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University of Huddersfield Repository

Fletcher, Simon, Longstaff, Andrew P. and Myers, Alan

Defining and Computing Machine Tool Accuracy

Original Citation

Fletcher, Simon, Longstaff, Andrew P. and Myers, Alan (2009) Defining and Computing Machine

Tool Accuracy. In: Laser Metrology and Machine Performance. Euspen Ltd, Euspen Headquarters,

Cranfield University, pp. 77-86. ISBN 978-0-9553082-7-7

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Defining and computing machine tool accuracy

S Fletcher, A Longstaff, A Myers

Centre for Precision Technologies, University of Huddersfield – UK

s.fletcher@hud.ac.uk

Abstract

Understanding machine tool performance is important for specifying or

comparing machines and for determining capability for production. Machine

accuracy has generally been described by the linear positioning accuracy and

repeatability of the axes. This specification neglects all other geometric effects

such as angular, straightness and squareness errors which can have a significant

effect upon the true precision capability of such machines. A more

comprehensive way to define a machine’s precision would be to specify the

accuracy for the full working volume of a machine tool, i.e. the Volumetric

Accuracy (VA) taking into account all geometric errors. Existing methods for

describing the volumetric accuracy recognised by the standards organisations

are the diagonal and step-diagonal methods. These are designed, in part, to be

rapid to reduce machine downtime but compromise accuracy and extensibility if

used in isolation. The reduction in accuracy is described in detail in the ISO

standard 230 part 6 [1]. This paper describes a definition of VA and a

methodology for calculating and reporting the performance of a 3-axis cartesian

machine tool that significantly reduces ambiguity compared to other methods

and supports a broader range of performance assessments.

Error measurement methods are discussed with respect to accuracy and test

time. An example model is discussed highlighting the ease with which 3D

positioning error can be calculated then methods for efficiently determining the

proposed volumetric accuracy. The method is extensible in that the data and

model describe the machine volume completely and therefore enables a variety

of performance assessments for machine comparison or process capability.

Examples are provided showing the difference in accuracy using variable

volume assessments and also an example part profile. The method also enables

easy calculation of the percentage contribution of each geometric component at

the volume positions most affecting the volumetric accuracy enabling targeted

correction with maximum performance benefit.

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1

Characterisation the positioning capability of machine tools can require a

significant amount of effort if it is to be determined for the entire working

volume or a subset of the volume. This level of understanding is required if the

production capability of the machine is to be assessed or for a more complete

comparison between machines. Traditional linear positioning and repeatability

figures have, and often still are, used to qualify a machine in terms of positional

accuracy. The additional factors, particular the other geometric errors (angle,

straightness and squareness) but also thermal errors are affecting many

manufacturing industries. Their identification using increasingly available

metrology such as Ballbars and laser interferometry systems is exposing a much

lower capability of the machines that is often initially communicated.

This shortfall in understanding can have a significant detrimental effect on

production systems when they fail to perform as expected and when there is

often large outlay associated with machine tools. There can be long term costs

associated with concessions and the effort required to maintain the accuracy

level compared to a machine that is over capable.

An increased awareness of the factors influencing machine capability has led

to an increase in the use of volumetric assessment methods for determining

volumetric accuracy (VA). This can be identified using direct measurement or

by an error synthesis method.

Direct methods usually involve either on-machine probing of a artefact such

as a ball or hole-plate with additional spacers or by a laser tracker which

combines measurement of distance and two angles (azimuth and elevation).

Generally, artefacts are small and therefore efficiency and accuracy diminishes

as multiple positions in a larger volume are measured and the data stitched

together. Laser trackers have a large range but the accuracy diminishes with

distance. The 3D coordinate uncertainty at a target distance of 10m was

estimated from the specifications of three popular tracker systems to range from

70 µm to 136µm.

The ISO 230 part 6 [1] standard provides instructions that have been used for

estimating VA from face and body diagonal tests which goes some way to

providing a more complete picture of the machine positioning capability. These

test are relatively quick to perform but they cannot provide an unambiguous

description of the magnitude of the individual contributing error components

affecting the tool point in each axis direction [2,3].

Error synthesis methods involve indirect determination of position error by

calculating the effect of the individual geometric error components using a

complete kinematic model of the machine [4]. This method enables full volume

assessment using readily available metrology applied thoughout manufacturing

industry such as laser interferometry, inclinometers and artefacts. This paper

describes a definition of VA and a methodology for calculating and reporting the

performance of a 3-axis cartesian machine tool using this method that

significantly reduces ambiguity and supports a broader range of performance

assessments.

Introduction

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2

Using the proposed methodology in this paper requires that all the individual 21

error components (3-axis machine) on the machine need to be measured and this

process can be laborious and time consuming depending on the equipment

available. Traditional laser interferometers such as the Renishaw’s ML10 or HP

systems are widely available either in-house or through measurement services

and these enable measurement of most of the errors in 1 to 2 days. More

recently, multi-degree of freedom systems have been developed such as the API

XD6 [8] which can measure all six errors on a axis simultaneously, significantly

reducing the measurement time to within 1-day even on large machines. Further

reduction in machine downtime is possible using the new software system

TRAC-CAL produced by ETALON [9] which, in conjunction with a standard

laser tracker or the newly developed LaserTracer, is reported to take between 2

to 4 hours to calibrate a medium 3-axis machine. The measurement principle

used by TRAC-CAL is solely based on the use of the laser wavelength and

multilateration to calculate all the individual geometric errors thus increasing the

accuracy beyond that capable using angle and distance normally used by a single

position tracker.

With these systems, it is becoming increasingly efficient to obtain the

detailed error data for the indirect volumetric accuracy determination.

3

Transferable model

Describing the rigid body kinematics using homogenous transformation matrices

has been shown to be a reliable method of calculating the 3D tool point error

using data about the individual error components. It has been commonly applied

for error compensation [4, 6, 7].

Avoiding the use of transformation matrices, a machine specific geometric

model can be created easily by adding the effects of each contributing error by

studying the machine geometry and applying a simple protocol. Each of the

measured 21 errors are considered and the effect determined. Figure 1 shows an

effect Ex resulting from an X axis angular error about a Y axis φy(x) (X pitch)

and movement of the amplifier axis Z.

Data collection for indirect method

Figure 1. Determining error effects

φy(x) – X rotation about Y (X pitch)

X

Z

Εx=φy(x) × Z

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Figure 2 shows a 3-axis model implemented in Microsoft Excel for a machine

with all axes moving the tool (example shown in figure 4). The X, Y and Z axes

are the Bottom (B), Middle (M) and Top (T) axes respectively. This hierarchy is

determined from the way the axes are stacked [7]. Most common machine tools

can be categorised into three distinct configurations based on this hierarchy:

1. All axes move the tool. wBMTt.

2. One axis moves the workpiece, two move the tool. BwMTt.

3. Two axes move the workpiece, one moves the tool. BMwTt.

Where t and w relate to the tool and workpiece respectively.

X error X linear Y in X Z in XX ab Y * ZY ab Y * Z

Xe331.98 -32.1931.3039.975

Figure 2. Geometric model in Excel for wBMTt configuration.

It is important that the effect of the all the errors are added together correctly

and therefore a model and measurement protocol is required. Generally, the

direction of an error is considered a positive error if the axis used to compensate

that error has to move in a negative direction. For linear and straightness errors,

shown by the left most diagram in Figure 3, this is simple. For angular errors,

the direction of an additional amplifier axis needs to be considered. The middle

diagram in figure 3 shows a B axis pitch error (B axis rotation about the T axis),

the effect of which is to produce an error in the B axis direction ε+B with a

magnitude which is a function of the M axis position. The indicated counter-

clockwise error is positive with positive movement of the amplifier. Exceptions

to this rule occur when there are two effects from an angle. An example is

shown in the right most diagram in figure 3. This is a B-axis roll error (B axis

rotation about the B-axis) where an effect occurs in the negative direction. This

requires a subtraction in the model as indicated in figure 2 in the Z error

equation.

Figure 3. Error measurement protocol diagrams

3.1

The final part of the protocol involves measurement offset positions. Due to

Abbé offset, the angular errors have a varying effect on the linear and

Measurement offset positions

X ab Z * Y XZsqr * Z XYsqr * M

61.5 63.35

Xe:

0-231

204.915

Y error

Ye

Y linearX in YZ in YX ab X * Z Y ab X * Z

-2.45

YZsqr * ZYe:

9.2 -38.554 -18.188 1.516558.05

-40.4255

Z error

Ze

Z linearX in ZYin Z

21.126

-X ab X * YZe:

9.2 -39.75921

11.567

T+

M+

+ve

ε+M

ε-T

M+

B+

+ve

ε+B

M+

B+

+ve

-ve

ε+M

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straightness depending on measurement location i.e the offset position of the

perpendicular axes.

B+

M+

T+

Top axis

Bottom axis

Middle

axis

Figure 4. Example machine configuration showing hierarchy

In theory it does not matter where the B axis is positioned for the M and T axis

measurements. In practice the B axis will be positioned at mid travel or at some

convenient location for mounting the optics. It is good practice to keep the B

axis position in the same location for all M and T axis measurements. The B axis

is not an amplifier for any angular error component and the axis offset is 0.

The M axis will probably be positioned at a convenient location for

mounting the optics for B and T axis measurements. The M axis position should

remain the same for all B and T axis measurements. The B axis position error

varies as a function of M axis position (and B axis pitch), also the T axis

position varies as a function of M axis position (and B axis roll) so the M axis

offset is the M position at which the B and T axis linear positioning errors are

measured. The M axis position must obviously remain the same for both linear

positioning measurements.

The T axis will probably be positioned at a convenient location for mounting

the optics for B and M axis measurements. The T axis position should remain

the same for all B and M axis measurements. The B axis position error varies as

a function of T axis position (and B axis yaw and M axis roll), also the M axis

position error varies as a function of T axis position (and B axis roll and M axis

pitch) so the T axis offset is the position at which the B and M axis linear

positioning errors are measured. The T axis position must obviously remain the

same for both linear positioning measurements.

3.2

Most measurement systems can record the data in an ascii file and therefore can

be imported easily into software. In the Excel example, position dependent error

data can be incorporated and referenced appropriately in the model. This very

quickly allows the 3D error to be determined for any position in the working

volume. This can immediately provide useful information for specific process

capability by comparing errors between positions representing simple

component features.

Loading error data

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4

In order to calculate the error for every position in the volume, then it becomes

necessary to use nested loops to efficiently carry out the calculations.

A software package was developed by Postlethwaite et al. [7] in order to

calculate the volumetric error from data obtained of the individual axis error

components. This commercial software uses universal geometric models in

order to simulate the effect of the geometric errors over the machine volume for

any 3-axis machine with at least one axis moving the tool. The largest error in

the specified volume is provided together with a breakdown of how each of the

individual error components contributes to the result. This can be a valuable tool

for determining where the significant contributors are and assignment of

corrective effort.

The calculation of this largest error is dependent on the offset positions of

the measurements due mainly to the fact that the angular errors have no

calculated effect at these positions. It is therefore possible to derive different

values if the measurements are carried out with different offset positions (and

therefore different amplification) or different reference points. The errors in the

volume are with respect to this offset position and therefore do not provide an

indication of error during production if the part or dimension datum’s vary,

which in production they invariably will.

Calculating volumetric accuracy

4.1

By comparing the difference between two vectors in the volume, we find the

errors that would affect production, for example between two features on a part

or from a datum to a feature. By comparing every point in the volume with

every other point we find all possible combinations of moves and the errors. For

a grid resolution of 21targets cubed, there are 9261 points in the volume and

almost 43million comparisons. This number can cause problems with memory

allocation if the results are to be stored for analysis and visualised. The process

can also take several minutes to complete.

A solution has been devised that significantly reduces the number of

comparisons by dismissing vectors in the volume that have a similar direction

but are smaller than some other. As the vectors are created, they are grouped

according to their direction. Two parameters are therefore required. The first is

the angular tolerance which determines whether each new calculated vector fits

an existing group or needs a new group. The second is a magnitude which is a

for comparing the magnitudes of each vector with the largest in a group.

The angular comparison must be sensitive to all 3 directions therefore the X,

Y and Z components of each vector are normalised and subtracted from the

group mean. The sum of the differences is then compared with the tolerance.

Capability assessment

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Figure 5 shows a plot containing all vectors calculated in the working volume of

a large horizontal machine. Each vector is represented by a cone having both

direction and magnitude. A low spatial resolution of 12*8*6 was used to

maintain visual clarity.

Figure 5: Cone plot showing volumetric error

The number of vectors is 819 which requires 334562 comparisons to find the

largest vector difference/ volumetric accuracy of 1306µm. The default

magnitude tolerance is 0.9 which gives the comparisons in Table 1.

Table 1. Reduction in vector comparisons

Parameter

None

0.05

0.1

0.2

Even with the parameter set at 0.2 the calculated volumetric accuracy was

within 1%. The significant 88% reduction in comparisons and therefore

calculation time means that even with high resolution simulations, more a rapid

result. A further consideration is the use of a histogram to show distribution of

the vector differences. The left hand chart in Figure 6 is the distribution without

reduction whereas the right chart is using the angular tolerance of 0.2. Although

the volumetric accuracy was correct, the distribution accuracy has diminished.

Records

819

365

192

118

Comparisons

334562

41472

25992

7200

≈% reduction

88

92

98

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Figure 6. Histogram of volumetric error

Increasing the magnitude tolernce also reduces the calculation time but adversly

affects the histogram. Generally a value between 0.8 and 0.9 gives a good

compromise.

The methodology is extemely robust and reductions from more than 10x106

to a few thousand comparisons can still give the correct volumetric accuracy.

4.2

The volumetric accuracy derived from the largest vector difference gives the

true machine production capability. This figure is often large and it is very

unlikely that two holes, for example, will be drilled at these positions. The

histogram also shows that most of the error in the volume is in the region of

250µm. Simply re-running the simulation with reduced traverse range can give a

capability more suited to relevant component sizes. Considering a reduction

from 12m x 4m x 2m (>100m3) to 6m x 2m x 1m, the volumetric accuracy

reduced from 1306 µm to 380µm.

Reduced volume

0 200 400600

Error (µm)

800 10001200 1400

0

2

4

6

8

10

12

14

16

Frequency (%)

0

200

400

600

Error (µm)

800

1000

1200

1400

0

5

10

15

20

Frequency (%)

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4.3

The error synthesis method also allows easy determination of tool point error for

specific production profiles such as that shown in figure 7. 3D vectors are shown

along the edge of the aerofoil and the X, Y, Z and vector sum errors shown on a

chart. It is usual for only one or two directions to be relevant for a particular

component feature. Tool paths can be approximated or retrived from a part

program as was the case in the example shown. A great deal of care must be

taken to consider all factors that could affect the part accuracy during real

production such as fixture or part datuming using probing etc.

Profile simulation

Figure 7. Tool path profile with error vectors and error chart

5

Combining the effects of the individual error components of a cartesian machine

tool can enable a machine specific model that allows a accurate determination of

the machine volumetric accuracy. In additon a thorough analysis can be made of

the machine capability for production. Great care must be taken to assure sign

convention. For full analysis of the volume a high-resolution spatial grid needs

to be used for comparison of error vectors. Amethod has been devised to reduce

computation overhead by grouping similar error vectors. This has a significant

speed improvement without affecting simulation exactness.

6

References

1. ISO 230-6: Test code for machine tools – Part 6: ‘Diagonal displacement

tests’, 2007.

2. Chapman M, ‘Limitations of Laser Diagonal Measurements’. Precision

Engineering 27, 2003, pp401–406.

3. Soons JA, ‘Analysis of the Step-diagonal Test’. Proceedings of the 7th

Lamdamap Conference, 2005, pp126–137.

4. Okafor A. C., Ertekin Y. M., ‘Derivation of machine tool error models and

error compensation procedure for three axes vertical machining center using

rigid body kinematics’, Int. J. of Machine Tools & Manufacture, 40, 2000,

pp1199–1213.

Conclusions

-150

-100

-50

0

50

100

150

900

1000

1100 1200

1300 1400

1500

1600

1700

Error (microns)

Y axis position

Z err

X err

Y err

Vect

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5. Wilhelm R.G., Srinivasan N., Farabaugh F., ‘Part form errors predicted

from machine tool performance measurements’. Annals of the CIRP, 46, 1,

1997, pp. 471–474.

6. Schwenke, H., Knapp, W. et al. ‘Geometric error measurement and

compensation of machines – an update’, CIRP annals – Manufacturing

Technology, 57, 2, 2008. pp660-675.

7. Postlethwaite, S.R. & Ford, D.G., “A practical system for 5 axis

volumetric compensation” Laser Metrology and Machine Performance IV,

pp 379-388, 1999.

8. Lau K., Ma Q., Chu X., Liu Y., Olson S., ‘An advanced 6-degree-of-

freedom laser system for quick CNC machine and CMM error mapping and

compensation’, Automated Precision, Inc., Gaithersburg, MD 20879, USA.

9. ETALON AG, ‘www.etalon-ag.com’.

10. Uddina M. S., Ibarakia S, Matsubaraa A., Matsushita T., ’Prediction and

compensation of machining geometric errors of five-axis machining

centers with kinematic errors’. Precision Engineering, 33,2009, pp194–

201.

11. ISO 230-3: Test code for machine tools – Part 3: ‘Determination of thermal

effects’, 2001.