 1198631198712

=



Δ119874119909= (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

Δ119874119910= (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

Δ119874119911= 12059011198641015840

119905+ 1205831

(35)


compensation is1198821015840(1198751015840119909 1198751015840

119910 1198751015840

119911)

1198751015840

119909= 119875119909+ Δ119874119909= 119875119909+ (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

1198751015840

119910= 119875119910+ Δ119874119910= 119875119910+ (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

1198751015840

119911= 119875119911+ Δ119874119911= 119875119911+ 12059011198641015840

119905+ 1205831

(36)



Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)


Sensor


HMI interface

Compensation value


Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller


AD

times

Machine tool

RS232

Part program


Theoretical value

FeedbackPC


Compensation model

USB



Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500


30

15

0

minus15

minus30

minus45

Time (min)





S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

minus10

minus20

minus30

minus40

Elon

gatio

n (120583

m)

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Resid

ual (120583

m)

2

1

0

minus1

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)


Ther

mal

yaw

(998400998400 )

(a)

4

3

2

1

0

minus1

minus2

minus3

Resid

ual (

998400998400)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(b)




0119909and 119863

0119910in 119883119884-direction


1198630119909=

Δ1198711

tan 120579119909

minus 1198631198711= 548659mm

1198630119910=

Δ1198712

tan 120579119910

minus 1198631198712= 508706mm

(34)



1198711= 1198631198712

=



Δ119874119909= (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

Δ119874119910= (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

Δ119874119911= 12059011198641015840

119905+ 1205831

(35)


compensation is1198821015840(1198751015840119909 1198751015840

119910 1198751015840

119911)

1198751015840

119909= 119875119909+ Δ119874119909= 119875119909+ (119863 + 54866) (120590

21205791015840

119909119905+ 1205832)

1198751015840

119910= 119875119910+ Δ119874119910= 119875119910+ (119863 + 50871) (120590

31205791015840

119910119905+ 1205833)

1198751015840

119911= 119875119911+ Δ119874119911= 119875119911+ 12059011198641015840

119905+ 1205831

(36)



Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)


Sensor


HMI interface

Compensation value


Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller


AD

times

Machine tool

RS232

Part program


Theoretical value

FeedbackPC


Compensation model

USB



Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500


30

15

0

minus15

minus30

minus45

Time (min)





S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ther

mal

pitc

h (998400998400

)


0

minus5

minus10

minus15

minus20

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

(a)

Time series

minus50 0 50 100 150 200 250 300 350 400 450 500

Time (min)

minus15

minus10

minus05

00

05

10

15

20

Resid

ual (

998400998400)

(b)


Sensor


HMI interface

Compensation value


Compensation wayPLC

FB2

FB3 Variables

Axis-determination

CNC controller


AD

times

Machine tool

RS232

Part program


Theoretical value

FeedbackPC


Compensation model

USB



Zth

erm

al er

ror (120583

m)

0 100 200 300 400 500


30

15

0

minus15

minus30

minus45

Time (min)





S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S3 measurement

30

20

10

0Xth

erm

al er

ror (120583

m)

0 100 200 300 400 500

Postcompensation

Time (min)


S4 measurement

Yth

erm

al er

ror (120583

m)

15

10

5

0

minus5

minus10

minus150 100 200 300 400 500

Postcompensation

Time (min)





4in


6 Conclusions





Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article thermal-induced errors prediction and ... · measurement and described a method to...

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