University of Huddersfield Repository
Fletcher, Simon, Longstaff, Andrew P. and Myers, Alan
Defining and Computing Machine Tool Accuracy
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
This version is available at http://eprints.hud.ac.uk/5554/
The University Repository is a digital collection of the research output of the
University, available on Open Access. Copyright and Moral Rights for the items
on this site are retained by the individual author and/or other copyright owners.
Users may access full items free of charge; copies of full text items generally
can be reproduced, displayed or performed and given to third parties in any
format or medium for personal research or study, educational or not-for-profit
purposes without prior permission or charge, provided:
The authors, title and full bibliographic details is credited in any copy;
A hyperlink and/or URL is included for the original metadata page; and
The content is not changed in any way.
For more information, including our policy and submission procedure, please
contact the Repository Team at: E.email@example.com.
Defining and computing machine tool accuracy
S Fletcher, A Longstaff, A Myers
Centre for Precision Technologies, University of Huddersfield – UK
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 . 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.
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  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 . 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
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  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  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
With these systems, it is becoming increasingly efficient to obtain the
detailed error data for the indirect volumetric accuracy determination.
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)
Εx=φy(x) × Z
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 . 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
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
Figure 3. Error measurement protocol diagrams
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
Y linearX in YZ in YX ab X * Z Y ab X * Z
YZsqr * ZYe:
9.2 -38.554 -18.188 1.516558.05
Z linearX in ZYin Z
-X ab X * YZe:
straightness depending on measurement location i.e the offset position of the
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
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.
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
Loading error data
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.  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
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
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.
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
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.
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
The methodology is extemely robust and reductions from more than 10x106
to a few thousand comparisons can still give the correct volumetric accuracy.
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.
0 200 400600
800 10001200 1400
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.
Figure 7. Tool path profile with error vectors and error chart
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.
1. ISO 230-6: Test code for machine tools – Part 6: ‘Diagonal displacement
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,
Y axis position
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–
11. ISO 230-3: Test code for machine tools – Part 3: ‘Determination of thermal