Precision Engineering
Journal of the International Societies for Precision Engineering and Nanotechnology
26 (2002) 412–424
A novel technique for calibration of polygon angles with
non-integer subdivision of indexing table
T. Yandayan a,∗ , S.A. Akgöz a , H. Haitjema b
a
The National Metrology Institute of Turkey (UME), TÜBITAK Ulusal Metroloji Enstitüsü, P.K.21, 41470 Gebze-Kocaeli, Turkey
b Precision Engineering Section, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Received 13 June 2001; received in revised form 8 March 2002; accepted 27 March 2002
Abstract
Polygons are basic angle standards for angle measurement, particularly used for calibration of rotary angular indexing, and for measuring
equipment such as dividing heads and tables. A main application in daily life is in bar-code readers. Calibration of such angle standards is
required for traceability and at the highest accuracy it is a responsibility of national metrology institutes. In order to investigate uncertainty
parameters on polygon calibration and to establish the capabilities of national metrology institutes, intercomparision measurements in the
name of EUROMET project 371 “angle calibration on precision polygons” between 12 European countries have been carried out. Two
precise polygons with 7 and 24 faces have been calibrated by the participants. Difficulties arose for precise calibration of seven-sided
polygon for those institutes, which do not have a high-resolution angle comparator or two autocollimators. UME, the National Metrology
Institute of Turkey, has applied an alternative technique for precise calibration of seven-sided polygon without using high-resolution angle
comparators (i.e., indexing tables or angle dividers) or two autocollimators. The technique is based on the circle closure principle. The
pitch and cumulative angles of the polygon are extracted from the angle measurement between some polygon faces (such as one and four
(1/4), analogous 2/5, 3/6, 4/7, 5/1, 6/2 and 7/3) the angle of which can be generated close enough by the indexing table. This means that
the polygon can be regarded as unfolded in seven 3-pitch angle intervals of 3 × 360◦ /7 ≈ 154◦ 17 , making up 1080◦ in total. The method
gives the differences between these seven intervals; with the closure condition (the sum must be zero) this gives all absolute angles. A
full uncertainty evaluation is given that is based on the model function which relates the measured values to the polygon angles. For the
calibration actually carried out, this yielded an uncertainty of 0.24 . Within this uncertainty the measured polygon angles corresponded
very well with the reference values of the intercomparison. The method is of use for laboratories which do not have a high-resolution angle
comparator (i.e., an indexing table or angle divider) or two autocollimators for the calibration of such angle standards.
© 2002 Elsevier Science Inc. All rights reserved.
Keywords: Polygon; Angle standards; Angle measurement
1. Introduction
Apart from dimensions, form and roughness, angle measurements are important in manufacturing. It is required to
use some kind of angular measuring instrument during the
manufacturing of angular parts, e.g., a dividing head, a rotary
table or a polygon. Precise manufacturing is also affected
by angle components. For instance, positional accuracy in a
machine tool can be affected by any small angular motion of
the moving carriage, giving an Abbe offset. The unwanted
angular motion in a measuring machine can also contribute
∗ Corresponding author. Tel.: +90-262-646-6355;
fax: +90-262-646-5914.
E-mail address: [email protected] (T. Yandayan).
to positional errors when probing components. It is, therefore, essential to measure these angular error components,
which are sometimes among the largest errors related to positioning accuracy in a machine tool or measuring machine.
The SI unit of the plane angle (the radian) is, in practice,
established by appropriate subdivision of a full circle. This
enables an error-free standard as the full circle is 360◦ .
However, in the case of practical angle measurement, reference and working angle standards, which are mainly angle
gauge blocks, indexing tables, circular scales and polygons,
are used. The polygons are the most robust and precise angle standards used by laboratories for transferring angular
measurements. They are discs made of steel or glass and
have equally inclined and optically flat reflecting faces.
They may have up to 72 faces but 6, 8, and 12 faces are most
0141-6359/02/$ – see front matter © 2002 Elsevier Science Inc. All rights reserved.
PII: S 0 1 4 1 - 6 3 5 9 ( 0 2 ) 0 0 1 2 3 - X
T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
common. For precise calibration of such angle standards,
two accepted measuring methods are the cross-calibration
and the two-autocollimator technique [1–7].
For the cross-calibration technique the rotary table must
have an angular indexing or measuring capability, whereas
for the two-autocollimator technique this is not necessary.
There are several possibilities for the number of measurements to be taken and their evaluation. Since all these techniques use complete closure (i.e., completing to 360◦ to
eliminate the errors), they are free from bias.
In the case of having one autocollimator and one rotary
table with an angular indexing or measuring capability,
which is the case for most laboratories, an applicable technique is the cross-calibration technique. The calibration
capability for polygon types in this case depends on the
smallest increment of the indexing table. Most common indexing tables (such as MOORE precision indexes [8]) have
a smallest increment of 15 of arc or 10 of arc. This is not
sufficient for polygons such as seven-sided ones as the interval angle cannot be generated by the indexing table accurate
enough to remain in the range of common autocollimators.
In order to calibrate such polygons, a novel technique has
been applied by UME, the National Metrology Institute
of Turkey, during intercomparison measurements carried
out between European countries. In this paper the applied
technique is presented with intercomparison measurement
results and evaluation of the uncertainty components.
2. Angle measurements
2.1. Units of angle
There are two kinds of angle definitions: plane angle and
solid angle [9]. Whilst plane angle is the ratio of two lengths,
413
solid angle is the ratio of an area to the square of a length.
Solid angle is used in theoretical calculations and is not
treated in this paper. Plane angle is generally expressed using
one of the two systems.
2.1.1. Sexagesimal system
This system dates from the Babylonians. The (◦ )-sign is
a hieroglyph of the sun, and is still in daily use, also for
engineering applications. The angle units in this system are
as follows:
1◦ = 1/360 of a full circle
1 = 1/60 of a degree
1 = 1/60 of a minute
2.1.2. Radian system (the SI unit of the plane angle)
This system is used generally in mathematics and for more
theoretical applications (as SI unit: 1 m/m), 1 rad is the angle
subtended at the center of a circle by an arc of length equal
to the radius. 2␲ rad equals 360◦ .
2.2. Angle calibration and generation
2.2.1. Subdivision of a circle
Suppose that one full circle is divided in to seven nominal
angle intervals, A, B, C, D, E, F, G and each has small
error such as a, b, c, d, e, f, g (Fig. 1(a)). Although each
of these errors may be different, the circle closure demands
that the sum of all angles be 360◦ , from this it follows that
the sum of errors must be zero, i.e., the average deviation
is zero. These seven unknown errors can be determined by
comparing eight intervals with angle interval of α i having
small error α i on the other circle in turn and measuring
Fig. 1. Explanation of the complete closure principle. (a) Circle 1. (b) Circle 2.
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T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
the seven differences x1 , x2 , x3 , x4 , x5 , x6 , x7 as shown in
Fig. 1(b).
Seven equations are obtained:
αi −a = x1 ,
αi −b = x2 ,
αi −c = x3 ,
αi −d = x4 ,
αi −e = x5 ,
αi −f = x6 ,
αi − g = x7
(1)
Using the fact that the sum of seven intervals on the same
circle is a complete circle (Fig. 1(a)), and thus the sum of
the seven errors must be zero, we can write:
a+b+c+d +e+f +g =0
(2)
summing Eq. (1) gives,
7αi − (a + b + c + d + e + f + g)
= x 1 + x 2 + · · · + x7
(3)
since a + b + c + d + e + f + g = 0, α i is given by
αi =
x1 + x2 + · · · + x7
7
(4)
with this, all deviations are known:
a = αi − x1
b = αi − x2
..
.
(5)
g = αi − x7
This is called the complete closure principle and this can be
applied to angle calibration standards.
2.2.2. Trigonometrical method
It is also possible to calibrate and generate an angle by
using the sine or tangent principle. Here the angle is calculated from two known lengths. However, it is only recommended for angles up to 45◦ . It is considered a very precise
method for small angles particularly less than 1◦ .
In the two autocollimator technique, α i angles as described in Section 2.2.1. are obtained by fixing the angle
between two autocollimators, which are positioned around
the circle, at a nominal angle of the polygon. Polygon intervals are compared with these angles and the results are
calculated with the aid of the complete closure principle. In
an extended version of this method, the autocollimators are
positioned at all angles of the polygon and the results for
each full circle are averaged.
In the cross-calibration technique, α i angles are obtained
from an indexing table. The important issue here is the repeatability of the indexing table. Polygon intervals are compared with indexing table’s intervals by autocollimator measurements. The polygon is positioned at all angle intervals
α i on the indexing table, and by combining these measurements the errors of both the polygon and the indexing table
are calculated using the complete closure principle.
Apart from these two methods, direct comparison, which
is the direct comparison of polygon intervals against indexing table intervals, can also be applied. However, it requires
pre-calibration of either the polygon or the indexing table
by one of the above mentioned methods.
In all methods, comparison of angle intervals is made by
optically probing the polygon faces using an autocollimator.
Fig. 2 illustrates the polygon calibration and angle intervals,
which must be determined [5]. The pitch angles of the polygon, α i are the angles between adjacent normals of the n
measuring faces. The pitch angle deviations α i are the deviations values of the pitch angles from the nominal angle
value αn = 360◦ /n. The angles between the normal N1 of the
first face and the other faces are the cumulative pitch angles
β i . The cumulative angle deviations β i are the deviations
of the cumulative pitch angles from their nominal values.
Determination of either α i or β i will be sufficient since
they are related to each other as shown in the equation given
in Fig. 2. As already stated, the closure principle demands
that the sum of all pitch angle deviations be zero.
2.3. Calibration of angle standards
Indexing tables, angle gauge blocks and polygons can be
considered as the most precise angle standards. Application
of the complete closure principle in the calibration of indexing tables and polygons produces a bias-free result. The
repeatability of measurements is essential for the final uncertainty which can be obtained, together with the traceability
of the measured deviations. Also the quality of the indexing
mechanism for tables and the flatness of the reflecting faces
of the polygon are of importance for such calibration process. Considering transfer, stability and use of the standards,
the polygons are the most suitable angle standards.
There are two techniques commonly available for polygon
calibration: cross-calibration and two-autocollimator technique [1–7] as mentioned before. They both use the complete closure principle for the calibration.
3. European Collaboration in Metrology
(EUROMET) project 371
A main objective of this project was to provide information about the uncertainty parameters of measurement
for the calibration of precision polygons according to the
document “ISO guide to the expression of uncertainty in
measurement” [10]. For this reason, it was recommended to
measure the circulated polygons in their inverted as well as
in their normal position.
Later, it was aimed by Consultative Committee for Length
(CCL) in Bureau International des Poids et Measures
(BIPM) to establish and confirm the capability of National
Measurement Institutes (NMIs) for Mutual Recognition Arrangement (MRA), carrying out ‘key comparisons’ which
establish measurement capabilities in essential (‘key’) fields
T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
415
Fig. 2. Definitions for polygon calibration.
and techniques [11]. EUROMET has declared the EUROMET project 371 as one of these “key comparison” with
the identifier EUROMET.L-K3.PREV in the BIPM key
comparison database.
The project covered an international comparison of angle
calibrations to be carried out on two precision polygons
with 7 and 24 faces, respectively. It was completed between
1996–2000. PTB (National Metrology Institute of Germany)
was the pilot laboratory.
The participating countries (and institutions) were:
Germany (PTB), Switzerland (OFMET), The Netherlands
(NMi), Finland (VTT), UK (NPL), Italy (IMGC),
Slovakia (SMU), France (LNE), Spain (CEM), Poland
(GUM), Turkey (UME), Czech Republic (CMI).The used
seven-sided polygon has the following specifications:
Pitch angle
Mirror size
Diameter
Material
Weight (including case)
Manufacturer
Identification number
360◦ /7 = 51◦ 25 42.857. . . 15 mm in diameter (limited
by the aperture of the
case)
60 mm
Glass
495 g
Rank Taylor Hobson
SP LE 5997
The following measurement results were reported for the
normal and inverted position of the polygon:
• The pitch angle deviations α i .
• The cumulative angle deviations β i .
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T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
Either one of two well-known method, one autocollimatorangle measuring tables or two autocollimators-rotary table
were recommended by the Pilot Lab. (PTB) for measurement of these parameters [5]. A mounting device delivered
with the polygon has been used for adjustments.
4. Alternative approach for calibration
of seven-sided polygon
The following equipment was available at the laboratory
of UME for calibration of the polygon:
Autocollimator
Type-model
Manufacturer
Range
Resolution
Accuracy
Serial no.
Möller-Wedel Elcomat HR 2-axis
Möller-Wedel Optical GmbH
±150 arc seconds
0.005 arc seconds
0.02 arc seconds with uncertainty of
0.03 arc seconds.
161
Indexing table
Type-model
Manufacturer
Resolution
Reproducibility
Serial no.
Moore 1440 precision indexing
with mechanical lift
Moore Tool Company
15
±0.1 arc second
281
It was impossible to apply the cross-calibration or the direct comparison method to the seven-sided polygon for UME
using the existing equipment. This is because the smallest
angle which can be obtained by a Moore precision index
with 1440 teeth is 0.25◦ (i.e., 15 or 900 arc seconds) and
does not allow to probe all seven polygon faces directly
within the autocollimator range. However, it is possible to
probe some faces such as face one and four (1/4), analogous
2/5, 3/6, 4/7, 5/1, 6/2 and 7/3. This is achieved by shifting
the autocollimator X-axis to accommodate both faces in the
autocollimator range (±150 arc seconds).
The pitch angles between the polygon faces were compared with the indexing table interval by rotating the indexing table between 0 and 154◦ 15 (always these two nominal
angular positions only) and measuring the rest of the angle
difference by the autocollimator. No intermediate or subsidiary table was used. The polygon setting was performed
by manually positioning the polygon faces on its mounting
device (Fig. 3) to the autocollimator axis. For each probing
of the faces, the following setting and leveling procedure
was carried out.
4.1. Preliminary adjustments
The recommended mounting device delivered with the
polygon (by the Pilot Laboratory, PTB) has been used in order to facilitate the polygon adjustments and to exclude measurements which may occur due to using different mounting
devices by the NMI’s. The mounting device is positioned
on the indexing table and adjusted by probing the mounting
shaft with a precision indicator as shown in Fig. 3. It was
aimed to adjust the mounting device to the axis of rotation of
the table. The eccentricity of the mounting shaft to the axis of
rotation of the table was about ±30 ␮m. After the base plate
of the mounting device is clamped tightly to the measuring
Fig. 3. Preliminary adjustments for the calibration of polygon.
T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
417
table, the polygon is placed on the mounting shaft and manually fastened by slightly tightening the securing screw.
The autocollimator has been adjusted as close as possible with its optical axis perpendicular and in true alignment
to the indexing table’s axis of rotation. It was also aligned
to probe the center of the polygon faces. Then the polygon
is adjusted using the adjusting screws of the mounting device in a plane perpendicular to the indexing table’s axis of
rotation so that pyramid errors of all measuring faces are
at least within ±10 arc seconds as recommended in the instructions [5]. This was achieved by probing the faces of the
polygon and recording the Y-axis values from the autocollimator display. Adjustment using the screws has been carried
out until the readings became within ±6 arc second. Later,
the optical path length between autocollimator and polygon,
which was kept as small as possible was shielded against air
turbulence and thermal effects with the aid of prolongation
tube made of paper. Finally, the polygon faces were measured in normal and inverted position using our new method.
The measurement conditions are as follows:
Laboratory conditions
Number of repeated
measurements
Number of polygon positions
relative to indexing table
in the seven intervals of
the polygon
(20 ± 0.5)◦ C,
50% humidity
10 (in each position
of the polygon)
7
4.2. Measurement procedure
The nominal pitch angle α n is 360/7 = 51◦ 25 42.857. . . .
The Moore table is rotated over nominal 51◦ 15 with
a very high reproducibility. The rest of the angle,
about 10 42.857. . . (642,8 arc seconds) cannot be measured by the Elcomat HR autocollimator as its range is
±150 arc second. However, if the faces 1/4, 2/5, 3/6, 4/7,
5/1, 6/2 and 7/3 are probed (i.e., about 154◦ 17 08.57 ),
154◦ 15 can be generated by the indexing table and the
nominal difference 2 08.75 = 128.57 can be measured
by the Elcomat HR autocollimator. This was the measurement approach of UME which enabled the calibration of
the seven-sided polygon (Fig. 4).
Fig. 4. Explanation of the novel technique for calibration of seven-sided
polygon.
The polygon faces were set such that the nominal autocollimator reading would be half of 128 which is ±64 .
Thus, where the autocollimator is used at similar nominal
readings all the time where in the end only differences are
relevant, the calibration errors of the autocollimator in the
range of 128,57 are minimized. After the adjustments and
leveling were performed, the readings for faces were taken
as given in Table 1.
Fig. 5 illustrates the measurements on the first pair of
faces 1/4. The same procedure follows with the pairs 2/5,
3/6, 4/7, 5/1, 6/2 and 7/3 after rotation (shift) of the polygon on the indexing table face-by-face. Thus, the used angle interval 154◦ 15 of the table is the same for all pairs
of faces and the indexing table error is eliminated (see
Eqs. (6)–(14)).
For each face pair measurement, the deviation from nominal difference 128.57 was calculated. As each face of the
polygon is probed twice and the circle is closed to complete
360◦ in this method, the average of the deviations shown in
Table 1
The readings taken from the autocollimator (in arc second)
Face 1
Face 4
Difference
Mean of differences
Standard deviation of difference values
over 10 measurements
64.120
64.110
–
–
64.125
−64.760
−64.760
–
–
−64.705
128.880
128.870
–
–
128.830
128.857
0.06
Deviation from nominal value = 128.857 − 128.571408 = 0.286
418
T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
Fig. 5. Measurement of a seven-sided polygon with the novel technique. (a) First measurement: Face 1. (b) Second measurement: Face 4.
Table 2 is the error of indexing table for that certain interval.
Mathematically this is derived as follows.
The 3-pitch angle intervals γ i (shown in Fig. 4) are measured by readings Mi of the autocollimator difference:
γi = Mi + θ
(6)
where θ is the unknown, but constant (can vary in repeated
operations due to random errors of the indexing table and
the manipulation) angle between the two positions of the
indexing table of 154◦ 15 nominal. Summation over all angles and
γi = 1080◦ (closure condition 3 × 360◦ ) gives
from Eq. (6)
7
1080◦
j =1 Mj
−
(7)
θ=
7
7
This is replaced in Eq. (6) and gives:
7
1080◦
j =1 Mj
γ i = Mi −
+
7
7
Finally, the deviations of the cumulative pitch angles β i
between the faces were calculated using the γ i values. Angle
1/4 (=β 4 ) was directly taken as γ 1 and the other cumulative
angles were calculated as following (Fig. 6):
β1 = γ1 + γ2 + γ3 + γ4 + γ5 + γ6 + γ7 − 1080◦
= 0◦ (trivial case)
(8)
β2 = γ1 + γ2 + γ3 + γ4 + γ5 − 720◦
β3 = γ1 + γ2 + γ3 − 360◦
(9)
(10)
Table 2
Processing of the readings taken from the autocollimator (in arc second)
Mi
Faces
Deviation
Deviation − average (Mi −
M1
M6
M4
M2
M7
M5
M3
1/4
2/5
3/6
4/7
5/1
6/2
7/3
0.286
0.805
0.216
−0.421
0.371
−1.455
−0.388
0.370
0.888
0.300
−0.337
0.455
−1.372
−0.304
Seven times indexing table error
−0.587 (sum)
Indexing table error
−0.084 (average)
0.000 (sum)
7
j =1 Mj /7)
γi
γ1
γ6
γ4
γ2
γ7
γ5
γ3
T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
419
Fig. 6. Determination of cumulative pitch angles β i .
β 4 = γ1
(11)
β5 = γ1 + γ2 + γ3 + γ4 + γ5 + γ6 − 720◦
(12)
β6 = γ1 + γ2 + γ3 + γ4 − 360◦
(13)
β7 = γ1 + γ2
(14)
The pitch angles α i and their deviations can easily be
calculated using the cumulative pitch angles β i and their
deviations (Fig. 2).
Another way of describing this method is as follows: The
polygon is regarded as unfolded in seven 3-pitch angle intervals of 3 × 360◦ /7 ≈ 154◦ 17 , making up 1080◦ in total.
The method gives the differences between these seven intervals, and with the closure condition (the sum must be zero)
this gives all absolute angles.
4.3. Evaluation of uncertainty
Earlier, uncertainties of angle calibration using circle
closure were analyzed by Tyler Estler [12]. The uncertainty
of the measurement has been calculated according to “ISO
Guide to the expression of uncertainty in measurement
(GUM)” [10]. The ISO Guide states: “The uncertainty in the
results of a measurement generally consists of several components which may be grouped into two categories according to the way their numerical value is estimated; Type A:
those which are evaluated by statistical methods; Type B:
those which are evaluated by other means”. It also states
that “imported values” or single results from equipment or
techniques, which have had prior evaluations of uncertainty,
should be treated as Type B uncertainty contributions.
4.3.1. Uncertainty budget
For setting up an uncertainty budget according to the
GUM, first the model function is needed which relates measurements Mi to the measurements γ i and β i . In the general case, taking n as the number of polygon faces, and ρ
as the number of pitch angle intervals measured in once,
Eq. (7), describing the special case n = 7 and ρ = 3 can be
generalized as:
n
ρ × 360◦
j =1 Mj
γi = Mi −
+
n
n
M
1
ρ × 360◦
j =i j
= Mi 1 −
−
+
(15)
n
n
n
assuming uncorrelated measurements Mi with standard uncertainty uM , it follows for the standard uncertainty in γ i :
n ∂γi 2 2
2
(16)
uM
uγ =
∂Mj
j =1
n
j =1 (∂γi /∂Mj )
2
can be calculated as follows.
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T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
Mi is one of the Mj starting from j = 1 to j = n. With,
1
∂γi
=− ,
∂M1
n
1
∂γi
=1− ,
∂Mi
n
=
1
∂γi
=− ,
∂M2
n
=
−


(n−1) times
2
+
(n − 1)
n2
1
u2M = 1 −
u2M
n
(17)
This shows that the uncertainty uγ in the 3-pitch angle
intervals γ (n = 7) are slightly less than the uncertainties
uM in the measurements M. However, when summing up
these angles to obtain the cumulative pitch angles β i , the
uncertainties uγ can not just be added quadratically as the
quantities γ i are correlated. To relate the cumulative pitch
angles to the uncorrelated measurements, the model function for a summation of angles γ i is written according to
Eqs. (8)–(14) (disregarding the constants 360◦ , 720◦ and
1080◦ ) generally up to ρ × 360◦ as:
γ i = γ1 + γ 2 + · · · + γ m
i=1
(in Eqs. (8) − (14), m = 1 . . . 7)
(18)
considering
γ i = Mi −
n
Mj
j =1
n
β=
m mρ × 360◦
= 1−
Mj
n
n
+
ρ × 360◦
n
n
m mρ × 360◦
Mj +
n
n
(19)
j =1
n
j =1 (∂β/∂Mj )
2
can be calculated using:
m
∂
∂β
1−
=
(M1 + M2 + · · · + Mm )
∂Mj
∂Mj
n
m
(Mm+1 + Mm+2 + · · · + Mn )
n
mρ × 360◦
+
n
−
∂β
m
m
∂β
∂β
= 1−
= 1−
,
,...,
∂M1
n
∂M2
n
∂Mm
m
= 1−
n
∂β
m
∂β
m
m
∂β
=− ,
= − ,...,
=−
∂Mm+1
n ∂Mm+2
n
∂Mn
n
Some book-keeping on the appearance of these terms
yields:
from Eq. (15),
j = 1 to m → m times,
j = m + 1 to n → (n − m) times
β can be calculated as follows:
m
j =m+1
For the uncertainty in β the partial derivatives to the measurements Mi must be taken:
n ∂β 2 2
uM
(20)
u2β =
∂Mj
Now
m
j =1
j =m+1

1 2
1 2
1 2
1 2

 2
= 1−
+ −
+ −
+ ··· + −
u

n
n
n
n  M
1
n
m
n
m
m Mj −
Mj
n
n
j =1
j =1
1−
Mj −
m
n ∂γi 2 2
=
uM
∂Mj
=
m
mρ × 360◦
Mj +
n
n
j =1
j =1
we find that
β =
m
+
n
Mj −
j =1
∂γi
∂γi
1
1
= − ,...,
=−
∂M3
n
∂Mn
n
uγ 2
m
thus we obtain:
n ∂β 2 2
uM
∂Mj
j =1


m 2 m 2
m 2
=
1
−
+
1
−
+
·
·
·
+
1
−

n
n
n u2β =
γi
i=1
(M1 + M2 + · · · + Mn ) ρ × 360◦
= M1 −
+
n
n
(M1 + M2 + · · · + Mn ) ρ × 360◦
+
+ M2 −
n
n
+ · · · + Mm −
(M1 +M2 + · · · +Mn )
+
n
ρ×360◦
n
m times

m 2 m 2
m 2 
 u2
+ −
+ −
+ ··· + −
 M
n
n
n
n−m times
T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
421
Fig. 7. The characteristic graph showing the uncertainties of cumulative angles uβ against the face number m.
m 2
m2
= 1−
m + 2 (n − m) u2M
n
n
m2
= m−
u2M
n
(21)
so for n = 7 and m = 1, . . . , 7 this gives uncertainties of
respectively 0.92, 1.20, 1.31, 1.31, 1.20, 0.92 and 0 times
the uncertainties in measurements uM . This means that the
uncertainty values for each β value will be different. The
characteristic graph given in Fig. 7 explains this. It shows
that uβ is maximum for m = n/2 in the case of largest
separation between two faces, m = 0 and m = n are the
trivial cases referring to differences between the same face
(no. 1).
As can be clearly seen in Table 2, the M values are used to
calculate the corresponding γ values using Eq. (15) and then
Eq. (19) is used to calculate the β values (e.g., Eqs. (8)–(14)).
Uncertainty values for each β value can be calculated using
Eq. (21). In Table 3 the uncertainty values for each β value
are also given.
Note that the uncertainty in the pitch angles uα = 1.20 ×
uM . It is apparent that m = 5 and m = 2 are valid for β 2 and
β 7 (see Eqs. (9) and (14)), respectively. uα can be calculated
as u2α = 1.43 × uM and uα ≈ 1.20 × uM using Eq. (21). As
β2 = α2 , (β7 = 360◦ −α 1 ), β 2 and β 7 are in fact pitch angles
(Fig. 6) and their uncertainty values are equal to each other
as shown in the Table 3. It can be concluded that Eq. (21)
is generally valid for the variance of the accumulated angle
in a divided circle.
Below we list the contributions to the uncertainty in the
pitch angle uα .
4.3.1.1. Type A. Repeatability: The standard deviation of
readings over 10 measurement results is 0.1 , so uMi =
0.1 . This means that the standard deviation
of the mean of
√
this value M =
Mi /n is 0.1 / 10 = 0.03 . In α the
standard uncertainty will be 1.2 × 0.03 ≈ 0.04 and the
degrees of the freedom is 10 − 1 = 9.
4.3.1.2. Type B.
1. Autocollimator errors (specified in its calibration certificate): The maximum deviation is 0.02 according to
the calibration certificate. Considering that it is rectangular distribution the degrees of the freedom is νi →
∞ and the standard uncertainty uAC in α is:
uAC = 1.2 × 0.02 /31/2 = 0.02
2. Uncertainty of the autocollimator error: The uncertainty associated with the calibration result is declared
as 0.03 in the calibration certificate with coverage factor k = 2 (95% confidence level). Considering that it
is normal distribution the degrees of the freedom ν i is
400 and the standard uncertainty uCE in α is:
uCE =
1.2 × 0.03
= 0.02
2
3. Contribution from polygon pyramidal errors: The reflected faces of the polygon are in practice not in the
square position in relation to the measuring plane by
small tilts referred as pyramid errors. In that case, the
measuring plane is the plane in which the sum of the
squares of the pyramid errors of all measuring faces is
Table 3
The uncertainty values for each β value
Mi
γi
βi
1
2
3
4
5
6
7
γ1
γ2
γ3
γ4
γ5
γ6
γ7
β4
β7
β3
β6
β2
β5
β1
Note: β1 = 0◦ is the trivial case, which explains that uβ = 0.
uβ
= γ1
= γ 1+ γ 2
= γ1 + γ2
= γ1 + γ2
= γ1 + γ2
= γ1 + γ2
= γ1 + γ2
+ γ3
+ γ3
+ γ3
+ γ3
+ γ3
− 360◦
+ γ 4 − 360◦
+ γ 4 + γ 5 − 720◦
+ γ 4 + γ 5 + γ 6 − 720◦
+ γ 4 + γ 5 + γ 6 + γ 7 −1080◦
0.92
1.20
1.31
1.31
1.20
0.92
0.00
×
×
×
×
×
×
×
uM
uM
uM
uM
uM
uM
uM
422
T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
a minimum. Due to 10 arc seconds pyramidal error, the
cross-talk between X- and Y-axes is experimentally verified as 0.1 arc second. This is also taken as rectangular
distribution which makes the standard uncertainty upy
due to this error in α:
upy =
1.2 × 0.1
= 0.07
31/2
and the degrees of the freedom ν i is 100.
4. Resolution of the autocollimator: The resolution of the
autocollimator is 0.005 . Considering that it is rectangular distribution this gives a standard uncertainty ur
in α of:
ur =
1.2 × 0.005
= 0.004
31/2
and the degrees of the freedom is νi → ∞.
5. Estimated uncertainty introduced by non-centrality of
beam on the polygon face: This is about 0.05 . Considering that it is rectangular distribution this gives for
the standard uncertainty uC in α:
uC =
1.2 × 0.05
= 0.04
31/2
and the degrees of the freedom is νi → ∞.
6. Difference between the normal and inverted polygon
position: The standard deviation of differences between
the normal and inverted position for α is measured as
uni = 0.05 and the degrees of the freedom is νi =
(7 − 1) = 6.
4.3.2. Combined and expanded uncertainty
The combined standard uncertainty, being the root sum
square of the all uncertainty contributions, is given by:
Uc(α) = [(0.04 )2 + (0.02 )2 + (0.02 )2 + (0.07 )2
+ (0.004 )2 +(0.04 )2 + (0.05 )2 ]1/2
= 0.11 (k = 1)
The effective degrees of freedom ν eff of the combined standard uncertainty from the Welch−Satterthwaite [10] is given
by
νeff
(0.11 )4
=
= 93
[((0.04 )4 /9) + ((0.02 )4 /∞)
+ ((0.02 )4 /400) + ((0.07 )4 /100)
+ ((0.004 )4 /∞) + ((0.04 )4 /∞)
+ ((0.05 )4 /6)]
Then the appropriate coverage factor can be taken as k =
2 from the t-distribution for the above value of effective
degrees of freedom. If the combined standard uncertainty is
expressed for confidence level of 95% with, the expanded
uncertainty in α is given by:
Ue(α) = 0.22 (k = 2)
For the expanded uncertainty in β we take as the maximum
uncertainty
1.31
Uα = 0.24
Ue(β) =
1.2
5. Results and discussions
The cumulative angle deviations of the seven-sided polygon reported by UME are given in Table 4 for normal and
inverted position of the polygon. As the cumulative angle
deviations relate to the datum face (Face 1), any bias error
in the measurement of this face enters as a constant into all
values of β i . In order to eliminate this dependence on the
error of a single face, the Pilot Laboratory (PTB) preferred
to relate the values of β i to the average of all faces. These
values are called reduced angle deviations βi r [13].
βi r = βi −
n
1
βk
n
(22)
k=1
where i = 1, . . . , n: the face number
The above definition states that ni=l βi r = 0 and the
deviation from the nominal angle between any two faces is
equal to the difference between their reduced angle deviations. The advantage of using the reduced angle deviation is
that the value relates a single face to all others while the cumulative angle deviation relates to the first face only. Thus,
the results of different laboratories can be evaluated less dependent on a single deviating result. Calculating β i in this
way would change, and further complicate Eqs. (19) and
(21), probably giving a slightly smaller value. As the differences will be small and because in use of the polygon the
results are usually referred to the first polygon face, we consider our used equations and uncertainties as most useful.
Table 5 illustrates the UME reduced angle deviations of
the seven-sided polygon, the weighted mean as a reference value of the EUROMET intercomparison for each face,
and the difference of the reduced angle deviations from the
weighted mean of the faces for normal position. Table 6
Table 4
Cumulative angle deviations of the seven-sided polygon reported by UME
(in arc second)
Polygon in normal position
Polygon in inverted position
Faces
Cumulative angle
deviation
Faces
Cumulative angle
deviation
1–1
1–2
1–3
1–4
1–5
1–6
1–7
1–1
β 1 = 0
−1.343
−0.271
0.370
−0.455
0.029
0.033
β 1 = 0
1–1
1–2
1–3
1–4
1–5
1–6
1–7
1–1
β 1 = 0
−1.302
−0.347
0.331
−0.395
0.092
0.007
β 1 = 0
T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
423
Table 5
Comparison of UME results with reference values of EUROMET intercomparison for normal position of polygon (in arc second)
Face number
Reference values
(weighted mean)
UME results
Difference of UME results
from the reference values
1
2
3
4
5
6
7
0.23
−1.08
−0.11
0.52
−0.19
0.38
0.25
0.23
−1.11
−0.04
0.60
−0.22
0.26
0.27
0.00
−0.03
0.07
0.08
−0.03
−0.12
0.02
Table 6
Comparison of UME results with reference values of EUROMET intercomparison for inverted position of polygon (in arc second)
Face number
Reference values
(weighted mean)
UME results
Difference of UME results
from the reference values
1
2
3
4
5
6
7
0.21
−1.08
−0.08
0.52
−0.19
0.37
0.24
0.23
−1.07
−0.12
0.56
−0.16
0.32
0.24
0.02
0.01
−0.04
0.04
0.03
−0.05
0.00
Table 7
Comparison of UME results with reference values of EUROMET intercomparison for mean of normal and inverted position of polygon (in arc second)
Face number
Reference values
(weighted mean)
UME results
Difference of UME results
from the reference values
1
2
3
4
5
6
7
0.22
−1.09
−0.09
0.52
−0.19
0.37
0.25
0.23
−1.09
−0.08
0.58
−0.19
0.29
0.25
0.01
0.00
−0.01
0.06
0.00
−0.08
0.00
Fig. 8. Comparison of EUROMET 371 reference values and UME results in the uncertainty limits.
424
T. Yandayan et al. / Precision Engineering 26 (2002) 412–424
shows the similar values for inverted position and Table 7
shows for mean of normal and inverted position [13]. It
should be noted that the weighted mean is calculated by the
Pilot Laboratory (PTB) using participant’s results and their
associated standard uncertainties. It is accepted as the reference value.
Participant’s results are checked against the reference values with their associated uncertainties. The closeness between the UME results and the reference values are evaluated considering the uncertainties. Fig. 8 shows a graph
plotted using the values in Table 7. It can be clearly seen
that the results agree with the reference values far within
the uncertainty limit; expressed as a standard deviation, the
difference is only 0.04 .
The difference between normal and inverted position is
quite small. The standard deviation of this difference among
seven faces is 0.05 .
All participants apart from UME have used one of
two known standard measuring methods, either the
cross-calibration or the two-autocollimator technique. The
reference values are calculated using all results. The agreement between UME results and reference values apparently
states that the new method applied by UME proves to be
working well for such kind of applications. It should also
be noted that the uncertainty of the applied method is also
reasonably small and is the fifth smallest uncertainty among
the 12 participants.
6. Conclusions
1. The new technique described in this paper for calibration of polygon angles with non-integer subdivision
of an indexing table can be used by those having no
high-resolution indexing tables/rotary tables, or wishing not to use any sub-indexing equipment.
2. Intercomparison results of UME with the new technique among the 12 European countries shows that the
technique works very well. It is seen from the results
that the measured polygon angles reported by UME
corresponded very well with the reference values of
EUROMET intercomparison and, thus, the validity of
the method is proved.
3. The new technique has given an uncertainty of about
0.24 . This is a favorable value both in an absolute
sense as well as when compared to the given participant
uncertainties in the intercomparison.
Acknowledgments
The authors thank Dr. Reinhard Probst of PTB for his
comments and support also for permitting publication of
intercomparison results in the paper. Also Nuray Karaböce
and Bülent Delibas are acknowledged for their kind help
and support.
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