Thermal Error Modeling Method for a CNC Machine Tool Feed Drive

Research ArticleThermal Error Modeling Method for a CNC MachineTool Feed Drive System

Kuo Liu1 Mingjia Sun2 Yuliang Wu2 and Tiejun Zhu2

1College of Mechanical Engineering Jilin University Changchun 130025 China2State Key Laboratory Shenyang Machine Tool (Group) Co Ltd Shenyang 110142 China

Correspondence should be addressed to Kuo Liu liukuo0727qqcom

Received 17 July 2015 Revised 18 September 2015 Accepted 20 September 2015

Academic Editor Ricardo Aguilar-Lopez

Copyright copy 2015 Kuo Liu et al This 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

Thedisadvantages of the common current thermal errormodelingmethods forCNCmachine tool feed drive systemswere analyzedsuch as the requirement ofmany temperature sensors to reach high accuracy and poor applicability of differentmoving states A newrobustmodelingmethod based on the heat transfer theory is proposed and the procedure of the thermal tests for a feed drive systemis presented Multiple regression method and robust modeling method based on the heat transfer theory were respectively usedto establish a thermal error model and a pointer automatic optimizer was used to optimize the parameters in the robust model Acompensation simulation was conducted under five different moving states using these twomodeling methods and the advantagesof the robust modeling method were proved Finally the compensation effect of the robust modeling method was verified under arandom moving state on a vertical machining center

1 Introduction

Currently mainly two methods are used to reduce CNCmachine tool thermal errors error prevention and errorcompensation [1] The error prevention method tries toeliminate or reduce the deformation of machines during thedesign or construction phase of the machine tool [2] such asscrew or nut cooling using thermally insensitive materialsand symmetric design The error prevention method cancontrol the thermal errors of machines to some extent buthas some disadvantages such as higher cost Moreover not allthe feed drive systems can be designed as a heat symmetricstructure The error prevention method compensates anychanges in dimensions due to thermal fluctuations Thismethod can be implemented during any designconstructionphase of themachine tool [3]The error preventionmethod isa ldquosoft technologyrdquo and inexpensive but the problem is howto generate opposite errors in a machine tool rationally at aspecial time and position

Many studies have developed compensation schemes tocounteract the thermal deformation of a feed drive systemSome studies have established thermal error models using

the multiple linear regression method [4ndash7] However themultiple linear regression method has poor robustness If theposition and speed of a feed drive system in an actual cuttingprocess is different from those of the modeling tests theprediction effects will be poor Other studies have establishedthermal error models using the artificial neural networkand other methods [8ndash11] An artificial neural network canonly provide a better effect when complete input and outputinformation are used Mistakes can occur if inaccurate inputand output information are used In summary the currentstudies established thermal error models by mainly using themathematical methods to predict thermal errors by readingthe temperatures of the critical points in real time The maindisadvantage of the existingmethods is that when themovingstate of the machine tool is changed the prediction effect willbe poor Moreover many temperature sensors are needed toobtain high accuracy resulting in a higher cost and lowersystem reliability

Some machine tool builders have applied thermal errorcompensation on CNC machine tools such as OKUMArsquosthermofriendly technology andMAZAKrsquos intelligent thermalshield Howevermost of them compensate the thermal errors

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 436717 6 pageshttpdxdoiorg1011552015436717

2 Mathematical Problems in Engineering

Temperature sensor 1


Figure 1 Placement of temperature sensors on the machine tool

caused by ambient temperature variation and spindle rota-tion For the thermal errors caused by ball-screw friction ballscrew or nut cooling is used instead of error compensation

According to the current studies and applications ofthermal error compensation a new robust modeling methodbased on the heat transfer theory is proposed In thismodeling method the thermal errors caused by ambienttemperature variation and ball-screw friction were calculatedseparately Based on the heat production heat conductionand heat convection theory the ball-screw temperature fieldat any time can be obtained to predict ball-screw thermalerrors Finally the developed method was compared withthemultiple regressionmodelingmethod through simulationand experiment

2 Testing of Thermal Errors

The thermal errors of a feed drive system were investigatedon the 119909-axis of a vertical machining center This machinetool used a cross-sliding table structure with one end fixedand one end supporting ball screws The control systemused is FANUC 0iMATE-MD the strokes of 119909119910119911-axis are850500540mm respectively and the maximum speeds are323230mmin respectively

Two temperature sensors whose tolerance is plusmn01∘C (5ndash45∘C) were placed on the nut and base near the bearingblock [12ndash14] as shown in Figure 1 Several experiments wereconducted to optimize the best placement of the temperaturesensors

Thermal errors were investigated using a dual-frequencylaser interferometer XL80 system as shown in Figure 2Importantly the ldquoexpansion compensationrdquo should be setat 20∘C to cancel the ambient temperature compensationfunction of the software

The tests were conducted under five moving states andthe test parameters are shown in Table 1

For example the procedure of thermal tests in state 1 isdescribed as follows

(1) Test the positioning error of 119909-axis in the range 0ndash700mm and record the values of temperature sensors1 and 2

(2) Let 119909-axis move in the range 210ndash490mm at8000mmmin for a period of time (sim10min)

Table 1 Parameters of thermal error tests

Speed(mmmin)

Range(mm)

State 1 8000 210ndash490State 2 6500 210ndash490State 3 15000 210ndash490State 4 8000 0ndash210State 5 8000 0ndash700

Figure 2 Investigation of thermal errors using a laser interferome-ter

(3) Stop moving Test the positioning error and recordthe values of temperature sensors 1 and 2

(4) Repeat steps (2) and (3) until 119909-axis reaches the heatbalance

(5) Let 119909-axis stop at a certain position to cool downTest the positioning error at intervals (sim10min) andrecord the values of temperature sensors 1 and 2

Based on the above tests thermal error curves (Figure 3)and temperature curves (Figure 4) were obtained In Figure 3the warm-up curves are marked in blue and the cool-downcurves are marked in red

Tests were conducted under states 2ndash5 in the samemanner

3 Multiple Regression Modeling Method

The multiple regression model is a multiple-input-single-output system The multiple regression method has someadvantages such as a simple modeling procedure When themoving state of a machine tool is constant a relatively highcompensation accuracy can be obtained The thermal errormodel established with the multiple regression method canbe described as follows

119864 = (119886119898119879119898

119899

+ 119886119898minus1

119879119898minus1

119899

+ sdot sdot sdot + 1198861119879119899+ 119887119898119879119898

119887

+ 119887119898minus1

119879119898minus1

119887

+ sdot sdot sdot + 1198871119879119887+ 119888) 119875

119909

(1)

where 119879119899is the real-time temperature of sensor 1 ∘C 119879

119887

is the real-time temperature of sensor 2 ∘C 119875119909is the real-

time position of 119909-axis mm 119886119894and 119887119894are the multinomial

coefficient of 119879119899and 119879

119887 respectively

Mathematical Problems in Engineering 3

116min

0min

minus10

0

10

20

30

40

50

60

70

Erro

r val

ue (120583

m)

7000 300 400 500 600100 200Position of x-axis (mm)

Figure 3 Thermal error curves of 119909-axis

Tem

pera

ture

(∘C)

Temperature of nutTemperature of base

17175

18185

19195

20205

21215

22

2000 4000 6000 8000 10000 120000Time (s)

Figure 4 Temperature curves of 119909-axis

119886119894and 119887

119894can be determined using the least square

method According to previous studies themaximumdegreeof polynomial 119898 le 4 Moreover Δ119879

119899and Δ119879

119887can be used

instead of 119879119899and 119879

119887in (1) to obtain the same result Because

they are both used to fit the slopes of the thermal error curvesthe only difference is the values of 119886

119894 119887119894 and 119888

4 Robust Modeling Method Based onthe Heat Transfer Theory

The thermal error119864 of a feed drive system can be divided intotwo parts119864

119890(caused by the changes in ambient temperature)

and 119864119898(caused by the nutrsquos movement) According to the

temperature superposition principle they can be superposed[15] that is the temperature response of multiple sources isthe same as the sum of the temperature responses of all thesingle sources

The screw is discretized into119872 segments and the lengthof each segment is 119871 as shown in Figure 5

M

L1 L2 L3 L

L

i LM

Px

Figure 5 Discretization of the screw

41 Errors Caused by the Changes in Ambient Temperature Ingeneral the change in ambient temperature is slowThereforethe change in ball-screw temperature caused by the changein ambient temperature is also slow Therefore 119864

119890can be

described as follows

119864119890= 120572 (119879

119887minus 1198791198870) 119875119909 (2)

where 120572 is the thermal expansion coefficient of ball screw120583m(m times

∘C) 119879119887is the real-time temperature of sensor 2 ∘C

1198791198870is the initial temperature of sensor 2 during the test ∘C

42 Errors Caused by Movement The temperature distribu-tion of each point in an object is known as temperaturefield which is not only the function of position but alsothe function of time [16] Because ball-screw axial thermaldeformation is themain source of accuracy loss in a feed drivesystem ball screw can be simplified to a one-dimensional bar

For a certain segment 119871119894of ball screw nut friction can

cause a temperature rise of 119871119894 119871119894will conduct heat to both

sides of 119871119894and simultaneously exchange the heat with the

surrounding airTherefore the thermal equilibrium equationfor 119871119894can be expressed as follows

119888119898Δ119879119871 119894= 119876119871 119894-1 minus 119876119871 119894-2 minus 119876119871 119894-3 (3)

where 119888 is the heat capacity of ball screw J(kg times ∘C)119898 is themass of a segment of ball screw kg Δ119879

119871 119894is the temperature

rise of 119871119894 ∘C 119876

119871 119894-1 is the friction heat production of 119871119894 J

119876119871 119894-2 is the axial heat conduction of 119871

119894to both sides J 119876

119871 119894-3is the heat convection of 119871

119894with the surrounding air J

421 Friction Heat Production For 119871119894 the total friction heat

production 119876119871 119894-1 is

119876119871 119894-1 = 119876119873 (4)

where119876 is the heat production of 119871119894after one friction J119873 is

the number of frictions of 119871119894

422 Axial Heat Conduction If the nut moves on 119871119894at time

119905 then the axial heat conduction 119876119871 119894-2 during (119905 119905 + Δ119905) can

be expressed as follows

119876119871 119894-2 = 120582119860

(119879119871 119894minus 119879119871 119894minus1

) + (119879119871 119894minus 119879119871 119894+1

)

119871

Δ119905(5)

where 120582 is the coefficient of heat conduction W(m times∘C) 119860

is the cross-sectional area of ball screw m2 119879119871 119894

is the tem-perature of 119871

119894at a certain time ∘C 119879

119871 119894minus1is the temperature of


119871119894minus1

at a certain time ∘C 119879119871 119894+1

is the temperature of 119871119894+1

at acertain time ∘C

For 1198711and 119871

119872especially

1198761198711-2 = 120582119860

1198791198711minus 1198791198710

1198711198980

Δ119905 + 120582119860

1198791198711minus 1198791198712

119871

Δ119905

119876119871119872-2 = 120582119860

119879119871119872

minus 119879119871119872minus1

119871

Δ119905 + 120582119860

119879119871119872

minus 119879119871119872+1

1198711198981

Δ119905

(6)

423 Heat Convection If the nut moves on 119871119894at time 119905

then the heat convection 119876119871 119894-3 during time (119905 119905 + Δ119905) can be

expressed as follows

119876119871 119894-3 = ℎ119878 (119879

119871 119894minus 119879119891) Δ119905 (7)

where ℎ is the heat exchange coefficient [17]W(m2times∘C) 119878 isthe heat exchange area of 119871

119894 119878 = 119860times119871 m2 119879

119891is the ambient

temperature ∘CΔ119879119871 119894can be obtained from (3)ndash(7)

The thermal change of ball screw is a dynamic processtherefore the temperature field of the ball screw also changesdynamically For 119871

119894119879119871 119894(119905+Δ119905) at time 119905+Δ119905 can be calculated

using 119879119871 119894(119905) at time 119905 [18]

119879119871 119894(119905 + Δ119905) = 119879

119871 119894(119905) + Δ119879

119871 119894 (8)

Therefore ball-screw thermal errors caused bymovementat a certain time 119905 can be expressed as follows

119864119898=

119872

sum

119894=1

120572 (119879119871 119894minus 119879119891) 119871119894 (9)

43 Total Errors of FeedDrive System Combining (2) and (3)the total thermal error model of the feed drive system can beobtained as follows

119864 = 119864119890+ 119864119898 (10)

A robust modeling method based on the heat transfertheory considers the dynamic process of a feed drive systemrsquostemperature field therefore even if the moving state of thefeed drive system changes an excellent compensation resultcan still be obtained

44 Identification of Parameters In the robust model someparameters are difficult to determine such as heat capacity 119888coefficient of heat conduction 120582 heat exchange coefficient ℎand heat production of 119871

119894after one friction 119876 A parameter

identification method is needed to determine these parame-ters The above mentioned parameters were optimized usingthe pointer automatic optimizer of the ISIGHT 50 softwareand the optimal values of 1198881015840 ℎ1015840 1205821015840 and 1198761015840 were obtained

Erro

r val

ues

Erro

r val

ues

of st

ate5

(120583m

)

minus8minus6minus4minus2

02

of st

ate1

(120583m

)

minus6

minus4

minus2

0

2

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000

Position of x-axis (mm)


Figure 6 Simulation results of M1 for state 1 and M5 for state 5

5 Simulations and Experiments

51 Simulations The compensation results of the multipleregressionmodel and robustmodel based on the heat transfertheory were compared using Matlab R2014a

A positioning error compensation was also included inthese twomodels considering the existing positioning errorsThe positioning error compensation was used to compensate0min errors using the following

119864119901= 11988941198754

119909

+ 11988931198753

119909

+ 11988921198752

119909

+ 1198891119875119909+ 1198890 (11)

511 Results of the Multiple Regression Model In a multipleregression model 119879

119899rsquos four-order polynomial and 119879

119887rsquos one-

order polynomial were used as shown in (12) because thecorrelation coefficient between 119879

119899and 119864 is larger than that

between 119879119887and 119864

119864 = (11988641198794

119899

+ 11988631198793

119899

+ 11988621198792

119899

+ 1198861119879119899+ 1198871119879119887+ 119888) 119875

119909 (12)

Themultiple regressionmodels were established based onthe data of states 1 and 5 M1 is the model established basedon the data of state 1 and M5 is the model established basedon the data of state 5

The compensation results ofM1 for state 1 andM5 for state5 are shown in Figure 6

The compensation results of M1 for states 2ndash5 are shownin Figure 7

512 Results of the Robust Model Based on the Heat TransferTheory A robustmodel based on the heat transfer theorywasestablished based on the data of state 1 The compensationresults for states 1ndash5 are shown in Figure 8

Figures 6ndash8 show that if the modeling state is the sameas the verifying state good compensation results can beobtained from the multiple regression model otherwise thecompensation results are poor However good compensationresults can be obtained from a robustmodel based on the heattransfer theory even if the modeling state is not the same as

Mathematical Problems in Engineering 5Er

ror v

alue

sof

stat

e2(120583

m)

Erro

r val

ues

of st

ate3

(120583m

)Er

ror v

alue

sof

stat

e4(120583

m)

Erro

r val

ues

of st

ate5

(120583m

)

minus10

0

10

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000minus20

020406080

0

50

100

0

50

100

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000





Figure 7 Simulation results of M1 for states 2ndash5

0246

Erro

r val

ues

of st

ate1

(120583m

)Er

ror v

alue

sof

stat

e2(120583

m)

Erro

r val

ues

of st

ate3

(120583m

)Er

ror v

alue

sof

stat

e4(120583

m)

Erro

r val

ues

of st

ate5

(120583m

)

minus4minus2

0246

minus2

minus10

minus5

0

minus15minus10

minus50

minus10minus5

05

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000

0 200 300 400 500 600 700100

100 200 300 400 500 600 7000

0 200 300 400 500 600 700100






Figure 8 Simulation results for states 1ndash5

Table 2 Moving states in compensation tests

Speed(mmmin)

Range(mm)

Time(min)

State 1 8000 0ndash280 6State 2 5000 490ndash700 10State 3 10000 240ndash450 5State 4 12000 0ndash700 2

signals

Motor


Position

Ball

FOCAS

Temperature acquisition

Compensation modelCompensation

Compensator (PC)FANUC 0i mate-MD

+ minus

screw

Base

Job program

Figure 9 Diagram of data acquisition and error compensation

the verifying states because the robust model considers themoving information of a feed drive system Therefore it canbe concluded that the robustmodel based on the heat transfertheory has stronger adaptability than a multiple regressionmodel

52 Experiments The advantages of the robust model basedon the heat transfer theory were verified by simulationreported in Section 51 In this section the compensationeffect will be verified through experiments

The reading of a feed drive systemrsquos position from CNCand the writing of compensation values to CNC are neededfor compensation experiments Fanuc Open CNC API Spec-ifications (FOCAS) was used to obtain these reading andwriting functions as shown in Figure 9

The 119909-axis was moved according to the moving statesshown in Table 2

Position errors were investigated using a laser interfer-ometer after each moving state and the results are shown inFigure 10

The simulation and test results in Figures 8 and 10show that good compensation results were obtained fromthe robust model based on the heat transfer theory underdifferent moving states

6 Conclusions

The disadvantages of existing thermal error models wereanalyzed and a new robust model based on the heat transfertheory was proposed Multiple regression and robust modelswere derived and used for simulations and experiments Theresults show that the robust model based on the heat transfertheory has better accuracy and robustness and can satisfy


Starting residual errorResidual error after state 1Residual error after state 2

Residual error after state 3Residual error after state 4

Resid

ual e

rror

(120583m

)

0

2

4

6

8

minus4

minus2

10

12

100 200 300 400 500 600 7000Position of x-axis (mm)

Figure 10 Compensation effects of the robust model based on theheat transfer theory

the actual application The advantages of this technology areas follows

(1) Only one temperature sensor is needed for a linearaxis in real-time compensation and the cost is low

(2) Themanufacturing accuracy of a singleworkpiece canbe improved

(3) The manufacturing consistency of the bulk of work-piece can be improved and the rejection rate can bereduced

(4) Machines do not need to warm up before themachin-ing Thus time and power costs can be saved

(5) Machines do not depend on a constant-temperatureworkshop and construction and power costs can besaved

Through the above analyses it can be concluded that thistechnology has excellent potential

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by National Science andTechnology Major Project of Peoplersquos Republic of China(2013ZX04011011)

References

[1] R RameshM AMannan andA N Poo ldquoError compensationin machine toolsmdasha review Part I Geometric cutting-forceinduced and fixture-dependent errorsrdquo International Journal ofMachine Tools and Manufacture vol 40 no 9 pp 1235ndash12562000

[2] M Mori H Mizuguchi M Fujishima Y Ido N Mingkaiand K Konishi ldquoDesign optimization and development ofCNC lathe headstock to minimize thermal deformationrdquo CIRPAnnalsmdashManufacturing Technology vol 58 no 1 pp 331ndash3342009

[3] 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

[4] J Zhu J Ni and A J Shih ldquoRobust machine tool thermalerror modeling through thermal mode conceptrdquo Journal ofManufacturing Science and Engineering vol 130 no 6 pp0610061ndash0610069 2008

[5] O Horejs MMares P Kohout P Barta and J Hornych ldquoCom-pensation of machine tool thermal errors based on transferfunctionsrdquoMM Science Journal pp 162ndash165 2010

[6] M Pajor and J Zapłata ldquoCompensation of thermal deforma-tions of the feed screw in a CNC machine toolrdquo Advances inManufacturing Science and Technology vol 35 pp 9ndash17 2011

[7] J Zhu Robust thermal error modeling and compensation forCNC machine tools [PhD thesis] University of Michigan AnnArbor Mich USA 2008

[8] C D Mize and J C Ziegert ldquoNeural network thermal errorcompensation of amachining centerrdquoPrecision Engineering vol24 no 4 pp 338ndash346 2000

[9] Z F Jin and P Wang ldquoNeural network-based thermal errormodeling in ball screwrdquo in Modular Machine Tool and Auto-matic Manufacturing Technique pp 67ndash70 2012

[10] M T Ozkan ldquoExperimental and artificial neural network studyof heat formation values of drilling and boring operations on Al7075 T6workpiecerdquo Indian Journal of Engineering andMaterialsSciences vol 20 no 4 pp 259ndash268 2013

[11] X-H Yao J-Z Fu and Z-C Chen ldquoBayesian networksmodeling for thermal error of numerical controlmachine toolsrdquoJournal of Zhejiang University Science A vol 9 no 11 pp 1524ndash1530 2008

[12] J Y Yan and J G Yang ldquoApplication of synthetic grey corre-lation theory on thermal point optimization for machine toolthermal error compensationrdquo International Journal of AdvancedManufacturing Technology vol 43 no 11-12 pp 1124ndash1132 2009

[13] T Zhang W H Ye R J Liang P H Lou and X L YangldquoTemperature variable optimization for precision machine toolthermal error compensation on optimal thresholdrdquo ChineseJournal of Mechanical Engineering vol 26 no 1 pp 158ndash1652013

[14] Y X Li J G Yang T Gelvis and Y Y Li ldquoOptimizationof measuring points for machine tool thermal error basedon grey system theoryrdquo International Journal of AdvancedManufacturing Technology vol 35 no 7-8 pp 745ndash750 2008

[15] J Y Xia BWu and YM Hu ldquoThe thermal dynamic character-istic of ballndashscrew under the variational multindashthermal sourcerdquoChinese Mechanical Engineering vol 19 pp 955ndash958 2008

[16] B J Liu Temperature field and thermal deformation of feedsystem on gantry machining center [PhD thesis] NanjingUniversity of Aeronautics and Astronautics Nanjing China2013

[17] A Verl and S Frey ldquoCorrelation between feed velocity andpreloading in ball screw drivesrdquo CIRP AnnalsmdashManufacturingTechnology vol 59 no 1 pp 429ndash432 2010

[18] G HolroydThemodeling and correction of ball-screw geometricthermal and load errors on CNC machine tools [PhD thesis]University of Huddersfield Huddersfield UK 2007

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Stochastic AnalysisInternational Journal of




Figure 1 Placement of temperature sensors on the machine tool

caused by ambient temperature variation and spindle rota-tion For the thermal errors caused by ball-screw friction ballscrew or nut cooling is used instead of error compensation

According to the current studies and applications ofthermal error compensation a new robust modeling methodbased on the heat transfer theory is proposed In thismodeling method the thermal errors caused by ambienttemperature variation and ball-screw friction were calculatedseparately Based on the heat production heat conductionand heat convection theory the ball-screw temperature fieldat any time can be obtained to predict ball-screw thermalerrors Finally the developed method was compared withthemultiple regressionmodelingmethod through simulationand experiment

2 Testing of Thermal Errors

The thermal errors of a feed drive system were investigatedon the 119909-axis of a vertical machining center This machinetool used a cross-sliding table structure with one end fixedand one end supporting ball screws The control systemused is FANUC 0iMATE-MD the strokes of 119909119910119911-axis are850500540mm respectively and the maximum speeds are323230mmin respectively

Two temperature sensors whose tolerance is plusmn01∘C (5ndash45∘C) were placed on the nut and base near the bearingblock [12ndash14] as shown in Figure 1 Several experiments wereconducted to optimize the best placement of the temperaturesensors

Thermal errors were investigated using a dual-frequencylaser interferometer XL80 system as shown in Figure 2Importantly the ldquoexpansion compensationrdquo should be setat 20∘C to cancel the ambient temperature compensationfunction of the software

The tests were conducted under five moving states andthe test parameters are shown in Table 1

For example the procedure of thermal tests in state 1 isdescribed as follows

(1) Test the positioning error of 119909-axis in the range 0ndash700mm and record the values of temperature sensors1 and 2

(2) Let 119909-axis move in the range 210ndash490mm at8000mmmin for a period of time (sim10min)

Table 1 Parameters of thermal error tests

Speed(mmmin)

Range(mm)

State 1 8000 210ndash490State 2 6500 210ndash490State 3 15000 210ndash490State 4 8000 0ndash210State 5 8000 0ndash700

Figure 2 Investigation of thermal errors using a laser interferome-ter

(3) Stop moving Test the positioning error and recordthe values of temperature sensors 1 and 2

(4) Repeat steps (2) and (3) until 119909-axis reaches the heatbalance

(5) Let 119909-axis stop at a certain position to cool downTest the positioning error at intervals (sim10min) andrecord the values of temperature sensors 1 and 2

Based on the above tests thermal error curves (Figure 3)and temperature curves (Figure 4) were obtained In Figure 3the warm-up curves are marked in blue and the cool-downcurves are marked in red

Tests were conducted under states 2ndash5 in the samemanner

3 Multiple Regression Modeling Method

The multiple regression model is a multiple-input-single-output system The multiple regression method has someadvantages such as a simple modeling procedure When themoving state of a machine tool is constant a relatively highcompensation accuracy can be obtained The thermal errormodel established with the multiple regression method canbe described as follows

119864 = (119886119898119879119898

119899

+ 119886119898minus1

119879119898minus1

119899

+ sdot sdot sdot + 1198861119879119899+ 119887119898119879119898

119887

+ 119887119898minus1

119879119898minus1

119887

+ sdot sdot sdot + 1198871119879119887+ 119888) 119875

119909

(1)

where 119879119899is the real-time temperature of sensor 1 ∘C 119879

119887

is the real-time temperature of sensor 2 ∘C 119875119909is the real-

time position of 119909-axis mm 119886119894and 119887119894are the multinomial

coefficient of 119879119899and 119879

119887 respectively


116min

0min

minus10

0

10

20

30

40

50

60

70

Erro

r val

ue (120583

m)



Tem

pera

ture

(∘C)


17175

18185

19195

20205

21215

22

2000 4000 6000 8000 10000 120000Time (s)


119886119894and 119887



119899and Δ119879

119887can be used

instead of 119879119899and 119879



119894 119887119894 and 119888







M

L1 L2 L3 L

L

i LM

Px



119890can be


119864119890= 120572 (119879

119887minus 1198791198870) 119875119909 (2)









119888119898Δ119879119871 119894= 119876119871 119894-1 minus 119876119871 119894-2 minus 119876119871 119894-3 (3)



rise of 119871119894 ∘C 119876







production 119876119871 119894-1 is

119876119871 119894-1 = 119876119873 (4)






119876119871 119894-2 = 120582119860

(119879119871 119894minus 119879119871 119894minus1

) + (119879119871 119894minus 119879119871 119894+1

)

119871

Δ119905(5)







119871119894minus1




For 1198711and 119871

119872especially

1198761198711-2 = 120582119860

1198791198711minus 1198791198710

1198711198980

Δ119905 + 120582119860

1198791198711minus 1198791198712

119871

Δ119905

119876119871119872-2 = 120582119860

119879119871119872

minus 119879119871119872minus1

119871

Δ119905 + 120582119860

119879119871119872

minus 119879119871119872+1

1198711198981

Δ119905

(6)




119876119871 119894-3 = ℎ119878 (119879

119871 119894minus 119879119891) Δ119905 (7)


119894 119878 = 119860times119871 m2 119879





using 119879119871 119894(119905) at time 119905 [18]

119879119871 119894(119905 + Δ119905) = 119879

119871 119894(119905) + Δ119879

119871 119894 (8)


119864119898=

119872

sum

119894=1

120572 (119879119871 119894minus 119879119891) 119871119894 (9)


119864 = 119864119890+ 119864119898 (10)





Erro

r val

ues

Erro

r val

ues

of st

ate5

(120583m

)


02

of st

ate1

(120583m

)

minus6

minus4

minus2

0

2

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000







119864119901= 11988941198754

119909

+ 11988931198753

119909

+ 11988921198752

119909

+ 1198891119875119909+ 1198890 (11)



119887rsquos one-



between 119879119887and 119864

119864 = (11988641198794

119899

+ 11988631198793

119899

+ 11988621198792

119899

+ 1198861119879119899+ 1198871119879119887+ 119888) 119875

119909 (12)







ror v

alue

sof

stat

e2(120583

m)

Erro

r val

ues

of st

ate3

(120583m

)Er

ror v

alue

sof

stat

e4(120583

m)

Erro

r val

ues

of st

ate5

(120583m

)

minus10

0

10

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000minus20

020406080

0

50

100

0

50

100

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000






0246

Erro

r val

ues

of st

ate1

(120583m

)Er

ror v

alue

sof

stat

e2(120583

m)

Erro

r val

ues

of st

ate3

(120583m

)Er

ror v

alue

sof

stat

e4(120583

m)

Erro

r val

ues

of st

ate5

(120583m

)

minus4minus2

0246

minus2

minus10

minus5

0

minus15minus10

minus50

minus10minus5

05

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000

0 200 300 400 500 600 700100

100 200 300 400 500 600 7000

0 200 300 400 500 600 700100








Speed(mmmin)

Range(mm)

Time(min)


signals

Motor


Position

Ball

FOCAS




+ minus

screw

Base

Job program








6 Conclusions





Resid

ual e

rror

(120583m

)

0

2

4

6

8

minus4

minus2

10

12












Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










116min

0min

minus10

0

10

20

30

40

50

60

70

Erro

r val

ue (120583

m)



Tem

pera

ture

(∘C)


17175

18185

19195

20205

21215

22

2000 4000 6000 8000 10000 120000Time (s)


119886119894and 119887



119899and Δ119879

119887can be used

instead of 119879119899and 119879



119894 119887119894 and 119888







M

L1 L2 L3 L

L

i LM

Px



119890can be


119864119890= 120572 (119879

119887minus 1198791198870) 119875119909 (2)









119888119898Δ119879119871 119894= 119876119871 119894-1 minus 119876119871 119894-2 minus 119876119871 119894-3 (3)



rise of 119871119894 ∘C 119876







production 119876119871 119894-1 is

119876119871 119894-1 = 119876119873 (4)






119876119871 119894-2 = 120582119860

(119879119871 119894minus 119879119871 119894minus1

) + (119879119871 119894minus 119879119871 119894+1

)

119871

Δ119905(5)







119871119894minus1




For 1198711and 119871

119872especially

1198761198711-2 = 120582119860

1198791198711minus 1198791198710

1198711198980

Δ119905 + 120582119860

1198791198711minus 1198791198712

119871

Δ119905

119876119871119872-2 = 120582119860

119879119871119872

minus 119879119871119872minus1

119871

Δ119905 + 120582119860

119879119871119872

minus 119879119871119872+1

1198711198981

Δ119905

(6)




119876119871 119894-3 = ℎ119878 (119879

119871 119894minus 119879119891) Δ119905 (7)


119894 119878 = 119860times119871 m2 119879





using 119879119871 119894(119905) at time 119905 [18]

119879119871 119894(119905 + Δ119905) = 119879

119871 119894(119905) + Δ119879

119871 119894 (8)


119864119898=

119872

sum

119894=1

120572 (119879119871 119894minus 119879119891) 119871119894 (9)


119864 = 119864119890+ 119864119898 (10)





Erro

r val

ues

Erro

r val

ues

of st

ate5

(120583m

)


02

of st

ate1

(120583m

)

minus6

minus4

minus2

0

2

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000







119864119901= 11988941198754

119909

+ 11988931198753

119909

+ 11988921198752

119909

+ 1198891119875119909+ 1198890 (11)



119887rsquos one-



between 119879119887and 119864

119864 = (11988641198794

119899

+ 11988631198793

119899

+ 11988621198792

119899

+ 1198861119879119899+ 1198871119879119887+ 119888) 119875

119909 (12)







ror v

alue

sof

stat

e2(120583

m)

Erro

r val

ues

of st

ate3

(120583m

)Er

ror v

alue

sof

stat

e4(120583

m)

Erro

r val

ues

of st

ate5

(120583m

)

minus10

0

10

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000minus20

020406080

0

50

100

0

50

100

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000






0246

Erro

r val

ues

of st

ate1

(120583m

)Er

ror v

alue

sof

stat

e2(120583

m)

Erro

r val

ues

of st

ate3

(120583m

)Er

ror v

alue

sof

stat

e4(120583

m)

Erro

r val

ues

of st

ate5

(120583m

)

minus4minus2

0246

minus2

minus10

minus5

0

minus15minus10

minus50

minus10minus5

05

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000

0 200 300 400 500 600 700100

100 200 300 400 500 600 7000

0 200 300 400 500 600 700100








Speed(mmmin)

Range(mm)

Time(min)


signals

Motor


Position

Ball

FOCAS




+ minus

screw

Base

Job program








6 Conclusions





Resid

ual e

rror

(120583m

)

0

2

4

6

8

minus4

minus2

10

12












Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119871119894minus1




For 1198711and 119871

119872especially

1198761198711-2 = 120582119860

1198791198711minus 1198791198710

1198711198980

Δ119905 + 120582119860

1198791198711minus 1198791198712

119871

Δ119905

119876119871119872-2 = 120582119860

119879119871119872

minus 119879119871119872minus1

119871

Δ119905 + 120582119860

119879119871119872

minus 119879119871119872+1

1198711198981

Δ119905

(6)




119876119871 119894-3 = ℎ119878 (119879

119871 119894minus 119879119891) Δ119905 (7)


119894 119878 = 119860times119871 m2 119879





using 119879119871 119894(119905) at time 119905 [18]

119879119871 119894(119905 + Δ119905) = 119879

119871 119894(119905) + Δ119879

119871 119894 (8)


119864119898=

119872

sum

119894=1

120572 (119879119871 119894minus 119879119891) 119871119894 (9)


119864 = 119864119890+ 119864119898 (10)





Erro

r val

ues

Erro

r val

ues

of st

ate5

(120583m

)


02

of st

ate1

(120583m

)

minus6

minus4

minus2

0

2

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000







119864119901= 11988941198754

119909

+ 11988931198753

119909

+ 11988921198752

119909

+ 1198891119875119909+ 1198890 (11)



119887rsquos one-



between 119879119887and 119864

119864 = (11988641198794

119899

+ 11988631198793

119899

+ 11988621198792

119899

+ 1198861119879119899+ 1198871119879119887+ 119888) 119875

119909 (12)







ror v

alue

sof

stat

e2(120583

m)

Erro

r val

ues

of st

ate3

(120583m

)Er

ror v

alue

sof

stat

e4(120583

m)

Erro

r val

ues

of st

ate5

(120583m

)

minus10

0

10

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000minus20

020406080

0

50

100

0

50

100

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000






0246

Erro

r val

ues

of st

ate1

(120583m

)Er

ror v

alue

sof

stat

e2(120583

m)

Erro

r val

ues

of st

ate3

(120583m

)Er

ror v

alue

sof

stat

e4(120583

m)

Erro

r val

ues

of st

ate5

(120583m

)

minus4minus2

0246

minus2

minus10

minus5

0

minus15minus10

minus50

minus10minus5

05

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000

0 200 300 400 500 600 700100

100 200 300 400 500 600 7000

0 200 300 400 500 600 700100








Speed(mmmin)

Range(mm)

Time(min)


signals

Motor


Position

Ball

FOCAS




+ minus

screw

Base

Job program








6 Conclusions





Resid

ual e

rror

(120583m

)

0

2

4

6

8

minus4

minus2

10

12












Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ror v

alue

sof

stat

e2(120583

m)

Erro

r val

ues

of st

ate3

(120583m

)Er

ror v

alue

sof

stat

e4(120583

m)

Erro

r val

ues

of st

ate5

(120583m

)

minus10

0

10

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000minus20

020406080

0

50

100

0

50

100

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000






0246

Erro

r val

ues

of st

ate1

(120583m

)Er

ror v

alue

sof

stat

e2(120583

m)

Erro

r val

ues

of st

ate3

(120583m

)Er

ror v

alue

sof

stat

e4(120583

m)

Erro

r val

ues

of st

ate5

(120583m

)

minus4minus2

0246

minus2

minus10

minus5

0

minus15minus10

minus50

minus10minus5

05

100 200 300 400 500 600 7000

100 200 300 400 500 600 7000

0 200 300 400 500 600 700100

100 200 300 400 500 600 7000

0 200 300 400 500 600 700100








Speed(mmmin)

Range(mm)

Time(min)


signals

Motor


Position

Ball

FOCAS




+ minus

screw

Base

Job program








6 Conclusions





Resid

ual e

rror

(120583m

)

0

2

4

6

8

minus4

minus2

10

12












Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Resid

ual e

rror

(120583m

)

0

2

4

6

8

minus4

minus2

10

12












Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Thermal Error Modeling Method for a CNC Machine Tool Feed Drive