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Enhancement of machine tool accuracy : theory andimplementationCitation for published version (APA):Theuws, F. C. C. J. M. (1991). Enhancement of machine tool accuracy : theory and implementation. TechnischeUniversiteit Eindhoven. https://doi.org/10.6100/IR362622

DOI:10.6100/IR362622

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ENHANCEMENT OF MAClllNE TOOL ACCURACY:

THEORY AND IMPLEMENTATION

F.C.C.J.M. THEUWS

ENHANCEMENTOFMACillNETOOLACCURACY:

THEORY ANDIMPLEMENTATION

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven,

op gezag van de Rector Magnificos, prof.dr. J.H. van Lint, voor een cornrnissie aangewezen door het College

van Dekanen in het openbaar te verdedigen op vrijdag 4 oktober 1991 te 16.00 uur

door

FRANCISCUS CORNELIS CA THARINA JOSEPHINA MARIA THEUWS

geboren op 16 juli 1964 te Luyksgestel

Dit proefschrift is goedgekeurd door de promotoren:

Prof.dr.ir. A.C.H. van der Wolf

en

Prof.dr.ir. P.H.J. Schellekens

Aan: Jozet,

mijn ouders,

opa en oma

I

SUMMARY

In this report the results are presented of a study on the enhancement of the accuracy

of a commercially available machine tool by software error compensation. First, the

problem field has been outlined by a qualitative analysis of the main error sources.

Based on this analysis the most significant error sources are identified as the

geometric errors and the thermal behaviour. The further study has been focused on the

development of a model by which the effect of these error sources can be

compensated.

In order to obtain a basis for modelling of the geometric structure, a classification

methodology of multi axis machine tools has been developed .

To assess the effect of the geometric errors, a mathematical model has been devised,

capable of prediction of the location error of the tool with respect to the workpiece as

a function of measurable geometric errors in the guideways. This model is applicable

to machine tools of arbitrary configuration. As an example, the mathematical model

has been elaborated for a five axis milling machine. The geometric errors of the

milling machine under investigation have been measured and transformed to

functional relationships. In this way a continuous description of the geometric error

structure of the machine tool is obtained The measurement set-ups and the method of

modelling the individual geometric errors are discussed.

The thermal behaviour is determined empirically. For this purpose a measurement

set-up has been developed, yielding simultaneous information on the drift of the tool

holder and the thermal distribution of the machine tool. Using this set-up a number of

measurements have been carried out on various locations within the working space

and with diverse load situations. This obtained results have been analyzed by

statistical methods and a relationship between the displacement and the measured

relevant temperatures has been developed. This relationship has been applied to

predict the thermal behaviour of the tool holder, based on information on the

temperature distribution of the machine tool.

n Summary

For both the geometric error model and the thermal model a compensation algorithm

has been devised, including the effects of the scale expansions.

The geometric error model has been partly implemented into the control system of the

milling machine for real-time compensation.

The thermal error compensations are computed by a coupled PC that continuously

monitors the thermal state of the machine tooL The calculated compensations are sent

to the control system, which compensates in real time for the thermally induced

errors. The magnitude of the actual compensations is dependent on the thermal

distribution of the machine tool and the position of the carriages.

The effectiveness of the compensation algorithms has been verified by a hole plate

measurement (geometric errors) and thermal drift measurements (thermally induced

errors). The results prove the suitability of the compensation methodology for

enhancement of machine tool accuracy.

As a final test on the use of the devised compensation, several workpieces have been

manufactured. The results show an enhancement of the practical achievable accuracy

on a commercially available machine tool by a factor 2.

Finally, conclusions are drawn with respect to the achieved improvement in accuracy,

and some recommendations are presented that indicate the areas for further

investigation.

m

SAMENV ATTING

Het doel van dit onderzoek is de verbetering van de nauwkeurigheid van een

produktiemachine door softwarematige correctie. Daartoe is een kwalitatieve analyse

gemaakt van de bronnen van afwijkingen die bij dergelijke machines optreden. Hieruit

is gebleken dat de afwijkingen in de geometrie en het thermisch gedrag de hoofd~

bronnen voor de onnauwkeurigheid vormen. Het verdere onderzoek heeft zich gericht

op de ontwikkeling van een model dat in staat is de effecten van deze bronnen te

beschrijven, zodat in een later stadium een softwarematige compensatie mogelijk is.

Als basis voor de modellering van de geometrische afwijkingen structuur, is een

classificatie methodiek ontwikkeld voor meer-assige produktiemachines.

Voor de evaluatie van het effect van de geometrische afwijkingen is een mathematisch

model ontwikkeld. Met behulp van dit model is men in staat om de resulterende

afwijkingen in de positie van het gereedschap, uit te drukken in de individuele

afwijkingen in de geleidingen. Het model is universeel toepasbaar op meet- en

produktiemachines. Als voorbeeld is bet model uitgewerk:t voor een vijf-assige

freesmachine. De individuele afwijkingen in de geometrie van deze machine zijn

gemeten en omgezet naar een functionele representatie. Hierdoor is een continue

beschrijving van de afwijkingen structuur tot stand gekomen. De toegepaste meet

opstellingen en de modellering van de individuele afwijkingen is toegelicht.

Het thermisch gedrag van de produktiemachine is empirisch bepaald. Hiervoor is een

meetopstelling ontwikkeld die informatie verschaft over de temperatuursverdeling en

de verplaatsing van de hoofdspil. Met behulp van deze opstelling zijn een groot aantal

metingen uitgevoerd, op diverse plaatsen in bet werkbereik en onder verschillende

belastingscondities. De verk:regen resultaten zijn geanalyseerd met behulp van

statistische methodieken.

IV Sarnenvatting

Hieruit is een relatie ontwikkeld die de verplaatsing van de gereedscbapshouder

beschrijft uit de meting van een aantal relevante temperaturen aan de machine. Deze

relatie is gebruikt om bet thermiscb gedrag van de machine te voorspellen ..

Zowel voor bet geometrische als bet thermische model is een correctie algoritme

ontworpen. De therrniscbe uitzetting van de linialen is hierin verdisconteerd.

Het geometrie model is gedeeltelijk geimplementeerd in bet besturingssysteem voor

real-time correctie.

Op basis van de beersende temperatuursverdeling van de machine, worden de

compensaties voor bet therrnisch gedrag berekend door een externe PC. Deze stuurt de

compensaties via een interface naar bet besturingssysteem van de machine, welke de

positie van bet gereedschap real-time corrigeert. De grootte van de compensaties is

afhankelijk van de positie van de sledes en van de momentane temperatuursverdeling

over de machine.

De effectiviteit van de compensaties is gecontroleerd met behulp van een gatenplaat

(geometrische afwijkingen) en drift metingen aan de boofdspil (therrniscbe

afwijkingen). De resultaten tonen aan dat de gekozen methodes zeer geschikt zijn om

de nauwkeurigheid van een produktiemachine te verhogen.

Als een laatste test op de praktische toepasbaarheid van de compensatie zijn diverse

werkstukken geproduceerd. De resultaten tonen aan dat een nauwkeurigheids

verbetering met een factor 2 haalbaar is.

Tot slot worden enige conclusies getrokken met betrekking tot de behaalde nauw

keurigheidsverbeteringen. Tevens worden enkele aanbevelingen gedaan waarmee

toekomstig onderzoek de toepasbaarheid van de methode kan optimaliseren.

v

CONTENTS

SUMMARY I

SAMENVATI1NG m

Chapter 1 GENERAL INTRODUCfiON 1

Chapter2 THE MAIN ERROR SOURCES 9

2.1. Geometric Errors 10

2.2. Thennally Induced Errors 13

2.3. Errors due to Finite Stiffness 17

2.4. aassification 19

Chapter 3 MODEL OF THE GEOMETRIC ERROR STRUCTURE 31

3.1. The Modelling System 32

3.2. The General Model 33

3.3. Elaboration of the General Model to the Type Dependent Model 42

3.4. Assessment of the Geometric Errors 48

3.5. Modelling the Geometric Errors 57

Chapter 4 DETERMINATION OF THE THERMAL BEHAVIOUR

4.1. The Methodology

4.2. The Measurement Strategy: Possibilities and Limitations

4.3. Results of the Measurements

63

64

69 72

VI Contents

Chapters DESIGN OF A CORREcnON ALGORI11iM 81

5.1. The Geometric Errors 82

5.2. The Expansion of the Scales 90

5.3. The Thermally Induced Errors 94

5.3.1. The Statistical Modelling Methodology 95

5.3.2. Development of the Modelling Procedure 99

5.3.3. The Modelling Methodology and Results 109

5.4. Implementation of the Correction Algorithm 113

Chapter 6 VERIFICATION OF THE CORRECTION ALGORITHM 125

6.1. The Geometric Error Conection

6.2. The Thermal Error Correction

6.3. Manufacturing of a Test-workpiece

Chapter 7 CONO..USIONS

APPENDICES

125

133

137

145

I Mathematical Elaboration of the General Model 149

ll Measurements and Results of Geometric Errors 161

III Correlation between Temperature Sensors 187

IV Results of the Modelling Procedure of the Thermal Behaviour 189

V Results of the Tests on the Probe System 193

CURRICULUM VITAE 195

ACKNOWLEDGEMENTS 197

1

Chapterl

General Introduction

In the manufacturing of complex products machine tools play an important role. Due

to several reasons one can observe a tendency towards higher accuracy at all levels of

production. As the driving forces for higher accuracy [1.1] could be listed the

requirement for:

• better performance and reliability of the products;

• miniaturization and integration of product-parts for weight and space

saving purposes;

• automatic assembly;

• active noise reduction of accurate parts in gearboxes, etc;

~ response to increased accuracy in other fields, for example electronics.

Taniguchi [l.Z] showed the development of the achievable accuracy in machining

over the last fifty years (figure 1.1) •. l:le predicts a further increase in achievable

accuracy for the next decades.

The curves presented by Taniguchi state the highest possible accuracy. The depicted

values can be achieved under ideal circumstances with an optimal designed machine

tool and skilled personnel. Therefore, it is possible that in practical situations, the

achievable accuracy may differ as much as an order of magnitude [1.3] from the

values as predicted by Taniguchi in figure 1.1. In this paper we will focus on machine

tools that are applied in normal machining, with typical inaccuracies varying from 5

to 200 J.!m.

2 Chapter 1

O.:lnm, (AtomiC Avera99-latt1ce dll"nension dimension d1stance) 1900 1920 1940 1960

Fig. 1.1. Taniguchi's prediction of achievable accuracy

Despite the wide range in achievable accuracy, the conclusion from Taniguchi's

report, that the requirement for higher accuracy is not yet satisfied, is still valid.

Forced by this requirement, the performance of machine tools is continuously being

improved. The classical way to increase the performance of a machine tool is to

enhance the behaviour of the mechanical structure. In terms of geometric behaviour

this implies the aim at faultless movements of the carriages and exact squareness

between the guides. For the improvement of the thermal behaviour various

experiments with thermal control [1.4], thermal invariant structures [1.5] and

compensating heat sources [1.6, 1.7, 1.8] are carried out. However, these methods of

improvement are costly and the physical limits will soon be achieved. For this reason,

new techniques to improve the overall behaviour of machine tools, are being

developed. With the aid of computer technology it is possible to compensate for the

errors existing in machine tools instead of avoiding them. However, this error

correction method requires a very thorough investigation of the machine tool's

behaviour and the factors influencing it. This is the fundamental reason for the

initiation of extensive research programs all over the world [1.9, 1.10, 1.11 1.12].

General Introduction 3

When the basic function of a machine tool is reviewed, it can be expressed as the

transformation of rough material into a usable product. If the product has specified

dimensions and tolerances, the function of the machine tool is to generate the product

with accompanying specifications within a given time and at given costs. A number of

influences on this transformation are responsible for errors in the dimensions of the

final product. An overview of the main influences that may disturb the fmal accuracy

of the product, is depicted in figure 1.2 [1.13].

Roomt~mperature\ E t 1 Radian heat.7 S~ue~~~

Convection beat

Driv .. ~ Bearing friction Cutting process

Coolant Internal !lydraulic Sources

Falhng chips Gearbox

Friction o1 Carriages

Program Errors

I Operator

Cutting Forces

Culling oondit.io0111 Lubrication

Rough material

Translation•

Environment

Fig. 12. Sources of errors in machine tools

. Rouch material Cuttmak tutt.in& conditions Forces" Lubrication

In this paper we will concentrate on the most influential error sources. These error

sources can be described as:

• Geometric errors due to imperfect movements of the carriages. In the production of the guides the manufacturer tries to avoid unwanted

motions of the carriages, for example rotations. But even with the present

production techniques the limits of achievable accuracy are restricted and

therefore the carriage will show certain deviations from the perfect behaviour.

These deviations will manifest themselves as position and orientation errors

of the tool with respect to the workpiece.

4 Chapter 1

• Thennal behaviour of the machine's structure.

Under the influence of several internal and external heat sources a machine

tool will demonstrate a certain thermal behaviour with, due to expansion of

the individual elements of the machine, an accompanying distortion. The

resulting errors in the position and orientation of the tool are at least of the

same order of magnitude as the errors caused by geometric errors [1.16].

Consequently, the analysis of the thermal behaviour is one of the main

objectives of this project.

• Finite stiffness of the elements of the machine.

One should make a distinction between the static and dynamic stiffness of a

machine tool. The static stiffness of the machine is important when the

machine is loaded with heavy workpieces, when heavy machine parts are

moving while machining or when large cutting forces are to be expected. In

this case the bending of machine parts will result in a deviation in the

position of the tool with respect to the workpiece, and as a result this will

introduce an error in the dimensions of the product. The dynamic stiffness

contributes merely to the surface roughness of the product and less to the

dimensional properties. Therefore, in this research project, it will not be taken

into account.

The main goal of this research is the improvement of the accuracy of a commercially

available machine tool by error correction. This is achieved by the implementation of

a correction algorithm into the control system. This error correction should comprise

the most significant errors. This goal requires a systematic investigation of the main

error sources in multi axis machine tools. In this thesis we will confme ourselves to

the thermally induced errors and the geometric errors because they contribute to more

than seventy percent of the resulting error of a machine tool [1.14, 1.15]. The finite

stiffness errors will be subject of future investigation.

The target group that could benefit from the results obtained during this project, are

mainly manufacturers of machine tools and suppliers of control systems. However,

also the users of machine tools can benefit from the presented methodologies for

checking the accuracy of their machine tools. In this way an indication of the

accuracy of a machine tool can be obtained and proper actions can be taken. This can

be very useful in, for instance, acceptance tests.

Generalmtroduction 5

IB our strategy the basic error condition of a machine tool is defmed by its geometric

errors. This condition will not change in short term operation. While machining, the

thermal behaviour of the machine tool will cause additional deviations of the tool with

respect to the workpiece. The effects of these error sources are assumed to be

independent and therefore treated separately. In Chapter 2 we will present a

qualitative analysis of the main error sources and explain the chosen nomenclature of

the geometric error sources. Furthermore a classification methodology for multi axis

machine tools will be presented. This classification enables us to specify groups of

machine tools with common error sources.

In order to investigate the effect of the geometric errors on the position of the tool

with respect to the workpiece, a description of the geometric error structure of multi

axis machine tools is necessary. For this purpose a mathematical model of the

geometric error structure is developed. This will be presented in Chapter 3.

If this mathematical modelling technique is applied to a machine tool, it is possible to

calculate the resulting error of the tool, with respect to the workpiece, based on the

results of measurements of the individual geometric errors. As an example this model

will be worked out for a typical milling machine. The necessary input parameters for

the model are obtained from a five axis milling machine. Through these measurement

results, piece-wise polynominals are fitted. In this way, a continuous description of

each individual geometric error, based on discrete measurements, is achieved.

With the knowledge of the value of each individual geometric error at every position

of the carriages. application of the mathematical model yields the error of the tool

with respect to the workpiece.

The second main error source under study is the thermal behaviour, which will be

discussed in Chapter 4. Several methods are possible to describe the thermal

behaviour but, taking the goal of this research into account, we have chosen for an

empirical approach. This approach implies the simultaneous measurement of the

actual thermal distribution of the machine's structure together with the relative

displacement of the tool. The applied measurement set-up and the obtained

measurement results will be discussed.

6 Chapter I

For both the geometric errors and the thermal behaviour a first version of an on-line

correction algorithm is developed, which will be described in Chapter 5. Due to lack

of computing time in the control system for calculation of correction terms, only the

most influential geometric errors are included in this correction. The necessary

correction formulas are determined by application of the mathematical model

described in Chapter 3.

From the measurements of the thermal behaviour a relationship has been determined

that describes the displacement of the tool in dependence of the thermal rise of several

temperature sensors attached to the machine tool. This relationship has been obtained

by application of statistical analysis techniques. An advantage of the applied

techniques is the possibility to reduce the number of necessary temperature sensors.

The reduction is based on the criterion of relevancy to the prediction of the

displacement of the tool.

To verify the effectiveness of the implemented correction algorithm a two stage test

has been applied. First, the geometric error correction has been checked using a

reference length. Secondly, the effectiveness of the total correction methodology is

verified by producing workpieces on a five axis milling machine with and without the

implemented software correction. The methods and results are discussed in Chapter 6.

Finally, conclusions and recommendations for future research are presented in

Chapter 7.

General Introduction

References

[1.1] Peters J.:

Metrology in Design and Manufacturing - Facts and Trends.

Annals of the CIRP, Vol. 26/2. p415-421, 1977.

[1.2] Taniguchi, N.:

7

Current Status in, and Future Trends of, Ultraprecision Machining and

Ultrafine Materials Processing.

Annals of the CIRP, Vol. 32/2, p573-580, 1983.

[1.3] Lo-A-Foe, T.C.G.:

Single Point Diamond Turning.

Dissertation, University of Technology Eindhoven, 1989.

[1.4] McClure R.:

Manufacturing Accuracy through Control of Thennal Effects.

Moore Special Tool Co. Inc, p185-197, Bridgeport, USA.

[1.5] Schunck J.:

Untersuchungen iiber die Auswirkungen Thennisch Bedingter Verformungen

auf die Arbeitsgenauigkeit von Werkzeugrnaschinen.

Dissertation, Technische Universitlit Aachen, 1965.

[1.6] Spur G.:

Thermal Compensation by Spindle Heating.

Leaflet of Production Technology Centre Berlin, Institute for Machine Tools

and Manufacturing Technology, Technical University Berlin, 1990.

[1.7] Sata T.:

Analysis of the Thermal Deformation of a Machine Tool Structure and its

Application.

Proc. of the 14th Int. MTDR conference, p275-280, 1973

[1.8] Sata T.:

Control of the Thermal Deformation of a Machine Tool.

Proc. of the 16th Int MTDR conference, p203-208, 1975.

[1.9] McClure R.:

Thermally Induced Errors.

Technology of Machine Tools, Vol5, p9.6.1-9.6.23, 1980.

[1.10] Quality of Automation.

Research Program of the National Institute of Standards and Technology.

Project NISTIR 89-4045, Gaithersburg, Maryland 20899, USA, 1988.

[Lll] The Science of an Advanced Technology for Cost-Effective Manufacture of

High Precision Engineering Products.

Research program of Purdue University, West Lafayette, ONR Contract No.

83K0385, USA, 1983.

[1.12] Development of Methods for the Numerical Error Correction of Machine

Tools, Bureau Communautaire de Reference, Brussels, november 1989.

[1.13] Theuws F.C.C.J.M.:

Nauwkeurigheidsanalyse van CNC-produktiemachines.

Lezing voor Mikrocentrum West, T.U. Eindhoven, WPA nr. 0670, 1989.

[1.14] Hemingray C.:

Some aspects of the accuracy evaluation of machine tools.

Proc. of the 14th Int MTDR conference, p281-284, 1973.

[1.15] Ferreira P.M.:

A Contribution to the Analysis and Compensation of the Geometric Error of a

Machining Center.

Annals of the CIRP, VoL 35/1, p259-262, 1986.

[1.16] Venugopal R.:

Thermal Effects on the Accuracy of Numerically Controlled Machine Tools.

Annals of the CIRP, Vol 35/1, p255-258, 1986.

9

Chapter2

The Main Error Sources

The manufacturing of a product can be described as a transformation from a blank

into a useful product. A number of error sources act on this transformation and are

responsible for the error in the dimensions of the product. In order to determine a

priority scheme for the analysis of the error sources it is important to classify the

effects of these error sources to order of magnitude. Out of bibliographical studies it

appeared that the geometric errors and the thermal behaviour are the main error

sources in machine tools. Together they cause over seventy percent of the total

resulting error of a machine tool [2.1, 2.2, 2.3].

The large amount of different designs of machine tools, all with their specific

applications, advantages and limitations, implicates the necessity of reducing the types

of machine tools under investigation. In choosing a suitable machine tool group for

testing the developed methodology for accuracy enhancement, the following must be

taken into consideration:

• the current trend towards products with increasing accuracy and complexity

will rise the demand for more flexible machine tools.

• the larger proportion of the practical problems with accuracy occur with

milling machines.

• the modelling of the geometric structure becomes more complicated as more

elements, especially rotary elements, are included.

• for correction purposes it is necessary that the machine tool is numerically

controlled.

Based on these considerations the target group of this research is defined as five axis

milling machines.

10 Ompter2

In the next section we will take a closer look at the main error sources i.e geometric

errors and thermal behaviour. Furthermore some attention is paid to finite stiffness

effects.

Finally a classification of five axis milling machines is presented to distinguish

machine tool groups with similar error sources.

2.1. Geometric Errors

In general machine tools, in particular milling machines, possess three perpendicular

linear axes of movement. In addition to these three axes, rotary axes can be mounted

on the machine tool. The three linear axes form a cartesian coordinate system which

allows the tool to be positioned at any place within the range of the axes. Normally

the linear movements along an axis of this system are performed by a carriage-guide

system. In the ideal situation the spatial position of the tool can be determined by the

positions of the carriages, presented by the corresponding measuring systems.

However, due to the imperfect geometrical shape of the guides of a machine tool, the

carriages will display erroneous movements. These erroneous movements will result in

an error of the position and orientation of the tool with respect to the workpiece. In

order to classify these erroneous movements we first present some kinematic

principles.

Basically, a body possesses six degrees of freedom that determine its location (i.e.

position and orientation) in space [2.4, 2.5]. These degrees of freedom are built up out

of three translations and three rotations. Consequently a body can reveal six sources of

error which result in another position and orientation of the body than expected. As a

carriage of a machine tool is basically a body in space with five degrees of freedom

suppressed, this theory does also apply to these elements. Application of this theory to

a linear carriage-guide system implicates that, due to imperfections in the shape of the

guide, the carriage will display straightness errors, rotations about all three axes and

an error in the position along the guide.

In figure 2.1 a linear carriage-guide system is depicted with its possible erroneous

movements.

The Main Error Sources 11

Fig. 2.1. Carriage-guide system with possible geometric e"ors

In order to avoid wrong interpretation of each geometric error a definition is required

of the nomenclature of the erroneous movements.

In principle any definition suits the purpose but for uniformity reasons we will confine

us to the German standard, lied down in VDI2617, Blatt 3 [2.6]. This standard uses

three characters to identify the individual geometric errors. Hereby the first,

lower-case, character represents the axis of movement, for instance "x" for the guide

in figure 2.1. The second, lower-case, character represents the type of geometric error

i.e. translation or rotation. The last, lower-case, character represents the axis along

which, or rotation about which, the geometric error is acting. For example, if this

notation is applied to the rotation of a carriage of the X-guide about the Y-axis, the

geometric error source is denoted as "xry".

It must be noted that the geometric error of the carriage in the direction of movement

(xtx, yty or ztz) does not find its cause in the guide but in the measuring system

attached to the guide. Thereby, in principle, it is not a geometric error. However, in

this paper it will be treated as an error in the geometry for simplicity reasons.

Also rotary elements are liable to erroneous movements. Similar to linear axes a

definition of the individual errors is necessary. In figure 2.2 a rotary element is

depicted with its geometric errors.

12 Chapter 2

This definition is again accordingly to the German standard VDI 2617, Blatt 3. The

identification of rotary axes of machine tools is defined in DIN 66217 and ISO 841

[2.7, 2.8]. According to this definition the rotary axis around the X-axis is denoted as

the A-axis, around the Y-axis as the B-axis and around the Z-axis as the C-axis. In

conjunction with the above presented definition of geometric errors for linear axes, the

geometric errors of rotary axes are also denoted by three characters. Hereby the first,

lower case, character represents the rotary axis of movement ("a", "b" or "c"), the

second, lower-case, character represents the nature of the error (i.e. "t" or "r"). The

last, lower-case, character represents the axis along which, or rotation about which,

the geometric error is acting (i.e. "x", "y" or "z"). As an example, the scale error of a

rotary element, which provides the rotation around the Z-axis, is denoted as "crz".

As the coordinate frame rotates along with the rotary element, additional definitions of

the directions of the X-, Y- and Z-axis are necessary. In this thesis is chosen to let the

direction of the X-axis of the coordinate frame attached to the rotary element,

coincide with the X-direction of the machine coordinate frame when the rotary

element is at its reference or zero position. If the rotary element starts its movement,

the coordinate frame XYZ rotates along with the rotary element. This defmition is

necessary to avoid problems in the definition of direction if two serial rotary elements

are applied.

z ctz

Y~x cty crz

ctx

Fig. 2.2. Rotary axis with its geometric errors


2.2. Thermally Induced Errors

In modern machining thermal influences are responsible for the largest part of the

inaccuracy of the product [2.3, 2.9, 2.10]. A possible approach to deal with this

subject is to describe the machining process as a three element system. Thereby the

elements are:

1) the machine tool's structure;

2) the workpiece;

3) the measuring systems of the machine tool.

Thermal influences act on all these three elements. We can distinguish six sources of

thermal influences, all acting on the three basic elements. The resulting thermal

situation of the machine tool can be separated into two types: the thermal rise and the

(variable) thermal gradients. The six sources of thermal influences are depicted in

diagram 2.1 as: heating or cooling provided by the room environment (1), heating or

cooling provided by various cooling systems (2), the effect of people (3), heat

generated by internal sources of the machine (4), heat generated from the cutting

process (5) and the initial thermal situation of the machine tool (6) [2.11]. The latter

thermal source does not act on the current thermal situation of the machine tool but

reflects the thermal history embedded into the machine's structure. This effect may be

caused by large fluctuations in the temperature of the environment.

, The thermal rise causes the three elements to expand. This would not affect accuracy

if all expansion coefficients are equally sized and no phase difference in the thermal

rise of the elements is present. However, in practical situations these conditions often

fall short so errors due to different expansions of the elements are induced.

The second thermal situation is much more difficult to comprise. The (quasi-)steady

state thermal gradients cause the machine structure to deform. In diagram 2.1 an

overview of the sources and their qualitative influence is depicted.

14

CONDUCTION

Uniform temperature olher than 20°C

Chapter 2

HEAT FLOW

Electronics Hydraulic friction

Motors lk Transducers Electrical

Frame stabilislq

Non-uniform temperature field

TOTAL THERMAL ERROR

Diagram 2.1. Overview of thermal effects {2 .11].

The room environment typically contributes to the unifonn expansion and the static

thennal gradients of the three basic elements. It is therefore advisable to use a

temperature stabilized room for precise machining [2.12]. In special cases, if the

coolant is applied for showering the total machine tool or internal cooling of the

machine's structure, the room environment and the coolant are the only sources that

may cause a unifonn temperature rise [2.11]. This is depicted by the unidirectional

arrow "-+" in the diagram.


Also the operators of the machine tool may contribute to thermal gradients in the

machine, though this effect is considered to be of minor influence.

The cutting process is, particularly in rough machining, a potential heat source.

However, previous research revealed circumstantial evidence that, due to the fact that

over 70% of the induced heat is absorbed by the chips, relatively low significance of

the cutting process on the thermal behaviour of a machine tool is to be expected, at

least while finishing [2.13]. This conclusion should be accompanied by the remark

that at this time there is no known method to measure, record and distinguish thermal

effects during real metal removal processes [2.14].

Finally, the heat produced by various sources within the machine tool contribute to

thermal gradients. In a machine tool there are a number of heat sources like drives,

friction in bearings, friction between guides and carriages, gearboxes, etc. (see figure

1.2). These heat sources generate a temperature field over the machine's structure

which is not steady state.

Although all· the heat sources mentioned above contribute to the thermal distortion of

the machine tool, the friction in the spindle bearings is regarded as the main source

[2.15, 2.16].

In order to verify this conclusion for the milling machine used for this research, whose

structure is depicted in figure 2.3, some experiments are carried out. These

experiments include the measurement of the thermal distribution of the machine tool.

Spindle head

Drive Y-axis

Fig. 2.3. Milling machine used for this research

16 Olapter 2

During these measurements the machine tool was loaded with a spindle speed of 6000

rpm and simultaneously the carriages were moved back and forth over the range of

the axes.

The results of the preliminary temperature measurements are depicted in figure 2.4.

An extensive discussion of the thermal behaviour will be presented in Chapter 4 and

Chapter 5.

..... .e ~

I ~ l ~

Thermal rise of machine parts, n:6000 lpl1l for 8 bours 70

65

60

55

50

45 Main drive

-----------------------------------------------------------

Drives of the X andY axes . .......... .. ....................................

. ·······

200~--~----~2~--~3----~4----~5~--~6----~,~--~8

Time (bours)

Fig. 2.4. Temperatures on a machine tool under continw:>us load

Out of these preliminary measurements it appears that. in accordance with the results

of previous research [2.15, 2.16], the spindle bearings and the main drive can be

indicated as the main internal heat sources.

Several approaches have been. made to model the thermal gradients behaviour

[2.17, 2.18] with varying success. But, while most of them concentrate on the steady

state gradients, in normal machining the thermal state of a machine tool will vary

constantly. This causes a thermal situation that is much more complicated to model.


An extensive approach to model this behaviour is the use of finite element and

analysis methods (FEA). Apart from the difficulties in the modelling procedure due to

the complex structure of present machine tools, several uncertainties in the model,

such as coefficients of expansion [2.11], radiation, convection and conduction, cause

the contribution of this methodology to be of limited practical importance [2.19, 2.20].

Therefore an empirical approach to determine the actual relationship between the

thermal distribution and the displacement of the tool holder remains necessary [2.21].

The PEA-method can be useful in the prediction where the temperature sensors should

initially be located [2.22] and in the development of a qualitative description of the

thermal behaviour of a machine tool.

2.3. Errors due to Finite Stiffness

In addition to the distortion of the machine tool due to thermal effects, a number of

forces act on the machine's structure. These forces cause the structure to deform and

consequently disturb the actual position of the tool. While machining a workpiece

three basic types of forces [2.23] are present:

1) the load of a workpiece on the machine tool;

2) the forces induced by the cutting process;

3) gravity forces due to the displacement of masses while moving the carriages of

the machine tool.

The type 1 forces are dependent of the weight of the workpiece put on the machine

tool. The effect of these forces is that the machine parts will bend and thereby cause a

change in the geometric errors. However, this change is relatively low in comparison

to the absolute values of the geometric errors and its effect on the position of the tool

holder is small compared to the thermally induced errors. Therefore these finite

stiffness errors are considered to be of minor influence in modern machine tools [2.1]

and will not be taken into account in this project.

The type 2 forces act directly between the tool and the workpiece, causing the tool

holder to deflect and thereby introducing errors in the dimensions of the product.

18 <llapter2

Several papers describe the effect of static forces on the machine's structure [2.24,

2.25]. But, with the assumption that no large depths of cut are allowed in the finishing

process, the static cutting forces are relatively small. Combined with the relative high

stiffness of the machine tool these forces cause no large deflections of the tool holder

[2.13, 2.26].

In addition to the effect of static cutting forces, two types of dynamic forces cause the

accuracy of the product to decrease. First, form errors of the blank product will

introduce varying cutting depths and thereby varying forces acting on the tool. This

causes the tool holder to deflect and the result of these deflections is a copy of the

form error in the end product with an improved accuracy of typically six times [2.27].

Secondly, the flexibility of the machine tool itself will vary while moving the

carriages. This causes a varying deflection of the tool holder due to the cutting forces.

However, with present constructions of machine tools the effect of this error source is

relatively small (2.27].

It should be stated that, with reference to the above mentioned error sources,

bibliographical studies have shown that the cutting process is complex to such an

extent that the commonly used determination of the stiffness of the tool holder,

together with relative simple force calculations, is not a suitable method to predict and

correct the deflection of the tool holder under the influence of cutting forces [2.28].

The type 3 forces are caused by movement of large masses. As a result of these

gravitation forces the geometric errors of one guide are dependent of the position of

one or more other guides. An example of this effect is depicted in figure 2.5.

Therefore, if the machine type indicates that a certain finite stiffness effect is to be

expected due to movement of masses, the measurement of the individual geometric

errors should be organized to comprehend these effects.

Within the scope of the project we will confine us to the investigation of the effect of

the type 3 forces. The description of the effects of the type 1 and type 2 forces is

scheduled to be subject of a sequel of this project.


Fig. 25. Possible effect of the movement of a carriage on the geometric error of a

guide due to finite stiffness

2.4. Classification

Based on the considerations denoted in paragraph 2.1 the target group of this research

is defined as five axis milling machines with perpendicular axes of rotation. It should

be noted however that the mathematical model presented in the next Chapter is

applicable to all types of milling machines.

As there are a lot of different types of fives axis milling machines, a classification of

commercially available machine tools is useful. With this classification a basis for

modelling the geometric structure can be achieved and type dependent finite stiffness

effects can be pointed out.

The classification of different types of five axis milling machines as presented below

is developed in order to simplify the identification of significant errors (2.29, 2.30].

Therefore the various types of machine tools are categorized into groups with the

same kind of errors (type dependent errors). This classification is based on the number

of possible movements of the tool holder [2.31].

20 Olapter2

In general, milling machines have three perpendicular linear axes. In addition to these

three linear axes, a five axis milling machine consists of two rotary axes. Also the

spindle of a milling machine can be defined as a rotary axis. However, the spindle

will not be included in this classification, since the errors introduced by the spindle

are not machine type dependent.

Because rotary axes are available as an option, the three linear axes constitute the

basic machine. In all cases the rotary axes will be be added to this basic machine and

therefore they will not interfere with the kinematic chain of linear axes. For this

reason a distinction is made between the basic machine and the additional rotary axes.

Basic Machine

The basic machine has three perpendicular linear axes. These linear axes represent the

kinematic chain of the basic machine. One end of this kinematic chain supports the

tool, at the other end the workpiece can be placed. The base of the machine is situated

within this kinematic chain. As measurements mostly are referred to the base of a

machine, a distinction between two chains is made:

• A - chain: this chain supports the tool;

• B - chain: this chain supports the workpiece.

In figure 2.6 an example is depicted of a milling machine with is kinematic chain

representation.

In the ISO 841 standard [2.8] the nomenclature for the axes of motion is defmed. In

normal cases the axis of rotation of the tool is designated as the Z-axis. However,

some modem machine tools are equipped with a swivelling tool holder, which allows

to mill in a horizontal as well as in a vertical plane. Then applying the ISO standard

to these types of milling machines leads to different viewpoints. It appears that the

last kinematic axis can be designated as the Z-axis as well as the Y -axis. To avoid

problems with the nomenclature, the axes are not named X, Y or Z but indicated as

horizontal (H) and vertical (V) axes of motion.


A-chain

Fig. 2.6. Example of A- and B-chain in a knee-type milling machine

As a machine consists of three axes, four classes can be discriminated with a different

number of axes present in a particular chain. Within these classes two groups are

distinguished, where the construction of the machine causes another type dependent

error structure.

In table 2.1 the resulting classification is presented. This table gives an overview of

the different types of milling machines and their specific errors. In figure 2.7 some

examples are depicted of the in table 2.1 presented classes.

Number of axes in: Some of the class dependent errors Examples of Classification during traverse of the axis machine types

A-Chain B-Chain without process forces (figure 2.7)

0 3 None

1 2 Class lA Be of: Knee-type B-Chain: V - H Table guide =F , Weight/Load) milling A-Chain: H Ram = F(Ram) machine

Class m Bending of: Fixed-bridge B-Chain: H -t H Table guide = F(Table, Weight/Load) milling A-Chain: V machine

2 1 Class ITA: Bending of: Fixed-column B-Chain: H Table guide = F(Table, Weight/Load) milling A-Chain: V -t H Column = F(Ram, Vertical slide) machine

Ram = F(Ram)

Class llB: Bending of: Fixed-bridge B-Chain: H Table guide = F(Table, Weight/Load) or travelling A-Chain: H -t V Column/bridge = F(Ram guide) column machine

3 0 Class rnA: Bending of: Travelling A-Chain: H -t H -t V Bridge guide = F(Bridge) bridge milling

Bridge = F(Ram guide) machine

Class illB: Bending of: A-Chain: H -tV -t H Column guide = F(Column) Travelling

Column = F(Vertical slide, Ram) column milling Ram = F(Ram) machine

"" : Sequence in kinematic chain F(E): Function of position of element H : Horizontal axis of motion V : Vertical axis of motion


lA

III A IIIB

Fig. 2.7. Examples of milling machines of table 2.1

Rotary Axes

Apart from three linear axes, a five axis milling machine consists of two rotary axes.

Taking two rotary axes, it is possible to distinguish three classes with a different

number of rotary axes present in a chain. In the next table this classification is shown.

24 Olapter2

i Number of rotary elements in:

Classification A-Chain B-Chain

aass 1 0 2 Workpiece table can rotate about vertical axis and swivel about horizontal axis

Oass2 1 1 Workpiece table can rotate about vertical/horizontal axis and tool bolder can swivel about horizontal axis

Class 3: 2 0 Tool holder can swivel about both horizontal and vertical axis

Table 2.2. Classification of rotary axes

The notation of the axis of rotation depends on the direction of the linear axes.

According to the definition in DIN 66217 and ISO 841 [2.7, 2.8] the rotary axis

around the X-axis is denoted as the A-axis, around the Y-axis as the B-axis and

around the Z-ax.is as the C-axis. For the presented classification this implies that no

general pronouncement is possible with respect to the presence of an A-, B- or C-ax.is

in the classes 1, 2 and 3.

Example of the Oassification

The Maho 700S is a five axis milling machine with three perpendicular linear axes

and two rotary axes with perpendicular axes of rotation (figure 2.8).

The basic machine, which consists of three linear axes, has one linear axis between

the tool and the base (A-chain) and two linear axes between the workpiece and the

base (B-chain). As the B-chain contains the vertical axis, this basic machine can be

classified as Class IA.

Both the kinematic A- and B-Chain contain a rotary element. Therefore the whole

milling machine can be classified as Class IA,2.


Fig. 2.8. A five axis milling machine (Maho 700S)

With the described classification a distinction can be made between different types of

five axis milling machines that manifest the same kind of errors (i.e. type dependent

errors). With the knowledge of the type dependent errors, the definition of a

calibration set·up and the error budget of the machine are significantly simplified.

As the classification distinguishes between different number of axes present in the A

and B-chain, it becomes less difficult to model the geometric error structure of the

milling machine by using this classification.

26 Olapter2

References

[2.1] Hemingray C.P.:

Some Aspects of the Accuracy Evaluation of Machine Tools.

Proc. of the 14th Int. MTDR Conference, p281-284, 1973.

[2.2] Ferreira P.M.:

A Contribution to the Analysis and Compensation of the Geometric Error of

a Machining Center.


[2.3] Venugopal R.:


Annals of the CIRP, Vol35/l, p255-258, 1986.

[2.4] Week M.:

Werkzeugmaschinen Band 2, VDI-Verlag DUsseldorf, 1981.

[2.5] Herreman G.:

Lasermeasurement Systems for Machine Tool Testing.

Technology of Machine Tools, Vol. 5, 9.8.1-9.8.14, 1980.

[2.6] VDI 2617, Blatt 3.

VDIIVDE Richtlinien, Genauigkeit vom KoordinatenmefJgeriiten,

Komponenten der MefJabweichung des Geriites, May 1989.

[2.7] DIN 66217:

Koordinatenachsen und Bewegungsrichtungen fi.ir Numerisch Gesteuerte

Arbeitsmaschinen.

December 1975.

[2.8] ISO 841:

Numerical Control of Machines - Axis and Motion Nomenclature.

ISO International standard, First edition, July 1974.


[2.9] Bryan J.:

International Status of Thermal Error Research.

Annals of the CIRP, Vol. 39/2, p645, Citation of Kaebemick, 1990.

[2.10] Sato H. et al:

Development of Concrete Machining Center and Identification of the

Dynamic and the Thermal Structural Behaviour.


[2.11] Bryan J.:



[2.12] Kurtoglu A.:

The Accuracy Improvement of Machine Tools.


[2.13] Tlusty J. and Koenisberger F.:

Specifications and Tests of Metal Cutting Machine Tools.

Proceedings of the Conference, 19th and 20th February, Volume 1, p63,

UMIST, Manchester, 1970.

[2.14] McClure R.:

Thermally Induced Errors.

Technology of Machine Tools, Vol5, p9.6-11, 1980.

[2.15] Bryan J.:


Annals of the CIRP, Vol. 39/2, p653, Reference to McMurthry, 1990.

[2.16] Bryan J.:


Annals of the CIRP, Vol. 39/2, p653, Reference to Janeczko, 1990.

28 Otapter2

[2.17] Bryan J.:


Annals of the CIRP, Vol. 39/2, p649, Reference to Trapet et al, 1990.

[2.18] Balsamo A., Marques D., Sartori S.:

A Method for Thermal Deformation Corrections of CMM's.


[2.19] Bryan J.:


Annals of the CIRP, VoL 39/2, p651, Communication with Hicks, 1990.

[2.20] Jedrzejewski J., Kaczmarek J., Kowal Z., Winiarski Z.:

Numerical Optimization of Thermal behaviour of Machine Tools.


[2.21] Bryan J.:


Annals of the CIRP, Vol. 39/2, p651, Communication with Spur, 1990.

[2.22] Bryan J .:


Annals of the CIRP, VoL 39/2, p651, Communication with Harary, 1990.

[2.23] Week M.:

Geometrisches Maschinenverhalten unter Statischer und Therrnischer

Belastung.

Fertigungstechnik, jg. 104, m. 1/2, p25-28, 1982.

[2.24] Week M.:

Mej3techniken zur Beurteilung von Werkzeugmaschinen.

Industrie-Anzeiger, nr 72, jg. 107, p154-157, 1985.


[2.25] Kersten A.:

Geometrisches Verhalten von Werkzeugmaschinen unter Statischer und

Thermischer Last

Dissertation, TH Aachen, 1983.

[2.26] Hocken Robert J.:

Machine Tool Accuracy.

Technology of Machine Tools, Vol. 5, 1980.

[2.27] Tlusty J.:

Machine Tool Mechanics.


[2.28] Tlusty J., Koenisberger F.:

Comment to the article of A. Kurtoglu.

Annals of the CIRP, Vol. 39/2, p722, 1990.

[2.29] Development of Methods for the Numerical Error Correction of Machine

Tools.

Bureau Communautaire de Reference,

Projectnr. 3320/1/0/160/89/8-BCR-NL(30), Brussels, November 1989.

[2.30] Spaan H., Theuws F.:

Oassification of Multi Axis Machine Tools,

WPA 1007, Technische Universiteit Eindhoven, Januari 1991.

[2.31] Draft Standard ASME B5 TC52, June 1990.

31

Chapter3

Model of the Geometric Error Structure

One of the major influences that determine the accuracy of multi axis machines are

systematic errors in the movements of the carriages i.e. geometric errors. A definition

for these errors is presented in Chapter 2. The main problem in this field is to

determine a relationship that describes the systematic error in the location (i.e.

position and orientation) of the tool, in dependence of the position of the machine's

carriages. Many studies have assessed the problem of describing this relationship

[3.1-3.7]. The applied methods range from correlation models, to trigonometric

analysis, to 'error matrix' representations. However, in recent reports a tendency

towards the use of rigid body kinematics can be observed This method yields, in case

of a three axis machine, a description of the location error of the tool as a linear

combination of 21 measurable geometric errors.

In this Chapter, we will describe a general methodology for the construction of a

model, which relates the various geometric errors to the location error of the tool. This

methodology is developed by the Metrology Laboratory of Eindhoven University

[3.8, 3.11] and will be applied to a five axis milling machine. With the elaboration of

the general model to a machine specific model, a useful tool is obtained for software

error compensation. The development and implementation of a frrst version of a

software correction will be discussed in Chapter 5.

The geometric errors, necessary for the input of the model, are measured on the

milling machine under research. The measurement set-ups and results are discussed.

Finally, we will apply fitting procedures using polynominals, in order to obtain a

continuous description of each error, based on discrete measurements.

32 Chapter 3

3.1. The Modelling System

In the modelling of the error structure of multi axis machine tools, several levels can

be specified. At the top of the modelling system (see figure 3.1) stands the general model [3.8]. This general model relates errors in the location of the tool, with respect

to the workpiece, to errors in the location of coordinate frames attached to succeeding

components of the machine (i.e. the geometric errors).

Elaboration of the general model for a machine tool type yields the type dependent

modeL This model contains the common properties of the error structures belonging to

machine tools of the same type. Thereby the type dependent model must be placed

below the general model. In the optimal situation the type dependent model can be

also used to store type dependent errors, for example fmite stiffness effects and

thermal behaviour. However, as the division into type dependent and individual errors

is a complex problem, the practical significance of this error classification is still

limited.

The result of the modelling methodology is the so--called individual model. This

model describes the error structure of an individual machine tool at a certain time and

place. With this model the machine's accuracy can be unambiguous assessed and, if

desired, improved by software error correction.

General model

I 1---f-------tl Structure of the machine I

f-----+---11 Type dependent errors j

I Type dependent model J

f--1----1: Individual errors I

Individual model I Fig. 3.1. Modelling system of the geometric error structure

Model of the Geometric Error Structure 33

3.2. The General Model

Introduction

The general model can be applied to multi axis machine tools, composed of rotary and

linear elements in an arbitrary serial configuration. It relates errors between the actual

and nominal location of the tool (with respect to the workpiece), to errors in the

location of coordinate frames attached to succeeding components of the machine.

Such errors describe the difference between the nominal and actual geometry of

machine parts enclosed by two frames. The number and position of the coordinate

frames is chosen such that there is one kinematic element, i.e. carriage-guide system,

between each two frames. This choice is adequate for application of this methodology

to machine tools, coordinate measuring machines and robots.

Starting with the global coordinate system 0 attached to the machine tool's foundation,

the orthogonal frames are successively numbered. As depicted in figure 3.2, a prefix is

added to this number. This prefix identifies the corresponding frame as being part of

kinematic chain 'a' from foundation to tool, or chain 'b' from foundation to workpiece

(as already stated in Chapter 2, this differentiation of the kinematic chain into two

separate chains is made for convenient assessment of the geometric errors). Two

additional frames 'wp' and 'tl' are introduced, which are attached to workpiece and tool

respectively.

34 Chapter3

IIIII IIIII

Drive System al

~--~~~--!~~~----~ Machine Foundation

(Body 0)

Fig. 3.2. Nomenclature of the coordinate frames attached to a multi axis machine with n + m kinematic elements

Mathematical Derivation of the Geoeral Model

The nominal relation between the homogeneous coordinates kp and 1p of a point p in

frames k and I respectively, can be described by a 4 x 4 transformation matrix k T1 [3.9]:

[3.1]

[3.2]

[3.3]


In this transformation the 3 x 3 matrix k~ describes the orientation of frame I with

respect to frame k. The 3 x 1 vector ktl contains the coon:linates of the origin of frame

1 in frame k.

The inverse transformation 1Tk =(kT~1) can be expressed as:

[3.4]

[3.5]

For a multi axis machine, composed of n kinematic elements in chain 'a' and m

elements in chain 'b', successive application of these transformations yields the

following expression for the nominal location wpTd of the tool coordinate system 'tl'

in the workpiece coordinate system 'wp'.

[3.6]

= Tb TT1

[ bkTbk l] rrn=l [ ak-lTak] anTtl wp m k=m -[3.7]

Contrary to this ideal or nominal situation, an actual machine tool possesses errors in

the relative location of subsequent frames, as well as in the location of the tool with

respect to the last frame 'an' of the kinematic chain. Because none of the

contemporary multi axis machines show an absence of Abbe ·offsets, the relevant

errors in the relative location of two subsequent frames are not limited to those in the

moving direction of the enclosed kinematic element (i.e. scale errors).

Consequently, all possible errors of a guide as defmed in Chapter 2 have to be taken

into account. For the location of frame k with respect to frame k-1 this implies:

• translational errors k-lekx' k-leky and k-lekz along the x, y and z axes of frame k respectively.

• angular errors k-l~' k-Itty and k-l~ about the x, y and z axes of frame k respectively.

36 Olapter3

In figure 3.3 an example of a two-dimensional carriage-guide system is depicted with

the two coordinate frames and the related errors.

Fig. 3.3. Carriage-guide system and possible e"ors

The nomenclature of the individual errors in the general model differs from the

definition as presented in Chapter 2. The reason for this departure are the lengthy

formulas that arise in the process of developing a general description of an error

structure. With the above presented notation, addition and multiplication of a variable

number of terms becomes relatively easy to summarize. However, once the general

model is worked out for a particular machine tool, the individual geometric errors will

be nominated accordingly to the definitions of Chapter 2.

In the analysis of the effect of angular errors on the machine tool's accuracy,

linearization is applied, i.e. cos{e)=l and sin(e)=e. Since the absolute values of these

errors are relatively small for the target group of machine tools, this approximation is

valid. Application of this approximation yields additive and commutative properties

for the various errors.


This condition results in the following relationship between the actual transformation

k-lT~ and its nominal k-lTk:

Rot[x. k-tf:ta:] · Rot[y. t-l~] · Rot[z. k-lb] [3.8]

[3.9]

0 "t-1~ k-1~ k-lekx

0 t-letz "t-1~ k-leky where: k-l8rk = [3.10]

-k-1~ k-lekx 0 k-letz

0 0 0 0

I : 4 x 4 identity matrix.

translation along the local x-axis by a distance k-lekx

Rot[x. k-tf:ta:]: rotation about the local x-axis by an angle k-tf:ta:·

Similarly. the actual location wp Tatl of the tool coordinate system with respect to the

workpiece coordinate system can be expressed as:

wpTatl = wpTtl (I+ wp8rd) [3.11]

0 - e wp tlz e wp tly wpetlx

wpetlz 0 - e wpetly where: 8rl =

wp tlx [3.12] wpt - e wpetlx 0 e wp tly wp tlz

0 0 0 0

38 Cbapter3

Here transformation wp Of tl contains the errors in the location of the tool coordinate

frame with respect to the workpiece. It consists of translational errors:

- [ e e e ]T ~ - wp tlx wp tly wp tlz '

defined along the x-, y- and z-axis of the nominal toolframe 'tl', and angular errors:

.e .• - [ e e e ]T wp-u - wp tlx wp tly wp tlz

about the x-, y-, and z-axis.

Successive application of relation [3.9], yields the following expression for the actual

location wp Tat! of the tool coordinate system with respect to the workpiece coordinate

system:

[3.13]

[3.14]

Transformation an 8rt1 contains the errors in the location of the tool with respect to the

last frame 'an' of the kinematic chain. For metal cutting machine tools, errors such as

spindle induced errors, tool misalignment, tool wear and thermal tool expansion can

be included in this transformation [3.10].

Note that the transformation from frame bm to the workpiece frame ~Tbm) is

separated from the error terms. This implies that errors between the workpiece and the

machine tool are not taken into account. The reason for this exclusion is that the

workpiece coordinate frame is actually generated in the machine coordinate frame, in

case of a fully machined workpiece. For partial machining, the errors in the location

of the workpiece frame are randomly distributed. Therefore no calculations and

corrections of the induced errors are carried out.


In the elaboration of relation [3.14], an approximation will be made by ignoring

higher order effects, consisting of the product of a matrix or with one or more similar

matrices. This approximation is valid, since the difference between the actual and

nominal machine structure usually does not significantly change the active arm of

angular errors and the direction in which the various errors act.

Combining relation [3.11] with relation [3.14] now yields the following expression for

the errormatrix wp 8r 11 in the relative location between tool and workpiece:

1

OT= wp tl - tlbm l [bmTbk bk-IorbkbkTan] anTtl+ k=m

n

liTO l [ OTak ak-lorak akTan] an Ttl+ an8ftl [3.15]

k=l

A more convenient description, which also provides more intuitive insight in the basic

error relationships, can be obtained by decomposing the error transformations of

relation [3.15] into their basic errors e and e. This procedure requires extensive vector

algebra [3.8], after which the angular and translational errors between tool and

workpiece, as formerly denoted in wp or tl' can be summarized in the 6 x 1 vector

wpEtl. This vector can be expressed as similarly denoted errors in the relative location

between succeeding frames:

m n

wpEtl =- l ( tlFbk bk-lEbk) + l ( tlF ak ak·lEak) +an Ell [3.16]

k=l k=l

[3.17]

This vector represents the errors of the tool with respect to the

workpiece, defined in the tool-frame.

40 Chapter 3

[3.18]

This vector represents the errors in the location of frame k with

respect to frame k-1, i.e. the geometric errors of the kinematic

element k.

0 1 (6x6)matrix

tlRk [3.19]

This matrix, the so-called F-matrix, denotes the effect of the errors

k-lEk' acting between the elements k-1 and k, on the resulting error

between tool and workpiece.

Here tl~ x tl~ denotes a 3 x 3 matrix whose columns contain the

vector cross product of vector tl~ with the respective columns of

matrix tl~·

The errors wpetl in the relative position of the frames attached to tool and workpiece,

are expressed as a linear combination of the transformation of the errors k-lek in the

relative position of two subsequent frames, and the effect of related angular errors

k-tEk with active arm kttr This active arm kttl is determined by the structure and dimensional properties of the machine tool. In the following section this will be

elaborated for a specific machine tool.

The errors wpetl in the relative orientation between tool and workpiece, are

transformed to the tool coordinate frame by a rotation tl~·

Finally, error anEtl in the relative location of the tool with respect to the last frame 'an'

of the kinematic chain, is added.

Note that the errors wpEtl are defined in the nominal tool coordinate system. For

correction purposes it is necessary to transform these errors to the machine coordinate

system. This can be implemented in relation [3.16], by either premultiplying each of

the 3 x 3 sub-matrices of t!Fk with the appropriate orientation transformation wpRtl' or

by backtransformation of the resulting error to the direction of the machine's axes.

As already discussed, the chosen nominal location of the various frames seriously

affects the efficiency of the final model. A generally useful model can be obtained by

placing the frames in the centroid of the various kinematic elements, with one axis

aligned with the respective axis of movement.


In the evaluation of the effect of geometric errors, we note a difference between

carriage-guide systems where the carriage moves on a fixed guide, and those where

the guide moves in a fixed carriage. For a system where the carriage moves on a fixed

guide, the position of the carriage has no part in the active arm of the rotational errors.

This in contrast with a system where the guide moves in a fixed carriage. Here the

position of the guide is an integrated part of the active arm for rotational errors,

induced in the carriage.

For the model this implies that a distinction has to be made between kinematic

elements whose corresponding frame moves with the carriage and those where this

frame is fixed relative to the guide (figure 3.4). This can be incorporated into the

model by introducing so-called shape and joint transformations. The shape

transformation k-lSk describes the relative nominal location between frames k-1 and k,

in case their respective kinematic elements are at home position. Joint transformation

Jk describes the nominal angular or translational movement of kinematic element k. In

accordance with the characteristics of the respective kinematic elements, application

of these transformations yields the following expression for the relative nominal

location k-l Tk between two succeeding coordinate frames:

• Moving ----+ Moving k-lTk = k-t8k 1k [3.20]

• Moving ----+ Fixed k-lTk = k-t8k [3.2l]

• Fixed ----+Moving k-lTk = 1k-t k-t8k 1k [3.22]

• Fixed ----+Fixed k-lTk = 1k-t k-t8k [3.23]

Fig. 3.4. Connection between two kinematic elements with a 'fixed' and 'moving'

coordinate frame

42 Chapter 3

3.3. Elaboration of the General Model to the Type Dependent Model

The methodology described will be applied to the five axis milling machine under

investigation depicted in figure 3.5.

Fig. 3.5. Five axis milling machine used for this research

This machine tool consists of one horizontal linear element and one rotary element

with a horizontal axis of rotation in chain 'a' from foundation to tool. Chain 'b' from

foundation to workpiece consists of two linear elements, one vertical and one

horizontal, and one rotary element with a vertical axis of rotation. In the first stage of

the modelling process, coordinate frames are located in the workpiece, the tool and in

the centroid of each joint. In figure 3.6 the schematic representation of this milling

machine is depicted.

Note that the length of the tool is implemented in the model by a variable named 'L'.

The frames located in the various kinematic elements can be characterized as:

• Frame wp: fixed

• Frame tl fixed • Frame b3 fixed

• Frame a2 fixed • Frame b2 moving

• Frame al fixed • Frame bl moving


As a machine tool actually generates a workpiece. the position of the coordinate frame

of the latter is unknown at the start of the manufacturing process. Therefore the

workpiece frame is thought to be at the same location as frame b3. However. the

inclusion of a frame wp into the model makes it very convenient for application to

measuring machines where the location of the workpiece coordinate frame is often

well defined.

660

1245

Zo

Xo Yo

515 1150

350 660

Pig. 3 .6. Schematic representation of the five axes milling machine under investigation

Application of the above presented formulas, and abbreviation of 'cos(q)' and 'sin(q)'

to 'cq' and 'sq' respectively, results in the expression of the nominal coordinate

transformations between succeeding frames.

44 Olapter 3

As an example the transformation matrix from frame 'tl' to frame 'a2' can be

calculated as:

32T11 = Fixed - Fixed = J32 32stl

= [~!~ -~4!~ g g l [A ? g gOO+Ll 0 0 1 0 0 0 1 -140 0 001 0001

[

cqa2 -sqa2 0 -(200+L)sqa2] _ sqa2 cqa2 0 (200+L)cqa2 - 0 0 1 -140

0 0 0 1

[3.24]

The calculation of all required transformation matrices is presented in Appendix I.

These transformation matrices can be used to express the nominal position of the

tool-frame relative to the workpiece-frame. As the elaboration of equation [3.16]

requires the construction of the F-matrices out of ul\ and tl'k for all kinematic

elements, the first step is to calculate the tlTk matrices. From these tlTk matrices, the

required nl\ and n'k matrices can be extracted (see equation [3.2]).

Application of expression [3.5] onto the obtained transformation matrices yields, for

example [3.24], the following matrix:

[ cqa2 sqa2 0 l T

= -sqa2 cqa2 0 , -32Rtr~ 0 0 1 =

_ [ -~~~ ~~~ g -(28o+L) l tiTa2 -- 0 0 1 140

0 0 0 1

[3.25]

This procedure can also be carried out on all other transformation matrices

(see Appendix 1) and yields eventually the transformation from frame 'wp' to

frame 'tl':

[

cqa2.cqb3 sqa2 cqa2.sqb3 = -sqa2.cqb3 cqa2 -sqa2.sqb3

-sqb3 0 cqb3 0 0 0

305.sqa2 + qbl.sqa2 + (-350+qb2).cqa2 l 305.cqa2 + qbl.cqa2 - (-350+qb2).sqa2- (200+L) 310- qal

1

[3.26]


The next step is the determination of the F-matrices that describe the effect of the

individual errors between the coordinate frames, on the total error between tool and

workpiece. From the calculated transformation matrices we can deduce that

(see equation [3.2]):

[3.27]

tlRwp contains the additional transformation of the rotary element b3. However, as we

state that the error between frame b3 and frame wp is zero, the obtained term for

equation [3.16] will automatically yield a contribution of nil. Therefore it is not

necessary to calculate the uf wp -matrix.

For the elaboration of the relevant F-matrices the vectors t1\c need to be calculated.

These calculations are fully described in Appendix I, below two examples are

presented.

= [=] ta2z [3.28]

[ 210.sqa2 l

= 210.cqa2 - (200+L) 805- qal [

talx l = taly talz

[3.29]

The matrix tlFk is defined as (see equation [3.19]):

The vector cross product of the vectors u\c and the matrix tl~ can be summarized as:

[

tkz.sqa2 -tkz.cqa2 tky l tkz.cqa2 tkz.sqa2 -tkx with k = a2, at, bl, b2 b3

-tkx.sqa2 - tky.cqa2 tkx.cqa2 - tky.sqa2 0 [3.30]

46 Chapter 3

Implementation of this relation into expression [3.19] yields the following general

F-matrix:

cqa2 -sqa2

0 tkz.sqa2 tkz.cqa2

(-tkx.sqa2- tky.cqa2)

sqa2 0 cqa2 0 0 1

-tkz .cqa2 tky tkz. sqa2 -tkx

(tkx.cqa2-tky.sqa2) 0

0 0 0 0 0 0 0 0 0

cqa2 sqa2 0 -sqa2 cqa2 0

0 0 1

The index k indicates the concerning coordinate frame, i.e. a2, al, b1, b2 or b3.

[3.31]

Application of relation [3.16], yields the following expression for the errors wlu and

wpetl in the orientation and position of the tool coordinate frame with respect to the

workpiece coordinate frame:

Orientation errors

[ cqa2 sqa2 0 l [ cqa2 sqa2 0 l [ cqa2 sqa2 0 l

+ -sqa2 cqa2 0 1ea2 + -sqa2 cqa2 0 0e 1 - -sqa2 cqa2 0 0;,1 0 01 3 0 01 a 0 01

[ cqa2 sqa2 0 l [ cqa2 sqa2 0 l

- -sqa2 cqa2 0 bl ;,2 - -sqa2 cqa2 0 b2;,3 0 0 1 0 0 1

Position errors

[

140. sqa2 -140. cqa2 -(200+L) cqa2 sqa2 0

wpetl = a2et1 + 140. cqa2 140. sqa2 0 -sqa2 cqa2 0

(200+L) . cqa2 (200+L). sqa2 0 0 0 1

[

(805-qa1)sqa2

+ (805-qal)cqa2

( -21 Osqa2)sqa2- {210cqa2-(200+L))cqa2

-(805-qa1)cqa2

(805-qa1)sqa2

(-210sqa2)cqa2-{210cqa2-(200+L))sqa2

210cqa2- (200+L) cqa2 sqa2 0

-210sqa2

0

-sqa2 cqa2 0

0 0 1

[3.32]

kl


- (825-qal)cqa2 [

(825-qal)sqa2

-(632.5sqa2 + qblsqa2)sqa2 - (632.5cqa2 + qblcqa2- (200+L))cqa2

- (825-qal )cqa2

(825-qal)sqa2

(632.5sqa2 + qblsqa2)cqa2 - (632.5cqa2 + qblcqa2- (200+L))sqa2

632.5cqa2 + qblcqa2 - (200+L) cqa2 sqa2 0

-(632.5sqa2 + qblsqa2)

0

- (685-qal)cqa2

-sqa2 cqa2 0

0 0 1

[

(685-qal)sqa2

-(700sqa2 + qblsqa2 + ( -350 + qb2)cqa2)sqa2 -(700cqa2 + qblcqa2 - ( -350 + qb2)sqa2 - (200+L))cqa2

-(685-qal)cqa2

(685-qal)sqa2

(700sqa2 + qblsqa2 + ( -350 + qb2)cqa2)cqa2 -(700cqa2 + qblcqa2 - ( -350 + qb2)sqa2 - (200+L))sqa2

700cqa2+qblcqa2-(-350+qb2)sqa2-(200+L) cqa2 sqa2 0

-(700sqa2+qblsqa2+(-350+qb2)cqa2)

0

- (310-qal )cqa2

-sqa2 cqa2 0

0 0 1

r

(310-qal)sqa2


-(310-qal)cqa2

(310-qal) sqa2


305cqa2+qblcqa2-(-350+qb2)sqa2-(200+L) cqa2 sqa2 0 -(305sqa2+qblsqa2+(-350+qb2)cqa2)

0

-sqa2 cqa2 0

0 0 1

47

[3.33]

48 Chapter 3

3.4. Assessment of the Geometric Errors

In the preceding paragraphs, a general model has been postulated which relates errors

in the actual location of frames attached to tool and workpiece, to errors in the

location of frames attached to succeeding components of the multi axis machine.

Furthermore, this general model has been elaborated to construct the type dependent

model, which describes this relationship for a certain machine.

As a final step in the development of the individual model, the relation between the

geometric errors and the position of the carriages has to be obtained. This requires an

extensive performance evaluation of the machine tool. The measurement set-ups and

the obtained results will be discussed in this section.

One of the problems in the assessment of the geometric errors is the physical

impossibility to measure in the center of the elements, as accordingly to the definition

of the position of the coordinate frames is required. This implies for the obtained

translational errors that a correction to the center of the elements is necessary. The

correction value is determined by the position of the measurement i.e. the influence of

rotational errors on the measured displacement. In figure 3. 7 a two-dimensional

example is depicted for the measurement of xtx. In this case, the bare measurement

data have to be corrected with the influence of the rotation e (i.e. e.D) in order to

obtain the true error xtx, defined in the center of the carriage.

Frame of measurement

/ Defined frame in model

Fig. 3.7. Example of a measurement ofxtx and the effect of rotations

In the next section the assessment of the bare measurement data will be discussed. As

the machine tool under research is placed in a temperature stabilized room with a

temperature of 23 ± 0.5 °C, this temperature is accepted as the reference temperature

for all measurements.


Software for Automation of the Measurements

As the number of measurements, required for the purpose defined in this project, is

very large, dedicated software is developed. This software has basically two main

tasks:

1) Control of the position of the machine tool;

2) Collecting and storing of measurement results.

In order to perform task 1, an interface has been accomplished with the milling

machine under investigation. This interface uses the RS232 port of an

IBM-compatible computer for data transport. The communication with the milling

machine runs under a specified protocol which is delivered by Philips Industrial

Electronics, Machine Tool Controls. With this interface, the computer can control

almost any feature of the machine tool. This allows convenient programming of, for

instance, the positioning of the axis for measurement purposes.

The second task of the program is to gather the results from the installed measurement

devices. As most modem instruments are available with an JEEE-interface, the

communication and installation facilities for all necessary measurement devices as a

Hewlett Packard laserinterferometer, Wyler electronic levelmeters, Keithley

temperature measurement equipment and Hottinger Baldwin inductive displacement

transducers, has been developed. With this interface facility, the computer aided

set-up and read-out of these instruments becomes relatively simple. Besides the IEEE

interfacing, the program is developed to have communication facilities through

RS232, ND converter and keyboard input to accommodate any measurement situation

that might occur.

50 Olapter 3

The program is completely menu-driven, which makes it convenient to use. A typical

measurement definition consists of the following steps:

• definition of the error to be measured, the instrument to be used, the

interface type and interface address (IEEE);

• software installation of the measurement instrument (if necessary);

• definition of the initial position of the machine tool;

• definition of the measurement sequence:

- definition of the start-position of the measurement;

- defmition of the axis and range of the measurement, the required

positions (linear steps back and forth, pelgrimstep or a number of

random positions);

- definition of the number of repeats of the measurement sequence

• definition of the file for data storage and the number of repeats of the

measurement.

After the above described steps have been completed, a measurement will be carried

out automatically, yielding a datafile and an accompanying measurement information

file. During the measurements all relevant information is displayed on the screen to

allow visual checking of incoming data. Before taking the actual measurement

reading, the program scans the reading of the measurement instrument until it remains

within a specified range. This procedure enables elimination of dynamic effects

induced by positioning of the machine tool.

In figure 3.8 a schematic overview is presented of the used measurement instruments

and the applied interfaces.

All the measurements described below are carried out by application of this software

package.

Model of the Geometric Error Stmcture 51

5-AXIS IIILIJNG IIACHINE

c z arx y

Fig. 3.8. Scheme with measurement instruments and interfaces

Measurements and Results

With the aid of the above presented program all 21 geometric errors, present in the

system of the three linear axes, have been determined. The measurement sequence of

these measurements is defmed over the entire range of the respective axis of

movement, with a measurement step of 10 mm. All measurements are carried out

back and forth over the range of the axis.

Below some measurement results are presented First, the measurement of xrz is a

typical example of a measurement set-up and obtained results. Secondly, a couple of

measurements are discussed that give cause to further investigation. An overview of

all measurement set-ups and accompanying results is presented in Appendix II.

In all the graphs presented below the bare measurement data are depicted. This

implies that no correction for thermal expansion or, in case of translational errors, a

correction for the influence of rotations is carried out.

52 Chapter 3

Rotation error xrz

The measurement set-up for xrz is depicted in figure 3.9. The reference interferometer

is mounted to the ram while the retroreflector is connected to the workpiece table,

which performs the movement in X-direction. The results of this measurement are

presented in figure 3.10. These results directly reflect the rotation error between the

coordinate frames of the X- andY-axis.

Laserhead

Workpiece table Angular interferometer

Fig. 3.9. Measurement set-up for xrz

Measurement of xn 8~----r-----~----~----~----,-----~----~

20 measurements

4

3

2

100 200 300 400 500 600 700

Position of the x-axis [mm]

Fig. 3.10. Error xrz versus position of the X-carriage


Rotation error xrx

For this measurement a set of electronic levelmeters is applied. The electronic

levelmeters are calibrated against a sine bar yielding a maximal inaccuracy of 0.5

arc sec.

The measurement set-up for xrx is depicted in figure 3.11. The reference levelmeter is

mounted on the ram, thereby eliminating the effect of rotation of the overall machine

structure, while the measurement levelmeter is placed on the workpiece table, which

performs the movement in X-direction. The rotation error does not depend on the

position of measurement, so the obtained results directly reflect the rotation error

between the coordinate frames of the X- and Y -axis.

Reference levelmeter

Workpiece table

Fig. 3.11. Measurement set-up for xrx

Execution of the described measurement yields the results that are graphically

depicted in figure 3.12. In this graph all bare measurement data are depicted that are

obtained by repeating the measurement 20 times. The rotation error of the machine is

defined in arcsec (1 arcsec "" 4.8e-6 rad).

In the results of xrx, individual peakvalues of the measurand can be observed with

magnitudes of the same order as the measured error. Plotting the measurement results

sequentially reveals that the peak error repeats equidistant Transformation of

measurement indices to time leads to the conclusion that the peakvalues repeat every

30 minutes. In figure 3.13 the measurement results of xrx are depicted together with a

quasi timescale. From an inspection of the machine constants it appeared that the

periodic peaks are induced by the lubrication pump of the machine tool, that is

activated every 30 minutes.

54 Olapter 3

Measurement of xrx

20 measurements

-2

-3

4L-----~----~----~----~----~----~----~ 0 100 200 300 400 500 600 700

Position oftbe x-ax.is [mm]

Fig. 3.12. Error xrx versus position of the X-camage

Measurement of xrx 2r---~----~----~----~----r---~----~----~

Bare dala

0

1f .. ! -1

~ -2

-3 Help scale : : ~

I I ' '

4 i !

0 s 7 8

Time [hours]

Fig. 3.13. Results of measurement of xrx sequentially and time indicator of 30 minutes


Scale error ztz

For this measurement a Hewlett Packard 5528 laserinterferometer is applied with

accompanying linear optics and an air sensor.

The measurement set-up is schematically depicted in figure 3.14a and in figure 3.14b

the actual situation is displayed. The interferometer is mounted on the workpiece

table, while the retroreflector is connected lo the ram of the machine tool. The

influences of the rotation zrx and 'li'f have to be eliminated from the obtained

measurement results. This yields the error ztz of the coordinate frame positioned in

the centroid of the Z-carriage.

Laserhead

Interferometer Retroreflector Workpiece table

Fig. 3.14a. Measurement set-up for ztz

Fig. 3.14b. Actual situation of the above presented scheme

56 Chapter 3

Execution of the described measurement yields the bare, uncorrected results that are

graphically depicted in the first part of figure 3.15.

In the results of ztz a clear form of hysteresis can be observed. This is not caused by

the hardware of the machine, but purely by a reproducing temperature field over the

Z-scale. In figure 3.16 the temperature on three positions of the Z-scale is depicted.

These temperatures were obtained during the measurement of ztz. The second graph

in figure 3.15 represents the error ztz corrected for the effect of the changes in the

temperature of the Z-scale. aearly the hysteresis has disappeared.

Analysis of the cause of the thermal problem leads to the conclusion that the hydraulic

installation warms up the Z-scale by radiation. In order to avoid this problem it is

therefore advisable to isolate the scale of the Z-axis.

O.o3 zTz, bare data

~ 0.02

';:' O.ol

~ l

-0.01 0 100 200 300 400 500 600

Position of the ~ge [mm)

0.03 z corrected data

0.02

~ iO.ol

Ji

-0.010 100 200 300 400 500 600

Position of the Z-amiage [mm]

Fig. 3.15. Error ztz versus position of the Z-carriage

Model of tbe Geometric Error Structure 57

Temperai!Ue of Z-scale during :<1z measurements 27.5.-----.---~----.----~--~-------,

Posllim of the Z-caniage

Fig. 3.16. Temperature of the Z-scale during measurement of ztz

3.5. Modelling the Geometric Etrors

The previous described measurements yield datasets consisting of discrete measured

errors. However, for correction purposes, as will be explained in Chapter 5, these data

have to be available as a set of support points with accompanying error values. To

obtain tbis form of the errors, an intermediate transformation into a continuous

description of the errors is necessary. This implies that fitting techniques are applied

to the datasets. Although most mathematical functions can be applied to describe the

general trend of an error, their use is limited for modelling the, often irregular, shape

of the remainder. That is, the behaviour of an error in one region of the domain may

be totally unrelated to the behaviour in another region. Polynominals, along with most

other mathematical functions, have just the opposite property: their behaviour in a

small region determines their behaviour everywhere. In the application of the

individual model as a basis for software error compensation, the use of special

functions is preferable. A group of functions that possesses this property to a lesser

extent, are the so-called piecewise polynominals [3.11]. Piecewise polynominals can

be described as a set of polynominals defined upon limited continuous parts of the

domain. The pieces join in the so-called knots, obeying continuity conditions with

respect to the function value itself and an arbitrary number of derivatives.

58 <lmpter3

The number and degrees of the polynominal pieces, the nature of the continuity

restrictions and the number and positions of the knots may vary in different situations,

which gives piecewise polynominals the desired flexibility.

Definition of Piece-wise Polynominals

A straightforward mathematical implementation of the continuity restrictions can be

obtained by the use of truncated polynomials, or "+"- functions, as basic elements in

the piecewise polynominal models. The"+"- function is defined as:

• u+ =u

• u = 0 +

if u > 0

if u ~ 0 [3.51]

In general, with k knots tl, .... , tk and k+l polynomial pieces each of degree n, the

truncated power representation of a piecewise polynominal p(x) with no continuity

restrictions can be written as:

n k n j

p(x) = 2 f3oj xi + 2 2 f3i/ x - ti ) + [3.52]

j=O i=l j=O

Note that the presence of a term fJij(x - ti)!, allows a discontinuity at ti in the j-th

derivative of p(x). Thus, different continuity restrictions can be imposed at· different

knots simply by deleting the appropriate terms. Normally, it is sufficient to ensure that

each model is continuous with respect to the function value and its first derivative.

An inherent problem in constructing the individual model is the unknown nature of

the errors to be described. The model's potential to accommodate irregular errors is to

an extensive degree determined by the number and position of the knots. If the

position of these knots are considered variable, that is: parameters to be estimated,

they enter into the regression problem in a nonlinear fashion, and all the problems

arising in nonlinear regression are present [3.11]. The use of variable knot positions

also carries the practical danger of overfitting the data, and makes testing of

hypotheses considering areas of structural change virtually impossible. Unless prior

information is available, we use a basic model which contains enough polynominal

pieces with a fixed length and a maximum allowable degree of two, to accommodate

the most complex error expected.


In the parameter estimation process, a stepwise regression procedure is implemented

to remove statistically insignificant parameters from the model. The reason for this

removal is twofold:

• including insignificant parameters hardly improves the model's quality of fit, but

increases the variance of the estimated parameters and response.

• identification of structural parameters enhances the diagnostic properties of the

individual model

Fitting the Measurement Data

The data obtained from the measurements, as described in Appendix IT, are averaged.

A least squares fitting procedure has been applied to this average data sets, using

piece-wise polynorninals. The piece-wise polynorninals are defined by the position of

the knots and the coefficients of the polynorninal for each interval. An example of a

result is graphically depicted in figure 3.17. This method yields a continuous

description of each geometric error, based on discrete measurements. This information

will be applied for software error compensation (Chapter 5).

Average zry and fitted po1ynominals 0.2.-----.----...-----..---...,.....------,.------,

.0.4

I .0.6

! .0.8

~ -1

-1.2

-1.4

-1.6

-1.8 0 100 200 300 400 500 600

Position of the Z-carriage [ mm]

Fig. 3.17. Example of fitting with piece-wise polynomina/s on error zry

60 Chapter3

In the graph depicted in figure 3.17, the dashed lines represent the averaged results of

a number of back and forth measurements. The solid line represents the fit through

these averaged measurement data using piece-wise polynominals.


References

[3.1] Love, W.J., Scarr, A.J.:

The Determination of the Volumetric Accuracy of Multi Axis Machines.


[3.2] Hocken, R.:

Three Dimensional Metrology.

Annals of the CIRP, Vol26/l, p403-408, 1977.

[3.3] Schultschik, R.:

The Accuracy of Machine Tools under Load Conditions.


[3.4] Duffour, P., Groppetti, R.:

Computer Aided Accuracy Improvement in Large NC Machine Tools.


[3.5] Busch, K., Kunzmann, H., Wlildele, F.:

Calibration of Coordinate Measuring Machines.

Precision Engineering, Vol. 7, nr. 3, p139-144. 1985.

[3.6] Donmez, A.:

61

A General Methodology for Machine Tool Accuracy Enhancement - Theory,

Application and Implementation.

Ph.D Thesis, Purdue University, 1985.

[3.7] Ferreira, P.M., Liu, C.R.:

A Contribution to the Analysis and Compensation of the Geometric Error of

a Machining Center.


[3.8] Soons, J.A.:

Modelvorming, schatting en correctie van de bij coordinatenmeetmachines

behorende afwijkingenstructuur.

Technische Universiteit Eindhoven, WPA-rapportnr. 0557, 1988.

62 Ompter 3

[3.9] Paul, R.P.:

Robot Manipulators: Mathematics, Programming and Control.

MIT Press, Cambridge, MA, 1981.

[3.10] Takeuchi, Y., Sakamoto, M.:

Analysis of Machining Error in Face milling.

Proc.of the 23th Int. MTDR conference, p153-158, 1982.

[3.11] Soons, J.A., Theuws, F.C., Schellekens, P.H.:

Modelling the Errors of Multi Axis Machines: A General Methodology.

Precision Engineering, Vol. 13, Accepted for publication, 1991.

63

Chapter4

Determination of the Thermal Behaviour

As stated in Chapter 2, thermal influences on the machine tool account for the largest

part of the inaccuracy of the product [4.1, 4.2, 4.3]. Although a number of heat

sources contribute to the thermal deformation of the machine tool, the friction in the

spindle bearings is regarded as the main source [4.4, 4.5]. This is verified for the

machine tool under investigation (see Chapter 2).

The key problem in this field is the determination of the relation between the

temperature distribution and the accompanying deformation. Two main approaches

can be taken to model this behaviour.

First, a complex and time consuming approach is the use of finite element analysis

methods (FEA). A number of researchers have applied this methodology on

(sub-)structures of machine tools [4.7, 4.7, 4.8]. Due to several reasons, as already

discussed in Chapter 2, the practical relevancy of this methodology for correction

purposes is still limited.

Secondly, and this approach is favorite to an increasing number of researchers

[4.9, 4.10], an empirical method can be applied to determine the actual relationship

between the thermal distribution and the displacement of the tool holder.

In this study we haven chosen the latter methodology for reasons of:

• no influence of approximations while modelling the structure (for

instance, coefficients of radiation and expansion, the shapes and

dimensions of the geometrical structure of the machine tool);

• the limited amount of time available for this topic. As the goal of this

study is the actual correction of a machine tool it necessitates a method

that has proven to yield practical results;

• the relationship obtained from the modelling between the displacement

and the thermal distribution reflects the actual distortion of the machine

tool, so for practical implementation no transformation is necessary;

• the extreme effort to apply the PEA-method, both financially and

mathematically, is not proportional to the limited goal of this study.

In order to determine the relationship between the thermal distribution and the

deformation of the machine tool, a measurement set-up is built, which is based on a

commonly applied principle [4.11, 4.12]. For correction purposes, the principal

interest is not the deformation of each machine component, but the displacement of

the tool with respect to the workpiece. Therefore, the measurement set-up is designed

to obtain the displacement of the tool holder in three orthogonal directions, and the

two relevant rotations.

The measurement of the thermal distribution is carried out by extensive temperature

measurement equipment.

In the next sections the different parts of the measurement set-up will be described.

Furthermore the obtained results will be depicted and discussed.

4.1. The Methodology

Set-up for Displacement Measurement

In the tool holder of the machine tool a cylinder is mounted. This cylinder is

machined on the machine tool under investigation in order to eliminate run-out errors

of the spindle. On the workpiece table a base is mounted with five contactless eddy

current displacement transducers. These transducers measure the displacement of the

tool holder with respect to the workpiece in three directions simultaneously. The

displacement transducers are calibrated separately in the actual measurement set-up

using a Hewlett Packard 5528 laserinterferometer. The results of this calibration

indicated an uncertainty of less than 1 Jim over the full range, after application of a

correction formula for systematic errors.

The transducers are linked to an amplifier (Hottinger Baldwin DMC9012A) that is

capable of reading all signals simultaneously.

Detennination of the Thermal Behaviour 65

The amplifier sends the obtained signals to a PC by means of IEEE interface.

In figure 4.1 a the measurement set-up for the determination of the displacement is

depicted. A photograph of the actual set-up is presented in figure 4.1 b.

Conladlea• d.t•plaeement triU1.d.u~n

Fig. 4.1 a. Scheme of set-up for displacement measurement

Figure 4 .1 b. Photograph of the set-up for the displacement measurement

66 Olapter 4

Set-up for Temperature Measurement

Scattered over the machine's structure a number of Pt-100 temperature sensors are

positioned. These sensors are fixed to the surface of the machine's structure, applying

heat conducting paste for assured thermal contact. The sensors are connected to a

scanner (Keithley type 706), capable of reading a maximum of 100 sensors. The

scanner is connected to a digital multimeter (Keithley type 196) which sends the

measured value to the PC by IEEE-interface. Each temperature sensor is calibrated

with an uncertainty of less than 0.1 °C, in a range from 15 to 50 °C, which is

sufficient for this purpose [4.13].

In figure 4.2 the position of each temperature sensor on the machine tool is depicted.

The choice of the position of each sensor is determined by:

• the capability of obtaining the thermal gradients in all directions. This

implies the sensors are located at the corners of an imaginary cube,

yielding the possibility to determine linear temperature gradients in three

orthogonal directions;

• a sensor density proportional to the importance of the heat sources;

• a minimum of three sensors on each measuring scale.

Fig. 4.2. Position of the temperature sensors on the machine tool

Determination of the Thermal Behaviour 67

From previous measurements (see Chapter 2) and bibliographical studies [4.14], it

appeared that the main spindle bearings, together with the main drive, are the most

important internal heat source. This implies for the measurement strategy that the

influence of the heat induced by the drives of the carriages on the thermal distortion

of the machine tool, will not be taken into account in this study.

For carrying out a measurement the surface of the cylinder is positioned on 0.5 mm of

the transducers. In this position the machine tool is loaded with a spindle speed over a

specified time. During the measurement the displacement of the tool holder and the

temperature distribution of the machine tool are obtained every 60 seconds. The

progranuning of the machine tool and the collection of the measurement data are

carried out automatically by a computer. Therefore the software package, as described

in Chapter 3, is extended. This software has basically three tasks:

1) Positioning of the axes so that a measurement can be carried out;

2) Control of the load sequence of the machine tool;

3) Collecting and storing of measurement results.

In order to perform tasks 1 and 2, the RS232 interface routine is applied.

The third task of the program is to gather the results from the installed measurement

devices. This is implemented by using an IEEE-interface to the DMC9012 amplifier.

The program is completely menu-driven, which makes it convenient to use. A typical

measurement definition consists of the following steps:

• software installation of the measurement instrument: number of

measurements per second and averaging number;

• definition of the initial position of the machine tool;

• definition of the measurement sequence:

- definition of the load condition (in general rpm);

- definition of the time span for the specified load;

- definition of the number of repeats of the measurement sequence;

• definition of the file for data storage and the number of repeats of the

measurement.

68 Chapter 4

After the above described steps have been completed, a measurement will be carried

out automatically, yielding a datafile and an accompanying measurement-information

file. During the measurements all relevant information is displayed on the screen to

allow visual checking of incoming data. In most cases the measurement instrument is

programmed to average a number of measurements, for instance 1200 measurements

are taken in 0.5 sec and the average is sent to the PC. This procedure eliminates the

effect of the vibrations of the machine tool on the measurement of the displacement.

Figure 4.3 presents an overview of the total measurement set-up and the applied

interfaces.

IBM-compatible computer

~X y

KEITHLEY Voltmeter &. Scanner

5 Transducers

Hottinger Baldwin DMC9012 Amplifier

Fig. 4.3. Scheme of the total measurement set-up

Detennination of the Thermal Behaviour 69

4.2. The Measurement Strategy: Possibilities and Limitations

The described set-up can be applied for measuring the displacement of the tool holder

in three orthogonal directions. In order to obtain a practical impression of the thermal

behaviour of the machine tool a measurement strategy is developed. With this

measurement strategy it is possible to obtain a thorough impression of the thermal

behaviour of the machine tool. Hereby various loads of the spindle speed can be

applied.

This strategy contains basically the following items:

1) the position of the measurement is limited to three positions along the

Z-ax.is, i.e. the ram extracted, centered and retracted;

2) the positions of the X- and the Y-ax.is remain constant during the

measurements, i.e. the X-carriage is put in the center (X=350 mm) and

theY-carriage at a coordinate Y=140 mm;

3) the position of measurement remains constant during the entire

measurement sequence;

4) all measurements are initiated from the same reference state;

5) no cooling liquid is applied during these measurements;

6) all measurements are carried out with a vertical milling head.

For the position of the Z-carriage the first statement has lead to the coordinates: 62,

314 and 566 mm. The number of measurement positions is chosen such that, with the

continuity of the deformation taken into account, a parabolic deformation can be

determined.

The choice of a fixed X-position is based on the assumption that the machine tool is

thermomechanically symmetric in X-direction. This implies that the position of the

X-carriage has no strong influence on the measured displacement. The latter

conclusion has been verified by taking three measurements with a variable X-position.

The measurement results of the displacement in Y- and Z-direction are depicted in

figure 4.4.

70 Chapter4

10 Measured d. lacement in Y -direction on different X- sitions

e Load : 6000 rpm 6 hours, 0 rpm 8 hours .:. i 5 ~ .~ -20 Q

-30 0 50 100 150 200 250 300 350 400 450

Tune[min)

Jacement in Z-direction on different X- ·lions

e .:.

I ~ g -20

-30 0 50 100 150 200 250 300 350 400 450

Time[min]

Fig. 4.4. Measured displacement in Y-direction on several positions of the X-axis

The reason for measuring with a fixed Y-position is twofold First, this position

reflects the major part of the positions during actual operation so the obtained results

correspond to the practical behaviour of the machine tool. Secondly, preliminary

measurements revealed the larger heat sources are present in the Z-axis, therefore this

axis will show the highest contribution to the measured displacement.

The 5th item of the measurement strategy requires some explanation. The milling

machine under investigation is equipped with a swivel head. This implies the vertical

milling head can be changed automatically by a horizontal milling head.

For the modelling of the thermal behaviour this procedure yields several problems.

First, the heat sources change dramatically as the bearings and the gearbox of the

vertical milling head are no longer driven. Secondly, the geometry of the machine,

and thereby its thermal behaviour changes as the vertical milling head is placed on top

of the Z-axis. Third, the position of measurement changes due to a new location of the

tool holder.

Determination of the Thennal Behaviour 71

These three problems has lead to the conclusion that a mixed modelling of a vertical

and horizontal milling head necessitates a different approach than the chosen

measurement set-up. Therefore we confme ourselves to the milling machine with the

head in vertical position.

Although the presented measurement strategy supplies a large amount of information·

concerning the actual displacement of the tool holder, some limitations are inherent to

the set-up and will be discussed below.

Limitations of the Measurement Strategy

Static modelling

All measurements are relative to one specific thennal reference situation. This implies

that the machine tool must be placed in a temperature stabilized environment in order

to get repeatable results. In this situation the repetition of a measurement while

applying the same load, yields identical measurement results.

Cooling liquid

Due to the nature of the measurement transducers, i.e. eddy current non-contact, the

use of cooling liquid causes difficulties in the collection of the measurement data. In

case of the used milling machine, the cooling liquid is pumped through the ram. It it

very likely that this has some effect on the thermal behaviour of the ram. In future

research, the effects of cooling liquid will be assesssed with the application of a

modified measurement set-up.

Moving carriages

The measurements are carried out with fixed carriages. This does not reflect the

normal situation of operation when the carriages move in order to generate a

workpiece. While moving the carriages, the thermal behaviour of the machine tool

might differ from the fixed situation.

72 Olapter4

However, as presented in Chapter 2 (figure 2.4) the movement of the carriages does

not induce structural heat sources compared to the main sources. Therefore the effect

of moving carriages on the thermal behaviour is restricted to a varying heat flow due

to the changing structure of the machine. The investigation of this effect is not carried

out in this study.

A main practical problem of implementing moving carriages into the model, is the

choice of sequence for the movements. If this sequence seriously effects the thermal

behaviour the modelling of this behaviour becomes highly complicated.

Cutting process

The effect of the heat induced by the cutting process is not taken into account by the

proposed measurement set-up. However, the influence of the cutting process on the

thermal behaviour on the total machine structure is regarded to be negligible in

finishing processes if the structure is insulated from falling chips [4.15].

4.3. Results of the Measurements

With the measurement set-up described a number of measurement cycles are carried

out, applying a spindle speed as load to the machine tool. During these measurements

the temperature correction of the manufacturer of the machine tool was put

non-active. In figure 4.5 the results of a measurement with a spindle speed of 5000

rpm for 6 hours, followed by a spindle stop of 8 hours, is presented. Only the

displacement of the cylinder is depicted. For the X- and Z--direction the displacements

is measured by the upper transducers of the measurement set-up.

As the standard available temperature compensation of the machine tool operates on

the information of the temperature of the spindle head, the temperature variation of

the spindle head is marked out on the abscissa. This choice enables us to draw

conclusions with respect to the suitability of one temperature sensor. On the ordinate

the measured displacement of the tool holder is plotted.

Determination of the Thermal Behaviour 73

The solid part of the graph represents the wanning up time, whereas the dashed part

represents the cool down period. The thick marks in the graphs represent the elapsed

time with a thickmark placed each 30 minutes.

During this measurement the spindle was positioned in the center of the working

space.

I -20

1 -40

! ~

0 5

20

Drift of the tool bolder in Y -direction

10 15 20 25

Temperature rise of the spindle bead (degree centigrade]

Drift of the tool holder in Z-direction ..... ._ ...........

30

I 0

I -20

-.. -.................. Cooling dowo

............................

u J!l .t -40 0

~ 0 5 10 15 20 25

Temperature rise of the spindle head [degree centigrade]

Fig. 45. Results of a drift measurement in Y- and Z-direction

Load situation: n=5000 rpm for 6 hours, n=O for 8 hours Solid line: warming up; Dashed line: cooling down

Thickmarks: time indication every 30 minutes

30

35

35

With the results depicted in figure 4.5 the necessity of an extended thermal model can

be clearly demonstrated. If a temperature rise on the spindle head of approximately

25 °C is measured, the displacement in Y-direction can be 40 as well as 52 JJ.m. The difference is determined by the wanning up or cooling down of the machine tool.

In the Z-direction this effect is even larger and the predicted displacement at a

temperature rise of 25 °C can even range as much as 40 Jlm. This difference, that may

look like a clear form of hysteresis, is caused by the thermal state of the other parts of

the machine tool.

74 Chapter4

These measurement results are obtained with one specific load (5000 rpm). In

practice, changing the load situation causes differences in displacement up to 50 Jlm

of the tool holder at a specific temperature rise of the spindle head.

From these observations we can conclude that one temperature sensor for correction

of thermal behaviour is absolutely insufficient.

Besides loading the machine tool with a spindle speed of 5000 rpm, measurements are

carried out with loads over the entire range of possible speeds (0-6300 rpm).

Figure 4.6 and 4.7 show some results with different loads. Note that the choice of the

abscissa is not restricted to a particular temperature sensor on the machine tool, but in

all graphs we have chosen for the temperature of the spindle head. Some

measurements show a discontinuity in the displacement measurement at the

switch-point from warming up to cooling down (in the graphs: the point where solid

line changes to dashed line). This effect is caused by forces exerted by the

deceleration of the actual spindle speed to zero.

Drift of the tool holder with a load of 1500 IJIIll for 6 hours 15.---~---r--~----r---~---r--~----r---~--•

............. ~ .. ···--·····~····-······:----.~~~~~~-10

e 2.

j ] .s ~

5

c -15 1 -20

..... -25

-30 Solid: Load= 1500 rpm Dashed: No-Load

: , __ _ -.. -~ .. ··.

-35~--~--~--~----~--~--~--~----~--~--~ 0 2 3 4 56 7 8 9 10


Fig. 4.6. Drift of the tool holder with a load of 1500 rpm

! t ~

10

s

.J -10 'CI

j- -15

f -20

1'-1 -25

Determination of the Thennal Behaviour

Drift of the tool holder with a load of 4000 rpm for 6 hours

-30 Solid: Load=4000 rpm Dashed: No-Load


Fig. 4.7. Drift of the tool holder with a load of 4000 rpm

75

Although loading the machine tool with a constant spindle speed provides a large

amount of information concerning the behaviour of the machine tool, practical

situations often display a rapidly changing load situation. To obtain an impression of

the machine tool's thennal behaviour in practical situations, it is loaded with a spindle

speed spectrum as defined in DIN 8602. This spectrum consists of spindle speeds

varying from zero to maximum spindle speed (n ), with steps of 25% of n . For max max the machine tool under investigation n is 6300 rpm. Each load situation is max maintained for 15 minutes. A graphical presentation of this spectrum is depicted in

figure 4.8. The displacement of the tool against the elapsed time is depicted in

figure 4.9.

76 Ompter4

Time (min)

Fig. 4.8. DIN 8602 spectrum

e .!.

~ .t:l

] j ~

l a -30

-35 0

Drift of tbe tool holder loaded with tbe DIN-spectrum

~ D ~ D ~ D B ~ ~

Time[min]

Fig. 4.9. Drift of the tool holder under load of the DIN-spectrum

All presented results are obtained in the center of the working space. The same series

of measurements are repeated for different Z-coordinates of the machine i.e. with the

Z-carriage extracted and retracted. These measurements yield a different behaviour

than those obtained in the center. An example of the displacement in Y -direction,

measured on different positions of the Z-carriage, is depicted in figure 4.10. Here a

load sequence is applied of 5000 rpm for 6 hours and no load for 8 hours. The solid

line represents the warm up period, whereas the dashed line depicts the cool down

period.

E' a. II

:li! i! 1 Jl '0

~ § ]. 0

10

0

-10

-20

-30

-40

-50

.6()

-70

-80

Detenninatioo of the 'Thermal Behaviour

0

Drift of tool holder in Y -<lirection with load of 5000 rpm

Solid: Load=SOOO rpm Dashed: No-Load

~ D a G D a D ~ B

Time [min]

Fig. 4.10. Drift of the tool holder on different Z-positions

77

All the described measurements reflect the displacement of the tool holder relative to

a specific reference situation. This implies that no historic parameters are included in

the modelling technique. In the next Chapter we will discuss the development of a

thermal model based on these measurements. It should be noted here that we will

confine us to the description of the static thermal behaviour i.e. the displacement of

the tool holder is related to the current thermal state of the machine tool, no historic

information is used in the modelling. However, in the near future related studies will

assess a type of modelling that includes thermal history.

78 Chapter 4

References

[4.1] Venugopal R.:



[4.2] Bryan J.:


Annals of the CIRP, Vol. 39/2, p645, Citation of Kaebernick, 1990.

[ 4.3] Sa to H. et al:

Development of Concrete Machining Center and Identification of the

Dynamic and the Thermal Structural behaviour.


[ 4.4] Bryan J.:


Annals of the CIRP, Vol. 39/2, p653, Reference to McMurthry, 1990.

[ 4.5] Bryan J.:


Annals of the CIRP, Vol. 39/2, p653, Reference to Janeczko, 1990.

[4.6] Brauning H.:

Ein Numerisches Rechenmodell zur Berechnung de Instationairen

Temperaturverteilung in Werkzeugmaschinen, Programmiert fiir

Elektronische Datenverarbeitungsanlagen.

Ph.D-dissertation, TH Aachen, 1972.

[ 4.7] Sata T.:

Dynamic Analysis of Machine Tool Structures by the Finite Element

Method.


Detennination of the Thermal Behaviour

[4.8] Zangs L.:

Berechnung des Thermischen Verhaltens von Werkzeugmaschinen.

Ph.D Thesis, TH Aachen, 1975.

[4.9] Kurtoglu A.:

The Accuracy Improvement of Machine Tools.


[4.10] Donmez, A.:

79

A General Methodology for Machine Tool Accuracy Enhancement- Theory,

Application and Implementation.

Ph.D Thesis, Purdue University, 1985.

[ 4.11] Draft Standard, Methods for Performance Evaluation of Computer

Numerically Controlled Machining Centers and Work Centers, ASME B5.52

Committee, Version 6.0, June 1990

[4.12] Week M.:

Werkzeugmaschinen, Band 4, Mef3technische Untersuchung und

Beurteilung.

VDI Verlag, DUsseldorf, 1985.

[4.13] Spaan H.A.M.:

Kalibratie van een 60-punts temperatuursmeetopstelling.

T.V. Eindhoven, WPA-rapportnr. 0942, oktober 1990.

[ 4 .14] Kersten A.:

Geometrisches Verhalten von Werkzeugmaschinen unter Statischer und

Themtischer Last.

Ph.D Thesis, TH Aachen, 1983.

[4.15] Hocken Robert J.:

Machine Tool Accuracy.


81

ChapterS

Design of a Correction Algorithm

The primary goal of this study is defined as the enhancement of machine tool

accuracy. For this purpose a correction algorithm is developed and implemented into

the control system. The correction algorithm is based on the measurements of the

behaviour of the machine tool and is split into the following three parts:

• the basic errors in the geometry;

• the linear expansion of the measuring scales due to temperature offsets

relative to twenty degrees centigrade;

• the thermal behaviour of the structure, based on the measured results.

The first part of the correction algorithm, i.e. the geometric error correction, is

dependent on the reference state of the machine tool and does not change rapidly. In

general the compensation values remain constant while machining.

The latter two parts, i.e. the thermal error correction, are dependent on the actual state

of the machine tool and can change rapidly. The values of this correction must be

updated while operating the machine tool.

For implementation of the correction algorithm into the control system, some

restrictions must be taken into account. These restrictions are mainly due to the timing

problem, i.e. the calculations of the correction value may take maximal the sample

time of the control system (15 msec).

In the next sections we will present the development of a geometric error correction,

derived from the mathematical model and adapted to the conditions of the control

system.

82 Chapter 5

Furthermore the development of a correction algorithm for thermal effects (scale

expansion and structure distortion) is presented. Hereby the scale expansions are

calculated by analytical formulas, whereas the distortion of the machine tool structure

is modelled by empirical measurements. For the latter we apply statistical methods.

The obtained results are presented and some efforts for improvement of the model are

discussed.

Finally, we will describe the implementation of the obtained correction algorithms

into the present available control system. This allows us to verify the effectiveness of

the compensation and implicitly the modelling procedure. The results of this

verification will be presented in the next Chapter.

5.1. The Geometric Errors

Conditions of the Control System

The mathematical model, presented in Chapter 3, can be applied for the correction of

the geometric errors. With the known error sources, the error vector E can be

calculated at any position of the machine's carriages. However, as the position of the

carriages can change rapidly, the calculation of the correction terms must be

performed every cycle of the control systems set-point generation. This implies that

every 15 msec a new correction vector must be available to the motion controller.

This is a main difference between the compensation of machine tools and measuring

machines: in case of measuring machines a compensation on the measured value can

be applied afterwards whereas for machine tools the set-point must be compensated.

Due to the maximum allowable computing time, the evaluation of the total model can,

at this moment, not be carried out. This, and some other, limitations on the possible

error correction methodology are imposed by the control system. It should be noted

that the presented restrictions are mainly due to the current hard- and software. New

developments are in preparation, so in the near future more of the possibilities of the

mathematical model can be utilized.

Design of a Correction Algorithm 83

The current limitations can be summarized as:

1) no evaluation of sine and cosine terms is possible due to lack of

time;

2) the orientation of the tool can not be corrected due to the above

mentioned reason and hardware limitations of the machine tool

used for this research;

3) a maximum of three correction functions, other than the correction

for scale errors, can be applied due to memory shortage;

4) the correction functions are allowed to be dependent on only one

axis. This implies that for instance a compensation in Z-direction

dependent of the position of the X- andY-carriage is not possible;

5) the corrections must be available to the control system in a

look-up table for speed purposes;

6} a maximum of 25 basepoints with accompanying compensation

values can be stored in look-up tables for each correction function.

The last condition necessitates a description of the resulting error function in a

look-up table with fixed gridpoints. The density of the gridpoints can be chosen freely

so the loss of information can be limited.

Elaboration of the Mathematical Model

The mathematical model of the machine tool under investigation will now be applied

to generate the correction algorithm for the geometric errors.

At this time, due to the second restriction imposed by the control system, the

correction for the error in the orientation of the tool is not possible, so we confine

ourselves to the elaboration of the position error. Using equation [3.33] as the basic

error model, we first eliminate the effect of the rotation elements on the tool position

by defining cos(q)=l and sin(q)=O. Secondly we eliminate the contribution of the

rotation elements on the error by designating their geometric error vectors (uE32

and

b2Eb3} to zero.

84 Chapter 5

This simplification yields the following equation for the position error wpetl of the tool

with respect to the workpiece:

[

0 -805+qal 210-(200+L) 1 0 0 l wpetl = 805-qal 0 0 0 1 0 0Eal

-{210-(200+L)) 0 0 0 0 1

[

0 -825+qa1 632.5+qbl-(200+L) 1 0 0 l - 825-qal 0 0 0 1 0 0Ebl

-(632.5+qbl- (200+L)) 0 0 0 0 1

[

0 -685+qal 700+qbH200+L) 1 0 0 l 685-qal 0 350-qb2 0 1 0 bl~2

-(700+qb1-(200+L)) -350+qb2 0 0 0 1

[5.1]

OEal = [ Oealx e 0 aly Oealz Oealx Oealy Oealz r [5.2]

OEbl = [ o\tx o\ty O~lz O~lx O~ly O~lz r [5.3]

b1Eb2 = [ bl~2x bl~2y b1~2z bleb2x bleb2y bleb2z r [5.4]

When the above derived equation is elaborated and the individual errors are

transformed from the general notation to the definition as presented in Chapter 2, the

following expression is obtained:

wpetlx = (-805+Z)zry + (10-L)zrz + ztx- (-825+Z)yry- (432.5+Y-L)yrz

ytx- (-685+Z)xry - (500+Y-L)xrz- xtx

wpetly = (805-Z)zrx + zty • (825-Z)yrx- yty- (685-Z)xrx

(350-X)xrz- xty

wpetlz = (-lO+L)zrx + ztz + (432.5+Y-L)yrx- ytz + (500+Y-L)xrx +

[5.5]

[5.6]

(350-X)xry - xtz [5.7]


The next step in the development of a correction algorithm is the adjustment of the

obtained formulas to fit into the control system.

First we exclude the straightness errors from the model. The reason for this exclusion

is that the absolute values of the errors prove to be very small [5.1].

Due to the fourth restriction of the control system the dependence on the error terms

must be reduced -to one axis of movement. This implies that cross terms as

"(-825+Z)yry" are impossible to implement directly. Consequent application of this

restriction leaves only one or two error components per direction for possible

correction.

The danger of the implementation of this limited set of corrections is that in the

original, uncorrected situation the effect of one error may be neutralized by the effects

of other errors. Implementation of only a part of the correction can then decrease the

accuracy.

Therefore we chose to simulate the effect on the position of the tool of all errors

simultaneously [5.2] and correct the machine tool for the obtained results. This

simulation must be carried out along suitable paths of the workspace, for instance in

the center.

For this simulation we applied the derived mathematical model for the geometric

errors. As input for the model the obtained measurement results can be used. Special

attention must be paid to the implementation of the squareness errors. As the

measurement results contain also influences of the rotation errors, each measurement

has to be corrected for the effect of other errors within the machine tool. This

correction is basically an evaluation of the mathematical model and requires some

computational effort. For the implementation of the squareness errors into the model,

they will be treated as offsets of rotations. Following this reasoning the squareness

error between the Y- and the Z-axis can be implemented in the error zrx by giving it

an offset of -2.0 arcsec, which is the recalculated squareness error.

Similarly, the squareness error between the X- and the Z-guide can be implemented in

the error yry by an offset of -0.1 arc sec. Finally, the squareness error between the X

and theY-guide can be implemented by an offset of +2.9 arcsec in yrz.

86 Chapter 5

Application of this simulation technique yields nine error functions that can be applied

for correction purposes: for each simulated path one function in the X-, Y- and

Z-direction. From these nine functions, the "scale-errors" can be used immediately for

correction. Because the control system is capable of storing a total of six correction

functions, three other suitable corrections need to be determined. These functions

represent the error in a direction other than the direction of movement

In figure 5.1 to 5.3 the simulated errors are depicted. It should be noted that in all

calculations a tool length (L) of 160 mm is applied.

Based on the simulations the following correction functions are preferred:

• the three simulated scale errors xtx, yty and ztz (these errors are

compensated to 20 °C);

• the simulated error zty;

• the simulated error ytx;

• the simulated error xty;

The latter three error functions will be indicated as 'straightness errors'.

xt0-3 Compensation values as function of tbe X -position 10

Position: X:0-700, Y=200,Z=300 8

6

I 4

i "' 2 0 -= = .. t u

-4 .• ..- xtz .-·

-6 0 100 200 300 400 500 600

Position of the X -carriage [ mmj

Fig. 5.1. Compensation dependent on the X-position

700

! ~ l -~ I

I 0 u


xlO·l Compensation values as function of the Y -position Sr---~--~--~--,---~--~--~---r--~---.

-S

-10

-15

Position: X=3SO, y .. o..soo, Z..300

Position oftbe Y -caniage (mm}

.·• .. •

Fig. 52. Compensation dependent on the ¥-position

xl(rl Compensation values as function of tbe Z.position 2r------r------~----~------,-------r------,

0\····-.. •. "'· .. -2 \

: \\ \\

\ ''\ ..

\ zty

-.. ···········-~-

...

········-.... ./

' / -12

····--.................................... -···---·· Position: X=3SO, Y=200, Z=0-600

-14

-16 . 0 100 200 300 400 500 600

Position oftbeZ-caniage (mm}

Fig. 53. Compensation dependent on the Z-position

87

88 Chapter 5

Evaluation of the Correction Algorithm

The final step towards geometric error correction is the transformation of the obtained

correction functions to look-up tables. This implies that the over the range of the axis

a set of gridpoints must be determined. Hereby the maximum number of basepoints is

prescribed.

With an allowable deviation of 1 J1m between the actual correction value, as depicted

in the above figures, and the real correction value, the number of intervals is

determined.

For the error xtx this approximation still yields an intetpolation error larger than 1

p.m. In the graph of the error a clear periodic behaviour can be observed. As the

machine tool is capable of compensating for cyclic errors, the simulated xtx is split

into a part that describes the general trend, and a cyclic part that is superimposed on

the trend.

In figure 5.4 the simulated error xtx is presented along with the trend and the periodic

term.

xl0-3 Splitting of xtx into a trend and a cyclic part 8.-----~----~----~----~----~----~----~

6

Cyclic term (period = 39.6 mm

Position of the X -carriage [mm)

Fig. 5.4. Splitting the simulated compensation xtx into an trend and periodic part


In figure 5.5 and 5.6 the calculated compensation functions are depicted together with

their approximated gridpoint representation.

Scale errors: model and gridpoint approximation

4

2

! j .. -4 ,j

l -6

-8

_______ 7!!,. _______ _.

-10

-12

-14 0 100 200 300 400 500 600 700

Position of the carriage

Fig. 55. Model of the scale errors and their gridpoint approximation

Errors xty, ytx and zty: model and gridpoint approximation

700

Position of the carriage

Fig. 5.6. Model of the straightness errors and their gridpoint approximation

90 Chapter 5

5.2. The Expansion of the Scales

Each carriage-guide system is equipped with a linear measuring system. Due to

temperature variations these scales will expand and thereby cause an error in the

position of the tooL In order to correct for the expansion relative to 20 °C, three

temperature sensors are attached to each scale. Unfortunately in the correction there is

a mixture of the thermal distortion, which is measured by the application of the

measurement set-up discussed in Chapter 4, and the thermal expansion of the scale.

As the reference state of the thermal distortion is set to 23 °C, this also effects a

rather complex correction strategy. The thermal error correction Atotal is built up out

of three separate terms:

- A distortion

- A distor. scale

- A exp. scale

this is the displacement of the tool holder due to the

deformation of the machine tool structure and the

expansion of the scales at the measurement position P 0

;

this is the expansion of the measuring scale at the

measurement position (P J· This expansion is due to

the temperature variations relative to 23 °C;

this is the true expansion of the scale on the actual

position (P A) due to temperature offset from 20 °C.

Contrary to the measurement position (P J which is fixed to one value, i.e. the

position of the carriages while measuring the distortion, the actual position P A may

vary while operating the machine tool.

The total thermal displacement is calculated by:

[5.8]

We will discuss below the determination of Adistor al and A al • . sc e exp. sc e The evaluation of the measured distortion of the structure to a correction model will

be presented in the next paragraphs.


The calculation of !J.di t ai is dependent on the measured temperatures of the s or. sc e

scale. A parabolic function is determined to fit the three measured temperatures. This

yields a function of the type:

Temperature= aX2 + bX +c [5.9]

The expansion on position P D with respect to the reference point can then be

calculated as

PD p 3 p 2

tJ.di ai = f a(aX2+bX+c-23)dX =a( a___Q + ~ + (c-23)P ) stor. sc e 0 3 2 D'

[5.10]

The calculation of !J. seal is dependent on the temperature distribution of the scale exp. e and on the actual position of the carriage (P A). The obtained parabolic function of the

temperature distribution can now be applied to calculate the expansion of the actual

position P A" This calculation is described by:

PA p3 p2

a ai = J a(aX2+bX+c-20)dX =a( a~+~+ (c-20)PA) [5.11] exp. sc e 0 3 2

Note that the expansion is calculated relative to the zeropoint of the machine's axis.

But, as only relative displacements can be compensated, the choice of the basepoint is

arbitrary.

In figure 5.7 an example of this modelling process is depicted. The upper part of the

graph depicts three measured temperatures, on position 0, 350 and 700 mm. Through

these data a parabolic function is calculated, yielding the coefficients a, b and c of

equation 5.9. These coefficients are applied to calculate the expansion at an arbitrary

position of the scale. In the lower part of the graph, the calculated expansion is

depicted as a function of the position of the carriage.

The basic effort for correction is the determination of the coefficients a, b and c of the

parabolic temperature distribution of each scale. Together with the actual position P A

these coefficients can be applied to calculate the expansion of the scale. Unfortunately

the control system has no time for evaluating a polynominal, so these corrections must

be available as look-up tables. The measurement of the scale temperatures and the

calculation of the look-up tables is carried out by an external PC and updated every 60

sec by RS232 interface. The compensation for expansion of the scales is schematically

depicted in figure 5.8.

92 Chapter 5

Exam le of the thermal distribution of a scale

WL-----~------~-----L------~----~------~----~ 0 100 200 300 400 500 600 700

Position on the scale [ mm I

2o.-----~--~Ca==k~ur~~Ied~e~x~a=n~si~o~n~$~(i~u~nc~nor·~n~o~f~th~e~~iti~·o~n~--.-----~

15

700

Position of the carriage [ mm I

Fig. 5.7. Temperature distribution and calculated expansion function of a scale

Tl TZ T3

Measurement of temperatures I ' ' ' t

Calculation of polynomina!

~ Calculation of compensation function c::i>

-!t Transformation of function

to look-up table

I ' ' ~

Actual compensation of the position of the carriage

Fig. 5 .8. Schematic representation of the compensation for scale expansion


The number and position of the gridpoints for the look-up tables are determined by

investigations of the possible correction profile. In figure 5.9 a number of temperature

profiles of the Y -scale and the calculated expansion are depicted. These temperature

profiles are measured every 45 minutes while loading the machine tool with 6000 rpm

for 6 hours, followed by a no-load situation of 8 hours. Clearly can be observed that

the necessary correction is nearly a linear function of the position of the carriage.

The gridpoints are calculated with an allowable difference between the ideal function

and the gridpoint correction of 1 IJ.m. This condition yields numbers of intervals for

the X- andY-axis of 80 and 60 respectively. The number of intervals for the Z-axis is,

besides by the expansion of the scale, also determined by the distortion of the total

structure and will be calculated in the next section.

~ ~ ~ a s ~ m a ~ a Position of the Y -carriage [ mm]

Position on the Y·scale [mm]

Fig. 5.9. Change of the calculated expansion of theY-scale under load of6000 rpm

94 Chapter 5

5.3. The Thermally Induced Errors

Description of the Measurement Sample

With the measurement set-up as described in Chapter 4, a large number of

measurements are carried out. From these measurements representative samples have

been extracted to serve as input for the determination of the relation between the

thermal distribution and the drift of the tool holder. These samples include loads

which cover the entire range of spindle speeds. The samples are built up by the

following sequences ('n' represents the spindle speed in rpm):

Measurement Load situation No-load situation

1 n=1500, 6 hours n=O, 8 hours 2 n=3000, 6 hours n=O, 8 hours 3 n=4000, 6 hours n=O, 8 hours 4 n=5000, 6 hours n=O, 8 hours 5 n=6000,6hours n=O, 8 hours 6 DIN 8602 --7 Product simulation --

Measurement 6 loads the machine with the spindle speed spectrum defined in

DIN 8602. The measurement sequence 'Product simulation' is built up by the

following spindle speeds and no-load periods:

n=4000, 10 min; n=O, 1 min;

n=3000, 5 min; n=O, 1 min;

This sequence is repeated twice.

n=5500, 30 min;

n=6000, 10 min;

n=O,

n=O,

1 min;

5 min.

As a measurement is taken every minute, the total sample contains 4776

measurements. Investigation of the obtained results reveal a continuously changing

thermal behaviour. Therefore it is allowed to reduce the amount of data by taking

every 5th measurement into the sample. This results in a dataset of 956 measurements,

whereas each observation contains the values of the measured displacement in three

directions, and 40 temperatures (39 attached to the machine tool and the air

temperature of the room).


These measurement samples were taken on three different Z-positions in order to

obtain a thorough description of the drift in dependence on the position of the Z.axis.

5.3.1. The Statistical Modelling Methodology

The goal of this part of the study is to determine a relationship between the relevant

temperature sensors and the measured displacement of the tool holder, based on

empirical obtained measurement results. The temperature sensors must be chosen such

that the predictive value of the model is optimal.

Several approaches are carried out in order to obtain the relation between relevant

temperature sensors and the displacement. First, preliminary calculations using least

squares fitting procedures are performed. With an intuitive selection of the

temperature sensors this method displayed fairly good results. However, as the

theoretical importance of a particular temperature sensor is unknown, a methodology

for relevance detection is desired. A possible approach is the use of statistical criteria

for elimination and calculation of the optimal model. Therefore the datasets are

examined with the use of a software package for statistical analysis of data (SAS).

This software package offers a wide range of utilities for regression analysis [5.3].

Regression Analysis

Taking a dataset and a direction of the displacement, the task of the regression

procedure is to determine:

1) the temperature sensors relevant for the description of the displacement;

2) an estimate of the coefficients of the predictors;

3) a statement on the confidence interval for future predictions on new

data.

96 Olapter s

As there is more than one regressor, this is a multiple regression problem. A general

notation of this problem is:

[5.12]

1\ with: - y: the estimated response variable (in this case the

displacement);

- xl' ........ , xK: the regressors in the total model (in this case the

temperatures on various locations of the machine tool);

- f;J0• /31' ......... f;JK: the unknown parameters of the model;

- e: the part of the results that are not explained by the model.

The first condition requires that the best suitable model can be extracted from the total

model, based on all available temperature sensors that are attached to the machine tool

(in this case 39). Because evaluating all possible regression models - i.e. the models

with one sensor, the models with all sets of two sensors, etc - is computationally very

extensive, various methods have been developed for evaluating only a small number

of subset regression models by either adding or deleting regressors one at a time [5.4].

These methods are generally referred to as stepwise-type procedures. They can be

classified into three categories:

1) forward selection;

2) backward elimination;

3) stepwise regression, which is a combination of 1) and 2).

In the next section these methods will be briefly described.

Forward Selection of Relevant Predictors

This procedure begins with the assumption that there are no regressors in the model.

An effort is carried out to fmd an optimal subset by inserting regressors into the

model one at a time. The first regressor selected for entry into the equation is the one

that has the largest simple correlation with the response variable y. Suppose that this

regressor is x r


This is also the regressor that will produce the largest value of the F-statistic for

testing significance of regression. This regressor is entered if the F-statistic exceeds a

preselected F-value, say FIN. The second regressor chosen for entry is the one that

now has the largest correlation with y after adjusting for the effect of the first

regressor entered (x1} on y. We refer to these correlations as partial correlations. They

are the simple correlations between the residuals .from the regression ~ = ~0 + ~1x 1 A

and the residuals from the regressions of the other candidate regressors on x1• say xj =

1\ 1\ a0 + a1ll' j=2, 3, ...... K.

Suppose that at step two the regressor with the highest partial correlation with y is x2.

This implies that the largest partial F-statistic is:

[5.13]

with: - SSR (x2 1 x1): Change in the sum of the squared residuals due to

the addition of regressor x2 with x1

already in the model;

- MSB: Mean square of 'the error between actual results

and model.

If this F-value exceeds FIN' then x2 is added to the model. In general at each step the

regressor having the highest partial correlation with y (or equivalently the largest

partial F-statistic given the other regressors already in the model) is added to the

model if its partial F-statistic exceeds the preselected entry level FJN The procedure

terminates either when the partial F-statistic at a particular step does not exceed FIN'

or when the last candidate regressor is added to the model.

The SAS software does not operate with an FIN value as treshold for entry into the

model, but with a Significance Level for Entry (SLE). This value represents the

probability that an F-distribution with p (=number of predictors} degrees of freedom

for the numerator and n-p (=number of observations minus number of predictors)

degrees of freedom for the denominator ( FP ), exceeds the obtained F-statistic. This n-p

value represents the accepted probability that a predictor is wrongly entered into the

model.

98 Chapter 5

Backward Selection of Relevant Predictors

Forward selection begins with no regressors in the model and attempts to insert

variables until a suitable model is obtained. Backward elimination attempts to find a

good model by working in the opposite direction. That is, we begin with a model that

includes all K candidate regressors. Then the partial F-statistic is computed for each

regressor as if it were the last variable to enter the model. The smallest of these partial

F-statistics is compared with a preselected value, F OUT' and if the smallest partial

F-value is less than F OUT that regressor is removed from the model. Now a regression

model with K-1 regressors is fit, the partial F-statistics for this new model calculated,

and the procedure repeated. The backward elimination algorithm terminates when the

smallest partial F-value is not less than the preselected cutoff value F oUT·

Backward elimination is often a good variable selection procedure if one would like to

start with including all candidate regressors, so no "obvious" candidates will be

missed.

There is however one disadvantage of backward elimination: if the predictors are

linear dependent, the model is singular and the procedure is terminated immediately.

Within the SAS-software package this problem is solved by terminating the current

model check and proceed with the next predictor.

Similar to forward selection, the backward selection procedure of the SAS software

does not operate with an F OUT value as treshold for removal from the model, but with

a Significance Level for Staying (SLS). This value represents the accepted probability

that a predictor is wrongly removed from the model.

Stepwise Selection of Relevant Predictors

Stepwise regression is a modification of forward selection in which at each step all

regressors entered into the model previously are reassessed via their partial

F-statistics. A regressor added at an earlier step may now be redundant because of the

relationships between it and regressors now in the equation. If the partial F-statistic

for a variable is less than F OUT' that variable is dropped from the model.

Stepwise regression in SAS requires two cutoff values, SLE and SLS. Frequently SLE

is chosen larger than SLS, making it more difficult to add a regressor than to delete

one.


Some Comments on Stepwise type Procedures

The most common criticism on stepwise regression algorithms is that complete

nonsense relationships can be determined. If a large amount of predictors is available,

it is statistically very likely that a combination of several regressors is capable of

predicting any response variable. Therefore it is of extreme importance to examine the

predictors on relevancy with respect to the response variable. However, in practical

situations the relation of individual predictors to the response variable is often difficult

to assess. In this case a statistical method to avoid the estimation of nonsense relations

is splitting the data into two randomly determined parts. One part of the data is used

for model building, the other part can be applied for validating the selected predictors.

A second disadvantage of stepwise procedures is that none of them guarantees that the

"best" subset regression model of any size will be identified. Part of the problem is

that it is unlikely that there is one "best" subset model, but probably several equally

good ones.

It should also be kept in mind that the order in which the regressors enter or leave the

model does not necessarily imply an order of importance to the regressors. It is not

unusual to find that a regressor inserted into the model early in the procedure becomes

negligible at a subsequent step. This is in fact a general problem with the forward

selection procedure. Once a regressor has been added it cannot be removed at a later

step.

5.3.2. Development of the Modelling Procedure

The goal of the procedure is defined as the determination of a adequate subset model

for prediction of the displacement of the tool holder, based on measured temperature

drift.

Therefore the datasets are all recalculated relative to the reference situation. This

manipulation has two effects. Frrst, one has to be sure that all measurements are

started at the same reference state, i.e. a temperature stabilized environment is

necessary. Secondly, the first measurement connects a zero temperature rise with a

zero displacement, i.e. in the model this implies that the interception term is

eliminated.

100 Chapter 5

The latter exclusion results in a regression equation of the fonn:

[5.14]

1\ with: - y: the estimated response variable (displacement of the tool

holder);

- /3p /31' ......• J3K: the coefficients to be estimated;

- x1, x2, ...... , ~: the temperature rises of the machine tool;

- e: the residuals i.e the part of the observed displacement of

the tool holder that is not explained by the model.

It is to be expected that the temperatures of the machine tool are highly correlated. To

avoid the main problem of forward selection, i.e. leaving redundant regressors in the

model, a stepwise selection procedure is applied. In Appendix ill the correlation

coefficients of an arbitrary dataset are printed, showing indeed strong correlation

between the individual sensors.

In the next sections we will present and analyse the typical problems that appear in

this application of regression analysis. The result of this analysis is a modelling

procedure that determines the relationship between relevant temperatures and the

measured displacement of the tool holder.

Data Splitting

As stated earlier one of the problems of stepwise regression with a large number of

predictors is the danger of obtaining nonsense relations. In order to overcome this

problem the datasets are split randomly into two equally sized parts. One part can be

applied for model building, the other can be used for validation of the significance of

the selected regressors.


Leverage Points

Leverage points are observations that have a strong influence on the estimated

coefficients of the regression equation. To detect the presence of leverage points in the

predictor space, a check can be carried out on the Cook's statistic for each

observation. Expressed in words the Cook's statistic of an observation i determines the

change in the estimated coefficients if observation i is deleted from the predictor

space. An observation with a large Cook's statistic does not necessarily possess are

large residual. In figure 5.10 a set of observations is depicted together with the

estimated regression model (solid line). Oearly can be observed that observation 8 has

a strong influence on the least squares estimation. The obtained model does not

describe the trend of the data properly although the residual of observation 8 is

relatively small. Elimination of observation 8 from the data yields the dashed line as

regression model, which fits the trend of the data adequately. The detection of

influential observations can be carried out visually only in the two dimensional

situation. In multiple regression the Cook's statistic can be applied for numerical

detection of these influential observations. A Cook's statistic larger than one indicates

an influential observation and should be deleted from the model.

Influential observation with a small residual 5.---~--.---~--~.r---.---~---r---T--~--~

4.5

4

3.5

3

2.5

2

1.5

0.5

' I

/~rrected fit ·:: ...

,ll

/,'+

'+

i i

17 ///

!

• Observ.8

0o~--o~.s~~--~L~S---2~--2~.5--~3----3~.5--~4---4~.5--~5

Predictor

Fig. 5.10. Influence of a leverage point on the estimated regression model

102 Chapter 5

Multicollinearity in the Predictor Space

In addition to a strong correlation between two individual sensors, there also exists the

problem of near linear dependencies between the temperature sensors in the dataset.

This problem is known as multicollinearity. This has strong effects on the selection of

regressors. The property of multicollinearity in the predictor space is depicted in

figure 5.11.

PC3 PC2

PC1

Fig. 5 .11. Data with strong multicollinearity

It is evident that all data points in the three dimensional space are grouped in the

shape of a cigar. This implies that a transformation to a new coordinate system with

one axis aligned along the main axis of the cigar, is very convenient for describing the

predictor space. If the cigar is perfectly rotation symmetric, the data orthogonal to the

main axis of the cigar are randomly distributed and no identification of a second main

axis is possible (the direction of PC2 and PC3 can freely rotate about PCl). In this

way we can describe the significant part of the predictor space with one new predictor

(PCl), which is composed of a linear combination of the three original predictors

(PCl= a Predl + b Pred2 + c Pred3). This procedure yields a dimension reduction in

the predictor space from three to one and provides the guarantee that the new

predictors show no linear dependencies between each other i.e. the predictors are

orthogonal (this implies that the correlation coefficients between the principal

components- PCl to PC3- are equal to zero).


This strategy is applied to the predictor space obtained after regression analysis. It

appears that strong multicollinearity is present in the data. A dimension reduction

from 15 to 4 is not unusual. Application of the principal component analysis to the

data sample shows that no structural loss of predictive power could be observed (see

figures 5.12 and 5.13).

Difference in predictive power after Principal Componeut analysis

! 8

I -10

>!. .5

I -20

;t -30

-50'-----'----'---~--'-----'---'----'----'--__j 0 ~ ~ ~ ~ ~ ~ ~ ~ ~

Time[min)

Fig. 5.12. Predictive power loss after application of principal components

20

10

! -~ i! ~ >- -20 .5

] -30

~ -40

-SO

-60 0

Difference in predictive power after Principal Componeut analysis

Uncorrected measurement

Load: DIN 8602-spectrwn

100 ISO ~ 250 300 350 400 450

Time[min)

Fig. 5.13. Predictive power improvement after application of principal components

104 Chapter 5

An advantage of the application of principal components analysis to the predictor

space is their characteristic to lower the absolute value of the coefficients of the

model. When normal regression analysis is applied to the split datasets it yields a set

of relevant predictors (=temperature sensors) and their estimated coefficients. These

coefficients show large absolute values which endangers the prediction power of the

model if during actual operation a temperature is measured wrongly or a situation

occurs that is not included in the model. Therefore minimizing the values of the

coefficients is desirable. Principal component analysis is capable of accomplishing this

effect without a structural loss of predictive power.

In order to illustrate this effect the obtained coefficients for the compensation in

X -direction are presented below, before and after principal component analysis.

Temperature sensor

Parameter estimate before PrincipalComponent analysis

Parameter estimate after PrincipalComponent analysis

T3 17 T8 T9 TIO Tll T13 T15 T18 T20 T22 T24 T28 T31

-18.626 2.939 9.006

-6.575 -15.035 11.928 5.204

-1.705 -12.998

4.870 -1.233 -5.493 23.960 10.138

1.572 0.946 0.739 0.121 0.699 0.101

-1.231 -0.569 0.204

-0.725 -0.614 -0.270 1.708 0.933

Non-equidistant Temperature Data

If we take a closer look at the distribution of the data points (see figure 5.14) we can

observe two phenomena:

1} on a time scale all measurements are obtained with equal spacing;

2) on a temperature scale the data points tend to cluster as the machine

starts stabilising, yielding relative few points at strong temperature

changes and a large amount in an almost stable situation.


10 Distribution of data ints versus time

e 0 ..:. 'a -10 .. ~ .!I li' Q

-40 0 100 700 800 900

Time[min)

10 Distribution of data

e 0 ..:. i -10

J -20

g -30

·~-.... ...... *++++++++ +++ ++

-40 0 2 4 6 8 10 12 14 16 18 20

Temperature rise of tbe spindle bead [degree centigrade]

Fig. 5.14. Distribution of data points on time scale and temperature scale

The second observation is intriguing. Because the applied statistical analysis software

operates with a least squares estimation of the parameters, the number of points in a

region directly influence the importance of that region. This implies that the software

will try to fit a temperature stable region with more effort than a rapidly changing

situation. In order to avoid this effect several experiments have been carried out with

weight factors added to the observations. This implies that the square of the residual

between the model and the observation is multiplied by a weight-factor [5.4]. In

algebraic notation this results for the solution of the least squares estimation in the

minimizing of :

with: - w.: the weight factor for observation i; 1

- y.: the i-th observation of response variable y; 1

1\ - y.: the modelled response for observation i.

1

[5.15]

106 Chapter 5

As a large weight of an observation is desirable in rapidly changing thermal situations,

the weight factors (w.) have been detennined by the thermal change between two I

measurements. For observation i this can be algebraically denoted by:

w.= 1

39

L Sensor(k). l

k=l

39

L Sensor(k)i-l k=l

with Sensor(k)i as the temperature of sensor k for observation i

[5.16]

In figure 5.15 two results are depicted of a measurement of the displacement of the

tool holder that is corrected for the thermal behaviour. The dashed line represents the

results where the coefficients of the thermal model are determined with the

application of weight factors, whereas the solid line represents the results of a

straightforward estimation of the correction coefficients.

Difference in predictive power of two pammetersets Wr---~----r---~----r----r----~--~----~

15

e ..:. c ~ g

1 ., '2 5 c 'a! 7:!. >< 1:l ;::.

0

Nwnber of observation

Fig. 5.15. Difference of results with and without weight factors on a new prediction


The results show no significant improvement of the fit From a statistical point of

view there is no reason why one should give more credit to one observation compared

to another and therefore these actions are omitted during the modelling process. But,

based on these studies, the lack of fit right after loading and unloading the machine

tool with a spindle speed can be explained by the feature of least squares estimation to

fit major part. of the observations.

Large Ousters of Outliers

In an obtained data sample outlying data points can be present because of errors in

recording, errors in transcription or the transmission, or an exceptional occurrence in

the investigated phenomenon. Although the developed modelling procedure is capable

of identifying and eliminating single outliers, the problem of clustered outliers can not

be detected. As shown in figure 5.16, clustered outliers can cause large deflections of

the true regression equation and therefore detection is necessary. This example depicts

the number of international phone calls made in Belgium over several years. It is clear

that the data contain heavy contamination from 1964 to 1969. Upon inquiring it turned

out that in these years another recording system has been applied.

The ordinary least squares solution is affected very much by these outliers. In order to

avoid the influence of outliers on the obtained solution, another algorithm is

developed [5.5] that does not minimize the sum of the squared residuals but the

median of the squared residuals. Algebraically this can be denoted by:

Mi . . di ( ")2 mnuze me an y.-y . 1

[5.17] 1

Geometrically this minimization corresponds to finding the narrowest strip that covers

half of the observations. This methodology is known as the Least Median of Squares

and yields the true regression equation from a dataset that is contaminated up to 50%

with bad measurements. Application of the LMS method to the data presented in

figure 5.16 yields a solution that corresponds to the pattern of the data points.

108 Olapter5

Number of international phone calls versus year 25

20

';;'

i 15

~ .. ., a 10 e ::! l'l " 5 "' 0

if

0 ........... Least Median of Squares fit

• 5 1950 1955 1960 1965 1970 1975

Year

Fig. 5.16. Number of international phone calls from Belgium in the years

1950-1971 with the Least Squares (dashed) and the Least Median of

Squares fit (solid)

In the multidimensional situation the presence of clustered outliers cannot be detected

visually and even with the current statistical software (SAS) it is very hard to

eliminate clustered outliers. There is however a methodology called TRADE (TRim

And DElete) which has however proven to be successful in most cases [5.6]).

To verify the presence of outliers in the actual datasets a program is applied

(PROGRESS) that does not operate with a least squares estimation but uses the least

median of squares.

Although the measuring process is believed to be very stable, calculation of the

regression parameters of the selected temperature sensors are carried out with

PROGRESS.

As expected the results gave no cause for further investigation on clustered outliers.


5.3.3. The Modelling Methodology and Results

Careful analysis of the described features of this regression problem, yields a

modelling methodology that contains the following sequences:

l) Split the data randomly into two equally sized parts, a model and a validation set;

2) Perform a stepwise regression procedure on the model part, applying a

significance level for entry and for staying of 99%;

3) Plot residuals against predict values and look for systematic patterns;

4) Check for individual outliers with the studentized residuals;

5) Check for leverage points with the Cook's statistic;

6) Check the normality of the residuals;

7) Validate the obtained model on the validation dataset by using the backward

elimination procedure with a significance level for staying of 99.9%;

8) Estimate the parameter values on the total dataset, using the selected regressors

after backward elimination;

9) Apply principal component analysis to the relevant predictors and perform

regression with those principal components with an eigenvalue larger than one;

10) Transform the regression coefficients of the principal components back to the

original predictors.

With the developed modelling procedure, three measurement samples (obtained at

three different positions of the Z-axis) have been evaluated. This yields 9 sets of

parameters linked to 9 sets of temperature sensors (3 times a set in X-, Y- and

Z-direction). The parameters are applied for prediction of the drift of the tool holder

on a new measurement sample.

In figure 5.17 to 5.19 the results are presented of an actual measurement in X-, Y

and Z-direction respectively, carried out on position 3 (Z-position = 566 mm), and the

residual after the measurement results have been corrected. A strong improvement can

be observed in the behaviour of the machine tool. In the next section some

experiments will be described that are carried out in the modelling process. These

experiments attempt to eliminate the observed malfunctions of the obtained model

with the intention to enhance the predictive power of the model.

110

10

~

~ >< .a

l .~

-20 0

-30

40 0 100

Chapter 5

Displacement of the tool holder before and after correction

200

Load: 4000 rpm 6 hours, 0 rpm 8 hours

Z-position: 566

300 400 500 600 700

Time[miD]

Fig. 5.17. Results of compensation in X-direction

0 8 .!!. -5 .j

~ -10

>- -15 .a I l

-20

iS

Displacement of the tool holder before and after conectioo

Load: 4000 rpm 6 hours, 0 rpm 8 hours

Z-positioo: 566

Time [mini

Fig. 5.18. Results of compensation in Y-direction

800 900

-40


Displacement of the 1001 holder before and afrer correction

Corrected results

Load: 6000 tpm 6 hours, 0 tpm 3 hours Z-posi.tion: 566

-SO 0 100 1!lO 300 400 500 600 700 800 900

Tune[min)

Fig. 5.19. Results of compensation in Z-direction

Development of the Compensation Functions

111

In order to calculate the compensation values at a position of the carriage that was not

included in the model, two methods are applied.

For a diversion of the actual position in the X- and the Y-direction from the

measurement position, the expansion of the corresponding scales is calculated and

added to the predicted distortion by application of equation [5.8].

For a different position of the Z-axis than included in the modelling, i.e. the

Z-coordinate is not 62, 314 or 566 mm, an interpolation strategy between the

measurement positions is necessary in order to obtain a continuous compensation as a

function of the Z-position. Since observations are available on three different

Z-positions, the possibilities are restricted to linear and quadratic interpolation.

In figure 5.20 the results are presented of an actual measurement of the displacement

of the tool holder in Y -direction. This measurement was carried out on a position of

the Z-carriage that was not included in the model (Z=188 mm). The compensation

values are calculated on each three measurement positions (Z=62, 314 and 566 mm).

Then an interpolation technique is applied in order to obtain the predicted

compensation on the actual position being Z=188 mm.

112 Chapter 5

The dashed line in figure 5.20 represents the residual with linear interpolation between

the calculated compensation values, whereas the solid line represents the results when

quadratic interpolation is applied.

Interpolation techniques: quadratic and linear 30.----.----.----.----.----.----r---~----r---~

20

e 10 ..:. .g il ·= "'i' > .s

"' " E 8 ]; 0

Load: DIN 8602-spectrum -50'-------'-----'-------'-----'------'-----'------'-----'----....J

0 50 100 150 200 250 300 350 400 450

Time[min]

Fig. 5.20. Results with different interpolation techniques

The Thermal Model for the Milling Machine under Investigation

The three data samples are modelled with the presented procedure. This yields

correction formulas for each direction of the form:

Adi . . stortion,t

n

= L cj Tempj

j=l

with: - i: the direction of compensation, i.e. X, Y or Z;

[5.18]

- n: the number of relevant sensors determined by the

SAS-procedure;

- C.: the calculated coefficient obtained from the modelling J

procedure;

- Temp.: the temperature of sensor j. J


As the measurement are carried out on three different Z-positions, the modelling

procedure yields nine compensation functions.

The (rounded) coefficients and temperature sensors for the center of the working

space are presented below. For a complete description of these functions we refer to

Appendix IV.

X-direction I Y -direction Z-direction

c Sensor c Sensor c Sensor

1.57 43 -0.32 51 1.97 43 0.94 47 0.26 52 1.91 44 0.74 48 0.65 53 2.13 45 0.12 49 -0.70 56 2.83 48 0.70 51 1.34 57 0.61 51 0.10 52 1.17 58 -1.11 53

-1.23 54 -2.21 39 1.16 54 -0.57 56 -4.05 50 0.35 57 0.20 59 2.80 34 0.43 58

-0.73 39 0.11 25 1.57 59 -0.61 50 -1.21 50 -0.27 29 -0.73 34 1.71 27 1.77 26 0.93 32 1.58 32

2.21 38 1.92 24

Position X=350, Y=140, Z=314 (in mm)

5.4. Implementation of the Correction Algorithm

The implementation of the correction algorithm is split into two parts. One part is

dependent on the reference state of the machine tool and does not change rapidly, this

is the geometric error correction. The other part is dependent on the actual state of the

machine tool and can change rapidly, this is the thermal error correction. In the next

sections the implementation of these two error corrections will be described.

114 Chapter 5

1be Geometric Error Correction

Based on the results of the geometric error modelling, six compensation functions are

available. These functions are transformed to basepoints and accompanying

compensation values. In this form the compensation functions can be stored directly

into the control system. During actual operation the control system checks the position

of the carriages, determines the required compensation of all six look-up tables and

corrects the position accordingly.

For the determination of the compensation values, the control system applies linear

interpolation between the two nearest gridpoint values. This method is graphically

illustrated in figure 5.21.

Compensation for geometric errors

Compensation curve

Actual position Position of the carriage

Fig. 5.21. Determination of the geometric compensation value

The Thermal Error Correction

The thermal displacement of the position of the tool holder due to temperature

variations at various points in the machine tool, can be calculated with the above

described modelling technique. To compensate in real time for the calculated

displacement, an improved temperature compensation has been devised, taking into

account the temperature of several parts of the machine, the position along the Z.axis

and the expansions of the X-, Y- and Z-measuring scales [5.7]. In order to verify this

method, an experimental set-up of the control system must be made that supports the

manufacturing of workpieces.


Because the control system is not yet equipped for taking in several temperature

values and no computing capacity in the system is available for calculations, the

necessary temperature measurements and computation of the required compensation

values are carried out by a coupled PC.

Each 60 seconds, the temperatures of N temperature sensors, which are connected to

the various points in the machine, are measured. From these temperatures, temperature

compensation values for the X-, Y- and Z-axis at a number of equidistant Z-positions

are computed. For the determination of these positions the following must be

considered.

In order to obtain a continuous description of the compensation as a function of the

Z-position a quadratic function is fitted through the compensation values at the three

measured positions. As the control system requires a look-up table as format for the

compensations, the obtained correction function must be transformed to gridpoints

with accompanying compensation values. In order to determine the number of

gridpoints, the maximum difference between the compensations on each-measurement

position has been calculated by applying a load of 6000 rpm in the most sensible

direction (Y). In figure 5.22 the results are depicted. The solid line represents the

difference in calculated correction between the positions Z=62 mm and Z=314 mm,

whereas the dashed line represents the difference between the positions Z=314 mm

and Z=566 mm.

Compensatioa diff""""""' between the three measusemeot positioos

20 /"·· •• , Difference between Z=314 and Z=566 . -. 10! ·---... ___ _ 0

Tune[min]

Fig. 522. Difference in correction between three measurement positions

116 Olapter 5

The method of compensating differs from the geometric error correction. Due to lack

of computing time there is no possibility to interpolate the compensation between two

gridpoints. Instead the nearest upper compensation value is applied for correction.

This compensation strategy is graphically elaborated in figure 5.23.

Compensation strategy for thermally induced errors

Actual position Position of the carriage

Fig. 5.23. Compensation strategy for thermally induced errors

One can observe that the maximum difference between the required compensations is

approximately 60 Jlm. If an error between the gridpoint representation and the actual

compensation function of 1 Jlm is allowable, and the distance is known by 314-62 = 252 mm, the interval length should be maximal 252/60 = 4.2 mm.

Over the whole range of the Z-axis (600 mm) this implies that minimal 600/4.2 = 143

intervals should be available.

The control system restricts the maximum number of intervals to 99, so therefore this

number is chosen for our correction. The loss of accuracy due to this restriction is

limited to an acceptable 1.5 f.Lm.

The obtained three sets (X-, Y- and Z-direction) of gridpoints with accompanying

compensations are sent from the PC to the temperature compensation memory of the

milling machine in the appearance of a temperature compensation table with a fixed

number of compensation entries.

The correction computed from this set hold for an arbitrary Z-position, but at one

fixed X-position and one fixed Y-position. To obtain the compensations at an arbitrary

X- andY-position, the expansion due to temperature of the X- andY-measuring scales

must still be accounted for.


The temperature compensation values for these expansions can be stored in two

additional temperature compensation tables, which hold the compensation values for

the X- and Y-axis due to expansion of the scales at a number of equidistant X- or

Y-positions, respectively. The resulting corrections for X andY must be superimposed

to the corrections as a function of the Z-position.

Since compensation changes necessitated by a changing machine temperature are fully

accounted for by the measurements and computation on the PC and the subsequent

refreshment of the temperature compensation tables in the milling machine,

temperature values read from the original installed temperature sensor are irrelevant

and must be ignored by the control system. In this set-up the milling machine only

handles the position dependent part of the temperature compensation.

The actual temperature compensation to be applied is dependent on the positions on

the X-, Y- and Z-axes. These positions may be rapidly varying functions of time;

hence, the positions must be read each sample time (15 msec). hnmediately after

reading the position, the CNC takes the appropriate compensation values from the

compensation tables and adds these values to the position signals from the measuring

system.

Implementation of the Correction Algorithm

In figure 5.24 the links of the experimental temperature correction system with it

environment is depicted. The following functional blocks can be identified:

1) a number of temperature sensors on the milling machine;

2) the data that represent the positions along the machine's axes;

3) the motion controller task, which handles the correction data;

4) the VDU (Video Display Unit) update procedure, which makes the

compensations available to the VDU.

118 Chapter 5

Motion controller

task 3)

Temperature Sensors 1)

Screen_ data

Fig. 5.24. Data context diagram of the experimental temperature correction system

In figure 5.25 the decomposition of the experimental temperature compensation

system is depicted. Processes 1, 2 and 3 are implemented in an ffiM-compatible

computer, which handles the temperature dependent part of the temperature

compensation. The other processes are implemented into the control system by the

Philips NC development department

This part handles the dependence of the temperature compensation on the position of

the machine's carriages.

The compensation table is the interface between the processes running on the PC and

the NC. It contains the compensation values for the computation of the position

dependent part of the temperature compensation. This table is updated by process 3

any 60 seconds to account for machine temperature changes.

The compensation table is read and split into a part that contains the dependence of

the X, Y and Z compensation on the Z position, a part that contains the dependence of

the X compensation on the X position, and a part that contains the dependence of the

Y compensation on the Y position.

The latter two parts represent the contributions of the expansion of the X and Y

measuring scales to the X and Y compensations, respectively.


PC (TUE)

Compensation_ table

Position

Screen_ data

NC (Philips)

T_Comp_Act

T_Comp_data

Fig. 525. Decomposition of the experimental temperature correction system

Both the X and the Y scales are provided with three temperature sensors; one on both

edges and one in the middle. It is assumed that there is a linear gradient between two

of these three sensors; this results in a compensation which is a quadratic function of

the position between the two sensors (see figure 5.26).

Since the measurements of the temperature and Z dependence were carried out at

fixed positions for X and Y, the expansion with respect to these positions should be

superimposed to the compensations, calculated dependent on the temperature and the

Z position, at the X and Y measurement positions.

120 Chapter 5

Temperature sensors

Yeasu:rl.ng ~

Temperature!Yl ...

1_---r=J

Ymin Ymid Ymax

Y_Comp(Y)

Fig. 5.26. Compensation due to temperature gradient along theY measuring scale

Format of the Compensation Table

The first temperature compensation table for sensor A is used to store the

compensation values. The representation for these coefficients must be compatible

with the table format as it is present i.e., the range of the coefficients is restricted

from -999 to 999.

Since the position of the carriages, and thereby the required compensation, may

rapidly change, the compensation must be calculated each sample time (15 msec). At

this moment there is no possibility to evaluate polynominals of any other functional

representation of the compensation within the given sample time, so all data have to

be transformed to look-up tables with a fixed number and position of gridpoints.

The table format for the Z dependent part of the temperature compensation is:

Sl

Sl

Sl

Sl

Sl

TO TO.l T0.2

T0.3

TO.l(Nz-1)

xxo XXI

XX2

XX3

X XNz-1

Y YO

Y Yl

Y Y2

Y Y3

Y YNz-1

zzo Z Zl

ZZ2

z Z3

Z ZNz-1


In this format XO, Xl, X2 represent the compensation values for the Z dependent part

of the temperature compensations for the X axis, and Nz is the number of gridpoints

on the Z-axis. etc.

The table format for the X and Y dependent parts of the temperature compensation is:

Sll TO xxo y YO

Sll TO.l XXI y Yl

Sll T0.2 XX2 y Y2

Sll T0.3 XX3 y Y3

I I

Sll TO.l(Nx-1) X XNx-1 y YNx-1

Sll TO.l(Nx) XXNx-1 y YNx

Sll TO.l(Nx+l) X XNx-1 y YNx+l

Sll T0.1(Nx+2) XXNx-1 y YNx+2

Sll TO.l(Nx+3) X XNx-1 y YNx+3

I I I

I

Sll TO.l(Ny-1) XXNx-1 y YNy-1

Ny and Nx are the number of gridpoints of theY- and X-axis respectively; here Ny > Nx.

In figure 5.27 an example is depicted of the correction values for the expansion of the

measuring scale of the X-axis.

TO TO,l TO.Z T0.3 TO.l(Nx-1) X-posiUon

Fig. 527. Example of compensation values and their notation for the X-axis

122 Chapter 5

Dosing of the compensations

The temperature compensation system is equipped with a "dosing mechanism" for the

compensations; compensation changes are fed to the motion controller increment by

increment. so as to avoid abrupt changes in compensations, which might leave visible

traces on the workpiece surface.

In case of dosing. the only allowed valued for T_Comp_data are: -1, 0 or 1 increment.

This results in a rate of compensation change of maximally 1 increment in 15 ms. The

actual rate of compensation change can be set. In figure 5.28 the dosing mechanism is

graphically depicted.

No dosing

Compensation

t TimE!

Temperature change

Dosing

t Time

Tempert~ture ch.arJ«e

Fig. 528. Dosing mechanism for temperature compensation


Future Developments

In order to fully utilize the possibilities of the geometric and thermal modelling

procedures, the control system will have to be adapted.

For the geometric correction this implies that a mathematical processor has to be

installed, capable of calculating the final model within the sample time of the motion

controller (15 msec). In this way the machine tool can take optimal advantage of the

available information concerning the geometric errors.

For the thermal correction the equipment for measuring temperatures will have to be

expanded The control system itself will have to be adapted to elaborate the thermal

model for the three measurement positions and determine the appropriate

compensation by interpolation techniques and calculation of the scale expansions.

These calculations must be performed in real time, whereas the updating of the

temperature information can be performed every 60 seconds.

With the amount of mathematical manipulations taken into account it will take some

time, investments and structural software design before this strategy can be

implemented. It should be noted however that investigations in this area are being

carried out and preliminary results are expected in the near future.

124 OJapter 5

References

[5.1] Spaan H.A.M., Schellekens P.H.J.:

Geometric Error Modelling of Machine Tools.

First Milestone Report of the BCR-project: Development of Methods for

Numerical Error Correction of Machine Tools, Draft Version, May 1991.

[5.2] Soons J.A.:

Personal communication with F.C. Theuws, 24 may 1991.

[5.3] SAS STAT User's Guide, Release 6.03, SAS Institute Inc, Cary USA, 1988

[5.4] Montgomery D.C., Peck, E.A.:

Introduction to Linear Regression Analysis.

Wiley Series in Probability and Mathematical Statistics, 1982.

[5.5] Rousseeuw P;J., Leroy A.M.:

Robust Regression and Outlier Detection.

Wiley Series in Probability and Mathematical Statistics, 1987.

[5.6] Dijkstra J.B.:

TRADE-regressie.

Eindhoven University of Technology, Computer Centre Note 46, april1990.

[5.7] De Ruiter, H.M.

Element Performance Specification of Experimental Improved Temperature

Compensation, Version 0.4.

Internal report, Philips Machine Tool Controls, march 1991.

125

Chapter6

Verification of the Correction Algorithm

The designed compensation algorithms for both geometric and thermally induced

errors are implemented into the control system. As fmal part of the study, the

effectiveness of the compensation, and thereby the modelling methodology, is

verified. This verification is split into three parts:

• verification of the geometric error correction;

• verification of the thermal error correction including the expansion of

the scales and the distortion of the structure;

• fmal indication of the achieved enhancement of production accuracy by

manufacturing a test-workpiece.

These three methods, and the results obtained, will be discussed in the next sections.

6.1. The Geometric Error Correction.

To verify the effectiveness of a geometric error correction, a number of methods are

available. For instance, one could consider the application of slip gages,

lasermeasurements, circular tests (i.e the comparison of a circular movement of the

machine to a reference circle [6.1]) or manufacturing workpieces. However, most

methods require either a complex set-up and evaluation software, which limits their

application to measuring machines, or include a cutting process, which disturbs the

thermal stable situation in which the geometric errors are defmed.

For checking the effectiveness of the geometric error compensation on the milling

machine under investigation, we have chosen for the hole plate method.

126 Outpter 6

This method is developed by the Physikalisch Technische Bundesanstalt (PTB) in

Braunschweig for calibration of coordinate measuring machines. However, if a milling

machine is equipped with a probe system, this method can also be applied to check

the geometric accuracy of these machine types. Fortunately, the milling machine

under investigation is provided with a Heidenhain infrared probe system, so in this

case the hole plate method can be applied for checking the effectiveness of the

geometric compensation.

The application of the hole plate method has the following advantages:

• it yields information of a number of points distributed in a plane;

• standard machine software routines can be applied

(circular measurement, etc.);

• the method is relatively easy to implement;

The main disadvantage of this method is that the information on the location of the

holes is restricted to one specific plane.

Description of the Hole Plate Method

In order to determine the geometric errors of a coordinate measuring machine, a

two-dimensional reference object is developed by PTB. This artifact consists of a grid

of boles, with an equal mutual distance. By measuring these holes, and knowing the

deviation from the nominal mutual distance, it is possible to evaluate the performance

of a coordinate measuring machine with respect to the geometric errors. When this

artifact is repeatably placed upon the machine, in such a way that the different

locations ··of the hole plate constitute the sides of a cube, and the hole plate is

measured with different probe-styli, it is possible to determine all 21 geometric error

components of a machine consisting of three linear axes [6.2].

As stated before, the hole plate has to be calibrated in order to be suitable for

calibration of coordinate measuring machines. This calibration is carried out by PTB.

Verification of the Correction Algorithm 127

Measurement Strategy

As the larger effects of the geometric errors are induced in the YZ-plane of the

machine tool, this plane is chosen as test plane for the effectiveness of the geometric

error compensation (see figure 6.1). The milling machine is programmed to measure

the outmost holes. The location of each hole is detennined by measuring back and

forth.

Fig . 6.1. Location of the hole plate for testing the geometric error compensation

First, the hole plate is measured without the geometric error compensation active.

Secondly, the geometric error compensation has been switched on and the

measurements are repeated. The results of both these measurements are presented in

the next paragraph.

Finally, as the control system limits the possible compensation method, a computer

simulation is carried out on the positions of the holes with the entire geometric error

model. This yields the errors that would be obtained if the total geometric error model

could be implemented.

As the measuring scales and the hole plate will expand due to temperature changes,

the temperature of these parts is measured during the experiments.

128 Chapter6

Measurements and Results of the Hole Plate Method

PTB supplied a hole plate with a grid of 9 x 9 holes. The nominal distance between

two holes is 50 mm. The actual distance is detennined by calibration at PTB. This

hole plate is measured on the milling machine, using a Heidenhain infrared probe

system.

The milling machine will detennine the location of each hole referred to the reference

hole. As the actual location of these holes is known, it is possible to detennine the

geometric error of the milling machine within a certain plane. The measured error is

not only composed of the geometric error of the milling machine, but also of the

repeatability of the used probe system. In order to detennine this repeatability, a test is

carried out with the probe configuration as depicted in figure 6.2.

~X y

Fig. 6.2. Used probe configuration to measure the hole plate

The following repeatability is found for this probe configuration:

2S ..,_ . = 1.4 J.lm; y-UJJection, center 28 z-direction, center = 7 ·5 J.lm.

The relatively large repeatability error in Z-direction is caused by offset in the

position of the probe tip, which should be in the centre beneath the switching probe

system. To reduce the effect of this random error, the position of each hole is

measured three times.


Before the measurement results can be evaluated, the bare measurements have to be

corrected for the expansion of the measuring scales of the milling machine and the

expansion of the hole plate itself.

As it is not possible to perfectly align the hole plate along one of the axes of the

milling machine, the measurement results will show an alignment error. This

alignment error is eliminated by a mathematical transformation of the measurement

results. By this transformation the same situation is obtained as would the hole plate

be aligned along one of the axis, which in this case is chosen as theY-axis.

In figure 6.3 the results of the hole plate measurement, after correction for alignment

errors and thermal expansion, are depicted (solid lines). Note that the depicted

deviations are only due to the errors in the geometric behaviour.

The same measurement set-up is applied to measure the hole plate with the geometric

error correction active. The measured results, compensated for alignment errors and

thermal expansion, are also depicted in figure 6.3 (dashed lines).

soo

400

! ·~ 300

N 8 200 i ~

100

0

Residuals of the bole plate measurement

1------i 10 um

- Only tbermal

0 100 200 300 400

Position Y ·axis [nun)

Fig. 6.3. Results of the hole plate measurement, after correction for alignment and

thermal expansions

Solid lines: No error compensation active

Dashed lines: Geometric error compensation active

130 Olapter6

In the evaluation of the measurement results it appeared that the thennal expansion of

the scales is a critical factor in the compensation. In order to illustrate this

observation, the results of a hole plate measurement with the geometric error

compensation switched on, but without compensation for the scale expansions, are

presented in figure 6.4. A clear deterioration of the geometric accuracy can be

observed One of the conclusions is therefore that the implementation of a suitable

compensation for the scale expansion · should precede the geometric error

compensation.

soo

400

I 300

-~ . N 200 J ·,; tf 100

0

Residuals of the bole plate measl!l'elllellt

.................... -... __ " -~1

' t . ' ~ ,......_; 10 urn ~

- Only geometric \

----- Geometric{Th~'. .J'Jr ..... -e- .............. .., ....... __ ·---·...... r

-too '----,o'----,1oo'-:---:-2oo'-:---,3oo'-:---,-400'-:---:-soo~

Position Y- axis [mm]

Fig. 6.4. Measured errors in the position of the holes

Solid lines: Geometric error compensation active

Dashed lines: Geometric and thermal error compensation active

The implemented geometric error compensation is a partial solution. The limiting

factor in this case, is the available computing time of the control system. ·An

interesting excursion in this context is the simulation of the full geometric error model

on the positions of the holes.

In the next section we will describe this simulation and present the results.


Verification of the Error Model with Simulations of the Hole Plate Method

A software package has been developed [6.3], which enables the calculation of the

error vector on any position within the range of the modelled machine, using the

individual model. This error vector is calculated with respect to the workpiece

coordinate frame, taking into account the dimensions of the tool In order to facilitate

simulations, the error vector can be determined as a function of the position of all

axes. Thus it is possible to evaluate the geometric performance of the modelled

machine.

One of the possibilities of the software package is to evaluate the geometric

performance of the modelled machine in a plane.

This software package is used for the simulations of the hole plate measurements.

400 ot•''.

350 \

300 \

1 250 t 200 ••

i 150 \.

~

Residuals of the bole plate measurement

1--t lOum

Simulation tolal model

100

50

....... Limited compensation

\. .. ......... ~.o-- ....................... .. ,. ........

0

0 100 200 300

Position Y • axis [mm]

Fig. 65. Residuals of an actual hole plate 11U!asurement

400

Solid lines: Simulation of expected residuals between model and

actual11U!asure11U!nt results

Dashed lines: Measured residuals with geometric error compensation active

132 Cbapter6

To simulate the measurements of the hole plate, the difference between the error

vector with respect to the reference hole and the error vector with respect to the

evaluated hole is determined.

The program is capable of calculation of this error vector for each hole, as it would be

measured by the milling machine.

These simulations enable direct comparison with the actual measurements carried out

on the milling machine under investigation. But. before comparing the results, the

actual measurements have to be corrected for the expansion of the scales and the

expansion of the hole plate. Also the alignment error of the hole plate along the axes

of the milling machine has to be corrected.

In figure 6.5 both the measurement results and the simulations are depicted. The

measurement results are obtained with the implemented geometric error compensation

active.

Conclusions

Before drawing any conclusions with respect to the effectiveness of the compensation

methodology, the uncertainty of the residuals is calculated [6.4].

The upper limit estimate of the uncertainty of the comparison, between simulations

and actual measurements, results in:

2Sresidual = 12.7 pm

The uncertainty in the actual measurement is mainly caused by the poor repeatability

of the used probe configuration.

An upper limit estimate of the simulated hole plate measurements yields:

2Ssimdation = 6.3 Jlrn.

Taking into account the upper limit estimates, it can be concluded that for the

comparison, depicted in figure 6.3, there seems to be a small difference between the

model and the actual error structure of the milling machine.


A major improvement of the geometric behaviour of the milling machine can be

observed. Already with the current limited compensation, the geometric accuracy of

the milling machine is enhanced by a factor 2.5. Furthermore, the simulated total error

compensation promises favourable results. Therefore, future research should focus on

the implementation of the total geometric error model into the control system.

The hole plate measurement proved to be a fast, accurate and highly informative

method. Some attention must be paid however to the orientation of the probe tip. As

most machine tools are not capable of orientating the tip in all directions, the

availability of star-type probes becomes highly desirable. With this option, the

application field of the hole plate method for machine tool accuracy testing could be

greatly extended.

6.2. The Thermal Error Conection.

In order to detect the drift of the thermally compensated machine tool, preliminary

experiments are carried out with the probe system in combination with the hole plate.

During these experiments the probe system proved to be very unreliable, yielding

measurement results that could not be explained by the thermal model. Some tests on

the functioning of the probe system are carried out (see Appendix V). These tests

revealed an unpredictable behaviour of the probe system used in combination with a

warmed-up machine tool. This is probably due to immediate expansion of the

(aluminium) probe system after tool exchange. This leads to the conclusion that for

verification of the thermal model, the combination probe system-hole plate is not a

suitable method.

We chose to apply the measurement set-up, described in Chapter 4, for verification of

the effectiveness of the thermal model.

First, the set-up is placed on a position within the workspace that is included in the

thermal model. The machine tool is loaded with a spindle speed, and the actual

displacement of the tool holder is measured, whereas the machine tool is real time

compensated. This compensation is calculated by a coupled PC and updated every 60

seconds.

The results of this measurement are depicted in figure 6.6.

134 Chapter6

Displacement of the tool bolder: With and without compensation

Time[min]

Fig. 6.6. Results of a real time compensated drift measurement, with and without

compensation active, position measured for development of thermal model

Dashed: Without compensation

Solid: Thermal compensation active

Secondly, the set-up is positioned at a location that was measured for the development

. of the thermal model. This position is chosen at the edge of the working space in

X-direction (X=l01), and between two modelled positions in Z-direction (Z=125),

thereby testing the validity of the calculated scale expansions and the applied

interpolation technique for different Z-positions.

The measurement results are depicted in figure 6.7.


Displacement of lbe lool holder: With and without compensation

300 350 400 450 500

T1111e [min]


compensation active, position at the edge of the workspace

Dashed: Without compensation

Solid: Thermal compensation active

Conclusions

The results of this verification show an improvement of the thermal drift dependent on

the position of the axis and the direction. In some directions a reduction of the

displacement of the tool holder up to a factor 3 can be achieved by implementation of

the compensation.

The compensation algorithm does also function at positions that were not measured

during the development of the modelling procedure.

From the results as depicted above, a deterioration with respect to the model

simulations as presented in Chapter 4, can be observed. A possible cause for this

effect can be the application of principal component analysis, which uses only the

larger part of the available information in the predictor space. Therefore an

experiment has been carried out with compensation values determined by the original

estimated model, i.e. applying the bare estimates of the model parameters.

136 Chapter6

The results on the position that was not measured in the model are depicted in

figure 6.8. An improvement of the predictive power, relative to the estimates of the

principal components analysis, can be observed.

Analysis of the measurement results revealed a strong similarity in the thermal

situations of the experiments. This implies that for repeating thermal situations the

original estimated parameters bear the highest predictive power.

However, as the predictions of the thermal model are actually used for compensation

of a machine tool, we prefer the application of the principal component estimates for

reasons of the robustness of the predictions against slight deviations from the

modelled situation.

In the next paragraph the results of a workpiece without compensation, with principal

component-based compensations and with bare estimate-based compensations, will be

presented to found this preference.

Displacement of the tool holder: With and without compensation


compensation active

Position at the edge of the workspace, not measured for the thermal

model

Compensation by model without principal component analysis


6.3. Manufacturing of a Test-workpiece

All previous measurements indicate an improvement of the accuracy of the machine

tool. However, for checking the factual relevancy of this enhancement it is important

to test the behaviour of the machine tool under realistic conditions, i.e. while

generating an actual workpiece.

In the selection process of the workpiece the following considerations are taken into

account:

• because the effects of cooling liquid on the thermal distortion are not

included in the model, the material must be machinable without the

application of cooling liquid;

• as the thermal behaviour of the machine tool is modelled on a flxed

position, the size of the workpiece should be realistic for this type of

machine tool;

• the workpiece should be easy to manufacture and measure, due to the

limited amount of time available for the project;

• the workpiece dimensions should incorporate the thermal distortion in

an three directions;

• as the changing of a cold tool into a warm machine seriously disturbs

the predicted displacement from the thermal model, the milling process

must preferably be carried out with one particular tool;

• in order to fully exploit the thermal model the workpiece must be under

process for several hours.

Based on these considerations a test-workpiece has been devised This workpiece is

depicted in figure 6.9. The applied material is brass, the tool is a round head cutter.

The process of manufacturing consists of rough machining of the product to 0.5 mm

of the fmal dimensions. Then the cutter mills circular-type paths with the altitude

changed after completion of each path. In this way the thermal distortion of the

machine tool in an three directions will copy on the flanks of the product. During

operation the temperature of the product is monitored continuously.

138 Chapter 6

I---- 110 -----l

--j r- 15

95

l-r-~~L-----~,.

!Or

~-----120------~

Fig. 6.9. Drawing of the test-workpiece

The workpiece is placed on a position in the center of the X-range, and with the ram

extracted. In this position, maximal distortions of the machine tool are to be expected.

A photograph which shows the actual milling of the workpiece and its position within

the workspace of the machine tool, is depicted in figure 6.10.

Fig. 6.10. Photograph of the actual milling of the test-workpiece


The results of the measurement of the lines along the workpiece flanks, manufactured

with and without software error compensation are presented in figure 6.11 and 6.12.

Blror ln Y -direction 0.08

0.06

0.04

0.02

I 0

~ .0.02 .$

! .0.04

.0.06

.0.08

-0.1

.0.12 15

Position relative to lbe center oflbe wodtpiece lmml

Fig. 6.11. Machine tool error copied on the lines 1 and 2, measured in Y-direction

Blror ln Y -direction 0.06

0.04

0.02

0

I .0.02

! .0.04 .$

! .0.06

.0.08

.0.1

.0.12

.0.14 5 10 IS 20 25 30 35 40 45 SO

Position relative to the center of lbe wotkpiece I mm I

Fig. 6.12. Machine tool error copied on the lines 3 and 4, measured in Y-direction

140 Chapter6

In the above described experiment the compensations are determined by using the

model with principal component estimates. In addition to this verification, a

test-workpiece is manufactured with the original estimates as coefficients for the

thermal model. The results of the measurements of the product are depicted in figure

6.13. The non-robustness of the original parameter-set is clearly demonstrated. With

respect to the uncompensated situation the results on flanks 1, 2 and 3 show a strong

improvement of the accuracy, whereas the error in flank 4 is seriously deteriorated.

As the experiment took place under identical environmental and cutting conditions, a

valid conclusion is that the original estimates for the parameters of the model are

sensible to small deviations from the modelled situation. Therefore they are not

suitable for compensation of an actual machining process.

Error in Y -direction. Compensaled wi1h original parameters

0.02

0

'i? .! ~.02 ~ .s j ~.04

~-06

~.08

~.1 '-----'---........L---....._ __ _.._ __ -:...~ __ __, 0 10 20 30 40 60

Position relalive to the center of the wodi:piece [mm]

Fig. 6.13. Machine tool error copied on the lines 1, 2, 3 and 4, measured in

Y-direction, compensation with original estimates of the

model-parameters

An interesting experiment is the verification of the validity of the model on a position

that was not measured during the thermal modelling procedure. Therefore two

workpieces are manufactured at an edge of the working space, i.e the position of the

X-carriage close to zero and a Z-position between two modelled positions. The results

of the measurement of the workpiece are presented below.

e .s .§ .5i

J

Fig. 6.14.

e .s .§ .5i

J

Fig. 6.15.


Euorin Y-direction, Position X=150 ,Z..180

0.02

0

-0.02

-0.04

-0.06

-0.08

-0.1

Position relative to the center of the workpiece [ mm)

Machine tool e"or copied on the lines 1 and 2, measured in Y-direction,

workpiece at the edge of workspace

Euorin Y-direction, Position X=150 ,Z..180 0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

-0.1

-0.12

-0.14 5

Position relative to the center of the workpiece [nun)

Machine tool error copied on the lines 3 and 4, measured in Y-direction,

workpiece at the edge of workspace

142 Chapter6

Conclusions

The final improvement in the factual machining accuracy depends on the position of

the axis, the reproducibility of the initial environmental situation and the size of the

workpiece.

Under practical conditions an improvement of the position accuracy can be achieved

of 1.5 to 8 times the original values.

The presented results apply to the manufacturing of relatively small workpieces. The

compensation for the thermal expansion and distortion of the workpiece is beyond the

scope of this project. Future research will be necessary to assess the thermal behaviour

of large workpieces under machining operations.

Verification of the Correction Algorithm

References

[6.1] Knapp W.:

The Circular Test for Testing NC-Machine Tools.

Published by S. Hrovat. ZUrich, 1987.

[6.2] Kunzmann H .• Trapet E .• Wiildele F.:

143

A Uniform Concept for Calibration, Acceptance Test, and Periodic Inspection

of Coordinate Measuring Machines using Reference Objects.

Annals of the CIRP, VoL 39/1, p561-564, 1990.

[6.3] Soons J.A.:

A Software Package for the Simulation of Geometric Error Models.

Internal development. University of Technology Eindhoven, june 1991.

[6.4] Spaan H.A.M .• Schellekens P.H.J.:

Geometric Error Modelling of Machine Tools

First milestone report of the BCR-project: Development of Methods for the

Numerical Error Correction of Machine Tools. Draft version, May 1991.

145

Chapter?

Conclusions

This project has attended to the enhancement of the accuracy of a commercially

available machine tool. Based on a qualitative analysis of the main error sources, the

study has been focused on the development of a model that compensates for the

effects of the geometric errors and the thermal behaviour.

For the assessment of the geometric errors a mathematical model has been devised,

capable of predicting the spatial deflection of the tool with respect to the workpiece,

as a function of the position of the carriages. This model is applicable to machine

tools of arbitrary configuration.

As a test the model has been elaborated for a specific machine tool, i.e. a five axis

milling machine. The geometric errors of this milling machine have been measured

and transformed to functional relationships.

The information obtained is implemented into the control system for real time

software error compensation. This compensation has reduced the geometric inaccuracy

of the milling machine from 30 to 15 Jlm. A hole plate measurement is carried out for

testing the accuracy improvement of the machine tool.

The results lead to the following conclusions with respect to the geometric errors:

• the mathematical model has proven to be of practical importance and is

· appropriate for the design of a geometric error compensation;

• the correction for the expansion of the measuring scales and the

workpiece should precede the geometric error compensation;

• an overall reduction of the inaccuracy to 50% of the original value is

achieved.

146 chapter?

The thermal behaviour is determined empirically. For this purpose a measurement

set-up has been developed, yielding simultaneous information on the drift of the tool

holder and the thermal distribution of the machine tool.

This information has been analyzed by statistical methods and a relationship between

the displacement and the measured relevant temperatures is obtained.

Also the thermal model has been implemented into the control system of the milling

machine. The compensation values are computed by a PC, which monitors the thermal

state of the machine tool, and sent to the control system. The control system uses the

supplied information for real time compensation of the position of the tool holder. The

value of the compensation is dependent on the position of the axes.

The thermally induced errors of a commercially available machine tool, which can

rise up to 80 J.Lnt, are reduced to a maximum of 30 J.Lnt.

With respect to the thermally induced errors we conclude the following:

• the statistical approach has proven to be successful as a ftrst concept for

the prediction of the thermal behaviour of a machine tool;

• for optimal performance of the model, the machine tool must be placed

in a temperature controlled environment;

• with the applied empirical method of modelling a practical

improvement of the thermal behaviour by a factor 1.5 to 4 can be

achieved;

• the magnitude of the thermally induced errors dominates the effect of

the geometric errors.

As a ftnal test on the practical relevance of the devised compensation methods, several

workpieces have been manufactured. Based on the obtained results we conclude that

the accuracy of a commercially available machine tool is enhanced by a factor 2,

thereby reducing the errors from an original 120 J.1ffi to a maximum of 60 J.lm.

Conclusions 147

Recommendations

The developed modelling methods have proven to yield an improvement of the

accuracy of a machine tool. Further enhancement of the performance of the modelling

methods can be achieved by investigation of related topics.

Below some recommendations are presented that indicate the areas of interest.

• The influence of cooling liquid on the thermal behaviour is not taken into

account in this study. As practical situations often require the use of cooling

liquid, it is advisable to study the effects induced by it.

• The thermal behaviour of a machine tool with moving carriages has not been

examined. Also this effect may disturb the heat flow through the components.

• The probe system proved to yield unreliable results in a heated up machine tool.

Future studies should assess this phenomenon.

• The effects of large, heavy workpieces on the geometric error structure might

cause considerable errors. With the use of strain gauges a workpiece-weight

dependent software compensation for the geometric errors can be devised.

• The current thermal model is static, i.e. no influence of history is taken into

account. By implementing historic parameters into the thermal model, one should

be able to model the response to quick variations in the thermal situation. As the

ability to determine the thermal behaviour of a machine tool in a not-controlled

environment is also dependent on the implementation of historic parameters, this

extension of the modelling methodology is strongly recommended.

• The described measurements of both the geometric and thermal behaviour are

time consuming and thereby costly. For practical application of the devised

modelling methodology, the measurement strategy and method should be

optimized.

• The possibilities of the geometric error model are not fully exploited by the

current control system. Further enhancement of the accuracy can be achieved by

adaptation of the control system for implementation of the full model.

148 Chapter?

• At this moment there is no known standard which supplies a practical guide for

measuring the geometric errors of rotary axes with a horizontal axis of rotation.

Future investigations in this field should assess this deficiency in the

determination of the geometric error structure of machine tools.

• The thermal behaviour of the workpiece, due to influences of the machining

process, will have to be investigated. By real time compensation of this

behaviour the manufacturing of large workpieces with a high level of accuracy

can be achieved.

149

Appendix I

Mathematical Elaboration of the General Model

In section 3.3.2 the following general equation is derived for the expression of the

location error of the tool as a function of the location errors of subsequent coordinate

frames:

m n

wpEtl = .l ( tiFbk bk-lEbk) + l ( tlF ak ak-lEak) + anEtl [ll]

k=l k=l

[1.2]

This vector represents the errors of the tool with respect to the

workpiece, defined in the tool-frame.

[I.3]

This vector represents the errors in the location of frame k with

respect to frame k-1, i.e. the geometric errors of the kinematic

elementk.

[I.4]

This matrix denotes the effect of the errors k-l Ek, acting between

the elements k-1 and k, on the resulting error between tool and

workpiece.

150 Appendix I

Here t1\: x 11~ denotes a 3 x 3 matrix whose columns contain the

vector product of vector u\: with the respective columns of matrix

tl~·

Below this formula will be elaborated for the milling machine under investigation.

Thereby the explanation of the different terms, as presented in section 3.3.2, is

supposed to be known.

The five axis milling machine used for this research is depicted in figure 1.1.

·~ y

Fig. 1.1. Five axis milling machine under investigation

This machine tool consists of one horizontal linear element and one rotary element in

chain 'a' from foundation to tool. Chain 'b' from foundation to workpiece consists of

two linear elements, one vertical and one horizontal, and one rotary element with a

vertical axis of rotation. In the first stage of the modelling process, coordinate frames

are located in the workpiece, the tool and in the centroid of each joint. In figure 1.2

the schematic representation of this milling machine is depicted.

Note that the length of the tool is characterized by introducing a variable 'L'.

Mathematical Elaboration of the General Model 151

The frames located in the various kinematic elements can be characterized as:

• Frame wp: ftxed

• Frame tl fixed • Frame b3 fixed

• Frame a2 ftxed • Frame b2 moving

• Frame a1 fixed • Frame bl moving

The frame of the workpiece is thought to be at the same location as frame b3.

660

1245

Zo

Xo Yo

515 1150

350 660

Fig. 1.2. Schematic representation of the five axes milling machine under investigation

152 Appendix I

Application of the above presented fonnulas, and abbreviation of 'cos(q)' and 'sin(q)'

to 'cq' and 'sq' respectively, results in the expression of the nominal coordinate

transfonnations between succeeding frames as:

a2Tu =Fixed -Fixed = Ja2 a2Stl

_ [ ~:i -~:i g g l [ ~ ~ g gOO+L l - 0 0 1 0 0 0 1 -140

0 001 0001

[

cqa2 -sqa2 0 -(200+L)sqa2] _ sqa2 cqa2 0 (200+L)cqa2 - 0 0 1 -140

0 0 0 1

[

1 0 0 0 l 0 1 0 -210 = 0 0 1 -665+qa1

0 0 0 1

0T31 = Moving ---t Fixed

[

1 0 0 350] 0 1 0 -95 = 0 0 1 495

0 0 0 0

0Tbl =Moving- Moving = 0sb1 Jbl

_ [ ~ ~ g ~~~.5] [A ~ g ~b1] - 0 0 1 515 0 0 1 0

0 0 0 1 0 0 0 1

[

1 0 0 350 l - 0 1 0 327.5+qb1 - 0 0 1 515

0 0 0 1

[1.5]

[1.6]

[1.7]

[1.8]


[

1 0 0 0 l 0 1 0-395 = 0 0 1-375

0 0 0 1

b3 Twp = Fixed - Fixed

[

cqb3 0 sqb3 0 l - 0 1 0 0 - -sqb3 0 cqb3 0

0 0 0 1

[

cqb3 0 0 1

= -sqb3 0 0 0

sqb3 0 l 0 0 cqb3 0 0 1

= 1b3 b38wp

[ 1 0 0 0] 0 1 0 0 0 0 1 0 0 0 0 1

153

[1.9]

[1.10]

[1.11]

These transfonnation matrices can be used to express the nominal position of the

tool-frame relative to the workpiece-frame. As the elaboration .of equation [1.1]

requires the construction of the F-matrices out of 11~ and t1\. for all kinematic

elements. the frrst step is to calculate the tiTk matrices. From these tl Tk matrices, the

required ttl\ and t1\. matrices can be extracted (see equation [3.2]).

154 Appendix I

Application of expression [3.5] onto expressions [1.5] to [1.11] yields the following

matrices:

T [100] 1R 2 = 0 1 0 ,

a a 0 0 1

==}tlat

T ORal

==}tlTO

= t1Ta2.a2Tal

[ cqa2 sqa2 0 - -sqa2 cqa2 0 - 0 0 1

0 0 0

[ 1 0 0 l = 0 1 0 ' 0 0 1

= tlTaratTO

[ cqa2 sqa2 0 _ -sqa2 cqa2 0 - 0 0 1

0 0 0

= tiTO'OTbl

[

cqa2 sqa2 0 _ -sqa2 cqa2 0 - 0 0 1

0 0 0

[1.12]

T [ 0 l -a1Ra2'al ta2 = -210

665-qal

[1.13]

210.sqa2 l 210.cqa2- (200+L) 805-qa1

1

T [ -~~0 l -ORarotal = -495

[1.14]

-350.cqa2 + 95.sqa2 + 210.sqa2 l 350.sqa2 + 95.cqa2 + 210.cqa2 - (200+L) 310- qal

1

632.5.sqa2 + qbl.sqa2 l 632.5.cqa2 + qbl.cqa2- (200+L) 825- qat

1

[1.15]

Mathematical Elaboration of the Geneml Model 155

~ t1Tb2 = tlbrb1Tb2 [1.161

[

cqa2 sqa2 0 700.sqa2 + qbl.sqa2 + (-350+qb2).cqa2 l _ -sqa2 cqa2 0 700.cqa2 + qbl.cqa2 - (-350+qb2).sqa2-(200+L) - 0 0 1 685 - qal

0 0 0 1

~ t1Tb3 = t1Tb2"b2Tb3 [l.l71

[

cqa2 sqa2 0 305.sqa2 + qbl.sqa2 + (-350+qb2).cqa2 l _ -sqa2 cqa2 0 305.cqa2 + qbl.cqa2 - (-350+qb2).sqa2-(200+L) - 0 0 1 310- qal

0 0 0 1

~ tlTwp = tlTb3"b3Twp [1.181

[

cqa2.cqb3 sqa2 cqa2.sqb3 _ -sqa2.cqb3 cqa2 -sqa2.sqb3 - -sqb3 0 cqb3

0 0 0

305.sqa2 + qbl.sqa2 + (-350+qb2).cqa2 l 305.cqa2 + qbl.cqa2 - (-350+qb2).sqa2- (200+L) 310- qal

1

The next step in the detennination of the F-matrices that describe the effect of the

individual errors. between the coordinate frames, on the total error between tool and

workpiece. From the above calculated transformation matrices we can deduce that

(see equation [3.2]):

[1.19]

11Rwp contains the additional transformation of the rotary element b3. However, as we

state that the error between frame b3 and frame wp is zero, the obtained term for

equation [1.1] will automatically yield a contribution of nil. Therefore it is not

necessary to calculate the tlF wp -matrix.

156 Appendix. I

For the elaboration of the relevant F-matrices the vectors tl~ can be calculated as:

tla2 = [ -(~+L) l = [;] [120]

[ 210.sqa2 l [mix l tl

1al

= 210.cqa2 - (200+L) = miy · [1.21] 805- qal mlz

[ -350.cqa2 + 305.sqa2 l = [E] tltO = 350.sqa2 + 305.cqa2 - (200+L) [!.22]

310- qat

[ 632.5.sqa2 + qbl.sqa2 l [~IX l n\1 = 632.5.cqa2 + qbl.cqa2 - (200+L) = ~ly [1.23] 825- qal ~lz

[ 700.sqa2 +qbl.sqa2+(-350+qb2).cqa2 l [~2x l u\z = 700.cqa2 +qbl.cqa2-(-350+qb2).sqa2-(200+L) = ~2y [124] 685- qat ~2z

[ 305.sqa2 +qbl.sqa2+(-350+qb2).cqa2 l [~3x l u\3 = 305.cqa2 +qbl.cqa2-(-350+qb2).sqa2-(200+L) = ~3y [125]

310- qat ~3z

The matrix tlk is defined as (see equation [1.3]):

tlFk [ t!Rk 0 l = ( tl~ X tlRk) tlRk

The outproduct of the vectors tl~ and the matrix tl~ can be summarized as:

[

tkz. sqa2 -tkz.cqa2 tky l tkz.cqa2 tkz.sqa2 -tkx with k = a2, al, bl, b2 b3

-tkx.sqa2- tky.cqa2 tkx.cqa2- tky.sqa2 0 [126]

Mathematical Elaboration of the General Model 157

Implementation of this relation into expression [1.3] yields the following general

F-matrix:

cqa2 sqa2 0 0 0 0 -sqa2 cqa2 0 0 0 0

0 0 1 0 0 0 [1.27] tkz.sqa2 -tkz.cqa2 tky cqa2 sqa2 0 tkz.cqa2 tkz. sqa2 -tkx -sqa2 cqa2 0

(-tkx.sqa2- tky.cqa2) (tkx.cqa2- tky.sqa2) 0 0 0 1

The index k indicates the concerning coordinate frame, i.e. a2, al, bl, b2 or b3

Application of relation [I.l], yields the following expression for the errors wpetl and

wpetl in the orientation and position of the tool coordinate frame with respect to the

workpiece coordinate frame:

Orientation errors

Position errors

[

140. sqa2 -140. cqa2 -(200+L) cqa2 sqa2 0

wpetl = 32etl + 140. cqa2 140. sqa2 0 -sqa2 cqa2 0

(200+L) .cqa2 (200+L). sqa2 0 0 0 1

158 Appendix I

+ (805-qal)cqa2 [

(805-qal)sqa2

(-210sqa2)sqa2-(210cqa2-(200+L))cqa2

-(805-qal)cqa2

(805-qal)sqa2

(-210sqa2)cqa2-(210cqa2-(200+L))sqa2

210cqa2-(200+L) cqa2 sqa2 0 l -210sqa2 -sqa2 cqa2 0 0E

111 0 0 0 1

[

(825~al)sqa2

- (825~al )cqa2

-(632.5sqa2 + qblsqa2)sqa2 - (632.5cqa2 + qblcqa2- (200+L))cqa2

- (825-qal )cqa2

(825-qal)sqa2

(632.5sqa2 + qblsqa2)cqa2 - (632.5cqa2 + qblcqa2- (200+L))sqa2

632.5cqa2 + qblcqa2 - (200+L) cqa2 sqa2 0 l -(632.5sqa2 + qblsqa2) -sqa2 cqa2 0 ol\1

0 0 0 1

l (685~al)sqa2

- (685-qal)cqa2

-(700sqa2 + qblsqa2 + ( -350 + qb4)cqa2)sqa2 -(700cqa2 + qblcqa2 - ( -350 + qb2)sqa2- (200+L))cqa2

-(685-qal)cqa2

(685~al)sqa2


700cqa2+qblcqa2- (-350+qb2)sqa2-(200+L) cqa2 sqa2 0 l -(700sqa2+qblsqa2+(-350+qb2)cqa2) -sqa2 cqa2 0 btl\z

0 0 0 1


r

(310-qal)sqa2

- (310-qal )cqa2


-(310-qal)cqa2

(310-qa1)sqa2

(305sqa2 + qblsqa2 + ( -350 + qb2)cqa2)cqa2 -(305cqa2 + qb1cqa2 - ( -350 + qb2)sqa2 - (200+L))sqa2

305cqa2+qblcqa2-(-350+qb2)sqa2-(200+L) cqa2 sqa2 0

-(305sqa2+qblsqa2+(-350+qb2)cqa2) -sqa2 cqa2 0

0 0 0 1

159

].,~, [1.29]

161

Appendixll

Measurements and Results of Geometric Errors

In the next sections the measurement set-ups for the geometric errors and the obtained

results are depicted. All results display the bare measurement data which implies that

no correction for thermal expansion, nor for influences of rotations, is carried out.

H.l. Measurements of the X-axis

Scale Error xtx

For this measurement a Hewlett Packard 5528 laserinterferometer is used with

accompanying linear optics. To eliminate the effect of temperature, pressore and

humidity of the air on the measored displacement, a so-called air sensor has been

applied. The laserinterferometer is calibrated by the Metrology Laboratory of the

University of Technology Eindhoven and thus traceable to the national standard of

length. The inaccuracy of this instrument is better than 0.05 + 0.5*L Jlm (L in m).

The measurement set~up is depicted in figure 11.1. The interferometer is mounted to

the ram of the machine tool, while the retroreflector is connected to the workpiece

table, which performs the movement in X-direction. The influences of the rotation xry

and xrz have to be eliminated from the obtained measurement results. This yields the

error xtx of the coordinate frame positioned in the centroid of the X -carriage.

162 Appendix II

Laser head

Workpiece table Interferometer

Fig. ll.l. Measurement set-up for xtx

Execution of the described measurement yields the results as graphically depicted in

figure 11.2. The error in the position of the machine, in this case xtx, is defined as:

Error "" True displacement - Assigned displacement [11.1]

Measurement of xtx

10 measurements 0.()25

I O.o2

I O.otS

' g 0.01

J o.oos

Position of the K-aJtis [nun]

Fig.ll.2. Error xtx versus position of the X-carriage

Measurements and Results of Geometric Errors 163

Rotation Errors xrx, xry and xrz

For these measurements the laserinterferometer, with accompanying angular optics, is

used for determination of xry and xrz, while the error xrx is determined by a set of

electronic levelmeters. The laserinterferometer as an angle measurement instrument is

calibrated by the Metrology Laboratory of the Technical University of Eindhoven

against a sine bar and thus traceable to the national standard of length. The inaccuracy

of this instrument is better than 0.2 arcsec. The electronic levelmeters are also

calibrated against a sine bar yielding a maximal inaccuracy of 0.5 arcsec

The measurement set-up for xrx is depicted in figure ll.3. The reference levelmeter is

mounted on the ram of the machine tool, thereby eliminating the effect of rotation of

the overall machine structure, while the measurement levelmeter is placed on the

workpiece table, which performs the movement in X-direction.

The rotation error does not depend on the position of measurement, so the obtained

results directly reflect the rotation error between the coordinate frames of the X- and

Y-axis respectively.

Reference levelmeter

J3k::: Measure levelmeter

'----------+--' Workpiece table

Fig. 11.3. Measurement set-up for xrx


depicted in figure ll.4. The rotation error of the machine is defined in arcsec

(1 arc sec = 4.8e-6 rad).

164 Appendix II

In the results of xrx, peaks can be observed with magnitudes of the same order as the

measured error. Plotting the measurements sequentially yields to the conclusion that

the peak error repeats every 30 minutes. In figure II.5 the measurement results of xrx

are depicted together with a quasi timescale. From an inspection of the machine

constants it appeared that the periodic peaks are induced by the lubrication pump of

the machine tool, that is activated every 30 minutes.

Meuurement ofxrx

Position of the x-axis (mm]

Fig. 11.4. Error xrx versus position of the X-carriage

Measurement of xrx 2r---~----~----r----T----~----~--~----.

Bare data

0 ~ ~ I

I -1

J -2

-3

I l I I ! l~ll I I i : f

-4 ! I i

0 2 3 4 s 6 7 8

Tune [hours]

Fig. 11.5. Results of measurement of xrx sequentially and time indicator of 30 minutes


The measurement set-ups for xry and xrz are depicted in figure ll.6 and II.7. The

reference interferometer is mounted to the ram of the machine tool while the

retroreflector is connected to the workpiece table, which performs the movement in

X~direction. The results of these measurements are presented in figure ll.8 and 11.9.

Also these results directly reflect the rotation errors between the coordinate frames of

the X- and Y -axis respectively.

Laser head


Fig.ll.6. Measurement set-up for xry

Laser head


Fig. II.7. Measurement set~up for xrz

166

2.5

2

1i' 1.5

I j

20 measurements

100

Appendix ll

Measurement of xry

200 300 400

Position of !be x..WS (nun]

Fig. II.8: Error xry versus position of the X-carriage

Measurement of xr:z

7 20 measurements

3

2

100 200 300 400

Position oftbex..WS [mm)

Fig. /1.9. Error xrz versus position of the X-carriage

500 600 700

500 600 700


ll.2. Measurements of the Y ~axis

Scale Error yty

For this measurement the HP5528 laserinterferometer is applied with accompanying

linear optics and the air sensor.

The measurement set-up is depicted in figure ll.lO. The interferometer is mounted to

the ram of the machine tool, while the retroreflector is placed on the workpiece table,

which perfonns the movement in Y -direction. The influences of the rotation yrx and

yrz have to be eliminated from the obtained measurement results. This yields the error

yty of the coordinate frame positioned in the centroid of the Y ~arriage.

Laser head

Fig.ll.JO. Measurement set-up for yty


depicted in figure n.ll.

168 Appendix II

Measurement of yty 0.035 ,...--...----..----r---.----...----,--.---...---....,---,

20 measurements O.o3

~ 0,025

l i 0.02 rl l O.ot5 . .. €. O.ot

I o.oos

~DM~-~-~-~-~-~-~~-~-~-~-~ o ~ a ~ a s a m ~ • ~

Position of the y..axis [mm)

Fig. II. II. Error yty versus position of the Y-carriage

Rotation Errors yrx, yry and yrz

Several instruments are applied to perform these measurements. First, for the

determination of the error yrx, we applied a Hewlett Packard laserinterferometer.

The set-up for this measurement is depicted in figure 11.12. The interferometer is

mounted to the ram of the machine tool, while the retroreflector is placed on the

workpiece table, which performs the movement in Y-direction. The obtained results

directly reflect the rotation error between the frame attached to the Y -axis and the

machine coordinate frame.


Laserhead

Fig. 11.12. Measurement set-up for yrx


depicted in figure ll.13. The rotation error of the machine is defined in arcsec.

Measorement of yrx

20 measorements

2

1.5

Position of they-axis lmm)

Fig.IJ./3. Error yrx versus position of theY-carriage

The measurement results of yrx show identical peaks as the results of xrx. This is also

caused by the lubrication pump that operates every 30 minutes.

170 Appendix IT

The measurement set-up for yry is depicted in figure IT.14. While this measurement

cannot be carried out by a laserinterferometer nor a set of levelmeters, two

straightness measurements are performed. The straightness measurement is performed

by movement of a straight-edge, in this case a calibrated surface of a squareness

reference block, along a displacement transducer. The reading of the transducer is a

measure of the error ytz with a contribution of yry.

+Z

®

Squareness reference

Fig. Il./4. Measurement set-up for yry

By choosing the set-up as depicted in figure ll.14 the effect of the error yry in the

measurement result will reverse sign between measurement 1 and 2. With the

necessary displacement in X-direction between the measurements the change in active

arm is known. The results of these measurements have to be corrected for the effect of

the error xrx on the orientation of the straight-edge. This error results in a

displacement Y*xrx which is not caused by theY-axis.


Taking the above procedure the error yry can be calculated as:

ytz(2)- ytz(l)- Y•xrx yry = ---------- [Rad] [11.2]

x(l)- x(2)

with: - ytz(2) and ytz(l) are the uncorrected results of the

straightness measurement;

- x(l) and x(2) are the positions of the X-carriage during the

respective straightness measurements;

- Y is the position of the Y -carriage during the straightness

measurements.

Execution of the described measurement yields the bare measurement results that are

graphically depicted in figure II.l5a. In figure II.15b the calculated error yry is

presented. The elaboration of these results for correction purposes will be discussed in

Chapter 5.

_ -0.98,---...,.---,-----,..-'Two'-"-"'-"'meas:;=uremen==;::;ts:..::o~f t=,~-....---.----,

io.985 j -0.99 8 ~.995 :a

~. %~:~

·~~=::,··-··•:::::::::::::::::::::::::----~ -1

1%1-Loos L___......,__.......,_x_=-'55._2_.69_1---" __ ...__....._ _ __._ __ ...__....:!1

SO 100 ISO 200 2SO 300

Position of the Y -caniage [mm)

Calculated

0

i -1 ! ~

-2

-3 Measurement not corrected for xrx influence

-4 so ~ ~ 200 s o ~ a ~ ~

Position of the Y ~arringe

Fig. II.15a. Measured e"ors ytz on two positions of the X-carriage

II.l5b. Calculated yry from straightness measurements

172 Appendix II

The measurement set-up for yrz is depicted in figure II.16. The reference

interferometer is mounted to the ram of the machine tool, while the retroreflector is

connected to the workpiece table, which performs the movement in Y -direction. Also

these results directly reflect the rotation errors between the coordinate frame attached

to the Y -axis and the machine coordinate frame.

Laserhead Angular retroreflector

Angular interferometer

Fig.l/.16. Measurement set-up for yrz

Measurement of yrz

20 measurements 3

2.5

2

50 100 150 200 250 300 350 400 450 500

Position of they-axis [mm]

Fig.ll.l7. Error yrz versus position of the ¥-carriage


II.3. Measurements of the Z-axis

Scale Error ztz

For this measurement the laserinterferometer is applied with accompanying linear

optics and the air sensor.

The measurement set-up is depicted in figure ll.l8. The interferometer is mounted on

the workpiece table, while the retroreflector is connected to the ram of the machine

tool. The influences of the rotation zrx and Wf have to be eliminated from the

obtained measurement results. This yields the error ztz of the coordinate frame

positioned in the centroid of the Z-carriage.

Interferometer Retroreflector Workpiece table

Fig. 11.18. Measurement set-up for ztz


depicted in figure II.l9.

In the results of ztz a clear form of hysteresis can be observed. This is not caused by

the hardware of the machine, but purely by a reproducing temperature field over the

Z-scale. In figure ll.20 the temperature on three positions of the Z-scale is depicted.

These temperatures were obtained during the measurement of ztz. The second graph

in figure II.19 represents the error corrected for these thermal effects. Clearly the

hysteresis has disappeared.

174 Appendix II

0.03 r----~----.---z!:..!T:.!:z.!z.bar~e~da~ta!..__.,.-----.----:=

v

o.oz ~ j 0.01 ol:--___ .,...,_,._..

-O.OlO'----l-'"00----200.__ ___ 300...._ ___ 400.._ ___ 500_._ __ ---..J600

Position oftbe~ge [mm]

0.03 .-------.-~--.----'z,_.T...._,corrected~~~d~ata~-.-----.------,

-O.Olo'----t ..... oo.,------2oo.._ ___ 3 ..... oo ____ 400.._ ___ soo_._ __ ___..J600

Position of the ~ge [mm)

Fig. 11.19. Error ztz versus position of the Z-carriage

Temperature of Z-scale during ZIZ measurements

400 500

Position of the Z-carriage

Fig. 11.20. Temperature of the Z-scale during measurement of ztz


Rotation Errors zrx. zry and zrz

Several instruments are applied to perform these measurements. First, for the

determination of the error zrx. we applied the laserinterferometer with the angular

optics.

The set-up for this measurement is depicted in figure II.21. The interferometer is

placed on the workpiece table while the angular retroreflector mounted on the ram of

the machine tool. The obtained results directly reflect the rotation error between the

machine coordinate frame and the frame attached to the Z-axis.

Laserhead

Angular interferometer Workpiece table

Fig. ll.21. Measurement set-up for zrx


depicted in figure II.22. The rotation error of the machine is defined in arcsec.

The measurement set-up for zry is depicted in figure II.23. This measurement is

carried out by a laserinterferometer and angular optics. The set-up is alike with the

one for the determination of zrx, but the optics are rotated about the Z-axis over

ninety degrees.

176 Appendix n

Measoremeut of zrx

20 measorements

-4 i ! -6

] -8

-10

-12

-14 0 100 200 300 400 500 600

Position of the z-Ws (nun)

Fig./1.22. Error zrx versus position oftlu! Z-carriage

Angular interferometer Workpiece table

Fig. 11.23. Measurement set-up for zry

By choosing this measurement set-up the results directly reflect the error zry. These

results are graphically depicted in figure II.24. The rotation error of the machine is

defined in arcsec.

Measurements and Results of Geomecrlc Errors 177

Measurement of 7Tf

15 measuremeJIIS

I ! -1

J -1.5

-2

-2..5 0 100 200 300 400 500 600

Position of the z-axis [mm)

Fig. 11.24. Error zry versus position of the Z-carriage

The measurement set-up for zrz is depicted in figure ll.25. The reference levelmeter is

mounted on the workpiece table, while the measurement levelmeter is connected to

the ram of the machine tool. Also these results (figure ll.26) directly reflect the

rotation errors between the coordinate frame attached to the Z-axis and the machine

coordinate frame.

Measure levelmeter

Workpiece table

l" ~ Reference levelrneter

Fig. 11.25. Measurement set-up for zrz

178 Appendix n

Meas11rement of zrz 2r-----~----~------~-----T------~----~

15 measurements

l.S

I o.s

~

-t o'-------:-:'loo:-:-----200~---:J00:-:>-:----:400"':-----:soo.L:------:'.600

Position of the z-llllis (mm)

Fig. l/.26. Error zrz versus position of the Z-carriage

ll.4. Determination of the Squareness. Errors

The squareness error between the carriages is determined by measurement of a

ceramic reference block. This block is calibrated on a 3D measuring machine and

possesses a squareness error of +1.7 arcsec. The measurements are carried out using

inductive displacement transducers. that are calibrated against a trace-able

laserinterferometer. This procedure yields two straightness measurements that, after

elimination of the pure straightness error by Least-Square fitting. can be used to

calculate the squareness error.

The squareness errors are included in the model as offsets of rotation errors i.e. they

will not be treated as separate geometric errors in the model.

Below the three different squareness measurements are described and tbe

measurement results presented.


• Squareness Error between the X- and Z-guide

For this measurement the reference block is placed on the machine table. The block is

supported on three points. The block is aligned along the X-axis to secure that the

displacement transducers remain in their calibrated range (0-2 mrn). First, the

displacement transducer is mounted on the ram of the machine tool and the reference

side of the block is measured, yielding the error xtz and the alignment error of the

block (figure 11.27 and ll.28).

X=O

Fig./127. Measurement set-up ofxtz (top view of machine tool)

-0.994

I -0.995

~ ! -0.996 'a "" ~ !J-0.997 ~

-0.998'

Measurement of XIZ

' Angle= +0.7 arcsec -0.999 :-' --~·::----::·-::------::7' ::------:-:-:· :------::7:' :-----::

0 100 200 300 400 500 600

Position oftbe X-<:aniage (mm]

Fig. /1.28. Measurement results ofxtz

180 Appendix n

Secondly, the other side of the block is measured yielding the error ztx and an error

that includes both the alignment error of the block, the squareness error in the block

and the squareness error of the machine tool (figure ll.29 and ll.30).

Z=800

Fig. 11.29. Measurement set-up ofztx (top view of machine tool)

Measurement of ztx

-0.993

l-0.994

1-0.995

"0 -0.996 ... i ~ -0.997

-0.998

Angle= +2.1 arcsec

-0.999soL_ __ l00'---15'-o--2oo'---"250'---'300'---"35'-o----"400'---'4SO'----:'soo

Position oftheZ-carriage [mm)

Fig. JI.30. Measurement results of ztx


Out of these measurement results the following conclusions can be drawn. The

positive angle between the best fitted line through of the measurement results and the

X-axis is explained by an alignment error of the block, whereas the block is rotated

about the Y -axis

The measurements indicate an alignment error (a) of 0. 7 arcsec.

The second measurement yields an alignment error (/J) of 2.1 arcsec. The total

uncorrected out of squareness, defined as the actual angle included by the guides

minus 90 degrees, is in this case calculated by: -{3-a, thus -2.8 arcsec.

However, the squareness error of the block, i.e. 1.7 arcsec, must be added to this result

to achieve the total squareness error of the Z-axis with respect to the X-axis. Applying

this correction yields a squareness error between the Z- and the X-axis of -1.1 arcsec.

• Squareness Error between the X- andY-guide

Again the reference block is supported by the machine table and aligned along the

X-axis. First, the displacement transducer is mounted on the ram of the machine tool

and the reference side of the block is measured, yielding the error xty and the

alignment error of the block (figure IT.31 and 11.32).

x-7oo x-o

Fig./1.31. Measurement set-up ofxty {front view of machine tool)

182 Appendix ll

Measurement of xty .0.994.-----,..------,----.-----.----.-----,

-t.ooto~.-__ _.too.__ __ 200_._ ___ 300_._ ___ 400...__ ___ soo..__ __ .....J600

Position of the X-amiage (mm]

Fig. //32. Measurement results ofxty

Secondly. the other side of the block is measured yielding the error ytx and an error


and the squareness error of the machine tool (figure ll.33 and ll.34).

Y=500

I

~ r+X ~ ) +Y

v

Y•O

Fig.ll33. Measurement set-up ofytx (front view of machine tool)


-1

rl.OOS

§ -1.01

1 !-1.015

t "' -1.02

-1.025

Measurement of ytx

Allgle = + 12.6 an:sec

-l.03o'---50'---1oo......__1..._50 __ 2..._oo __ 250...__300.....__3 ..... 50--400........._ __ 450.____,soo

Position of lhe Y ~(nun]

Fig. 11.34. Measurement results ofytx

183


positive angle between the best fitted line through the measurement results and the

X-axis is explained by an alignment error of the block whereas the block is rotated

about the Z-axis.

The measurements indicate an alignment error (a) of -0.1 arcsec.

The second measurement yields an alignment error (/J) of + 12.6 arcsec. The total

uncorrected out of squareness is in this case calculated by: -fJ-a, thus yielding -12.5

arcsec. However, the squareness error of the block, i.e. 1.7 arcsec, must be added to

this result, so the total squareness error of the Y-axis with respect to the X-axis is

-10.8 arcsec.

• Squareness Error between the Y- and Z-guide

Again the reference block is supported by the machine table. The block is now aligned

along the Z-axis. First, the reference side of the block is measured, yielding the error

zty and the alignment error of the block (figure ll.35 and ll.36).

184 Appendix 11

r

+Y

Fig. /135. Measurement set-up ofzty (side view of machine tool)

Measurement of zty

-1

Angle= -5.4 arcsec

-1.oo5 o~...-___ 100..__ ___ 200_.__ ___ 300_.__ ___ 400__._ __ _.soo ___ .....~600

Position of the Z-caniage [mm]

Fig. /1.36. Measurement results of zty

Secondly, the other side of the block is measured yielding the error yTz and an error


and the squareness error of the machine tool (figure ll.37 and II.38).


Y=O

Fig./1.37. Measurement set-up ofytz {side view of machine tool)

Measurement of ytz ..0.988 .-----.----.--..,.---...----,,---...,----,..--.,..---,

-0.99

..0.992

I..o.994 ~ 1..0.996

'S ..0.998

f -1

.l.(X)l

-1.1)04 Angle: -3.1 arcsec

-l.00650'---'100-:---l...L50--2-'-:00--250':---300-'---3:-':50--400-'---450.L---'500

Position of theY -caniage [mrn]

Fig./138. Measurement results ofytz

185


negative angle between the best fitted line through the measurement results and the

Z-axis is explained by an alignment error of the block. The measurements indicate an

alignment error (a) of -5.4 arcsec.

The second measurement yields an alignment error (/3) of -3.1 arcsec. The total

uncorrected out of squareness is in this case calculated by: a-/}, thus yielding -2.3

arcsec. However, the squareness error of the block, i.e. 1.7 arcsec, must be subtracted

from this result, so the total squareness error of the Y -axis with respect to the Z-axis is

-4.0 arcsec.

187

Appendix ill

Correlation between Temperature Sensors

Under continuous loads the temperatures on different positions of the milling machine

show high correlations. As an indication the Pearson correlations coefficients between

the first 7 and all 39 sensors are presented below. The underlined numbers indicate a

correlation coefficient higher than 0.9.

T1 T2 T3 T4 T5 T6 T7

T1 1.00000 0.22412 0.80631 0.85934 0.92512 0.20121 0.74612

T2 0.22472 1.00000 0.26862 0.94411 0.281!:1:6 0,9738~ 0.55587

T3 0.80631 0.26862 1.00000 0,92352 0.2!:1:616 0.2!112 0.39556

T4 0.85934 0.94411 0.22352 1.00000 0.2.68~1 0.25214 0.63041

T5 0.22512 0.28146 0.24616 0.2.6841 1.00000 0.2827~ 0.59166

T6 0.90721 0,9738~ 0.2!:1112 0.252H 0,9821~ 1.00000 0.59601

T7 0.74612 0.55587 0.39556 0.63041 0.59166 0.59601 1.00000

T8 0.53919 0.33097 0.17405 0.44286 0.38045 0.40409 0.2!:1824

T9 0.53690 0.29990 0.12947 0.39437 0.34642 0.37288 0.94451

TlO 0.21144 0.70482 0.52172 0.64282 0.72082 0.71513 0.81580

Tll 0.20112 0.68606 0.50009 0.62458 0.70288 0.69615 0.81375

T12 0.87033 0.62819 0.43587 0.57226 0.64813 0.64414 0;81027

Tl3 0.86390 0.61746 0.42356 0.56256 0.63786 0.63433 0.81397

T14 0.79670 0.52747 0.33143 0.49452 0.55033 0.56004 0.84546

T15 0.82880 0.57423 0.38129 0.54763 0.59644 0.60346 0.88421

T16 0.75183 0.46766 0.26797 0.43763 0.49328 0.50747 0.82232

T17 0.72829 0.43907 0.23865 0.41601 0.46664 0.48402 0.82430

T18 0.84807 0.60648 0.41414 0.57565 0.62315 0.61738 0.21816 T19 0.85414 0.62911 0.44269 0.60907 0.64810 0.65035 !},94152

T20 0.76802 0.49481 0.29065 0.47590 0.51853 0.52108 0.21223

188 Appendixm

Tl T2 T3 T4 T5 T6 n T21 0.76709 0.49945 0.29767 0.48592 0.52426 0.53197 0.22975

T22 0.66355 0.36652 0.16280 0.35802 0.39813 0.40586 0.79323

T23 0.70271 0.41130 0.21299 0.38374 0.43986 0.45734 0.77102

T24 0.22682 0.71947 0.53927 0.65978 0.73470 0.72590 0.82770

T25 Q.25815 0.81111 0.65631 0.77289 0.81951 0.80554 0.88884

T26 0.23073 0.72668 0.54815 0.66203 0.74008 0.73001 0.81534

T27 0.26023 0.80293 0.64354 0.74930 0.80890 0.79136 0.86387

T28 0,94297 0.2266.1 0.25266 0.23011 0.28118 0.97218 0.58504

T29 0.2557£1 0.99335 Q.23881 0.23232 0.28320 0.96526 0.62883

T30 0.60629 0.76164 0.80892 0.80017 0.74481 0.80374 0.53335

T31 0.65715 0.82006 0.87066 0.88520 0.81920 0.83173 0.53102

T32 0.88055 0.28852 0.22820 0.22204 0.97642 0.2151:!1 0.46617

1'33 0.88328 0.28221 0.21221 0.25132 0.28260 0.21123 0.51339

T34 0.72430 0.92118 Q,97!i32 0.87276 0.2012~ 0.~180 0.28607

1'35 0.74369 0.92838 0,97516 0.22322 0.22508 0.92223 0.39416

1'36 0.78006 0,94418 0.97~62 0.22210 0.24221 0.26224 0.40084

T37 0.73953 0.21225 0.94122 0.24131 0.22155 0.23128 0.46977

1'38 0.74010 0.92626 0.21222 0.20161 0,92226 0.24121 0.33774

1'39 0.69958 0.88817 0,93572 0,92616 0.89946 0.20602 0.43654

Appendix IV

Results of the Modelling Procedure of the

Thermal Behaviour

189

The three data samples are examined by the modelling procedure as described in

Chapter 5. This yields correction formulas for each direction of the form:

n

= L cj Tempj j=l

with: - i: the direction of compensation, i.e. X, Y or Z;

[IV.l]

- n: the number of relevant sensors determined by the

SAS-procedure;

- Ci the calculated coefficient obtained from the modelling

procedure;

- Temp.: the temperature of sensor j. j

As the measurements are carried out on three different Z-positions; the modelling

procedure yields nine compensation functions. The (rounded) coefficients Cj, and the

relevant temperature sensors Tempj (see figure 4.2), for the three measurement

positions are presented below.

190 Appendix D

Position: X=350. Y=l40, Z=62 (in mm)

X-direction Y -direction Z-direction


0.08 48 1.93 41 0.98 42 -0.10 49 -3.91 47 0.91 45 0.17 53 -7.66 48 1.50 46 0.03 54 1.05 51 0.08 52

-0.69 55 0.96 54 -0.51 53 -0.98 57 0.14 56 0.08 55 -1.06 58 0.64 57 -0.17 57 -0.41 50 0.16 58 -0.18 58 0.34 27 -1.53 59 -1.36 50 0.17 28 -3.89 50 0.68 29 0.13 32 -1.16 34 1.23 37 0.50 37 2.37 29 1.54 23 0.49 23 -2.11 30 1.18 35 0.41 38 3.07 25 1.90 21

0.74 27 1.71 22 -0.73 38 -1.00 36

Position: X=350. Y=l40, Z=314 (in mm)

X-direction Y -direction Z-direction


1.57 43 -0.32 51 1.97 43 0.94 47 0.26 52 1.91 44 0.74 48 0.65 53 2.13 45 0.12 49 -0.70 56 2.83 48 0.70 51 1.34 57 0.61 51 0.10 52 1.17 58 -1.11 53

-1.23 54 -2.21 39 -1.16 54 -0.57 56 -4.05 50 0.35 57 0.20 59 2.80 34 0.43 58

-0.73 39 0.11 25 1.57 59 -0.61 50 -1.21 50 -0.27 29 -0.73 34 1.71 27 1.77 26 0.93 32 1.58 32

2.21 38 1.92 24

Results of the Modelling Procedure of the Thennal Behaviour 191

Position: X=350, Y=l40, Z=566 (m mm)

X -direction

c Sensor c 0.27 42 -2.01 49 2.30 44 0.29 43 1. 71 51 2.38 45 0.26 48 1.46 52 3.53 49 0.22 49 0.56 54 -0.17 52

-0.15 51 -3.49 50 -1.10 53 -0.26 54 1.52 31 -1.10 54 0.06 56 1. 71 37 0.99 39 0.14 58 -1.47 50 0.03 60 0.37 25

-0.15 39 2.16 27 -0.97 50 1.47 28 0.14 34 1.43 32 0.04 26 1.86 24 0.26 27 2.48 21 0.24 31 0.28 28 0.23 32 0.28 35 0.33 21

193

AppendixV

Results of the Tests on the Probe System

As the experiments of the verification of the thennal compensation yielded

unexplainable results, some tests are carried out on the reproducibility of the probe

system. For these tests a warmed-up machine tool is used whereas the probe is

exchanged just before the moment of measurement. With the application of the probe

system the position of the workpiece table is determined on four positions of the table.

Between two measurement cycles a compensation value is sent to the control system

to check the correct incorporation into the measurement results. In table V.l the

registered positions by the probe system are presented. The values in brackets denote

the actual compensations between two measurements. The last column represents the

instructed compensation value.

No compensation Compensated No Compensation Compensation

-72.099 -72.101 (-2) -72.141 (-39) 42 -71.988 -72.186 (-198) -72.195 (-9) 208 -72.055 -72.236 (-181) -72.189 (47) 136 -72.063 -72.245 (-182) -72.197 (48) 136

Table V.i. Registered Y-positions with and without compensation, warmed-up

machine tool

One could conclude from this table is that either the compensation algorithm of the

control system is malfunctioning, or the probe system itself is not repeatable.

Therefore an experiment is carried out in a thennally stable situation. The probe

system is instructed to measure the position of the workpiece table four times, with a

tool exchange cycle between each measurement. Then a compensation table is sent to

the control system and the measurement cycle is repeated.

194 Appendix V

The measured positions are presented in table V.2. The compensation values are

identical to table V.l.

No compensation Compensated

-72.140 -72.142 -72.142 -72.100 -72.101 -72.102 -72.175 -72.175 -72.176 -71.969 -71.968 -71.968 -72.169 -72.170 -72.170 -72.033 -72.033 -72.034 -72.178 -72.180 -72.180 -72.043 -72.043 -72.042

Table V.2. Registered Y-positions with and without compensation, thermally stable

machine tool

The first conclusion from this experiment is that the repeatability of the probe system

is within 2 pm. Furthermore it appears that the compensations are adequately carried

out by the control system.

Therefore the thermal influences on the probe system are the only causes that

disturbed the actual reading of the probe system. as showed in table V.l.

A plausible explanation for this effect is the exchange of a relative cold probe system

into a warm tool holder causing the (aluminium) probe to expand very rapidly.

At this moment no further investigations of this problem will be carried out. In this

study we confme ourselves to the conclusion that the probe system is not suitable for

the verification of the thermal error compensation.

Name: Born:

Marital status:

Edncatioo:

1976- 1982

1982- 1987

1987- 1991

Miscellaneous:

Curriculum Vitae

Franciscus Comelis C.J.M. Theuws

July 16th 1964 in Luyksgestel, The Netherlands

Unmarried

VWO Rythoviuscollege, Eersel.

195

Study for Master's degree at the University of Technology

Eindhoven, faculty of Mechanical Engineering.

Doctoral study at the University of Technology Eindhoven.

Subject of the study is the enhancement of machine tool

accuracy. This study has been carried out at the Metrology

Laboratory of Eindhoven University and was funded by the

Foundation for Technical Sciences (STW).

• Chairman of the Technical Committee for Metrology of the

Dutch Calibration Organisation (NKO) since march 1991.

• Co-author of the NK()..guideline concerning acceptance

checks for coordinate measuring machines.

• Teacher at the post-graduate course on quality control since

1987.

197

Acknowledgements

The study described in this thesis has been canied out at the Metrology Laboratory of

the University of Technology in Eindhoven. The study has been funded by the

Foundation for Technical Sciences (STW), which I would like to thank for the

opportunity of realizing this project.

I would like to express my thanks to all staff members and students, who worked in

the Metrology Department during this study, for their cooperation and pleasant work

climate.

In particular I want to thank Henny Spaan and Hans Soons for their combination of

enthusiasm and brilliance that greatly contributed to the success of this project.

Furthermore, the permanent staff members Frits Theuws, Adriaan ·de Gilde, Harry

Sonnemans and Klaas Stroik should be mentioned for their practical support and

interest in this study.

Special thanks go to both the promotors Prof.dr.ir. A.C.H. van der Wolf and Prof.dr.ir.

P.H.J. Schellekens for their support and perception during this study. Also I wish to

state my appreciation to the members of the main promotion committee, Prof.dr.ir. J.J.

Kok and Prof.dr.ir. H.F. van Beek, for their positive contribution to this thesis.

Also I am much obliged to the members of the dining club 'Do not be embarrassed',

Dr.ir. T.S.G. Lo·A·Foe and Dr.ir. P.J. Bolt, for the many mind refreshing evenings.

A special word of thanks goes to Ir. J.A.W. Hijink, who was very helpful in the design

and manufacturing of the test workpieces, and to J. Cauwenberg who built the

temperature measurement set-up.

Last, but certainly not least, I want to thank my girlfriend Jozet for her patience and

encouragements that were always a stimulus to complete this study.

STELLING EN

behorende bij het proefschift van

F.C.CJ.M. Theuws

1 In tegenstelling tot meetmachioes kunnen produktiemachines een onnauw

kcurigheid in de aangegeven positie bezitten die onder de uitgegeven resolutie

van hct mcetsysteem ligt.

Dit proejschrift

2 Het relateren van de therrnische drift van produktiemachincs aan 6tn

tcmpcratuursmeetpunt leidt niet tot eenduidige resultaten.

Dit proefschrift

3 De vcel gcbruikte methode om met behulp van modellen de verspanings

krachten te bcrckcncn, en deze samen met de gemeten stijtheid van het

gereedschapswerktuig tc gebruiken voor correctle van de vcrplaatsing van de

hoofdspil, is slecht::; beperkt inzetbaar.

J. Tlusty, Annals of the CTRP, Vol 39!2, 1990

4 Voor het bcrcikcn van een hoge nauwkeurigheid met verspanende werktuigen

dicnt men het produkt niet alleen in een opspanning te bewerken maar, in

verband met de optredende therrnischc cffccten, ook met een gereedschap

Dil proejschrift

5 De onzekerheid in de lineaire uitzettingscoi!fficient van een materiaal en een

thennisch onbepaalde constructie beperken de mogelijkheden tot het analytisch

voorspellen v;w het thermisch gcdrag van constructies ten zeerstc.

J. Bryan, Annals of the CTRP, Vo/39/ I. 1990

6 Doordat thcrmische effecten een grote invlocd kunnen uitoefenen op de

herhalingsnauwkeurigheid, dient het gebruik van een tastsysteem bij ccn

bewerkingsmachinc voor de bepaling van de afmetingen van een produkt met

grote tc:rughoudendheid te worden toegepast.

Dit proefschri/t

7 Als cerstc aanzet voor een softwarematig gccorrigeerde produktiemachine is de

compensatic voor de thermische expansie van de linialen en van het produkt een

veelbelovende optic. Ecn geschikte bevestiging van de Hnialen is hiervoor

echter noodzakelijk. Dit proejschrift

8 Bij de verificatie van de geschiktheid van ecn model, waarin niet aile

optredende effecten zijn verdisconteerd, dit;;nt men crop te achten dat men niet

bezig is een nieuwe dimensie aan het model tot;; te voegen.

9 Wanneer men de heersende bureaucratie bij de overheid in ogenschouw neemt,

is de vaak geuite bewering dat ambtenaren niet hard hoeven te werken om de

geleverde prestatie neer te zetten een contradictio in terrninis.

10 Een voordeel van een krachtig computersysteem is de mogetijkheid om sncl

vcr.mderingen in het model te evalueren. Hierdoor wordt echter ook de

beschikbare tijd voor gedachtensprongen kleiner, waardoor het risico onstaat dat

de onderzoeker zijn model optimaliseert naar de: op dat moment ter beschik.king

staande gegevens.

11 Al zijn de: mathc:matische modellen nog zo universeel, iedert;; implc:mc:ntatie

c:rvan is unic:k_

Eigen ervaring, 1987-1991

12 Door de grote hoeveelheid onbepaaldheden tijdens een promotie-ondcrzoek, is

een projectmatige aanpak slechts zeer beperkt tot:pasbaar.

13 Het ontbreken van vaste aanvangstijden heeft op het rijgedrag van menig

T.U.-medewerker een wel ~eer rustgevende invloed.

Eigen observaties, 1982-/99/

14 Het verminderen van de financiele middelen t.b.v. onden;oek zal tot resultaat

ht:bben dat veel ontwikkelde modellen alleen in theorie functioneren.

15 Het instellen van "dead-lines" bij promotie-onderzoek heeft enkel ~in wanneer

deze term letterlijk wordt genomen.

16 Uit de inhoud van het pakket stel]jngen dat door ccn promovcndus aan zijn

proefschrift wordt toegevoegd zijn eerder de: frustratic:s dan de eruditie van de

auteur af te leiden.

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