一项高精度数控车床热误差补偿的研究外文文献翻译、中英文翻译
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Int J Adv Manuf Technol (2002) 19:850858Ownership and Copyright 2002 Springer-Verlag London LimitedA Study of High-Precision CNC Lathe Thermal Errors andCompensationP.-C. Tseng and J.-L. HoDepartment of Mechanical Engineering, National Chung-Hsing University, Taichung, TaiwanThis study is addressed at the thermal deformation errorsresulting from temperature rise that contribute to 40%70%of the precision errors in machining at a turning centre, andproposes an economic, accurate, and quick measurementmethod. It also investigates the thermal error differentialsbetween static idle turning and in the actual cutting environ-ment. The temperature measurement units are intelligent ICtemperature sensors with correction circuits. The A/D cardextracts and transforms data and saves data in the computerfiles, and the displacement sensor measures the displacementdeviation online during cutting. The temperatures and thedeviation of thermal drifts so obtained are used to establishthe relationship function using multivariable linear regressionand nonlinear exponential regression models, respectively.Finally, this paper compares software compensation methodsfor the thermal-drift relationship. As proven by experiments,the software compensation method can limit the thermal errorof a turning centre to within 5?m. Moreover, the softwarecompensation for the thermal error relationship using a singlevariable nonlinear exponent regression model can reduce theerror by 40% to 60%.Keywords: Compensation method; Deformation; Regressionmodel; Sensor; Thermal errors1.IntroductionUsing systematic methods to reduce the CNC machine toolerrors, improve product precision and thus increase the valueadded to the products has become an important area of study.This study targets the thermal deformation errors resulting fromtemperature rise that contribute 40%70% of the precisionerror in machining in a turning centre. Further, the studyproposes an economic, accurate, and quick measuring methodfor obtaining the temperature rise and thermal drift values, inorder to establish the relationship between geometric error andCorrespondence and offprint requests to: Dr. P.-C. Tseng, CIM Labora-tory, National Chung-Hsing University, 250 Kuo-Kung Road, Tai-Chung City, Taiwan 40227. E-mail: pctseng?.twtemperature for a turning centre. The use of the softwarecompensation method to improve the precision of machinedproducts is the subject of this paper.According to Bryan 1,2, the thermal errors of a machinetool arise from six sources, i.e. heat produced in the cuttingprocess, energy lost in the mechanical system, the coolingsystem, change of room temperature, operation of the machinetool, and thermal memory effect. Although much research onthermal errors has been carried out since 1967, little improve-ment in the thermal errors of machine tools was achieved until1990. The solutions to the thermal errors of machine tools areconcentrated on the following three aspects: reduction in theheat sources, control over the direction of heat transmission,and design of a thermally robust structure. Some manufacturershave produced machine tools with air bearings. Since thecomponent parts inside the bearing do not have direct contactwith each other, the configuration provides a solution to theproblem of generating heat from friction in the traditionalroller bearing. Although there has been work on thermal errorsof machine tools since 1960, most of the literature is devotedto CNC machining centres 37 and few papers mentioned theturning centres 810. This is because thermal error measuringmethods and techniques applied to machining centres cannotbe used for turning centres. The reasons can be summarisedas follows:1. The difficulty of measuring equipment set-up. With thenormal machining dimensions, it is not easy to set up theequipment for the error measurement in a turning centreowing to the limited space. Therefore, error measurementis more difficult for a turning centre than for a machiningcentre. In addition, the lathe turret tends to produce inter-ference and cause collisions.2. Symmetric structure. The H-type turning centre is a sym-metric structure, so its thermal error effect is not as obviousas that from a V-type machining centre, which in turncauses extra difficulties in building an accurate model ofthermal drift. According to reports by machine tool manufac-turers, the axial errors of a lathe within 100200 ?m shouldbe corrected on the basis of the relationship between tem-perature and displacement.3. Measurement benchmark. It is hard to find a measurementbenchmark for a turning centre, which is different from theHigh-Precision CNC Lathe Thermal Errors851machining centre. Hence, laser interferometer equipment hasto be used. However, it is still not easy to make thecorrection by laser interferometer equipment because ofunparalleled-axes scale errors.In the literature related to compensation methods for turningcentres, only the analogue method developed by Domez 11is close to completion. It adds analogue compensation signalsto the driving signals of the servo motor in order to realisethe compensation. However, this work targets compensationusing an open-loop controller and is thus not appropriate tothe currently more popular, but more restricting FANUC con-troller. The reason is that the FANUC controller uses digitalservo signals without the involvement of other external signalsin the feedback loop. Therefore, finding a quick measurementand compensation method are desired technologies.This study will be based on quick measurement of therelative values of temperature increase and thermal deformation,establish the relationship between geometric errors and thetemperatures of a turning centre, and then use a softwarecompensation method to improve product machining precisionand add value to the turning centre.The functions of the system are described in brief below:1. Use the IC-AD590 temperature sensor to measure the tem-perature, and the A/D card to extract data and transform sig-nals.2. Use the LTO2S circular displacement apparatus on the latheto measure the displacement.3. Establish the thermal error model based on actual cutting.4. Establish the thermal error model using linear and nonlinearregression analysis.5. Compare and develop simple thermal-error compensationmethods for a turning centre.2.Deformation Impact on MachiningPrecision and the CountermoveIn order to simplify the thermal error model for a turningcentre, early work shows that the focus may be placed onsuch critical component parts as the spindle and the servo axisof tool carriage that have direct impact on the precision of themachining parts 12,13.2.1Spindle Thermal Drift at the Turning CentreAn H-type spindle bearing, has a radial bearing both at thefront and at the rear of the spindle to resist the cutting force.In order to avoid the thrust from radial turning, the back ofthe spindle is equipped with a thrust bearing. Thus, there isonly one set of thrust bearings that can prevent the spindledisplacement resulting from the increase in temperature in theradial direction on both end bearings. This design places alimit on the rear part of the spindle while leaving room forthe front part to extend, thus causing the following three casesof spindle thermal deformation due to temperature rise:1. Only axial forward extension occurs. The reason for thistype of deformation is the axial temperature rise of thespindle housing or the spindle, and the Z-axial displacementof both against the machine datum.2. The spindle has simultaneous axial and radial thermal dis-placement. The radial thermal displacement could be upward(Y-axis) or horizontal (X-axis), the reason for which is thatthe spindle box has a thermal displacement in two directionssimultaneously and is combined with the spindles Z-axisthermal displacement.3. The divergence of spindle axis centre is caused by thermaldeformation due to the unequal bearing temperature rise atthe front and back bearings. This can be so serious that itmakes high-speed running very dangerous. This error ishard to compensate for.2.2Thermal Drift of Servo Axis at the TurningCentreThe turret is inserted on a ball screw, and the positioningprecision, straightness, and perpendicularity have become themajor determinants of the actual location of the cutting tip,together with the stick slip friction, the static and dynamicrigidity of the guideways, and the servo motors sensitivity tosmall incremental feeds. The ball screws temperature willincrease with frequent fast moves, or heavy-loaded cutting, orheat transmission from external sources, and the bearings onthe two fixed ends will cause deformation and then bendingof the ball screw. Therefore, the cutter turret has not onlyparallel movement in two directions, but also produces Abbeoffset errors. Fortunately, it will only cause a parallel movementof the cutter end in two directions to the rotational axis of asingle-point cutter, which can be compensated for using HTMcomputation. The errors will not cause a deviation of theworking point. However, the errors caused by ball screwdeformation and bending will cause a dynamic cutting tool inC-axial machining to overcut the component part or produceerrors when shaping component parts.The servo axes of the turning centre are comprised of twoservo motors each with ball screws. One is the X-axis and theother is the Z-axis. The error is the combination of the two.The thermal error is more complicated than that for a singleshaft. Therefore, the selection of appropriate bearings andspindle shaft provides the optimal solution to improving ballscrew errors. With regard to the thermal errors of the turningcentre caused by the components with high friction, high-speedrotation or movement (e.g. transmission axis, linear slide, high-speed spindle), there are three ways to improve this:1. Prevent heat transmission.2. Reduce or prevent thermal deformation.3. Compensate after thermal deformation.The approach of individual machine tool manufacturersdepends on individual technical level.2.3Other Errors of the Turning Centres1. Quasi-static errors. Slowly changing and increasing errorsafter using machines after delivery, including geometricerrors, kinematics errors, the errors caused by external852P.-C. Tseng and J.-L. Hoattachments,inappropriatemechanicassembly,unstablematerials, and improper sensor apparatus.2. Dynamic forces errors. The vibration of the mechanicalstructure attributable to sources of dynamic forces causeerrors on the surface of the component part.3. Impact of cutting tool deformation on precision machining.The proportion of the heat transmitted to the machine toolis small compared to the total amount of heat generatedduring cutting. However, it still causes the temperature torise owing to the small volume and thermal capacity of thecutting tool. When the cutting tool temperature rises theheating causes a volume expansion and eventually causesmachining errors.For static error improvement of a machine tool, HTM canbe used to represent the errors caused by an imperfect rotatingaxis in a turning centre lathe 14. A perpendicular Z-axisradial movement exists with respect to the rotating axis. Thedeflection from inaccurate clamping of components, axialmovement parallel to the Z-axis, and oscillation with an errorin terms of the angle from the Z-axis all cause inaccuracy inthe relative position of the moving parts. A description of theerrors and moving parts of these moving objects may berealised by converting and computing the objects movementfeatures relative to a datum coordinate. Such a conversionrelationship can be represented by 4?4 HTM. The product ofHTM refers to continuous coordinate conversion.0TN?Nm?1Tm? T1T2T3%TN(1)There are six errors in the degrees of freedom of the spindleof a turning centre. ?zis the revolution angle of the Z-axis;?xis the error of parallel movement of the X-axis; ?yis theerror of parallel movement of the Y-axis; ?zis the error ofparallel movement of the Z axis; ?xis the revolution error ofthe X-axis; and ?yis the revolution error of the Y-axis. T1represents a parallel movement to the X-axis by x; T2representsa parallel movement to the Y-axis by y; T3represents a parallelmovement to the Z-axis by z; T4represents a revolution of theX-axis by qx; T5represents a revolution of the Y-axis by qy;and T6represents a revolution of the Z-axis by qz. Hence, theHTM of the spindles errors can be written as:0TN?Nm?1Tm? T1T2T3T4T5T6(2)For the high-speed spindle of a turning centre, ?zis afunction of time, denoted by ?z(t), due to the high-speed ofthe revolution feature. Because ?z(t) changes very fast, theresulting errors cannot be compensated for immediately. Theerrors of the spindle of a turning centre can thus be simpli-fied as:0TN?Nm?1Tm? T1T2T3T4T5(3)The error HTM for the spindle of the turning centre afterthe computation with inputs of the revolution and parallelmovement matrices can be represented by:?XYZ1?error?XYZTX1Y1Z1?XYZ1?(4)The turning centre has two servo axes, the X-axis and theZ-axis, while the X-axis is mounted on the Z-axis. Basedon the HTM after coordinate conversion, the following errorconverting matrix of the cutting tools actual and ideal positionscan be obtained:?xt?yt?zt?actual?XtYtZt?actual?XtYtZt?ideal(5)For the turning centre, the actual machining error is thetotal of the cutting tools error and the component parts error:?X?Y?Z?1?REAL?xt?yt?zt1?actual?XYZ1?error(6)Micro adjustment when assembling the machine tool cancorrect the static errors and reduce the modelling and measuringtime, and make on-line improvement by tracing the systemserrors over time. However, the thermal deformation errors ofa machine tool, as reflected in Eq. (5), cannot be correctedby static compensation. The complicated relationship betweentemperature and deformation must be derived from experiments,and improved by on-line real-time measurement.3.Experiment Equipment and theMeasurement System3.1Temperature MeasurementThere are many types of temperature sensor 10. This experi-ment uses on IC sensor AD 590 with an output of 1 ? AC?1to measure the temperature. It has good linearity, widepower range (?4 V?30 V), and is small and easy to installso that no compensating circuit is needed. It is able to test atemperature range of about ?55C150C. The locations fortemperature measuring in this experiment include spindle, ballscrew, turret, cutting space, and room temperature. For thespindle as a whole, all the bearings are sources of heat. Theball screw affects the positioning precision and the turret hasan impact on the cutting tools temperature when cutting.Consequently, all these items should be measured. The distri-bution of the eight locations is shown in Table 1.3.2Displacement Measuring SystemTo implement a simple, easy, fast and accurate method toobtain highly reliable data, a RENISHAW-LTO2S lathe usingHigh-Precision CNC Lathe Thermal Errors853Table 1. The number and location of the parts for temperature measure-ment.NumberLocationNumberLocationCh1(T1)2nd Bearing onCh5(T5)End of spindlespindleCh2(T2)3rd Bearing onCh6(T6)1st Bearing onspindlespindleCh3(T3)Ball screwCh7(T7)TurretCh4(T4)Cutting spaceCh8(T8)Roomtemperaturea circular displacement measurement was hired. This apparatusconsists of an OMM (optical module machine), an OMP(optical module probe) and an MI12 interface. The measuringprobe relies on a clamp to fix it on to the turret. When theMI12 is activated, the OMP sends and the OMM receivesinfrared signals so as to reach a balanced state. When the probetouches the component part, the internal resistor connected tothe base of the probe changes and cuts off the infrared signalsfrom the probe. This output end is connected to the skip signalof the machine tool so that the machine tool will be informedthrough the PMC (programmable machine controller) to stopexecution of the current NC code. By MACRO, the machinetool will also be able to read the mechanic coordinates of theX- and Z-axis immediately before the skip.In order to ensure the accuracy and certainty of the LTO2Sexperimental data extraction, the contact measurement wasimplemented 50 times on the basis of the correction criteriausing a standard circular gauge (50 mm, precision with 1 ?m)to locate one-way recurrence and was examined with standardcompasses. The results show that the average errors on boththe X-axis and the Z-axis are within 1.5 ?m, implying goodstructural rigidity of the probe and high repeatability of thecomputation of the MACRO code.Additional analysis of cumulative errors is required of thecutter change of position because the probe is fixed on thetool carrier. The machine in the experiment is a Model GCL-2L (Goodway Machinery Co.) whose positioning precision inthe X- and Z-axis after compensation can reach 1 ?m. Accord-ing to the data obtained by measuring the same fixed datum,the average error of cutter change on the X-axis is 2 ?m andon the Z-axis is 3 ?m.3.3Data ExtractionThehardwarearchitecture ofthedataextractionsystememploys an A/D 812 (Yuan-Hwa Electronic Co.) card, workingwith an interface code written using Boland C?, to convertthe analogue signals of temperatures measured by an IC-AD590sensor into digital signals before saving them in the file. Theexperiment shows that the highest temperature of the spindlebearing of the machine tool is about 50C. Any higher tempera-ture will destroy the bearing. Let 50C equal an IC-AD590output voltage of 5 V, so the maximum input voltage is set tobe ? 5 V, with a linearity of 25 mV (10V/4096). For thetemperature sensor, in order to balance and reduce the voltagedecrease caused by long-distance transmission and interferenceby external voltage signals, all the signal lines are isolatedlines of the same length.3.4Experiment FrameworkOn the FANUC-OT controller used by the GCL-2L CNCturning centre it is easy to set the working coordinates. Theoverall method of the research is that a revolving speed metertests the revolving speed of the spindle. The temperaturemeasurement unit extracts the analogue signals of the tempera-tures by the IC-AD590 and converts them through the A/Dcard before the signals are saved in a personal computer file.The displacements measured by the displacement measurementunit are sent directly to the personal computer by means ofan RS232 interface and saved in the file. Linear and nonlinearregression analyses are eventually carried out using SPSS 15on all the information about temperature and displacement toobtain the relationship equation.4.Analysis of Thermal Drift Behaviour forthe Turning CentreThe thermal deformation analysis models for the turning centrecan be divided into static analysis models and dynamic analysismodels. The static analysis creates high temperature onlythrough movement of the X- and Z-axes and the spindlerevolution, and then further studies the relationship betweentemperature rise and thermal displacement. The study of therelationship between temperature rise and error based on realturning is called dynamic analysis. Generally, only the staticmodel has been studied because it is easy to create a hightemperature to measure and analyse, considering only a fewvariable coefficients. On the other hand, the dynamic modelconsiders much more sophisticated variables. For example, thetemperature distribution of the whole turning temperature site,the cutting tools wear and tear in the turning process, machinevibration, heat from the friction between cutting tool andcomponent part, and error undercutting real forces, and so on.This paper is an extension of 10, using S45C medium carbonsteel as the test material, and actually measures the thermaldeformation during real turning by experiment. It develops aset of systematic experimental methods, and proposes a seriesof modelling techniques for fast analysis of thermal errorsand compensation methods applicable to a real processingenvironment which has the requirement for high precision.4.1Static Thermal Drifts of the Turning CentreA standard testing rod is inserted into the chuck to measurethe deformation of the testing rod by changing the spindlespeed. To ensure circularity, the flatness of the end face, therepeatability of the LTO2S probe, and the errors caused bythe environment of the standard testing rod, the radial andlength deformation are measured by the average of three fixedpoints from the angles with 120 movements. The experimentwas carried out at 500, 1000, 1500, 2000, 2500 and 3000854P.-C. Tseng and J.-L. Hor.p.m. cutting speeds. The results show the distribution oftemperatures and thermal errors for the static experiment issimilar to the bearings temperature increase relationships fromthe empirical equation in the bearings technical manual. Thatis, the higher the revolving speed, the higher the temperaturerise; however, beyond a certain high revolving speed, thetemperature rise gradually approximates to a fixed level. Figure1 shows the standard static relationship of temperature riseand thermal errors.In the static analysis model, the location with the highesttemperature rise is at the tail end of the spindle. Its temperaturerises with the increase in the spindle speed, but eventuallyapproximates to a stable level. The maximum temperature levelis lower than 50C. The temperature increase at other locationsis not so obvious. From the experiment, in the static model,the displacement of the standard testing rod in the X- or Z-direction is very small, i.e. close to zero after deduction ofthe cutter change errors. The data from the static analysisshow that it is not necessary to build a thermal error modelfor static or light machining. But does this mean that thelathes thermal distribution or temperature rise will not affectthe precision of machining? Real cutting will be carried outto test the relationship between thermal or temperature riseand errors in idle running and actual cutting.4.2Thermal Errors of Actual Cutting at theTurning CentreIn order to provide the same datum for machining and measur-ing, pre-experiment adjustment and correction must be doneaccurately, and appropriate control over material and theenvironment is also required. The following assumptions aremade:1. A large amount of cutting liquid is used in the machiningprocess and the thermal deformation between cutting tooland component parts in the machining process is ignored.2. The X- and Z-axial offset errors of the ball screws arealready corrected by the controller parameters.Fig. 1. The standard thermal drift with 3000 r.p.m. without real cutting.3. A disposable cutting tool is used and the tool wear isignored.4. The cutting tools positioning errors are already corrected.5. The errors caused by cutter change are deduced by theaverage value in the experiment.6. Environmental impact is ignored, for instance, magneticfield, change in humidity, and vibration.7. The heat transmission is considered uniform and a consistenttemperature is maintained for the machining site.The experiment uses the G92 fixed turning C-code for thereal cutting. The feed value F for turning refers to the suggestedvalue of 0.25 mm rev?1? 0.35 mm rev?1for S45C mediumcarbon steel 16. In order to study the impact that the feedvalue has on spindle temperature rise, this part of the experi-ment uses the spindle speeds of 500, 1000, 2000 and 3000r.p.m. relative to the static experimental environment, and thedepth of cut (DOC) is fixed. The experimental results showthat when the depth of cut and the speed are fixed, and thefeed speed varies (0.25 mm rev?1? 0.4 mm rev?1), the increasein the spindle temperature is almost the same. This means thatwithin the normal feed speed range, the feed speed has aweak association with the spindle temperature rise. The errordistribution after the temperature stabilises (about 90 min) isabout 3 ?m, without much variation.If the revolution speed S is fixed, the feed value F seemsnot to have much impact on any change of temperature riseand thermal error. In order to prove this phenomenon, thedepth of cutting is changed to DOC = 0.1 mm, revolutionspeed S = 500 r.p.m., and feed F ? 0.250.4 mm rev?1. Theexperiment results in the same amount of temperature rise ofthe spindle, and the thermal errors increase to 6 ?m.Now that the feed F ? 0.4 mm rev?1is fixed, the depth ofcutting varies as DOC ? 0.5 mm, 1.0 mm, 2.0 mm, and5.0 mm, compared to the early experimental results, the depthof cut is found to be closely associated with temperature riseand thermal deformation. The deeper the cut, the larger thethermal error. This phenomenon holds true until DOC ? 2 mmwhen it stabilises. The time for the temperature to reach thestable level is relatively shortened to 45 min. When the speedS ? 3000 r.p.m., DOC ? 5 mm, the temperature rise andthermal error reach the maximum, where the thermal errorcould be as large as 9 ?m. The results are shown in Fig. 2.The following conclusions are drawn based on experi-mental results:1. Within the normal feedrate range, the feedrate is veryweakly associated with the rise in spindle temperature.2. Revolution speed S and the depth of cut have a close andpositive relationship with the rate of temperature rise andthe thermal error. The larger the values of revolution speedS and the depth of cut, the faster the rise in temperatureand the larger the thermal error.3. The temperature rise and thermal error have a correspondingrelationship to each other, therefore increase of speed S andthe depth of cut can be applied to fast modelling techniques.4. A maximum thermal error of 9 ?m exists when the revol-ution speed is 3000 r.p.m. and the depth of cut DOC= 5.0 mm.High-Precision CNC Lathe Thermal Errors855Fig. 2. The thermal drift at fixed F ? 0.4 mm rev?1, S ? 3000 r.p.m.,DOC ? 5 mm without real cutting.5. The relationship between temperature and thermal errorunder the conditions of idle running is quite different fromthat under the conditions of actual cutting. The model fortemperature and thermal error can only be developed bymeans of actual cutting.6. In the experiment, the thermal error is mainly caused bythe spindle deformation when the spindle is heated.4.3Thermal Error Model for SpindleBy analysing the temperature rise at different points in actualcutting, it is found that the temperature change at all the pointsmeasured does not contribute to the overall thermal errors.Therefore, the key to cost-effective, simple, easy, and fastmodelling is to locate the temperature points with the largestcontribution and use these points as the source of the thermalerrors caused by the rise in the spindle temperature.4.3.1Multivariable Regression AnalysisIn regression analysis, the larger the correlation coefficient, themore consistent the relationship between the two. The leastmean squares error represents the distance between the pre-dicted values in the regression and the actual values. Thesmaller the error, the shorter the distance between the predictedvalues and the actual values. The correlation coefficient andthe least mean square indicate the physical correlation ordistance between the regression values and the actual valuesmeasured.Regression analysis of X-axis.A regression analysis is runwith the temperature and thermal error at individual and multi-variable temperature levels, and the correlation between eachtemperature level and thermal error is observed. As shown bycomprehensive analysis in Table 2, the thermal error is notcaused by any single temperature level. The most highlycorrelated temperature level combination indicated by the corre-Table 2. Variable correlation comparison in regression for thermalerrors on X-axis.RegressionCorrelationLeast squaresvariablecoefficienterrorT10.73070.00641T20.81460.00702T30.75240.00789T40.53130.00672T50.55140.00684T60.57380.00690T70.50890.00868T80.73520.00636T1T80.85360.00588T1T2T3T80.87690.00502T1T2T3T40.80130.00563T1T2T3T4T50.76310.00571T1T2T3T4T5T60.79510.00535T1T2T3T50.82880.00521T1T2T3T5T60.83280.00508T1T2T3T70.80410.00510T1T2T4T7T80.81430.00528T2T3T4T80.79640.00569T2T3T7T80.79900.00560T3T4T5T60.78570.00539T1T2T7T80.80110.00548T1T3T5T7T80.81220.00552lation coefficients and least squares is T1, T2, and T3, as shownin the equation below:?X? 8.6702 ? 0.2167T1? 0.4419T2(7)? 0.1508T3? 0.0676T8Regression analysis of Z-axis.Z-axis regression analysis isdone by directly using a combination and permutation of manytemperature levels. In Table 3, the most highly correlatedtemperature combination indicated by the correlation coef-ficients and the least squares is T1, T2, and T3, as shown inthe equation below:?Z? ?16.11748 ? 0.44665T2? 0.23106T3(8)? 0.28244T84.3.2Nonlinear Regression AnalysisAccording to the experimental data, the working environmentof light cutting (DOC ? 0.5 mm) has an impact on thespindles temperature rise similar to static idle running, wherethe thermal errors exhibit a linear relationship. When DOC ?0.5 mm, the trend of thermal errors can be represented by anexponential model. The simplified on-line compensation modelcan compute the displacement at a single temperature level.Hence, the relationship equation between temperature level andthermal error for rough machining is obtained from the nonlin-ear regression model.X-axis regression analysis.To obtain the Pearson correlationcoefficient and the least squares for the temperatures measuredand the thermal errors, the strongest correlation is found at T2,with the smallest least squares. The relationship equationbetween T2and the X-axis thermal error is thus established.856P.-C. Tseng and J.-L. HoTable 3. Variable correlation comparison in regression for thermalerrors on Z-axis.RegressionCorrelationLeast squaresvariablecoefficienterrorT10.79070.00411T20.81980.00409T30.74110.00451T40.59670.00399T50.51220.00521T60.58130.00549T70.60110.00558T80.77440.00519T1T80.78920.00604T1T2T3T80.79610.00598T1T2T3T40.86940.00511T1T2T3T4T50.80110.00591T1T2T3T4T5T60.77140.00583T1T2T3T50.81070.00575T1T2T3T5T60.82330.00613T1T2T3T70.86020.00513T1T2T4T7T80.79930.00600T2T3T4T80.84590.00549T2T3T7T80.80130.00581T3T4T5T60.81210.00563T1T2T7T80.78330.00588T1T3T5T7T80.79950.00566When DOC ? 0.5 mm, the X-axis thermal error predictionmodel is depicted by the following equation, with the tempera-ture at T2as the basis, and offset as the constant.?X? ?0.31516 T2? 9.56763(9)When DOC ? 0.5 mm, the relationship between the tempera-ture at T2and the thermal error obtained from the exponentmodel of the nonlinear regression analysis is illustrated asbelow:?X? ? 7.33639 exp?2.11215T2?(10)Z-axis regression analysis.When DOC ? 0.5 mm, with thetemperature at T2as the basis, the Z-axis thermal error predic-tion model is displayed as:?Z? 0.31045 T2? 9.34764(11)When the DOC ? 0.5 mm, the relationship between thetemperature at T2and the thermal error obtained from theexponent model of the nonlinear regression analysis is asshown below:?Z? 8.95611 exp?5.08025T2?(12)5.Verification of the Thermal Error ModelThe on-line compensation operating procedures are shown asFig. 3. The OMP part of the LTO2S measurement system ismounted on the tool carrier. The measurements are based onrecalling the cutting tools NC codes. The LTO2S measurementFig. 3. The direct on-line compensation operating procedures.system can be used for direct on-line compensation for spindlethermaldeformationinprocessing.InexecutingLTO2Smeasurement codes, every action is written in the MACROlanguage, and then codes are called by G65 order to performthe measurement. The difference between the actual dimensionsmeasured and the expected dimensions for processing can besaved in the correction parameters to offset the cutting toolwear, and sent to the personal computers through the RS232transmission interface. When carrying out the next step inmachining, by the means of a direct call for correction, thatis, real-time on-line compensation for error, the machiningerror caused by heat and temperature rise of the spindle canbe minimised. If there is no such equipment, then the relation-ship between the spindles temperature rise and thermal errormust be established before the machine is shipped out of themanufacturers site. The focus of this paper is the softwarecompensation for thermal errors. This refers to predicting thethermal error from the spindle temperature rise, transmittingthe error values through an RS232 serial communication port toan external computer (or single chip) to correct the controllersparameters to offset the cutting tool wear, and call for correc-tion parameters to realise prompt compensation.Actual turning will be used to establish the different resultsbefore compensation and after actual on-line compensation. Inorder to compare the before and after results, the turningenvironment for testing refers to the thermal error experimentfor actual cutting mentioned earlier. The error compensationprocedure flowchart is shown in Fig. 4.5.1Empirical Test of the Multivariable RegressionModelApply the thermal errors obtained from multivariable regressionanalysis, and the single part temperatures measured in the realcutting environment, to the Eqs (7) and (8), to obtain the X-and Z-axis error values. Then put these error values into thecorrection parameter for the cutting tool wear, and call thisHigh-Precision CNC Lathe Thermal Errors857Fig. 4. The error compensation procedure flowchart.parameter in turning to execute on-line compensation. Asshown in Fig. 5, the real-time compensation based on thethermal errors obtained from multivariable regression analysiscan reduce the average errors to below 60%. Notably, in theearly stage of compensation, because the distance constant inthe compensation equation in the multivariable regressionmodel is a regression constant under different temperatures,the initial value of the temperature is the room temperaturewhen the machine tool is turned on. This constant will causeover compensation, requiring the temperature differential tobe compensated.5.2Empirical Test of the Nonlinear RegressionModelCombine the thermal errors obtained from the linear andnonlinear regression models, execute on-line compensation bythe relationship Eqs (9)(12). In the real cutting environment,when DOC ? 0.5 mm, the X- and Z-axis thermal errortemperature relationship equations are replaced by Eqs (9) and(11). When DOC ? 0.55 mm, the X- and Z-axis thermal errortemperature relationship equations are replaced by Eqs (10)and (12). The experimental results are shown in Figs 6 and7. Real-time compensation by executing the thermal errorcompensation equations combining the single-variable linearand nonlinear regression models can reduce the average errorsto below 40%. When DOC ? 0.5 mm, the test results whenthe cutting first starts are the same as those described inSectio
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