基于广义贝叶斯法则的机床热误差分析

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Jun.2013机 床 与 液 压Hydromechatronics Engineering Ve1.41 No.12DOI：10.3969/j.issn.1001-3881.2013.12.013Thermal Error Analysis in M achine Tools Based onGeneralized Interval BaysruleXIE FengyunSchool of Mechanical and Electronical Engineering，East China Jiaotong University，Nanchang 330013，China1.IntroductionAbstract：In traditional Baysrule of probability calculation，epistemic uncertainty and aleatoryuncertainty are not distinguished.Recently，a generalized interval Baysrule(GIBR)based onthe model interval WaS proposed.In GIBR，aleatory uncertainty is represented as probabilitymeasure：epistemic uncertainty is captured by interva1.To demonstrate the efectiveness ofGIBR，a case study of thermal error analyzing in machine tools of bearing is presented.The re-suits show that the GIBR has a borer performance to characterize thermal errors than that of thetraditional Baysrule。

Key words：thermal error，uncertainty，generalized interval Baysrule(GIBR)Thermal errors affect the accuracy of the work。

piece which is produced by on the metal·-cutting ma-chine tools considerably.Peklenik proposed the per。

centage of error from thermal effects will account from4O% t0 7O%.Recently.plenty of research has beendone on thermal errors.such as thermal error com。

pensation，therm al eror modeling，and prediction ofthermal deformationf 1-2 1.Although these methodsare processed by precise value or precise probability，it does not differentiate aleatory uncertainty that is a。

rised from the physical indeterm inacy of the worldand epistemic uncertainty is extracted from the uncer。

tain knowledge about its actua1 state. Recently.ageneralized interval Baysrule(GIBR)[3]which isbased on the model interval was proposed.In GIBR。

aleatory uncertainty is represented as probabilitymeasure；epistemic uncertainty is captured by inter-ReSeived：2013-01-22Jiangxi province natural science foundation(201 14BAB206003)；key laboratory of the ministry of education for vehicles and equip-ment(09JD03)XIE Fengyun.E-mail：xiefyun### 163.cornva1.Model interval can solve aleatory uncertainty andepistemic uncertainty simultaneously，and the calcu-lation could be simplifed in applications by Kaucherarithmetic[4]。

In this paper，the method to analyze thermal er-ror based on model interval probability is proposed。

The prior probability of temperature of machine toolsbearing is obtained by experiments in some interva1。

The posterior probability of unobservable therm al er-ror of machine tools bearing can be obtained by GIBRin some interva1.The result is a form of the model in。

terval probability that can improve the reliability ofmaking decision。

eralized interval theory is an extension of the classi。

cal BayesRule.Modelinterval5-6]and it is anextension of the classical interval with better algebraicbased on the Kaucher arithmetic.A model interval isnot constrained by lower and upper probability.Forinstance，in clasical interval，[4，7]is valid inter。

val and caled proper，[7，4]is invalid interval andcalled improper. However. they are allowable inmodel interva1.The relationship between proper andimproper intervals is established with the operator du-XIE Fengyun：Thermal Error Analysis in Machine Tools Based onGeneralized Interval Baysrule 63al as dual[x，x]：[x， ].The GIBR is stated thatP(E l A) p(A f Ef)P(Ef)dualp(A l E )dualp( )(1)Where，E (i1，， )are mutually dioint eventpartitions of sample space Q andand the boldface symbols are statedinterval probability。

Based on the GIBR，theassimilation can be solvedn∑p(E )1，i1the form of modelproblem of informationPrior probabilities andlikelihoods are constructed or solicited.If data are a-vailable，we may calculate interval probabilities bystatistical methods.If no data are available，domainexpeRs may give estimates of interval probability.Ifno knowledge is available at all，P [0，1]can beused。

Suppose the states of one or more variables 1， ， z at Scale X are not directly observable.Instead，the system can be observed via the variable Y corre-sponding to the unobservable Y at Scale Then P( 1，， z I Y)is obtained as：p(xl，， z I Y)p(x ， )f p(YI Y)P(Y f ， I dy) p(yl y)p(y l )p( ， 1)dydx。 f)(2)Where，P(Y l l，， 1)for variables l，，龙l atscale X and Y at scale Y，P(Y l Y)for observable Ycorresponding to Y，and the prior estimate for P( 1，， )[3]。

3.Thermal error experimentsIn order to experimentally study the thermal er。

ror for machine tools bearing， a quasi high-speedfeed system experimental bench(HUST.FS-0 1)wasset up.AVM1 82 linear grating scale with a resolution0f O.5 m is preferred to measure positioning errorsof different positions of the bearing.The Ptl O0 ther。

mal resistance with a measuring range of 0-150qCand corresponding intelligent thermal resistance mod-ulators are used to measure the temperatures of meas。

uring points. The experimental testing system isshown in Fig.1。

The signals of temperature sensors are obtainedby Advantech PCI1716L DA&C cards.Therm al er-rors were evaluated by the ACCOM softed with the VM1 82 grating scale providedby the Heindenhain Company[7].Experimental da。

ta was processed by wavelet de-noising.The tempera·ture of beating housing and the therm al error of bear-ing are shown Fig.2 and Fig.3，respectively。

Fig.1 Schematic diagram of testing system9·08·58.0蠢 7.57.06.50 20 4O 60 80 l00 12O 140Rum time/rainFig.2 Temperature of bearing housing20 4O 6O 8O 100 120 l40Runl time/minFig.3 Thermal error8 of bearing4.Thermal error model and result analysisAccording to form ula 4 we can obtain thatp(e I !)± f!! ： ]t I dp(e)p(Tt t)p(t le )p(e。)e)p(e)p(rl tc)p(t。I ec)p(e。)](3)dal[p(TI tp(rl )p(where，the posterior probability forP(eIT)，the pri-or thermal eror estimate for P(e)，the prior condi-tional probability forP(tl e)and P(tI e。).P(e。)1-dualp(e)，P(t。le)1-d酩口fp(tI e)，P(t。l e。)1-dualp(tle。)。

The temperature is divided into two intervals(-。，8]and(8，∞)，the thermal eror is di-vided into other four intervals， (- ∞，30]，(-∞，50]，(50，60]and(70，∞).For ex。

∑ 知∞ ∞如 如 ∞ 0dⅢr言0巴0葛d昌 LLHydromechatronics Engineeringample.assume the observed temperature iS T >8，and thermal error is e>70.the problem i8 how tocalculate P(e>70I T>8)and P(e>70I T>8)。

According to the classical Bayesrule。the datawas divided into 10 group of uniform spacing asshown in Fig.2 and Fig.3，P(e>70 lT>8)could beobtained as shown in Tab.1。

terval probability P(e>70 I T>8)is calculated byformula 3 and the results are shown ifl Tab.2. InTab.2，Interval division 1group 2 from Tab.2.Intervalis made of group 1 anddivision 2i8 made ofgroup 1，3 and group 2，3 from Tab.2.Interval divi-sion 3is made of group 1，3，4 and group 2，3，4from Tab.2.The prior interval probability and likeli-hoods will be obtained by the same cyclic iteration。

Tab.1 P(e>70 l T>8)based OI3 precise probabilityTab.2 P(e>70I T>8)based O13 model interval probabilityCompare the results between Tab.1 and Tab.2，the amplitude of variation of posterior probability byGIBR is smaller，and thermal error of the posteriorprobability is more accurate than that of the tradition-al Baysrule.In addition，with the group increas-ing，P(e>70IT>8)is close to real posterior proba-bility value 0.464 8 that can be obtained by recordeddata.In addition，the posterior probability P(e I T)will be updated instantly close to true value via GIBRwhile more prior data are being collected. Further·more，the posterior probability also will be closer tothe true value of engineering while prior intervalprobability and likelihoods are segmentation. Th eposterior probability calculation of therm al error isuseful for real-time monitoring of the system healthand reliability。

XIE Fengyun：Thermal Error Analysis in Machine Tools Based onGeneralized Interval Baysrule 655.ConclusionsUncertainties widely exist in processing of ther-mal error problem.The traditional therm al eror cal-culation by precise probability has some limitations，such as representing epistemic uncertainty，and in-consistency in the context of subjective probability。

In this paper，the proposed model interval is to char-acterize therm al eror of machine tools bearing.Mod-el interval has a capable of processing epistemic un-certainty and aleatory uncertainty simultaneously.Todemonstrate the proposed model interval，the posteri-or probability of machine tools bearing is calculatedby GIBR.The results show that the proposed methodhas a better perform ance to calculate the posteriorprobability of therm al eror. With two uncertaintycomponents considered simuhaneously，model inter-val form can improve reliability of the resuhs and pro-vide reliable basis for engineering decision-makingIn future work，some new algorithms of processed pri-or data are required，for instance，based on modelinterval and Hidden Markov model(HMM)may beconsidered。

基于广义贝叶斯法则的机床热误差分析谢锋云华东交通大学 机电学院，南昌 330013References：[1] XIE Fengyun.A Characterization of Thermal Error forMachine Tools Bearing Based on HMM J 1.MachineTool&Hydraulics，2012，40(17)：31-34。

[2] zHANG Ting，LIU Shihao.Overview of Thermal ErorCompensation Modeling for Numerical Control Machine[J].Machine T001＆ Hydraulics，2011，39(1)：122- 127。

3] WANG Y.Muhiscale Uncertainty Quantcation Basedon a Generalized Hidden Markov ModelJ 1.ASME Jour。

r4] Kaucher E.Interval analysis in the extended intervalspace IR[J].Computing Supplementa，1980，2：33- 49。

[5] Gardenes E，Sainz M A，Jorba L R，et a1.Modal inter-vals[J].Reliable Computing，2001(2)：77-111。

[6] Hu Y M，XIE F Y.An Optimization Method for Train-ing Ge neralized Hidden Markov Model based on Ge neral-ized Jensen Inequality[C]//Proceedings of the 9th Inter-national Conference on Informatics in Control，Automa·tion and Robotics(ICINCO 2012).2012：268-274。

[7] JIN Chao，wu B0，Hu Youmin.Wavelet Neural Net-work Based on NARMA.L2 Model for Prediction of Ther-mal Characteristics in a Feed System[J].Chinese iour。

摘要：在传统的贝叶斯法则概率计算中，偶然与认知不确定性没有区分，而在广义贝叶斯法则中，偶然不确定性表述为概率测量，认知不确定性通过区间来描述。以机床轴承热误差分析为例说 明了广义贝叶斯法则概率应用的有效性。与传统的贝叶斯法则进行了比较♂果表明：用广义贝叶斯法则能更好地描述 热误 差。

关键词：热误差；不确定性；广义贝叶斯法则中图分类号：TH133；TP391