NBSIR 74-532

NIST

PUBLICATIONS A1110M 02M307 REFERENCE

A Method of Calibrating

Two-Dimensional Reference Plates

Charles P. Reeve

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

July 26, 1974

Issued January 1980

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

QC

100

.U56

#74-532

1971

NBSIR 74-532

A METHOD OF CALIBRATING

TWO-DIMENSIONAL REFERENCE PLATES

Charles P. Reeve

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

July 26, 1974

Issued January 1980

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

Contents

Page

1. Introduction 1

2. The Physical Model 2

2.1. Reference Plate 2

2.1.1. Properties 2

2.1.2. Coordinate System 2

2.2. Measuring Machine 2

2.2.1. Capacity 2

2.2.2. Coordinate System 2

2.2.3. Location of Points 4

2.3. Setup Procedures 4

3. The Mathematical Model 4

3.1. Parameters 4

3.1.1. Description 4

3.1.2. Initial Estimates 5

3.2. Measurement Equations

3.3. Method of Solution 8

3.4. Error Analysis ^"

4. The Incomplete Mathematical Model H

-I o 5. Example L "

6. Conclusion -^2

Acknowledgement

Figures 3, 20

Table 1 14

Appendix A

Appendix B

18

Bibliography

A Method of Calibrating Two-dimensional

Reference Plates

by

Charles P. Reeve

1. Introduction

The Dimensional Technology Section recently began investigating the

problem of precisely measuring two-dimensional reference plates. The

study was motivated by two factors:

(1) The awareness of a growing need for this type of calibration

service.

(2) The acquisition of a sophisticated three coordinate measuring

machine

.

After the hardware was secured the immediate goal of the investigation

became the development of a measurement algorithm. In particular, an algorithm was needed which would estimate both the mean and standard

deviation of the relative (x,y) coordinates of the points on the reference plate. Additionally, it was desired to place as few restrictions

as possible on the placement of the plate on the measuring machine and

to include some machine parameters in the mathematical model if possible.

A search of the literature revealed one model which incorporated redundant measurements and polar coordinates. An examination of the structure of this model was helpful in developing a model in rectangular

coordinates

.

The mathematical model which was developed is nonlinear and has a high level of redundancy. It lays no claim to being the ultimate in so- phistication for this complex measurement problem, but is felt to be adequate. This report is concerned with the attainment of this immediate

goal. The material is divided into two main parts, the physical model

and the mathematical model. The physical model is not meant to be

strongly emphasized here and therefore is described briefly. The main

thrust is the presentation in detail of the mathematical model and its

data reduction algorithm.

The ultimate goal of the investigation is to develop a measurement

process which is under statistical control. On the way to attaining this

goal, the measurement process must be applied enough times to see if un- modeled systematic errors are significantly large and can be identified

and accounted for. Possible sources of such errors are machine scale

inaccuries, temperature effects, and bending effects, to name a few.

Hopefully, future reports will evolve as progress is made.
2. The Physical Model

2.1. Reference Plate

2.1.1. Properties

A "two-dimensional reference plate" is a rigid and flat surface on which a number of points, all in the same plane, are defined. The plate

is made so that the surface containing the points is parallel to the surface upon which the plate is resting. The current facility is capable of

measuring two types of plates, grid plates and ball plates. The points

on a grid plate are defined by the intersection of the centers of straight

lines which are engraved on the surface. The points on a ball plate are defined to be at the center of each ball.

2.1.2. Coordinate System

The plate coordinate system is coplanar with the points and is desth ignated by (x,y) where the coordinates of the i point are (x^,y^) . The

axes of the system could reasonably be defined in a number of ways. The

system which was adopted defines the coordinate axes in terms of the two

points p and q which are arbitrarily chosen unless otherwise specified by

the user. The x axis is defined to pass through points p and q while the

y axis is defined to pass through p and be perpendicular to the x axis.

Thus,

x = y = y =0.

P P q

The value of x

q is defined to be positive and the positive direction of

the y axis is defined such that it forms a left-handed coordinate system

which is compatible with the present machine coordinate system (see fig 1)

.

2.2. Measuring Machine

2.2.1. Capacity

The present facility has a measuring range of 48 inches on the X axis

and 24 inches on the Y axis . The vertical clearance is 12 inches . Any

plate which has a horizontal dimension greater than 32 inches will not

clear the measuring machine in the Y direction and thus cannot be normally

rotated to the different positions required during measurement.

2.2.2. Coordinate System

The measuring machine coordinate system lies in the horizontal plane

and is denoted by (X,Y) . As shown in fig 1, the X axis is defined to be

2
MEASUREMENT OF i** POINT IN

POSITION

FIGURE 1

3
colinear with the X' machine axis of measurement and the Y axis is defined

to be perpendicular to the X axis. The Y

1 machine axis of measurement is

out-of-parallel with the Y axis by a small angle a. The Y and Y' axes intersect the colinear X and X' axes at the same point. The coordinates

tti th

of the i point in the j plate position are given by (X^.,Y^).

2.2.3. Location of Points

The points on a grid plate are located by centering each intersection

of two lines under the crosshairs of a microscope either manually or automatically. The precision with which the points can be located is dependent on the quality of the lines and the precision of the instrument which

locates the intersections.

The points on a ball plate are located by centering each ball on the

vertical axis of a spindle while the spindle is rotating around the equator of the ball and giving a readout of the horizontal deflection. The

ball is considered to be centered when the deflection is minimized. The

precision with which the ball can be centered is dependent on the roundness of the ball, the sensitivity of the indicator, and the skill of the

operator.

2.3. Setup Procedures

The measurement algorithm calls for the reference plate to be measured in m different positions where m >_ 3. The degree of translation of

the plate between positions is basically irrelevant, but the angles of

rotation are important. Each rotation should be made approximately 360/m

degrees (2ir/m radians) clockwise so that the plate will be measured in m

equally spaced positions.

Before making any measurements, the procedure for rotating the plate

should be thoroughly understood so there is no danger of it being unexpectedly obstructed by any part of the measuring machine. In cases where

the geometry is such that the plate cannot be measured in equally spaced

intervals, it is permissible to choose an alternate method of spacing.

Note that setting the angles of rotation by eyesight gives sufficient

accuracy

.

3 . Mathematical Model

3.1. Parameters

3.1.1. Description

The mathematical model is an abstraction of the known parameters of

the physical model. Parameters whose existence or form is unknown are initially assumed to have a negligible effect and are put in the category

of "random measurement error".

4
The angle a which is associated with the out-of-perpendicularity of

the measurement axes has already been described. The angles ^_ , ^

represent the angles of rotation of the plate in its m positions. The

angles are measured clockwise from the \\> = 0 position which occurs when

the x axis of the plate coordinate system is aligned with the X axis of

the machine coordinate system. Also associated with the m plate positions

are the 2m parameters X .. , .... X and Y , , . . . , Y which taken pair- pi pm pi pm

wise are the coordinates of point p with respect to the machine coordinate

system.

Since the values x , y , and y were defined to be zero, they are

P P q

not considered to be parameters of the system. The remaining 2n-3 co- ordinates of the n points relative to the (x,y) coordinate system are the

n-1 values excluded) and the n-2 values y, , . . . , y

l n p

J 1J n

(y^, y^ excluded). To simplify later expressions, let the entire set of

2n+3m-2 parameters be designated by the vector 8 where

6 - (a X . Y . % . . . X Y i> x. y. . . . x y )

' pi pi 1 pm pm m 1

J l nJ n

~ (3 1

S 2 '•• 8 2n+3m-2

)T

'

A complete set of measurements gives the 2mn observations \0..> and

where i = l,n and j = i,m. The system then has 2mn-2n-3m+2 degrees

of freedom.

3.1.2. Initial Estimates

The nonlinear mathematical model is solved by an iterative process

which requires initial estimates for all parameters. Estimates for the

values X . and Y . are given by

PJ PJ

X

(0) = 0

X

. and Y

(0) = 0

Y

. PJ PJ PJ PJ

where j = l,m. The corresponding i\> . can be estimated by

\ qj pj /

where the value of k^ is dependent on the signs of the numerator and denominator as follows

5
num den

+ +0

+ 1

1

+ 2

Since a is normally very small, it can be estimated zero. The remaining

2n-3 plate coordinates can be estimated from the measurements made with

the plate in the first position by

CO) . rnx _C0), na AO), rnY v (0)> . .CO) x. - (0 ±1 - X

p]_ )cos ^ + C0

i;L - Y

p]_ ,sxn ^

-iff)- »> - < - ^)Cos »>.

3.2. Measurement Equations

X Y

Let 0.. and 0.. denote the observed and, if necessary, corrected co13 ij th th ordinates of the i point in the j position relative to the CX,Y)

coordinate system. Then, as shown in fig 1,

0?. = b + X . + a + e:. and

ij PJ iJ

Y Y

0. . = (c + Y .) sec a + e.

.

ij PJ iJ

where b = x. cos K - y. sin . ,

c = x. sin Ui . + y. cos <l»

. , and

i J i 3

a = Cc + Y .) tan a. PJ

X Y The values e.. and e , . are independent error values from a distribution

iJ iJ

2 with mean zero and variance a . The 2mn observational equations are given by
Y

= f

X

. Y

0

Y

. 13

+ eY

. 13

where i = 1, n and j = 1, m. The regression function f is defined according to the value of i as follows:

(1) if i ^ p and i j* q,

f^. = x. cos i> . - y. sin ib . + X , 13 1 31 3 P3

+(x. sin ib . + y. cos i> . + Y .) tan a

Y

f . . = (x . sin tLi . + y . cos ib . + Y . ) sec a .

(2) if i = p,

f

X

. = X . + Y . tan a

13 P3 P3

,Y

f . . = Y . sec a . 13 P3

(3) if i = q : v

f . . = x . cos i> . + X . + (x . sin . -I- Y . ) tan a

13 1 3 P3 1 3 P3

Y

f . . = (x . sin 4> . + Y . ) sec a . 13 1 3 P3

In vector notation

S - ? (a, X

pl , Y

?1 , *L, x

n> yn

) + e

= ? (3) + e .

7
3.3. Method of Solution

This nonlinear model can be solved by the Gauss-Seidel method as described by Himmelblau [5]~. The model is first linearized by expanding

it about initial guesses for its parameters in a truncated Taylor series.

The approximations are given by

(0)

and

2n+3m-2

f = Cf ) + Y

2n+3m-2

= (£

^

+ &

where 6^ = 3^ - 3^^ • The 3^ are the initial estimates for the parameters, and the subscripts on the functions and partial derivatives indicate they are evaluated using those initial estimates. (The expressions

for the partial derivatives are given in Appendix A.) Let

^Figures in brackets designate listing in the bibliography of the paper.
Then the linearized model is given by the mn pairs of equations,

_ 2n+3m-2/ 3ff

.

v 2n+3m-2f 3f

where Var [(z

X

.)_J = Var [(z

Y

.)J = a

2

. Let

ij 0 i] 0

2 o =

•

•

•

••

•

(0)

1

(0)

2

, and

-CO)

> 2n+3m-2

1.1 36 2n+3m-2/ 0

3f

Y

. u.

36 2n+3m-2/0

The least squares estimation then takes the form

E Cz

Q

} = T

Q

6 CO) where

Var Cz

Q

) - a I .
The normal equations take the form

T- T

Q

« C0) = T' Z

Q

.

After solving for the vector of unknowns 5 , the new approximations for

the parameters are given by

gd) . 3 (0) + 6 C0)

_

The calculation is then repeated with the new approximations substituted

into z^ and T^ to give the estimates 6 . The iterations are continued

according to the recursion relation

3

(k+l) m g (k) + 5 (k)

(k

Q

) until for some k=kg, the vector 6 is sufficiently near zero. The final

estimates are then given by

(k +1) (k ) (k ) 3 =3 +6 .

The number of iterations needed is, of course, dependent on the close- ness of the initial approximations to their true values and on the strictness of the convergence criterion. One criterion is that for some small

value oj,

4 <k>

< a) for i=l,2n+3m-2.

(The computer algorithm for storing the matrices and solving the normal

equations is given in Appendix B.)

3.4. Error Analysis

The variance-covariance matrix of the estimates is given by

(Tk

Tk

)_1 = \ • 0 0 0

The predicted values of the observations are given by

Ck

Q

) z, = T, 6

10
and the deviations by

d =

The estimate of a is given by

s = yj d'd/Umn-Zn-Sm+Z)

and the approximate standard deviation of the estimates by

i o

The estimate of the standard deviation of a single measurement , s

,

is likely to include effects from systematic errors such as machine scale

inaccuracies, thermal expansion, and elasticity of the plate and machine.

With a machine of high quality and a carefully controlled environment,

these errors may be small enough to pass as random errors. In that case,

the uncertainty of the calibrated values can be taken to be the three

standard deviation limit for random error,

If additional systematic errors are determined to be significant, the

estimated limit, E^, to these errors can be added giving a total uncertainty of the estimates of

Plates whose dimensions exceed the measuring range of the X' and Y

1 axes can still be calibrated if each point on the plate can be measured

in at least two positions. The complete model is modified to become an incomplete model where the measurements of off-scale points are omitted.

In the matrix T and vector z, the row elements corresponding to omitted

measurements are set to zero, or in effect, the omitted measurements are weighted zero. The calculations then proceed as usual except that the

degrees of freedom are reduced. Suppose that t is the number of times a

4. The Incomplete Mathematical Model

11
point is omitted from measurement, then the total number of measurements

is decreased by 2t, thus the degrees of freedom are reduced to

2mn-2n-3m+2-2t

.

5 . Example

This measurement process has been successfully implemented in the

calibration of ball plates. One of the first test calibrations involved

a plate with 34 one-half inch balls mounted in an irregular pattern within a 30 x 18 inch rectangle. The plate was measured in four positions

roughly 90° apart, but not every ball could be measured in each position.

The incomplete model was used with the following parameters

n = 34 (number of points)

m = 4 (number of positions)

t = 10 (total number of points omitted)

df = 174 (degrees of freedom)

P = 1 (point used for origin)

q - 11 (other point used for x axis)

.

The computed standard deviation of a single measurement was 18.8

microinches. The numbers in table 1 show how the estimates for six

selected parameters converged during three iterations. If the initial

estimates had not been so close, it may have taken an additional iteration to converge . The convergence criterion was

I 5 J < .0000005 for i = l,2n+3m-2.

x —

The uncertainty of the computed values was taken to be the three

standard deviation limit for random error.

6. Conclusion

The idea of using redundant measurements to calibrate two-dimensional

plates came about after reading of the experience that Brown [2] had with

his two-dimensional comparator which made radial measurements. The incorporation of his ideas into a system with X and Y measurements seems to have

been successful.

As of now this method has been applied to three different ball plates

.

The standard deviation of a single measurement has ranged between 15 and

18 microinches with the standard deviation of the ball coordinates being

12
slightly less. If this level of precision can be maintained then this

method will have significantly increased the precision with which ball

plates can be calibrated. Hopefully, the same will hold true for other

types of two-dimensional plates.

I wish to acknowledge three people who were of great assistance to me in the development of the methods described in this paper. I thank

Dr. Fred Johnson of the Applied Mathematics Division, a one-time associate of Duane Brown, for being kind enough to share with me the ideas

behind the sparse matrix algorithm. I also thank Dr. Chris E. Kuyatt of

the Optical Physics Division for his most helpful suggestions on linearizing the model and solving by iteration. I finally thank Dr. John A.

Simpson, also of the Optical Physics Division, for his initial ideas

which got this project off the ground, and for his encouragement which

helped me through the early stages of its development.

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14
Appendix A

Partial Derivatives of the Regression Functions

The partial derivatives of the functions fT. and f . . are given in

terms of the original parameters of the problem as follows where j=l,m:

Cl) i=l,n where i^p and i^q

3ff, r ... . f v 2

ii = (x. sin \b . + y. cos iJj . + Y .) sec a 7T i J 1 j PJ 8a

3f

X

. b —£L = 1

PJ

c. 3y^" = tan a

Pj

d. - = -x. sin . - y. cos . + x. tan a cos i> . - y. tan a sin to. 3^ i j i J 1 J 1 3

3f

X

. e. - X^ = cos Tb . + tan a sin ii

.

3x

±

y

j j

3 f

X

. f . = -sin ty^ + tan a cos

i

y

3f . .

3f

PJ

Y

(x. sin 4> . + y. cos + Y .) sec a tan a 3a v i

r j PJ

3f

Y

.

PJ

"Ii. 3y^" = sec a

i. -, ^ = x. sec a cos . - y. sec a sin ib J 3^. i J 1 :

15
Y

3f . . k U

x Y

3f

.

= sec a sin lii

.

J

i « = sec a cos ii>

.

J

C2) i=p

3f

X

. 3a

• =: Y . sec a

PJ

3f?.

b . —Il = i 3X . X

PJ

3f

X

. c "gy^" = tan a

PJ

= 0

3f

X

. d. -^L

J

3f

Y

. 1J e - - Y . sec a tan a 3a pj

3f

Y

. 3X . U

PJ

3f

Y

12. _

3Y

PJ

y

3f

= sec a

(3) i=q

3f

X -.1 = (x. sin H>. + Y

pj

) sec

2 a

16
tan a

x. (-sin to. + tan a cos

l 3

cos to . + tan a sin to . 3 . 3

(x . sin to . + Y . ) sec a

i J PJ

0

sec a

x. sec a cos to. i 3

sec a sin to . .

17
Appendix B

Computer Reduction using Sparse Matrix Algorithm

The normal equations which were given in section 3.3 can be written

where t instead of ' is used to indicate transposition. Let N = T T and

b = T

t z. Then the normal equations are given by

where T is 2mn x (2n+3m+l)

,

N is (2n+3m+l) x (2n+3m+l)

,

z is 2mn x 1,

b is (2n+3m+l) x 1, and

5 is (2n+3m+l) x 1.

Recall that the three parameters x , y , and y were excluded from

p V 'q

the original model. For the purpose of the computer reduction, it is

more convenient to reinsert these in their proper columns in the T matrix and set all elements of those columns equal to zero. When the N matrix is computed, the three rows and columns corresponding to the three

parameters will contain only zeros. The three diagonal elements are then

set equal to one. The three corresponding elements in the T

t

z vector are automatically forced to be zero. The result of this manipulation is that

the estimates of x^, y , and y^ are forced to be exactly zero, and they

appear in their natural order in the computer printout.

The solution to the normal equations is given by

If the normal equations are formed and solved by the usual full matrix

methods on the computer, then the storage required for the six vectors

2 and matrices given above is 24n + 220n + 364 for m = 4 (which is generally the case) . The storage can be computed for several values of n

as shown in the following table.

N <5 = b

6 = N'H) = Mb.

n storage required

4 1,628

20,864

262,364

3,928,364

25

100

400

18
Obviously for large values of n the storage requirements get out of hand.

Fortunately, the matrices T and N are sparse, that is, they have only a small percentage of nonzero elements. If they are properly partitioned

then their submatrices which are of block diagonal form can be compacted

so that only the blocks are carried in storage. An example is shown in

figure 2 where n = 8 and m = 3. The matrix T can be written as T = (T T)

where T is 2mn x (3m+l) and T is 2mn x 2n. Then

N

N

and

The normal equations can then be written in the partitioned form

where N is a (3m+l) x (3m+l) matrix with full first row column

and m 3x3 blocks,

N is a (3m+l) x 2n full matrix,

N is a 2n x 2n block diagonal matrix of n 2x2 blocks,

and the 6 vector is partitioned to be compatible with the partitioning

of N.

/k m

>

Let N~^" = M 4 |

. The equations for the partitioned inverse are

M = (N - NN V)"1

M = -MNN"

1

, and

M = N"

1

+ N

-1^

19
h-T-i- [t][f]

COMPACTING OF

SPARSE MATRICES

FOR COMPUTER

STORAGE
The equations for 5 and <S are

<5 = Mb + Mb

= M(b - wTH) , and

5 = mH + Mb

= N"

1^ + N^^^CNN-1

^" - b) .

The sequence of computations and the required storage are given below. The matrices which are carried in compact form are bracketed.

1 . Input

1.1 It]

1.2 [T]

1.3 z

2. Compute N and b

2.1 N = [T

t

]lT]

2.2 N = [T

t

] [TJ

2.3 [N]= [T^IT]

2.4 b = [T

t

]z

2.5 b = [T

t ]z

3 . Compute M

3.1 [A]=[N"1

]

3.2 B = N[A]

3.3 C = BN

1"

3.4 C = N - C

3.5 M = C"

1

Dimension

2mnx4

2mnx2

2mnxl

(3m+l)x(3m-fl)

C3m+l)x2n

2nx2

(3m+l)xl

2nxl

Storage

8mn

4mn

2mn

Subtotal: 14mn

9m +6m+l

6mn+2n

4n

3m+l

2

2n

Subtotal: 9m +6mn+9m-!-8n+2

(store in N)

C3m+l)x2n

(3m+l)x(3m+l)

6mn+2n

9m +6m+l

(store in C)

Subtotal: 9m +6mn+6m+2n+l

21
4. Compute <S

4.1 D = [A]b 2nxl

4.2 E = ND (3m+l)xl

4.3 5 = M(b - E) (3m+l)xl

5 . Compute 5

5.1 F = B

1^ 2nxC3m+l)

5.2 E = E -b

5.3 5 = D + FE 2nxl

6 . Compute diagonal elements of M

6.1 G = diag(FB)

6.2 diag (M) =

diagQN"1

]) + G

Subtotal:

Subtotal:

2nxl

(store in G)

2n

3m+l

3m+l

6mn+2n+2

6mn+2n

2n

6mn+4n

2n

Subtotal; 2n

Additional storage

for inversion of

the C matrix: (3m+l)x(3m+l)

Grand Total Storage:

9m +6m+l

2 7m +32mn+2 7m+18n+6

Note that the inversion of N involves only the simple inversion of several 2x2 matrices.

The following table compares the storage required for the sparse

matrix algorithm with the storage required for the full matrix algorithm

for several values of n with m = 4.

storage required ratxo

4

25

100

400

sparse

1,130

4,196

15,146

58,946

full

1,628

20,864

262,364

3,928,364

sparse/full

.694

.201

.058

.015

22
Additional storage is required by the computer algorithm for some single variables and a few other vectors, but it is only a small fraction

of the above amounts. Using the sparse matrix algorithm, the present

computer facilities at the National Bureau of Standards can handle values of n at least as large as 400.

This method of partitioning and solving the matrix equations is

given in a form similar to the one which was worked out by Brown [1]

.

There can be no doubt of its value because of the tremendous savings

in computer storage that it affords.

23
BibliographyBrown, Duane C, A Solution to the General Problem of Multiple Station Analytical Stereotriangulation, AFMTC TR 58-8,

February 1958.

Brown, Duane C. , Computational Tradeoffs in the Design of

a One Micron Plate Comparator, Presented at the 1967 Semiannual Convention of the American Society of Photogrammetry

and The American Congress on Surveying and Mapping, St. Louis

October 2-5, 1967.

Draper, N. R. , and Smith, H. , Applied Regression Analysis,

John Wiley & Sons, Inc., 1968, Ch. 10.

Goldberger, Arthur S., Econometric Theory, John Wiley & Sons,

Inc., 1964, pp. 27-28.

Himmelblau, David M. , Process Analysis by Statistical Methods

John Wiley & Sons, Inc., 1970, Ch. 6.

Mood, A. M. and Graybill, F. A., Introduction to the Theory

of Statistics, 2nd Edition, McGraw-Hill Book Co., 1963.

24
NB5-M4A (REV. 7-73>

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 74-532

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

A Method of Calibrating Two-dimensional Reference Plates

5. Publication Date

July 26, 1974

6. Performing Organization Code

7. AUTHOR(S)Charles P. Reeve

8. Performing Organ. Report No. NBSIR 74-532

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Tasle/Work Unit No. 2321120

11. Contract /Grant No.

2. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

NBS

13. Type of Report 3t Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or leaa factual summary of moat significant information. If document includes a significant

bibliography or literature survey, mention it here.)

With the acquisition of a three coordinate measuring machine the Dimensional

Technology Section began investigating the problem of precisely measuring twodimensional reference plates. The algorithm which was developed for this measurement process is highly redundant and incorporates a nonlinear mathematical model.

Both the mean and standard deviation of the relative (x»y) coordinates of the points

on the reference plate are estimated.

The long term goal of the investigation is to use this algorithm as a tool in

developing a measurement process which is under statistical control. On the way to

that goal certain problems and pitfalls associated with two-dimensional measurements

are sure to arise and will be examined. It is hoped that some light will be shed

on areas such as long term stability of measuring machine performance, measuring

machine characterization, temperature problems, and the stability of two-dimensional

standards.

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) Calibration; convergence criterion; Gauss-Seidel method;

incomplete model; least squares estimation; measurement algorithm; nonlinear mathematics

model; sparse matrix algorithm; standard deviation; systematic errors; two-dimensional

reference plates.

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