M APAN-Journal of Metrology Society of India (March 2016) 31(1):31–41
DOI 10.1007/s12647-015-0157-x
O R I G I N A L P A P E R
M. Arif Sanjid* and K. P. Chaudhary
Standards of Dimension Division, CSIR- National Physical Laboratory, Dr K S Krishnan Marg, New Delhi
110012, India
Received: 10 June 2015 / Accepted: 21 October 2015 / Published online: 30 November 2015
Metrology Society of India 2015
Abstract: In the field of dimensional metrology, angle blocks are calibrated using a precision indexed rotary table in conjunction with an autocollimator. Only few national metrology institutes (NMI) of developed counties has sophisticated automatic setup of indexed table, autocollimator. NMIs use individual analytical model for angle block calibration. Based on the analytical model, NMIs has exclusively written software to generate calibration results. Some NMIs of developing economics manually operate indexed rotary table, autocollimator for angle calibration. The readings of indexed table, autocollimator are recorded manually. The results are calculated manually or on excel sheets. Frequently, operator commits mistakes in determining the sign of angular deviation. At National Physical Laboratory-India, a generic analytical mathematical model is devised for angle block calibration. Software entitled ‘‘Calibration of angle gauge’’ is developed.
The software takes readings of indexed table, autocollimator to calculate the angular deviation of angle block. Instead of direct coding the generic analytical mathematic model, the software is written with logical basis of the generic analytical mathematic model. A set of measurement readings of angle blocks are used as reference data to validate the software and analytical model.
Keywords: Angle blocks; Software; Validation; ISO 17025; Calibration
1. Introduction
An angle block (AG) is hardened steel block having its two lapped working faces (measuring faces) inclined at a nominal angle in degree of arc, minute of arc and second of arc
denominations [ 1 ]. Angle blocks are usually supplied in sets
containing a series of angle blocks having different nominal
angles [ 2 ]. Precision indexed table is a small rotary table,
which is more accurate for generating precise angles over a full circle. One such table, the Moore’s precision indexed rotary table (MT), has 1440 radial serrations on each of its upper and lower eccentrically coupled circular plates (these are engaged by a piston cylinder slide fit assembly) and are indexed through multiple of  of arc with a total of angular deviation from its indexed position by \ 0.2 arc-second [
3
,
4 ].
Usually, indexed rotary table is calibrated to determine its angular deviations of each indexed position. An autocollimator is an optical instrument, which is used to measure small angles with very high sensitivity. The autocollimator
*Corresponding author, E-mail: sanjid@nplindia.org
projects a beam of collimated light. An external reflector reflects all or part of the beam back into the instrument where the beam is focused to a photo-detector. The autocollimator measures the geometric plane angle between the emitted beam and the reflected beam [
5
,
6
].
Periodically, a dual axis electronic autocollimator
(ELCOMAT 2000) of NPLI is compared against ‘‘Angle
Comparator WMT 220’’ at Physikalisch Technische Bundesanstalt
, Germany for traceability to SI unit [ 7 – 9 ]. MT is
calibrated using electronic autocollimator applying closure method/sub division method [
10 – 12
]. At NPLI, MT is manually operated in conjunction with an autocollimator, to calibrate angle blocks. Software entitled ‘‘Calibration of
Angle Gauges’’ calculates angular deviation of angle block by narrating each step of calculation. Software is written in visual basic version 6. Graphical user interface of software has fields to enter measurement readings.
A generic mathematical model is devised on basic principle of calculations of angular deviation of angle block. The proposed software is validated using a reference data set of measurements. Finally, a pseudo code of software along with explanation is given in annexure.
123

32 M. Arif Sanjid, K. P. Chaudhary
2. Method of Measurements
An angle block of nominal size 9 of arc is calibrated using
direct comparison method [ 13
,
14
]. AG is firmly fixed at the centre on MT using holding clamps. The autocollimator
(ELCOMAT 2000) is aligned with a working face ( say face A ) of angle block. The emitted optical beam from autocollimator strikes normal to the surface. Figure
1
shows this configuration; clamps are excluded for the photograph. The angular position ( I a
: M a
) of Moore’s indexed rotary table and measured value of angle ( A a
) at autocollimator constitutes reading on face A . MT is rotated till the other working face ( say face B ) of angle block reflects the emitted optical beam back into the autocollimator. The new angular position ( I b
: M b
) of MT and angle
( A b
) detected by autocollimator will be the reading on face -
B . Here the values I a
, I b are read from the main scale
(degree of arc) and values M a
, M b scale (minute of arc) of the MT.
are read from the Vernier
Angle block is measured in four orientations to eliminate the bias due flatness of working faces and pyramidal
error due angle block orientation on MT [ 15
]. It is common practice of minimizing the effect of pyramidal error by measuring AG in normal orientation as well as in inverted orientation (bottom up). The orthogonal axis misalignment should be \
0.2 % [ 15
–
17 ]. Figure
2
demonstrates procedure to obtain the proposed four orientations of angle block. The results obtained in the experiment are given in
Table
1
.
3. Calculation of Angular Deviation
Angular deviation a of angle block from its nominal angle
N equals to measured angle minus nominal angle, plus systematic errors. Equation
1
summarizes systematic position error d A t
( d A ) of MT (autocollimator) and random errors due to flatness variation d A
F
, pyramidal error variations d A
P
[ 17
]. Equation
2
gives the angular deviation a in terms of measured angular deviation D a and sum of errors.
X d ¼ d A
F
þ d A
P
þ d A þ d A t
ð 1 Þ a ¼ D a
X d ð 2 Þ
3.1. Manual Calculation
Manual evaluation of measured angular deviation of angle gauge is described in three steps.
Step 1 : compare the numerical values of degree denomination of readings obtained on faces of AG and arrange them in ascending order. Then compare the numerical values of minute’s denomination of these readings. If the numerical value in minutes denomination column is larger than the above written values add 59 min and 60 s in the respective columns. At same time reduce 1 of arc from the larger value in column of degree denomination.
The same procedure can be adopted for arc second case.
This sexadecimal subtraction results the measured rotation of MT. The sexadecimal subtraction process is itemized in Table
2 .
Fig. 1 Direct comparison angle block calibration setup at NPLI
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Validation of Software Used for Calibration of Angle Block at CSIR-NPL, India
Fig. 2 Orientation of the angle block on MT
Angle Block viewed from aperture of Autocollimator
Face A
Face B
Face A
33
Face B
Face A
Face A
Face B
Face A
Face B
Table 1 Measurements of angle gauge in four orientations
Orientation of AG
R 1
R 2
R 3
R 4
Indexing table reading on
Face A ( I a
: M a
67 :30
0
67 :30 0
67 o
:30 0
67 o
:30 0
)
Autocollimator reading on
Face A of AG ( A a
1.55
00
460.45
00
84.85
00
588.90
00
)
Indexing table reading on
Face B (Ib:Mb)
256 :30
0
238 :30 0
238 :30 0
256 :30 0
Autocollimator reading on
Face B of AG ( A b
)
2.20
00
459.60
00
86.60
00
587.05
00
Step 2 : compare the measured rotation of MT obtained in the step 1 with 180 of arc. If the value of minutes or second portion is larger than the above written value, spread the 180 as 179 , 59 min of arc and 60 s of arc.
Perform the sexadecimal subtraction to obtain measured angle of AG.
Step 3 : compare the measured angle block value obtained in the step 2 with nominal value of angle block. Perform the sexadecimal subtract to quantify angular deviation of angle block from it nominal value
N .
3.2. Generic Analytical Model
The analytical equation
3
can be used to determine angular deviation a
of angle block [ 17
].
a ¼ I b
0
þ
I
0 b
I
0 a
I
0 a
I b
0
180 o
I
0 b
I a
0
I
0 a
I b
0
N
I
0 a
180 o ð A
180 o b
A a
Þ X d
ð 3 Þ
Here I b
0
, I a
0 represents degree of arc of MT readings associated with face–B, face–A of angle gauge. The minute
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34 M. Arif Sanjid, K. P. Chaudhary
Table 2 Manually calculated results of four orientation of an angle block
Degree of arc
Minute of arc
Second of arc
ID
Normal orientation—R1
Start from Face B
Finishes at Face A
256
67
Measured rotation of Moore’s indexed rotary table (obtained by subtraction i.e. R1
Angle to be reduced to get acute angle of AG
Measured value of AG (e.g. M1
BA
Nominal value of AG
A
180
)
B
R1
A
)
Angular deviation of AG from its nominal value N
Normal orientation—R2
Start from Face A
Finishes at Face B
238
67
Starting Face A (238 of arc is spread as 237 of arc: 59 min: 60 s then added position wise) 237
189
180
9
9
0
67
170
179
Finishes at Face B
Measured Rotation of Moore’s indexed rotary table (obtained by subtraction i.e. R2
A
R2
B
)
Angle to be reduced to get acute angle of AG (180 of arc is spread as 179 of arc: 59 min:
60 s)
Measured Rotation of Moore’s indexed rotary table (Re writing in descending order)
Measured value of AG (e.g. A
180
M2
AB
)
Nominal value of AG
Angular deviation of AG from it nominal value N
Inverted orientation—R3
Start from Face B
Finishes at Face A
Measured Rotation of Moore’s indexed rotary table (obtained by subtraction i.e. R3
B
R3
A
)
Angle to be reduced to get acute angle of AG (180 of arc is spread as 179 of arc: 59 min:
60 s)
Measured Rotation of Moore’s indexed rotary table (Re writing in descending order)
Measured value of AG (e.g. A
180
M3
BA
)
Nominal value of AG (9 of arc is spread as 8 of arc: 59 min: 60 s)
Angular deviation of AG from it nominal value N
170
9
9
0
238
67
171
179
171
8
8
0
Inverted orientation—R4
Start from Face A to
Finishes at Face B
Starting Face A (256 of arc is spread as 255 of arc: 59 min: 60 s and appended)
Finishes at Face B
Measured Rotation of Moore’s indexed rotary table (obtained by subtraction i.e. R4
A
R4
B
)
Angle to be reduced to get acute angle of AG (180 of arc is spread as 179 of arc: 59 min:
60 s)
Measured value of AG (e.g. M4
AB
A
180
)
Nominal value of AG (9 of arc is spread as 8 of arc:59 min: 60 s)
Angular deviation
256
67
255
67
188
180
8
8
0
30
30
0
0
0
0
0
30
30
89
30
59
59
59
0
0
0
30
30
0
59
0
59
59
0
30
30
89
30
59
0
59
59
0
2.20
1.55
0.65
0
0.65
0
0.65
459.60
460.45
519.60
460.45
59.15
60.00
59.15
-
0.85
0
0.85
86.60
84.85
1.75
60.00
1.75
58.25
60.00
1.75
587.05
588.90
647.05
588.90
58.15
0
58.15
60.00
1.85
R2
A
R2
B
R2
A
R2
B
M2
AB
A
180
R1
B
R1
A
M1
BA
A
180
M1
N
M1-N
M2
M2
N
AB
M2-N
R3
B
R3
A
M3
BA
A
180
M3
M3
N
M3-N
R4
A
R4
B
R4
A
R4
B
M4
AB
A
180
M4
N
BA
M3-N of arc denomination M b is converted into degree of arc and added to degree of arc denomination I b e.g.
I b
0
= I b
?
M b
/
60.
A b
( A a
) represents autocollimator readings associated with face–B (face–A) of angle gauge respectively.
A generic analytical model is devised to accommodate measurement readings, nominal value of angle blocks in denominations degree of arc: minute of arc: second of arc.
Equations
4 ,
5
are two sign functions devised to use in the analytical model.
S 1 ¼ Sgn ð I bD
I aD
Þ ¼
½ I bD j I bD
I aD
I aD j
ð 4 Þ
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Validation of Software Used for Calibration of Angle Block at CSIR-NPL, India
Table 3 Calculation of results using analytical formulae ( N
D
= 9 ; N
M
= 0
0 and N
S
= 0
00
)
Calculation R 1 R 2
[ I bD
I aD
]
| I bD
I aD
|
S1
(| I bD
I aD
| 180 )
|| I bD
I aD
| 180 |
S2 da
D
= 60 9 60 9 (|| I bD
I aD
| 180 | N
D
) da
M
= 60 9 S1 9 S2 9 ( M b
M a
N
M
) da
S
= S1 9 S2 9 ( A b
A a
N s)
D a as per equation
189
189
1
9
9
1
0 00
0 00
0.65
00
0.65
00
171
171
1
9
9
1
0 00
0 00
0.85
00
0.85
00
R 3
171
171
1
9
9
1
0 00
0 00
1.75
00
1.75
00
35
S 2 ¼ Sgn ð j I bD
I aD j 180 o Þ ¼
½ j I bD j j I bD
I aD j 180 o
I aD j 180 o j
ð 5 Þ
The sign function S1 inverts the subtraction operation whenever the subtraction happens from lesser value to larger value. The sign function S2 assigns a positive sign when the AG on MT is arranged such that its value adds to
180 during rotation of MT. Conversely, results a negative sign when the rotation of MT is \ 180 in AG measurement. In the following equations I bD
( M b
), I aD
( M a
) represents degree of arc (minute of arc) denomination of MT readings on the respective facets of AG. Nominal angle N of AG under test is ramified into N
D minute: N
S second denominations for ease.
degree: N
M da
D
¼ ð j j I bD
I aD j 180 o j N
D
Þ ð 6 Þ da
M da
S
¼ S 1 S 2 ð M
¼ S 1 S 2 ð A b b
A
M a a
N
M
Þ
N
S
Þ
ð 7 Þ
ð 8 Þ
D a ¼ 60 60 da
D
þ 60 da
M
þ da
S
ð 9 Þ
Here da
D
, da
M and da
S represents angular deviations in degree of arc, minute of arc and second of arc denomination respectively.
D a is angular deviation from nominal size of angle block in second of arc. Table
3
give the results calculated for four orientations measurement using analytically derived formulae.
3.3. Automatic System
In the integrated automatic calibration setup, the readings of automatic indexed rotary table [
18 – 20
] and autocollimator are dealt in second of arc. For the convenience of operator, these reading will be displayed in degree: minute: second denominations on computer screen. Sophisticated indexed rotary tables have additional reading scale of arc second. The seconds scale readings of indexed table are S b
,
S a
. A generalized algorithm is described in Fig.
3 .
The measured rotation is obtained by subtracting modulus of angle reading obtained on the faces of AG under test. Absolute function is used to calculate the modulus angle. Then the measured angle will be evaluated by reducing 180 9 60 9 60 arc second. The measured angular deviation will be measured angle minus nominal value of angle block in arc second.
4. Software
The graphical user interface (GUI) of software is shown in
Fig.
4
. The GUI has ‘‘Enter’’, ‘‘Evaluator’’ buttons and three sets of data entry fields. The operator can fill the measurement readings and nominal value of angle block into these fields. The units of the fields are degree of arc, minute of arc and seconds of arc. The GUI is designed to depict each step of calculation. The nominal rotation 180 of arcs is spread as
179 of arc, 59
0 of arc and 60
00 of arc. The angular deviation of angle gauges is evaluated by subtracting the nominal value of angle block and the same will be visualized.
The software is written in two modules. Module 1 will be executed by clicking the ‘‘Enter’’ button. The measurement readings can be entered into software by clicking the enter button. Module 2 will calculates angular deviation by clicking the ‘‘Evaluator’’ button. The pseudo code of modules is described in annexure. Two consecutive subtractions are involved in the calculation of angular deviation of AG. During subtractions, negative value may results in the columns of degree: minute: second. The negative sign is removed by adding 60
00 and incidentally subtracting
1
0 from the preceding position (i.e. minute’s position).
Similarly, the negative sign of minutes’ position is removed by adding 60
00 and reducing 1 unison of arc.
The data type of degree of arc is positive integers in the range of 0–360; the set D
I
= {0, 1, 2, 3, 4, 5 … 360}. The data type of minutes of arc is also positive integers in the
R 4
189
189
1
9
9
1
0 00
0 00
1.85
00
1.85
00
123

36
Fig. 3 Flowchart of calculation of automatic system
M. Arif Sanjid, K. P. Chaudhary
a
a
a
a
b
b
b
b
a
r
a
a
b
a
a
b
b
b
b
b
a
m a
r
N
N
N
s
D
N
N
S
m a
N s
Fig. 4 Graphical user interface of the software range of 0–45 the set D
M
= {0, 15, 30 and 45}. The data type of second of arc is float in the range of
D
S
= { 1050.000, 1049.999, 1049.998, 0.002,
0.001, 0.000, ?
0.001, ?
0.002
… ?
1049.999, ?
1049.998,
?
1050.000}. The autocollimator measuring range is
± 1050.000 s of arc and its resolution is 0.001 s of arc.
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Validation of Software Used for Calibration of Angle Block at CSIR-NPL, India
Fig. 5 Scheme of validation analytical method, manual method and proposed software
Input data vector
(240 Measurements)
Manual calculation of α =
M(R m
)
Proposed
Software results α
= S(R m
)
Empirical calculation of α
= E(R m
)
37
Visual cross checking
Analytical verification
Visual Intercomparison
Not Possible
Yes, Possible
Both visual comparison results with other method and visual cross verification its own calculations is possible
5. Validation of Software
Validation of software is an essential technical requirement
of ISO/IEC 17025 [ 21
]. The validation involves proof of
three necessary and sufficient criteria [ 22
–
24 ].
Criteria 1 The software must accept all possible set of measurement results as its inputs. The input data vector is
R m
=
V R m,
(
A
D a
I
, D
M, n
[ R m
.
D
S
) should satisfy the closure property i.e.
Proof To implement the first criterion, input text box fields are coded to accept data which is subset of R m
. The software prompts ‘‘Enter valid Data’’ message whenever the user tries to enter data other than the subset of R m
.
Criteria 2 The software should result correct angular deviation for all possible combinations of input data.
Proof The specification for prismatic angle gauges,
Indian Standard 6231:1971 (reaffirmed 1998) recommends two combinations of angle block sets. Fifteen angle blocks were chosen from the both sets for calibration. The measurements of these angle blocks in four orientations at four quadrants of MT are chosen as reference input data.
Criteria 3 The angular deviation result of software should be algebraically same as manual, analytic results i.e.
a
S
= S ( R m
), a
M
= M ( R m
), a
A
= A ( R m
). The functions S (),
M () and A () corresponds to software, manual and analytical functions responsible for resulting angular deviation a
S
, a
M and a
A respectively. Then A D a = a
S
= a
M
= a
A
.
Proof The third criterion is proved by comparing the results of three methods using the reference input data.
Figure
5
depicts the scheme of implementation. 240 measurements are feed to the proposed Software, proposed analytical model and manual method of calculations.
Direct comparison method uncertainty of measurement of angle block from literature is given in Table
4
[
25 ,
26
]. The uncertainty due flatness of angle gauges is
0.01
00 of arc. This is inherent to the angle blocks. One cannot achieve measurement uncertainty better than
0.01
00 of arc. It is limiting value of uncertainty of measurement. The difference in results of the calculations among these three methods is less than this limiting value 0.01
00 of arc.
Different machines implement different analytical models internally. Those cannot be analyzed mathematically as their code is available readily [
27 ]. Then they can
be numerical validated by comparing results with pro-
posed software results [ 28
,
29 ]. The proposed software
results are compared with those results obtained manual method.
123

38 M. Arif Sanjid, K. P. Chaudhary
Table 4 Conventional example of direct comparison method uncertainty of measurement
Source of uncertainty Estimate Limits Type/Probability distribution
Uncertainty component u
1
(AC calibration) u
2
(Air turbulence) u
3
(AC resolution) u
4
(Flatness variation of AG) u
I
(PIT Positional Error) u p
(Pyramidal Error)
Combined uncertainty ( k = 1) u c
Expanded uncertainty ( k = 2) U e
–
–
– k /6
Bold values indicate intrinsic uncertainty of AG
0.03
00
± 0.025
00
± 0.025
00
Type B/Normal 0.015
00
Type B/Rectangular 0.014
00
Type B/Rectangular 0.014
00
± 0.05
l m Type B/Rectangular 0.028
l m
± 0.15
00
Type A/Rectangular 0.08
00
± 0.29
00
Type A/Rectangular 0.17
00
Sensitivity coefficient
Uncertainty contribution
1
1
0.015
00
0.014
00
1 0.014
00
0.35
00
/ l m 0.010
00
1
1
0.080
00
0.17
00
0.192
00
0.38
00
Degree of arc of freedom
Infinite
Infinite
Infinite
Infinite
32
32
51
6. Conclusions
The software entitled ‘‘calibration of angle gauges’’ developed at CSIR-NPL, India is validated to meet technical requirements of ISO/IEC 17025. First, the software is verified by code reviewing. The arithmetic’s implemented in the software involve mostly subtractions and modulus functions. Therefore numerical accuracy of the calculation
is least affected [ 30
]. The software narrates each step of calculation on the GUI to aid visual cross checking.
Therefore, there will not be any sign ambiguity. One can use analytical model to calculate the angular deviation on a
C program console. Automatic systems suffer inability of validation though it is easy to implement. Using this software, one can compare results of such systems to pick
software glitches [ 31
,
32 ]. Students, researcher can make
use of pseudo code given in annexure to develop their own version of software.
Acknowledgments Authors pay sincere thanks to the director,
CSIR-NPL, India for his encouragement.
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Validation of Software Used for Calibration of Angle Block at CSIR-NPL, India
Appendix: Coding of the angle block software
Module 1: Enter the measured reading of the angle block
39
123

40
Module 2: Calculation of angular deviation
M. Arif Sanjid, K. P. Chaudhary
123

Validation of Software Used for Calibration of Angle Block at CSIR-NPL, India 41
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