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Accuracy improvement for three axis CNC

machining centres by geometric, load and

thermal error compensation

D. G. Ford, S. R. Postlethwaite

Precision Engineering Centre, University ofHuddersfieldEmail: d.ford. @hud. ac. uk

Abstract

Based on an indirect identification pre-calibrated technique it utilises a uniquealgorithm, which allows the system to compensate for the geometric errorcomponents of any normal orthogonal machine tool configuration.

The rigid body model is modified to include for the non-rigid effects and anovel technique for reducing workpiece errors caused by the thermal distortionof a Computer Numerical Control (CNC) machine tool is introduced

The methodology and validation of the universal rigid body compensationmodel with its machine specific non-rigid body and thermal effects suitablyintegrated are demonstrated Other elements such as: calibration methodology,dynamic measurement, non-rigid effects/compensation, and thermal distortionmodels are the subject of further papers leading to the overall '

Nomenclature

X, Y, Z: Position co-ordinates of the axes.ex(x),ey(y),ez(z): X, Y, Z axis linear positioning errors.ey(x),ex(y): X, Y straightness errors in the XY plane.ez(x),ex(z): X, Z straightness errors in the XZ plane.ez(y),ey(z): Y, Z straightness errors in the YZ plane.4>x(x),(j>y(y): X,Y axis roll errors.<j>y(x),<|>x(y): X,Y axis yaw errors.<t>z(x),((>z(y): X,Y axis pitch errors.

Transactions on Engineering Sciences vol 23, © 1999 WIT Press, www.witpress.com, ISSN 1743-3533

114 Laser Metrologv and Machine Performance

0xy,8xz,0yz: Squareness errors in the XY, XZ, YZ planes.Ex, Ey, Ez : Actual error movement of X, Y, Z axes.(brfay): §x(x) can be represented as one component relating to rotation aroundthe X axis, as a function of X and Y position.(fcvfc.y): §y(x) can be represented as one component relating to rotation aroundthe X axis, as a function of X and Y position.j)z(y.jc): <|>z(y) can be represented as one component relating to rotation aroundthe Y axis, as a function of X and Y position.(fcCv.jc): tyxfy) can be represented as one component relating to rotation aroundthe Y axis, as a function of X and Y position.j)v(v.x): <)> Xv) can be represented as one component relating to rotation aroundthe Y axis, as a function of X and Y position.0jcv(jc.v): XY squareness changes as a function of the X and Y axis positions.0jcz(y.z): XZ squareness changes as a function of the Y and Z axis positions.Gyzfy.z): YZ squareness changes as a function of the Y and Z axis positions

1 Introduction

The ultimate objective is to develop a universal error compensation system thatis capable of enhancing the accuracy of CNC machine tools. The compensationsystem should:• Have the capability to compensate for systematic position errors produced

by all the major sources of machine tool inaccuracy.• Operate in real time and perform dynamic compensation in order to provide

true error correction during machining operations.• Operate without inhibiting or degrading the operation and performance of

the CNC controller and machine tool.• Be transportable so it can be used with a wide range of CNC controllers and

servo systems.• Be flexible and have the capability to be applied to all machine tool types

and configurations.• Be cost effective in order to provide an affordable and therefore acceptable

solution to the problem of machine tool accuracy.

2 Geometric Model for all Machine Configurations

Using vectors to represent the movement of the rigid bodies making up themachine structure, the motion of the machine can be represented by akinematics chain. A geometric model which defines the actual motion of themachine tool's axes as a function of its fundamental geometric errorcomponents with respect to a reference co-ordinate frame is derived and used tosynthesise the actual positioning error at the tool-point for any location in theworking zone.


Laser Metrology and Machine Performance 115

Ford et al* reported that it is relatively easy to combine three sets ofequations to provide a single geometric model that can be used for the BeaverVC35 three axis machine configuration shown in Figure 1.

Ex = ex(x) + ex(y) + ex(z) + <|>y(y).Z + <j>y (x).Z + Bxz.Z + Gxy.Y + <(>z(y).Y

(1)Ey = ey(y) + ey(x) + ey (z) + #(x).Z + #(y). Z + Gyz.Z - (j>z(x).X

(2)Ez = ez(z)+ ez(x) + ez(y) - $x (x).Y - <|>y(x). X - <j>y(y).X (3)

Omm

-450mm

-400mm

Omm

Figure 1: Block diagram showing the Beaver VC35 machine configuration

This universal geometric model forms the heart of the algorithm, used by thecompensation system, to generate the correction values that are applied to theaxes of the machine tool.

3 Non-Rigid Considerations for the Vertical Machining Centre

Blake developed a measurement strategy for the separation of geometric andnon-rigid errors. Table 1 illustrates a summary of the measurements taken on a


116 Laser Metrology and Machine Performance

Beaver VC35 machine. Ford et aP also outlined this measurement strategy andits validation against machines of the same type and configurations. Blaketreated the various components of the model for the Beaver VC35 machine(equations 1,2,3 above) as standard geometric components, which will have apossibility of being affected by non-rigid error changes. Firstly changes wereseen for both X and Y linear positioning, however, these were linked to angularerror changes (Table 1). Therefore linear positioning errors are not effected forthis machine, remaining as the standard geometric error component. Therewere no non-rigid effects apparent in any of the straightness measurements;therefore they remain in the model unmodified.

Finally changes were observed in the squareness data. In the originalgeometric model, the squareness data was entered as a constant value. Asimilar approach to this can be used, with the magnitude of the "constant"changing as a function of the corresponding axis positions (e.g. for X - Ysquareness, the X and Y axis positions, Qxy(x, y)).

TABLE 1 Summary of measurements from the Beaver VC35 machine

X

; \%v.

z

TOTAL

PitchYawRollLinear positioningHoriz. straightnessVert, straightness

Ash -;\/ 'y'"& V'- ''%' .a# ;_ ,\y /Linear positJ0M%Boriz. sW0#eseVert, staigkt8es$

PitchYawRollLinear positioningStraightness in XStraightness in Y

XYS#aie&e&sYZ Scpaieiiel&3E&pimewss

21

-& arc sees5 arc sees-20 arc sees55pm4pm-10pm

We GS'v%{r;;; ; $x;;r-2am$e&&x% /;-4< \' ;"':\:'-2pmT'% '- //

;4m V'%\:Y4 arc sees4 arc sees

25pmL5pm2pm

13 arc sees7 arc: sees ,45 ate sees

13 arc seesNegligible2.5 arc sees-50pmNegligibleNegligible

3me'Se 'X y \ -\ :^ 3'sm ' - "r r ' " " ''-%2aac$ecs./' \'\ v J

]5 j# y "I// f:,: <%€#giWe y '"/: \W0i#K

NegligibleNegligible

NegligibleNegligibleNegligible

25 am sees:)2afcsecs \ ' ' /_ "\5 arc sebs

i



Combining all of these error components into the geometric model, to create anerror model, results in the equations 4 to 6. Underlining of the equation portionindicates the changing error components.

Ex = ex(x) + ex(y) + ex(z) + <bv(v,x) .Z + <)>y (x).Z + 9xz(v.z) .Zy.y).7

= ey(y) + ey(x) + ey(z) + ([ucfoy) . Z + jucfy.x). Z + Gyzfy.z). Z -

(5)

T (6)

Thus all of the significant non-rigid error components have been insertedinto the error model for the Beaver VC35 machine as matrix elements.

4 Thermal Considerations for the Vertical Machining Centre

Allen" developed the compensation technique for the Beaver VC35 verticalspindle machining centre. Postlethwaite et al detailed the measurement strategyand the thermal algorithm development.

The analysis of this machine showed that the spindle was the major source ofheat. In addition the head slide exhibited a significant and rapidly changingposition independent thermal distortion in the Y axis direction (i.e. Y axis offsetunaffected by the axis position). The thermal compensation technique wastherefore applied to the head-slide to model the Y axis distortion.

In order to understand the nature of the head-slide distortion to facilitate thegeneration of the temperature and distortion models the internal structure of theheadslide was investigated. Figure 2 shows the internal structure.

The spindle is supported by two spindle bearings, the bearings beingmounted in two horizontal webs. It can be concluded from this that movementof the spindle bearings defines any movement of the spindle. Therefore if thecompensation models are able to calculate the expansion of the head-slide alongtwo lines, LI and L2, in line with the spindle bearings then movement of thespindle nose, L3 could be determined.

To achieve this spot temperature measurements were required along thelines LI and L2. A temperature model was required to calculate the temperaturedistribution along these lines from the spot temperature measurements. Finallya distortion model was required to calculate the expansion along LI, L2 and soL3 from the temperature distributions.



Ll

L2

I fcn

.1 En

Figure 2: Internal Structure of Head-slide

4.1 Measurement of Temperature and Distortion DataTo generate and test the temperature and distortion models, measurements ofheadslide temperature and distortion were required. The thermal imagingcamera was used to measure the temperature distribution across the whole of theside of the head-slide.

To measure head-slide distortion non-contact displacement transducers withan accuracy of ljj.m were used Three transducers were used; two in line withthe spindle bearings and one on a test mandrel mounted on a spindle.

For the measurement phase the spindle was run through a simple duty cycle,running at 3000 rpm for iVa hours, and cooling with the spindle stopped for afurther 1% hours. During this duty cycle thermal images and displacementreadings were recorded at regular intervals.

As well as providing a continuous picture the camera can record a sequenceof temperature distributions for off-line analysis. The images are stored as amatrix of temperatures with a spatial resolution of one pixel. Using amathematical analysis software programme in MATLAB^ temperature anddistortion models explained in sections 4.2 and 4.3 were derived.

4.2 Generation of the Temperature ModelThe temperature model requires parameters that can be optimised to fit thetemperature distributions produced by a particular machine. The temperaturebetween two sensors is given by:T(x)=ao+ai%» (7)Where x = distance along measurement line

n = derived constant model parameterT(x> = temp, at x along measurement line

Also ao = T(o> and ai = (Tp - T(o>) / L (8)Note: ao is a constant depending on which pair of lines is under consideration.Where L = length of measurement line

T(o) = temperature at x = 0



T(L) = temperature at x = LThis model is used to calculate the temperature distributions along a line

from three temperature measurements with different curve fits being appliedbetween pairs of sensors. Estimating the parameters manually, even with the aidof the thermal images, is a laborious and time consuming process and routineshave been developed within MATLAB to automate the process parameterestimation for the temperature model.

4.3 Distortion ModelThe distortion model is used to calculate the expansion of the head-slide, alongtwo lines LI and L2, from knowledge of the temperature distribution along thelines. Various forms of distortion model, ranging from complex finite-element-based models to simpler free expansion based models, have been investigatedThese investigations showed that for machine tool structures a simple free-expansion -based distortion model could produce results of comparable accuracywith that of the more complex models. This is probably due to the high stiffnessof machine tool structures withstanding any thermal stresses producedThe use of a free expansion model greatly simplifies the implementation ofthermal compensation . Based on free expansion the distortion of the lines onthe head-slide is a function of the temperature along the line, the length of theline and the coefficient of thermal expansion for the head-slide material. Usingthe temperature model already developed the free expansion between twosensors a distance L apart is given by:

L

AL = ocjaoo (9)

<x= Coefficient of thermal expansion for head-slide materialL = Length of measurement lineao,ai are constants derived from the thermal model ( equation 8)n = derived model constant derived by curve fitting the model against themeasured data Equation 7.

^ (10)

I n+1 JEquations of this form could be used to calculate the expansion along the

lines LI and L2 on the head-slide, but for thermal compensation it was essentialto determine the expansion along L3, in-line with the spindle nose (Figure 2).In general terms the expansion along the line L3 is given by:

+ asAI (11)where a:, and ag = constant model parameters



5 Compensation System and Analysis Software (CSS)

Four packages of software are associated with the operation of the CSS. Theseare:5.1 Rigid Body Compensation SoftwareThis software generates compensation values based on a rigid body model forall three axis configurations. The compensation software has the ability todefine the machine configuration; process the raw measured error data,geometric compensation tables for use in the correction algorithm, and generateand display the compensation data dynamically. The software is menu driven,and guides the user through the sequence of operations required to set up andactivate the error compensation. The software has been written using BorlandTurbo C V2.0 within the DOS environment for use with a high performancePersonal Computer (PC).

5.2 Dynamic Data to Geometric Error Conversion SoftwareThis software converts data measured dynamically using laser interferometerequipment into meaningful geometric error data, thus providing a high speed,high resolution machine tool calibration facility. The data is input to thesoftware as a standard Renishaw ASCII error file. The development of thesoftware was covered by Postlethwaite et al*.

5.3 Machine Tool Error Analysis And Display SoftwareThis software package complements the compensation software previouslydescribed Based initially on the rigid body geometric model the software willsimulate the effect of single or multiple error components throughout themachine volume. The results of the simulation will be presented numericallyand/or graphically in a meaningful and unambiguous way. It will provide andaid to the understanding and analysis of machine tool errors. The developmentof this software was covered by Postlethwaite et al\

5.4 Thermal Distortion SoftwareThis software package performs two important discrete functions. First,software is used off-line to complement the thermal imaging equipment byutilising a model to derive the positioning of the sensor transducers to be usedfor on-line compensation at the machine. The positions are derived from thethermal images of the machine whilst running through typical duty cycles.

The temperature and distortion software for on-line use contains uniquemodels which are capable of reading from the thermal sensors and deriving thedistortion correction needed to modify the rigid / non-rigid kinematics model asshown schematically in Figure 3. This software will be contained within the on-line computer and is written using Borland Turbo C V2.0 DOS environment foruse with a high performance PC. The development and validation of thethermal distortion software was covered by Allen*, Allen et al*, andPostlethwaite et al*.



6 Implementation of the Error Compensation SystemThe block diagram of the compensation system is illustrated in Figure 3. Thesystem consists of the four core software packages described earlier viz. thekinematics model, geometric error conversion, error analysis and display, andthe temperature / distortion software either contained within a high performancepersonal computer (PC) or a open architecture CNC controller.

Machine calibration

Rigid/non-rigidkinematics model

From temperature

Transducers

Universalmachineselection

Output moduleanalogue or digital

From encoders Position correction

Figure 3: Block Diagram of the Compensation System

7 Conclusions• Geometric error, head-slide thermal error, and work-piece distortion

improvements of 10:1, 9.3:1, and 32.5:1 respectively were achieved on theBeaver VC35 machine '*.

• Figure 4 illustrates the 9.3: 1 improvement for the head-slide thermal error• Figure 5 illustrates the improvement made on a single-axis using

compensation.

• A well designed machine with finer resolution and reduced repeatabilitycould achieve an even better performance from applied compensation. The



limiting correction factor applied to each axis is the uni-directionalrepeatability for that axis and for the Beaver machine it is circa 2 micronsper axis.

• The ultimate objectives listed in the Introduction have been met in somecases but not in others. We can conclude that Universal geometric rigidbody models are possible. Certainly for 3-axis applications and varioustypes and configurations we have achieved this goal. It is even feasible thata general algorithm for non-rigid is also possible but in the time availableto date the limitation is that insufficient machines have been tested in orderto confirm. However, machine specific non-rigid elements can beincorporated into the Universal rigid body algorithm relatively easily.

• All validations above also showed significant improvement on the dynamicperformance as measured by ball bar metrology equipment. A pair ofcontouring axes set to prescribe a circle gave a greatly improved circulartrace showing that the geometric errors had been largely eliminated .

c

eQS

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280

Time (minutes)

Figure 4: Plot comparing Compensated and Uncompensated Movement at theSpindle Nose for the machine.

Measuring strategies for the separation and identification of errorcomponents have been developed. Much research effort has been put in toreduce the training and calibration time. It is cost effective and thereforeprovides an affordable and acceptable solution to the problem of machinetool accuracy.The dynamic compensation offered can operate in real time duringmachining operations. It can perform without inhibiting or degrading theoperation and performance of the CNC controller and machine tool. It hasthe capability to compensate for systematic position errors produced by allthe major sources of machine tool inaccuracy. It is flexible and has thecapability to be applied to all machine tool types and configurations and is



transportable so it can be used with a wide range of CNC controllers andservo systems.

• A thermal tracking and correction system has been developed which iscapable of calculating position independent and position dependentcorrection values for each axis. In all error tracking techniques developedthe thermal errors are calculated from sensor measurements. The system isable to acquire data from a range of sensors.

• The tracking techniques used and any models required for a machine willbe developed off-line and then incorporated into the compensation system.The system is able to cope with the different models developed for eachmachine. Since the models for each machine will be different thecombination of sensor readings used with each model will also be different.The system is configurable to use its sensor inputs in different combinationsfor different machines.

Comp.

Figure 5: Example of a linear positioning correction on the machine

8 References

1. Ford DG, Postlethwaite SR, "Error Compensation applied to highprecision machinery", Proceedings for LAMDAMAP 93, Laser Metrologyand Machine Performance /, Computational Mechanics, pp. 105-112,ISBN: 1-85312-241-6, July 1993.



2. Blake MD, "Investigation into Load Effects on Machine Tool Accuracy",MPhil Thesis, University of Huddersfield, 1995.

3. Ford DG , Postlethwaite SR, Blake MD, "The Identification of Non-rigidGeometric Errors in a Vertical Machining Centre", Publication submittedto the LMech.E Part B , Accepted Dec 1998 awaiting publication. Copy ofpaper can be obtained from the University of Huddersfield

4. Allen JP, "A General Approach to CNC Machine Tool Thermal ErrorReduction", PhD Thesis, University of Huddersfield, 1997.

5. Postlethwaite SR, Allen JP, Ford DG, "The Use of Thermal Imaging,Temperature and Distortion Models for Machine Tool Thermal ErrorReduction", Proc. LMech.E, Vol 212 Part B pp.67l~679,Dec 1998.

6. Postlethwaite SR, Ford DG, "Dynamic Calibration of CNC MachineTools". Int. Journal of Machine Tools and Manufacture, Vol.37, No. 3, pp287-294, 1997.

7. Postlethwaite SR, Ford DG, "Geometric error analysis software for CNCMachine Tools", Proceedings for LAMDAMAP 97, Laser Metrology andMachine Performance ///, Computational Mechanics, pp.305-315, ISBN:1-85312-536-9. July 1997.

8. Allen JP, Postlethwaite SR, Ford DG, "Practical Thermal ErrorCorrection for CNC Machine Tools", Proceedings of theASPE ConferenceAnnual Meeting, Monterey, California, pp. 648-653, Nov. 9-14, 1996.

9. LINK Project Completion Report. Design of High Speed MachineryProgramme. Project Title: Super High Accuracy Feedback Transducer(SHAFT) for Single/Multi-axis Applications. Grant Ref: GR/61602. (Restricted)

10. Postlethwaite SR, "Electronic Based Enhancement of CNC MachineTools", PhD Thesis, University of Huddersfield, 1992.

Acknowledgements: Funded by two EPSRC grants GR/61602 andGR/L05624.


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