NBSIR 75-750

The Calibration of Indexing

Tables by Subdivision

Charles P. Reeve

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

July 1975

Final

U S. DEPARTMENT OF COWMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 75-750

THE CALIBRATION OF INDEXING

TABLES BY SUBDIVISION

Charles P. Reeve

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

July 1975

Final

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson. Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director
s.
Contents

Page

1. Introduction 3

2. The Method of Subdivision 3

3. Preparation for Measurement , 6

4. Complete Closure Design 6

5. Partial Closure Design 13

6. Sources of Error 19

7. Applications of Designs 19

8. Example 26

9. Conclusion 26

Acknowledgement 29

Figures , 5, 7

Tables 21, 25, 27, 28

References 30

Appendix A 31

Appendix B 34

2

1. Introduction

The indexing table plays a vital role in the calibration of angle

standards [1,2, 3]^. An object v/hich is wrung or clamped to an indexing

table can be rotated through certain angles very precisely. The smallest angular increment varies with different indexing tables but is most

commonly one degree. The deviation from nominal of any angular interval

in a high quality indexing table is usually no more than 0.25 second,

and the short term repeatability of any setting is usually less than

0.05 second.

In most angular calibrations the indexing table plays one of three

roles

(1) Is used simply to rotate an object through some nominal angle

whose precise value need not be known.

(2) Is calibrated simultaneously with an angular standard in a routine calibration process.

(3) Is used in the calibration of an angular standard where some

of its angles need to be known precisely beforehand.

The second role is seen in the calibration of polygons. For example,

a 30° polygon mav be calibrated by comparing each of its twelve angles

to the twelve 30 angles of the indexing table and then computing a least squares solution for the 24 unknowns. The third role occurs when

it is impossible or impractical to employ a self-calibrating algorithm

such as in the calibration of a small angle block (less than 15°) by

direct comparison to a known interval of an indexing table. In that case it is necessary to do a preliminary calibration of the indexing

table angle. One way of accomplishing this is by applying a measurement algorithm usually called the "method of subdivision". The

mathematics of this method are presented in great detail.

2. The Method of Subdivision

Angles on the same indexing table cannot easily be compared with

each other, so it is more convenient to calibrate two indexing tables

(denoted by A and B) simultaneously. That way each angle on one table

can be compared with several on the other to give a redundant set of

observations. This idea is incorporated in the method of subdivision

which consists of two types of measurement designs, "complete closure"

and "partial closure". The complete closure design is used to subdivide

"'"Figures in brackets indicate literature references at the end of the

paper.
the entire 360 degrees of each table into Dq equal segments A^,

A and B. , . . . , B respectively where

, E = 360° . 1=1 i=l :

Then by partial closure one segment on each table, say A^^ and B^^, is

• • • • subdivided into n, equal segments A, , , , . , A and B, , . , . , B

1 J. n^^ 1 ^ n^^ respectively where

^ ^1

^A. =A^and E»i = »r

i=l i^l

• • •

Similarly A^ and B^^ can be divided into n2 equal segments A^^,

A and B, , . . . , B respectively where

n2 1 n2

^2 ^^2 A. = A^ and J] B . = B^.

i=l i=l

-;vp:...-:.- .

This process can be continued until the desired level is reached.

As an example, if it were desired to know the value of the 0° - 1°

interval on each table, they could be calibrated using the complete

closure design with n^ = 12 and two partial closure designs with n^^ = 6

and - 5. Thus each 30^ interval would be calibrated, then 0° - 30° on each table would be subdivided into 5** intervals, and the 0° - 5°

interval on each table would be subdivided into 1° intervals. This

subdivision is illustrated in figure 1,

It fjhould be noted that only those increments which are common to

both tables can be calibrated. For instance, if table A has a smallest

increment of one degree and table B has a smallest increment of ten

minutes, then the smallest increment which can be calibrated on each

table is one degree.
0 30

/

\

/

/

0/5

\

\

\

/ \

i 1 \—

/ \

/ \

/ \ —I

1 1

i ^

30

H (- 330 0

Figure 1

MIRROR

HOOD

AUTCCOLLIMATOR

INDEXING

TABLES

SURFACE PLATE

Figure 2

5
3. Preparation for Measurement

In preparation for measurement table A should be mounted on table B as nearly concentric as possible, It is then helpful if this assembly

is mounted concentrically on table C which is either an indexing table

or a rotary table. This table is not to be calibrated but serves only

to rotate the other two tables between sets of measurements as required

by the measurement design. A mirror is then mounted on table A approximately at the center. An autocollimator is mounted so that the face of

the mirror is centered in its field of view. Adjustments are made so that the autocollimator reading is near the center of its scale when

each of the tables is set in its zero position. The autocollimator

should be adjusted so that it reads horizontal angle only.

If table C is not available then either an adjustable mirror must

be used or the autocollimator mupt be shifted between sets of measurements. The whole assembly should be clamped to a surface plate as shown

in figure 2. The process of clamping the tables should be done with a minimum of distortion, A hood should be constructed over the autocollimator and mirror so that no outside light can interfere with the autocollimator reading.

The assembly is then ready for the measurement process v*'hlch involves only the appropriate rotations of the tables. Tables A and B are always rotated in opposite directions through equal nominal angles.

The observed change In autocollimator reading is then equal to the

difference between the true values of the two angles. This is illustrated in figure 3 for the angles A^^ and B^^ which are nominally N with

deviations a^^ and respectively,

4. Complete Closure Design

Let tables A and B each be initially subdivided into n intervals of

360/n degrees, and let the deviations from nominal of the intervals be

denoted by the vectors a = (a.in. . , a )* and 3 = (3,in,,.&)'

respectively. If every a were compared v/ith every 6 the result would be

2 n measurements on 2n unknowns. In cases of large n this may require

more measurements than would be practical. If every a were compared

with only m of the 6's, where m<n, then there would be mn measurements

on 2n unknowris» so for m > 2 there would be redundancy in the system.

A convenient measurement algorithm is to take n blocks of m differences

of the form

6
Pos. I Pos. 2

MiRROR y//////

AUTO

COLLIMATOR

READING

y I = A + €

READING

y2 = A + (N-i-a,)-(N + )S,)+€2

Figure 3.

7
^1 - ^1

a - 3

m m

^3

-

uH-l nH-2''

"1

-

m-'l iB'-2

(note that 3. , " 3 , ) where with each succeeding block the o subscript

Zn-l n-i , is increased by one and the 3 subscript by two. The i block of n

differences is generated by taking m+l observations according to the

following scheme: '

^il ° \ ^ ^1

yj2 • \ + «i - ^21-1 ^2

13 \ * «i "i+l - ^2i-l - ^2i ^ ^3

i.mfl " + «i + . . . + ^'i^m-l - ^2i-l - • — -

^2i-hn-2 * "»^1

where is the initial reading of the autocollimator and the £*s are independent error values from a distribution whose mean is zero and

whose variance is a2. The subscripts of 3 are reduced modulo n. The

complete closure design will be denoted by C(n»m).

Before proceeding, a word should be said about sign convention.

This model assumes that the two tables are numbered in opposite directions,

and that increasing the angle of table A gives a positive deflection of

the autocollimator. If the tables do not conform to this convention,

they can be made to conform by reversing the assignment of the angles

or by reversing the sign of the observations or both.

th The new random variables for the 1 block, z. = (z. ,» . . . z. )*,

i 11* ' im • are formed by

8
xl i2 il X 2x-l 2 1

^i2 ^iS - ^i2 = - h± "3 - "2

^im ^ijHrl-l ^im °'i+m-l ^2i-hn-2 ^m,

or in matrix notation z, = My. where

X X

"-1 1 0 • • 1 0 0 0"

0 -1 1 0 0 0

M = •

•

•

•

•

• 0 0 0 -1 1 0

. 0 0 0 • • < 0

--1 1.

- Var(y^)

2 a

•iiH-l' then Var(zi> = a

2 -1 0 0 • 0 0"

-1 2 -1 0 0 0

V = MM' 0 -1 2 -1 0 0

m ••

•

•

. 0

•0 0 0• -1 2_

m

Let z = (z^ z^ . . form

. z^) ' . Then the least squares estimation takes the

E(z) = X

9
where

X =

D

m n

D

tn n

D P'""^

. m n

-DP°

m n

-D P^

m n

m n

and D = m (16 ] where I is the m x m identity matrix and

m ia,n-m m

6 is the m x (n-m) zero matrix, and P is the n x n permutation

m,n-m n matrix which is given by

0 10

0 0 1

Ll 0 0 OJ

The full variance-covariance matrix of the observations is given by

the nm X nm block diagonal matrix

V

m

W = V

m

Then

W

V

-1m

m

Vrl

m

,-1

where

m

10
,-1

nH-l

m m-1

m-l 2 (m-1)

m-2 2(m-2)

m-2

2 (m-2)

3 (m-2)

2 (m-1) m-1

m-1 m

The normal equations (incorporating the restraints

n n

^ a_j^ = ^ = 0) take the form

i=l i=l

X'W~''"X

i'

0

i 0

0 4

0 0 0

a

3^1

LX2J

0

- 0 .

where and are Lagrangian multipliers entering in the minimization

process and i = (1 1 . . . 1) ' . (The normal equations are developed

further in Appendix A in a form suitable for computer programming.)

The estimates are given by

-1

a x'w"

4 0

X'W~-^z

3 0 4

0

C a b X'w"-^2 h

4' 0 0 0 a' c d 0

LxJ . 0 4' 0 0. 0

- b' d c 0

where C is the variance-covariance matrix of the estimates.

11
The predicted values are given by

and the deviations by

d = z - z

.

The estimate of o is given by

a = If d'W d/(inn-2n+2)

and the standard deviation of the estimates by

the covarlance of a and 3, is given by

and a j+n. j+n'

X

J

a = a C.

i, n+j

For the cumulative values let

k k

12
The standard deviations are given by

i=l j=l 13

and

k

E 'n+i, n+j

There is no error entering from the restraints since they

represent an exact relationship.

5. Partial Closure Design

Let A. and B. be two calibrated intervals of the same nominal

angle on tables A and B respectively and let their observed deviations

from nominal be given by m^ and m^ respectively. Let the corresponding

variance-covariance matrix of the values be given by

S =

0 a

"^A Vb

o o m . m„ m„ A B B

The intervals may be subdivided into n segments denoted by

* * * • d = (a. . . . a )' and 3 = (3, . . . 3 )* by using the partial closure

± n ± n design under the restraint that

n

1=1 1=1

Each a can be compared to each 3 so that there are n measurements

of the 2n unknovms. One measurement algorithm which is convenient

forms the 2n-l groups of differences

] n

a - 3 1 1 n-1

^^2 - n

^1

-

n-1 n-1

La - 3 n n

?2

"

^3

-

n n-1-

, . . . , (a - n

13
where the central block has n differences and the adjacent blocks

decrease in size by 1 until the end blocks have only 1 difference.

Each block of k differences requires k+1 measurements. For example

the central block is generated by n+1 measurements according to the

following scheme:

y = A +

•^nl n 1

n2 = A + a - 3, + e,, n 1 1 2

y . , = A + a + . . . + a - e. ^n,n+l n 1 n 1

. . - 3 + e .

,

n n+1

where A is the initial reading of the autocollimator and the e's

n are independent error values from a distribution whose mean is zero

2 and whose variance is a . This partial closure design will be denoted

by P(n). The new random variables = (z^^ . . . z^^) ' are forijted by

nl n2 nl 1 1 2 1 r • o^fi;;-. V = - \2 = "2 - ^2 ^3 - '2

nn n,n+l nn n n n+1 n

or in matrix notation z = My where

. n -^n

M

-110

0-1 1 .

L 0 0 0

0 0

0 0

-1 Ij

2 2 Since Var(y ) = a I then Var(z ) = a V where V is defined

n n+1 n n n

as in the previous section. '
Let z' = (z' z' . . . z' , ) be the complete set of difference

1 Z Zn - i

measurements. The size of the vector z^ is k x 1 if k <_ n 4nd (2n-k) x 1

if k > n. Corresponding to each z^^ is a variance-covarlance matrix

which is k X k if k<^n and (2n-k) x (2n-k) if k > n. For

k = 1,2, and 3 the matrices take the form

^2 - [-1 1]' ^3=

The least squares estimation takes the form

E(z)=x(^)

where

X =

and D,

n

D , pl

n-1 n

1 n

->2

-D P«

n n -D

n-1

In 1

= [\ \,n-k] where I, , 6, , , and P are defined as in the

k* k,n-k' n previous section. The variance-covariance matrix of observations is

given by

15
where W is a block diagonal matrix whose blocks vary in size from

1 X 1 to n X n. Then

where V is the same as in the previous section. For example.

^1 2 2 3 Ll 2j' ^3 4

3 2 1

2 4 2 12 3.

» • • •

The normal equations (incorporating the restraints

n

/! a. = m. and 7_] 3. = m^) take the form

i=l i=l

16
i 01

0 i

i' 0 0 0

0 i' 0 0 m

B J

where X^, and i are defined as before. (The normal equations are developed further in Appendix B in a form suitable for compute

programming.)

:er

The estimates are given by

i o"

-1

X'W^-^z

0 i

*• 0 0 0

0 i' 0 0

C a

a' c

b' d

X'W'-^Z

m.

where C is the variance-covariance matrix of the estimates.

The predicted values are given by

(i)

and the deviations by

d = z - z.

The estimate of a is given by

a = ^d*w"^d/(n^-2n+2).

Since the random error entering through the restraints is non-zero it

must be taken into account when computing the total standard deviation

of the estimates. The complete expression for the variances and

covariances of the estimates is given by

Var \

• / = C + (a b) S (^i). Thus

17
-2 -2 2 ^2 2 -2

a* = a C. + a + b. a + 2a^b^a ,

- a. XI i m. 1 ni„ i i m,m„ ' X A B A B

-

: f

Op = C. . . . + a^_^ + h^.o^ + 2a. .b.. a

3j J+n, j+n j+n ra^ j+n ra^ j+n j+n m^m^ ,

and the covariance of and 3^ is i ; : • >'1<:a .

-2 -2

a* • =0 C. . . + a. a . ,o + (a.b .

, + a , .b.)a

a^a^ i,n+j i n+j i n+j n+j i m^m^

^ ^2 ' ' '

• ^ • - '"-W'T ? ' '

^ + b b _^.a : M ? i n+J m„ . _ ^ ; * >

For the cumulative values, let

k k •

1=1 ^ " 1=1 ^

The variances are given by

= E E c. . + ( E aj a -f f Vb.") 52 +

\ 1=1 j=i \ i=i y '"a \ iti V "b

2^ E aV E b.^ a , and a? = 9^ |; . C.^ +

\A 7 Vb \ 1=1 j=l ^-^^'3+"

Iw'^^A nli'w\ ' Hil^^-I?^-j Vb

18
6. Sources of Error

In the mathematical models just described the only errors accounted

for were the random errors of m.easureraent and the propagation of these

errors into the following series. The error of a single measurement

which was given as a is the combination of random errors from three sources

1) random setting error in indexing table A

2) random setting error in indexing table B

3) random error of the autocollimator reading

Since some of the physical parameters of the system are unknown and

cannot be modeled, there are systematic errors in the measured values.

These probably are small, but it has not been determined if they are negligible. Factors such as the autocollimator reading being biased

by vertical angle, imperfections in the mirror, effects of varying

temperature, and foreign particles in the setting mechanisms of the

indexing tables can influence the accuracy of calibrated values. There

is evidence that over a short period of time, such as a week, that these

effects are negligible in comparison to the random errors.

There are several reasons why long term changes in the indexing

table angles may not be negligible. If the table is moved from one location to another it may be clamped do\m in a different manner thus

resulting in slight distortions. The grease which lubricates the

tables may collect small amounts of dust over long periods of time and

as this dust works its way into the teeth of the tables it causes a variability in the setting of different angles.

Over a period of years the values of two indexing tables at NBS

have been observed to drift slowly, but whenever a table was dismantled,

cleaned, and reassembled there was usually a significant change in the

angular values. The conclusion to be drara is that over short periods

of time systematic errors can probably be ignored while over longer

periods of time, such as several months, they become significant and it

is better to recalibrate the table before using it.

7. Applications of Designs

As described in section 2, a combination of the two types of designs

allows the calibration of intervals of various sizes by subdivision.

However, each design can be useful by itself. UTien the complete

closure design is used alone, it subdivides the two tables into n equal angles each. The standard deviation of all the angles on indexing

table A are equal, and likewise for indexing table 3.

19
Table 1 lists the standard deviation coefficients, and k^, of

the angles for all values of n up to 36 and for all appropriate values

of m. Note that the standard deviation coefficients for the angles

on indexing tables A and B are not always equal for a given design. The

standard deviations of the individual angles are given by s^ = k^. s and

i

s„ = k s where s is the estimated standard deviation of a single

1

measurement. . .

The standard deviation coefficients for the cumulative angles are also given. They are the maximum values of the k's which correspond

to all possible sums of consecutive angles.

Table 1 is given primarily as an aid in determining which value of

m to use when subdividing two indexing tables into n equal parts each.

Assuming that there is a desired upper limit for the standard deviation

of each angle (s ) and that an approximate value of the observed

standard deviation (s) is known, then any value of m is acceptable

whose corresponding k value is less than the ratio s /s. For example, max

given the parameters s « .03, ^^^^ = .02 and n = 24, then k <_ .667;

therefore, m^6 for individual values and m ^ 10 for cumulative values.

If s = .05 then k < 1.667, and m>3 for individual values and m>7 for

max _ ' — —

cumulative values. ' >

The least squares solution to each complete closure design is

restrained by the fact that the sum of the angles on each table is

exactly zero, therefore there is no error entering the measurement

process from that source.

Table 2 lists the standard deviation coefficients, k = k. = k„, of

A B

the angles which are calibrated by the partial closure designs for values

of n up to ten. In this case the standard deviation coefficients are

the same for the corespondlng angles on indexing tables A and B, that

is, k, = k„ for each position. As in the first case s = s„ = ks, A B a 3

but now there is an error term entering from the restraint since the

values of the angles being subdivided are not known exactly. This error

is spread out over the smaller angles and generally will be small com- pared with the random measurement error within the series.

The combination of a complete closure design and one or more partial

closure designs gives a flexible system which should be adequate to

economically measure any possible combination of indexing table angles

which one could imagine. The only requirement for using these tools is

a modest amount of imagination. An example is given in the next section.

20
T AHLK I .

TAHIF-J \ AMP n (• VL I H"^ ATh r« W-FTM rH'Pf.rTT Clf'SlfRP nrstrMS.

. . . 1 N'!> rV I DM \L. . . . . , CIlMI'l \T] VC .

,

M K( A ) K< •» ) MAV K( \ ) \fA X K(

,707 ,70'' ,•^07

4 « "S 54 ,654 ,6 54 ,654

e .77S , TTc; . 77=: .'75

5 4 ,63 1 .6 33 .633

a c> , 5C 2 ,5^2 . ^9? .50

6 3 ,845 ,78'' .86"' .9 1 3 e 4 , 68 7 , 6 77 ,6P7

ft 5 .577 C T ^ .577

6 6 , 544 , 545 ,544 ,545

R , OP ^ 1,1^3 1,414

R 4 • ~* 1 o e ,830 ,881

fl , 'S 1 o

c r> £ ,596 , 6?5 .6?^'

8

c c /\ , 50 ,•^04 ,5 50 ,554

B

c: /> , 5 00 ,500

8 8 .477 ,4"'B • 4 ''7 ,4-8

9 3

4 r\ "5 1,001 • V 2 3 1 ,296 ! ,589

«; 4 • ^4 1 . ov ^ . 856 ,956

9 5

^ A • o ^ 0^ n cr .605 , 6 ^3 ,680

^> • 555 C I. T • r , 5 4 ,565

<; 7 • 5 0^ ,5 0*? . 5 C<^' ,509

o 8 ,4^1 . 4 T I .471 .471

9 • 45 1 • 45 1 .451 ,451

10 3 1.040 ,Q4Q 1 .42C 2 ,025

to 4 , 'SB , 764 .990 1 ,258

10 b; • 6 31 ,6 06 ,7 06

1 0 6 • 560 . 5c*6 .609

10 ,514 ,5 09 ,5 20 .520

1 0 8 • 479 .481 . 4 ''9 ,481

1 0 ,44'' , 44 ,44"' .4 4"'

1 0 1 0 ,430 ,4 30 ,430 ,430

1 2

"» I , 1 ? 1 1 , 0?7 1 .

I ^ 4 ,82 7 .80 2 1,201 1 .626

1 2 ,64 8 .619 .811 .94 6

1 I 6 ,57a .567 ,66? .'I 0

12 7 .<^1 c ,51 0 , 54 .565

1? 8 .4H"' ,4 83 , 4 9<" .503

1 2 o ,45 5 ,45? , 4 .4 5 8

I ? 1 0 , 4 .4^1 ,43?

I ? 1 1 ,408 .408 .4 08 ,405

1 ? 12 , 3°** ,39^ . 394 ,395

21
T\RLF 1 , ( CflMT. >

STAMDAffD nPVT ATIrtK C«EFFIC TENTS <^ F MOLtS INOEXINr;

TARLFS A AND W CALIBRATED WITH dMPLtTt CLi°>SUPE DESIGNS.

INDIVIDUAL, . , . . « CUMIUATI Vi: . , . » N ' Kl A 1 Kl H 1 MA < ivt A 1 MA X Kl

1 5 3 1 • 2 3 I 1.1 2 • d3 ^ I 3

1 5

A

1 * ,} 3'? » O ' 1

I 5 5

1 O A

1 , 2 o A

1 5 o

C ft o e; "3 • ' -5 3 , 15 7 ,528 .517 ,e^27 .688

15 8 .487 ,480 .565

I o V

A A

^ a o r

.-l OA

1 5 1 0

A 'a ^ ,4 33 A '3 n . ** 3 CJ A 'A O /< A O

1 s 1 I , »8- 1 3

1 5

t o 1 ^ X. O CI . 3VD Q ^ O Q , 3Vr5

I T? I -3 £1 A 1 n n

1 b

^ A d "3 & c e ,5 O !5 A fi?

i KI 3 1 T>

cr er '3 c e: .300 "3 'ir tr » 3 DO 2 « 300

1 o J I . 3 c V 1 5 A r» 1 , c

u

c » i oO O » c. _ O

t QI o A« . V 3 ' o rt n

I n

c . O Qi» nU .oo^ 1 • 1

1 "7 n

1 A o R C. T < "JIT

1 A "? c;

1

9 n » > 1 . c O \J

I 9

ooA a Q . O >J 1

1 o I O . *» o 1 , rj Oc ^ o 18 10 ,4 35 .434 .4T7 ^ .495

1 8 1 1 .414 .412 .4 38 ,443

I 8 1 2 .3 Q"' • 4 1 1 , *4 1 ^

I 8 1 3 .381 • 3o0 ~V £1 O . 3 88 17 Q O

1 8 1 4 , 36P .36 8 • 3 71 ~a "T ^

1 8 1 5 « 356 5 cr cr

1^ 355 "IS ^

• 3 Of "3 cr T>

I 8 I 5

Tl A A

, 3 4 *»• *9 A C£ • 345 T> y« A

I 8 1 r T "a ^ • 3 3 3 .333 "2 ~3 "TJ "2 "J T

1 8 1 8 , 325 "tl £» 32 6 . 3 tio .3 26

20 3 I • 39 1 1 . 30 4 3 • 4 :)0 o , 2 Ob

20 4 , 9 70 • 94 1 2 • 1- 85 - 3 , 7?6

2 0 5 , 7 J 3 • 6

1

I e V 5 . ? » (J U

20 6 .606 . 596 .^^*58' • 1 • « 1 0

20 7 • 5 37 . 524 ,"^58 • 9 76

20 8 ,4'95 " . 4 V

1

i . , 6 44

20 9 ,4 62 i , 4*^6 . 5 '5 6 ,622

20 I 0 ,4 37 .435 .5 00 .535

20 1 1 ,415 .41 2 . 454 «,469

20 12 ,397 .397 ,4?2^ .430

20 13 , 382 .380 , 397 ,399

20 1 4 .368 ,368 ,3"'7'- .3 79

20 15 .356 .355 • 3 ei .361

20 1 6 , 345 . 345 .3 47 .348

20 1 7 .335 .334 ,3 36

5 (

22
TABLH 1 . ( CHVT. » STANDARD DFVTATIflN CflEIFTCTFNTS e»F AKCLFS HN INDEXTVG

TABLES A AND H CALinPATED WITH CHMPIFTF ClrtSURF DBSTOV^?.

... IVDIVIDU AL. . ... .CUMIJLATl VF . . . , N M Kf B J MAX K 1 A 1 MA X

2 0

4 CI 1 a^ o c

• 325 , 3? o

^ c • 325 • 3 26

2 O I 9 • 316 , 3 I c # 3 1 t , 5 I 0

^ /I 20 2 O"Tl /\ • 3 09 ,310 . 3 Ov » Jl 0

^ A 3t e ^ I • 507 I ,42 4A AAA 0 0 A A

?4 4 1,004 2 . 7 85

^ , 696 1 . r 02 2,658

O A o .022 • oil 1.1 vr> 1 • Moo

^ A / C A C . 34 ? • o3 1 1 , 2 r 0

ots . OOU A O ^ » / .1 O

A K A cr. a . O 1 f OAZ4 I o f J f C A Q .03/

O A t « 1 i A t "T A 1 • «*! 3

A O O , 3

O A 1 d

T( G C ^ Q a A c: o A Q 0

9 A « -a 1 .3 ft "5 . J O J3 . oo 1 A 1 Q A '3 A « 3*

9 A 1 A . 30V» A a • J QV J

A 0 9

9 A % cI O

"3 C ^ « 3"30 ^ c c , 3 b o • 3 '3 -3 T A. 24 16 • 345 • 345 ,357 .358

24 17 . 335 .334 . 342 .343

1 o • 3 2 O #330 "a T A ,331

I V • 3 1 8 3 I r

Tl '5 0 • 3 2

"3 *> A

2 A- • 3 1 O • 3 1 Q

"ail

• 3 1 1 2 1 ^ A "a 3 O 3

"a A • 30 2

-4 /> -3 • 3 03 T /\ "a • 3 03

2 2 • 2 9o 2 Vo ^ 0 <^ £. ,296

A 2 3o o • 2 ov • 289 • 2 89 • 2 89

O A 24- 24 • 2 8 3 • 2 83 0 0 »

. 2 83 ,283

30 3 1 .665 1.588 6,0 86 1 I ,635

30 4- 1.121 1 • 09 3 3 . 782 7 ,085

30 5 .766 ^ ~B , 7 33 2,120 3 .75*5

30 6 ,643 ^ Tt . 63 2 1,552 2 ,650

30 r . 556 .542 1,113 1 • 768

30 8 , 507 ,503 ,9 01 1 ,359

30 9 ,469 .462 .7^1 1 ,021

3 0 1 0 .442 .440 ,6^3 ,837

30 1 I .419 .415 . 5 *^3 ,683

30 1 2 ,400 • 399 ,5'*1 ,592

30 13 .384 ,381 .458 ,515

30 14 .370 • 369 ,426 ,466

30 15 .357 .356 .3<^9 ,422

30 16 . 346 ,345 .3-'7 ,393

30 I'' .336 .335 .358 .367

30 18 . 326 , 326 .343 .349

30 19 .318 • 31 .?~0 ,333

30 20 ,310 ,310 ,319 ,321

23
TABLE 1 , ( C«KT. )

STAND/SRD DFYTATIrtN COEFFICIENTS KF AN'OLFS iNDFYlNrj

TABLES A ANP H C'^LIBt'ATEO WITH C'»MPLETF ClOSURF npsTHNS,

• • « TNOlVIDtJ AL. • . • • .CUMriATI VF, , N M K( A 1 HAX K( A ) Mix K(

30 21 .303 .302 ,3 09 ,310

2 2 • 296 . 29 6 . 3 00 .3 01

30 23 .290 .290 ,293 *,293

30 24 .284 .284 .286 ,286

30 25 .278 ' .278 ,2 79 ,27q

30 26 .273 ,27 3 .2 73 ,274

30 27 ^ .268 .268 ^ .268 .268

30 2« .263 .263 .263 .263

30 29 .258 .258 .258 ,258

30 30 .254 .254 ,254 ,254

36 3 1 .810 I .737 7.956 I 5 .379

36 4 1.203 1.175 4.919 9 ,348

36 5 .801 .768 2.715 * ,991

36 6 .664 .653 I .965 1 ,499

36 7 .567 .552 t .377 2,335

3^ a .515 .509 ! .096 I ,780

36 9 ,473 .4^6 ,867 1 ,328

36 10 .445 • 442 ,737 1 ,073

36 I t .421 .417 ,629 , ,862

36 12 .401 .400 ,560 ^ .7 31

36 13 .385 .382 .5 02 ,621

36 14 .370 .370 • 62 ,548

36 15 .358 .35o ,428 : ,4 87

36 16 .34 6 .346 .4 02 ,4 45

36 17 .3 36 .335 379 .4 09

36 I s .327 . 326 ,36! .381

36 \9 .318 • 317 .3 44 ,3 58

36 20 .31 O .310 .331 ,34 0

36 21 .303 .302 ,319 ,3r»4

36 22 .296 .29 6 .3 09 .31?

36 23 ,290 .290 .299 ,301

36 24 .284 .284 .291 .292

36 25 , 278 ,278 .284 .284

36 2 6 .273 ,273 .277 .279

36 27 ,268 .268 ,271 .2 71

36 28 ,263 .263 .265 .266

36 29 ,259 .259 ,260 .260

36 30 ,254 .254 . ? 56 .256

36 31 ,250 .250 .251 .2^1

36 12 ,247 .247 .247 .247

36 33 .243 .24 3 ,243 ,24 3

36 34 .2 39 .2 39 .2 39 .239

36 35 ,236 .236 • 236 ,236

36 36 .233 .233 .2 33 .233

24
STANOAP''* DFVMTIMM CO F I C f FN 1 S F AMGLFS

f^N TVDFXT>Jf7 T\HL'^.S \ KH '^ n CALl U^\T\IJ) WT-p'

PARTTAT. CLOSnot; nFRTOMS W"FPl- K(A)"K(n>.

f>rtp N"? N"4 N"5 N"^

1 ,''<^\ .6""' .'?^01 ,=546 .504

2 ,"^91 .-^^ft .-SOI .55?

3 - .f"'"' .66'' ,5«)P ,549

4 - - ,f-01 ,601 .54q

5 - - - .546 .'=;5?

6 - - - - ,50 4

PrtS N-7 N-« N«9 N-10

1 .471 .443 ,420 ,400

? .F 1 3 ,482 . 456 .4.-»4

? .511 ,4«0 ,454 ,43?

4 ,510 .479 .453 .4?!

5 .511 ,47C! .453 ,431

6 ,513 ,480 , 453 ,431

7 .471 ,482 ,454 ,431

8 - .443 ,45«« .432

o - - .420 .434

10 - - - .400

25
These designs are applicable to other areas where angle measurements are involved. The complete closure method is applicable to the

calibration of a polygon. When a polygon is substituted for the

mirror its exterior angles are calibrated. If the polygon has twelve

sides or less it is not too burdensome to take the full number of

measurements. For a twelve sided polygon the C(12,12) design could be

used requiring (12 x 13) = 156 measurements, a reasonable number.

However, if a 36 sided polygon were calibrated by the C(36,36) series

it would require (36 x 37) = 1332 measurements which normally would

not be economical. Instead, a design such as C(36,6) could be used

requiring a more modest (36 x 7) = 252 measurements. In cases where n

is large and m is small the error in cumulative values greatly exceeds

the error in the individual values. This may be an important factor in

polygon calibrations, so the value of m must be carefully chosen

according to the required precision.

Rotary tables may be used in place of indexing tables, but the

observed standard deviation will be much larger because rotary table

setting errors are generally much larger than indexing table setting

errors. ,

8. Example

The first application of the method of subdivision was in connection with the absolute measurement of a set of angle blocks. Angular

intervals on each indexing table which needed to be calibrated were

0° - 60°, 0° - 30°, 0° - 15°, 0° - 5°, and 0° - 1°. On table A these

angles were denoted by a^+a2, a^, a^+a^+OLy a^^ and a'-j^ respectively. The

corresponding angles on table B were similarly denoted using 3 instead

of a. The most natural subdivision scheme seemed to be complete

closure on the 30° intervals and two partial closures on 5° and 1°

intervals. The three designs chosen were C(12,6), P(6) and P(5).

The observed y values from the thre^ series are given in table 3.

The computed values for the individual angles and their corresponding

standard deviations are given in table 4 along with the observed

standard deviation of a single measurement, s, and the degrees of

freedom, df.

9. Conclusion

Computer programs have been written for the calibration of indexing

tables by the methods described in this paper. The programs have been

tested and successfully implemented. As a result several angle calibrations have been made more efficient by reducing both the time

required to make the measurements and the time required to reduce the

data.

26
TABl !• 3.

T HP F 1- ^Sl'KKVbMTS., Y, lAKFK IN TV* V-

. C 1 BRA 1 I •'^N rh 1\0E\1NC TAUT C CI

1 A n 1^ r T A AN D b , M A ?=! 1.1 ''' f ^1 bS ' S

1 A K. 1 IN l\ J /WAT rrr7C IN SFCf^NOS),

tit S I C C 1 f3 OA " • 1 ' . " 7,96 8.1^ 8,2. 8 • ! 8

O • ' V O » '3 6. -^9 ^ .91 85

Cf 15 fi- \ I .< , c > ^ A "4 7.00 6.91 * . "'8 R2

T

•-, lie 5 .50 5. ?7 . 4 4 5, 3

m " r I r ~ » c o ,

' o , or 5,6 1 5.58 5.5 4 • 4c

CI. '^ 'Sl' Kh O . O ** 5,57 5.53 5.47 5 . 5.27 5,12 5,01 5.11 5.35 5 • ? 9

4 .79 . 4, 67 4 . -^2 4 ,90 4 .05 4, 86

5.59 n,60 5, 53 5.73 5, 30 ,69 5. 38

5.6 3 ^.67 3 . 90 3 ,94 3, 8C ^ ,59 3. 66

3.31 3.61 3, 67 3.54 3.29 1 .10 - • 44

4,16 4 , 4,13 3 , 84 3,95 ^.11 4. 1 5

SFPTFS 2 4 .40 4.37

4.50 4. 49 4 . w4

P( 6 ) 3 .09 3.08 i . 06 3 . 09

4 , 06 4.09 , 4 , 0 •? 4.0 1 3. cf-- PARTIAL 3.52 3,50 3,4-3 3,4 0 3,47 3 .4?

CL«SUPF 6 ,66 6.63 6, 6-; 6,58 6. = 1 6 .48

4 , 88 4 , «5 4. H'< 4. 71 ^ . 6 6 4 .61

5.18 5.15 5,1:' 5.04 4,9<"

5.88 5,88 5, 8'^ 5.75

5.20 5,13 5,0-

5.4 4 5 .37

SI P IKS 3 3. ye 3.78

3.51 3. 47 3,45

P( 5 ) 4 ,63 4,58 4 , 65 4.3-'

3.99 4.00 1, 96 3.97 3,^-6

PA RTT AI 6,0 3 5,99 6. 07 6,0? 8 ,98

CL 6SURF 5.16 5.15 5.14 f. 1 6 5.18

6 . 38 5. 35 5. 37 5.

5.4 1 5.38 5.38

4 , 35 4,72

27
T A !1 L f- 4 ,

TAHT.FS A AVSn H (VALUF.R TV SKCf^VDS).

. 0 /5 0

I NTb VV M

10<

^0-

90-

I ?o.

1 50-

1 BO-

?1 0'

?40'

270-

700'

?30.

30

60

90

120

1 50

1 flO 210

240

270

?00

310

0

T\rtLF A tSri MATT.

-.11"'

- . 247

. I 01

.!.052

- .0#>5

-.091

-.061

-.013

.217

.042

TARIF H

I- ST T MAT K:

, 017

.019

.01 1 -.021

- .064

,04C

- , 00ft

.004

-.022

.018

- , 039

,015

<=lTr> DFV

.019

. 01 9

. 019

• 019

.019

. 01 c .019

. 01 9

. 019

. 019

.019

. 01 9

FhHirs 2

.010/26

0-5

5-10

1 0-lP

16-20

20-25

2f -30

.020

- , 020

.012

-.025

• .036

.044

.002

.0 04

. 042

.030

.014

.02 1

. 01 ^> .017

,016

.016

.01

.016

STRiFf; 1

. 01! /I 7

0- 1 1- 2

3- 4

4- 5

,004

,016

.015

,003

,01 B

.005

.023

.OIH

.O06

,004

, 01 ft ,019

,019

,019

,01 ft

28
Ackaowledgement

I wish to thank Mr. Ralph Veale of the Dimensional Technology

Section for acquainting me with the various ways in which indexing

tables are used in calibrations and for suggesting areas where subdivision schemes might be applicable. I especially thank Mr. Joseph M.

Cameron of the Office of Measurement Services for reviewing the paper

and for offering several helpful suggestions on improving it.

29
References

1. Cook, A. H. , The Calibration of Circular Scales and Precision

Polygons, British Journal of Applied Physics, Vol. 5, 1954,

ppk 367-371.

2. Hume, K. J., Metrology with Autocollimators , Hilger and Watts, Ltd.,

London, 1965, Ch. 6, 7.

3. Moore, Wayne R. , Foundations of Mechanical Accuracy, the Moore

Special Tool Co., Bridgeport, Conn., 1970, pp. 201-250.

4. Zelen, Marvin, Survey of Numerical Analysis, edited by John Todd,

McGraw-Hill, 1957, Ch. 17.

30
Appendix A

Reduction of the Normal Equations of the Complete Closure Model

The normal equations given in section 4 can be expressed in a form which is useful in programming the model for the computer. Instead

of performing the matrix multiplications the normal equations are formed

directly so the amount of memory needed is substantially reduced. The

product X'W is given by

X'W

-1 n m m

-P^ D'V

^ n mm

P^~^D'V"^

n mm

-P^-^D'V"^

n mm

. . P-^D'V

^ n m m

. . -P^D'V"^

n m m _

. M

.-M

where M. and M. and n x m matrices. Now

1 1

(P^'V h .. = n m m ij

-1

m

n—m,m

) . . = (P T ) . . ij n nm 13

(T ) . nm x+a,j

where the subscripts aje reduced modulo n. the observations, X'W" , is then given by

The coefficent matrix of

(M, ) . . = (T ) , ^, . and

Tc xj nm x+n-k+l,j

^Vij = ^Vi4^-2k+2,j ^^^^^ i = 1'^' j = 1'°^' ^

=

31
Let Q and Q be n x 1 vectors defined by

M

n -M

n_

n

where each z. is an m x 1 vector of observations. Then Q and Q can

be expressed by

n m n m

(Q) and

n m

k=l 3=1

n m

where i = l,n.

The product X'W is given by

X'W""'"X

n-1 n-k - , n-1 n-k _^

y P D' V'^ D P'^ -F P D' V D P"^^ n mm m n n m m m n

n-1 n-2k

-V P D' V D P"^ y P""^" D' V D P^* n mm m n n m m m n L k=0 k=0

n-1

,n-2k , 1 _^ _2k

N -N

-N' N

where N,N and N are n x n matrices,

32
Now ( P^D'V'^D V . = ( n m m m n /ij \ n

-1

m m , n—

n-m,iii n-m,n-in

n/ 13

, where the subscripts are reduced

n nm n /ij nm i+a , j -b

modulo n. The matrix of normal equations, X'W ^X, is then given by

n-1

(N)

(N)

13

j^^Q nm i+n-k,j-k'

n-1

(U ) . . , and nm i+n-k,i-2k' k=0

n=l

L(U ) 01 . where i

, „ nm i+n-2k,i-2k

k=0 '-^ i = l,n; j = l,n.

After augmenting with the two restraints the normal equations take

the form

N

• -K i 0

" — — a

" q"

•N' N 0 i -Q

i' 0 0 0 \

0

0 i' 0 0

_ 0_

33
-a Appendix B

Reduction of the Normal Equations of the Partial Closure Method

The normal equations in section 5 can be treated in the same manner as those in section 4. The product X'W"-^ is given by

x'w

^1 \' D' V"-'- n n

p"""-^ D' , v"-^, . . . p' d: n n-1 n-1 n 1

-P'nilD' v/ ... -P^ D' V""*" n n n n-1 n-1

M, . . . MM , . . . M1 n n-1 1

-M, . . . -M -M - . . .-M, LI n n-1 1 _

where M^^ and M^ are n x k matrices. Now

(D' V . a a ij

r y-1 n

a

6

n-a,a

(T and na ij I

(pa

p, v"-^). . n a a ij

(P T ) . . n na ij na x+a,j

where the subscripts are reduced modulo n. The coefficient matrix

the observations, X'W is given by

34
Let Q and Q be n x 1 vectors defined by

-Q

M, . . . M M , . . . M,

1 n n-1 1

-M, . . . -M -M , . . . -M,

1 n n-1 U 'n+1

'2n-lJ

where the z, are vectors of observations of size k x 1 if k < n

k

and (2n-k) x 1 if k > n. Then Q and Q can be expressed by

n k n-1 k

n k

k

n-1 k

n k n-1 k

k=l 3=1 k=l j=l

- E E (^„R)m.j <^k)j E E

1=1 1=1 k=l 3*=1

where i l,n.

35
The product X'w'^^X is given by

x'w -"-x [N -n]where

N =

^-i ^k ^ ^n-k ^n-k ^n-k ' k=l k=l

T D' V^^ D, P"""^ + V P""'^ D' V \ D , , and

k k k n n n-k n-k n-k'

N = k=l k=l

N = " k -1 n-k

^'^ y; p*" D' v:^ d, p"" + v\ d , . n k k k n-k n-k n-k k=l k=l

Now (P^ D' D, P^),,

n b b b n ij = (Pn

6

b b,n-b

_^n-b,b ^n-b,n-b

n 13

(P^ U , P"").. = (U , . n nb n xj nb i+a,j-c

where the subscripts are reduced modulo n. The matrix of normal

equations, X'W~ X, is then given by

n n-1

(N)

k

E (ViJ 2 <"n.n-k>i-Hn-kJ-k

n n-1

xj nk x,j-n+k n,n-k i+n-k,j*

36
where i = l,n; j = l,n.

After augmenting with the two restraints the normal equations

take the form

N -N i o' a r n

Q

-N' N 0 B

• -Q

i' 0 0 0 ^1

. 0 i' 0 0_ .^2.

where m. and m^ are restraint values from the previous series.

USCOMM-NBS-OC

37
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PLIULK ATION OR RI-PORT NO. NBSIR 75-750

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLI-: AND SUHTITLK

The Calibration of Indexing Tables by Subdivision

5. Publication Date

6. Performing Organization Code

7. AUTHOR(S) /-.V , r, T> Charles P. Reeve

8. Performing Organ. Report No. NBSIR 75-750

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS r DEPARTMENT OF COMMERCE |

WASHINGTON, D.C. 20234 J

10. Project/Task/Work Unit No.

1|. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, Slate, ZIP)

Same I-..'

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less (actual summary of most si^ilicant information. If document includes a significant

bibliography or literature survey, mention it here.)

The indexing table plays a vital role in the calibration of angle standards at the National Bureau of Standards. It is often useful to know the

values of several different intervals on the table as precisely as possible. The usual measurement designs call for a complete intercomparison of

two tables which are being calibrated simultaneously. Two new types of

measurement designs have been developed which call for only a fraction of

this effort. One type of design subdivides the entire circle on each table into equal segments, and the other design subdivides these into smaller

segments. Together the designs form a flexible system for calibrating any

desired intervals on the two tables. Formulae are given for the propagation of random errors through all levels of subdivision. An example is

included

.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Angle Standard; autocollimator; calibration; complete closure design; indexing table; least squares estimation; partial closure design; standard

deviation; subdivision; variance-covariance matrix.

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