NBS

PUBLICATIONS

NBSIR 79-1758

The Calibration of a Roundness

Standard

A111D4 0231b5

Charles P. Reeve

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

June 1979

1

‘ S. DEPARTMENT OF COMMERCE

C

TIONAL BUREAU OF STANDARDS

100

U56

79-17 5^

C . 2

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<*T*-ii Swrsau of $‘ ari

SEP 2 0 1979

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NBSIR 79-1758

^OAOO

U "5 to THE CALIBRATION OF A ROUNDNESS

STANDARD

Charles P. Reeve

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

June 1 979

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps. Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director

Contents

1. Introduction 1

2. Preliminaries 1

2.1. Roundness Measuring Instruments 1

2.2. The Polar Graph 2

2.3. Methods of Determining Roundness 2

3. Procedure for Taking Data 5

4. Mathematical Model 8

4.1. Parameters 8

4.2. Measurement Equations 9

4.3. Restraints 10

4.4. Method of Solution 11

4.5. Simplified Expression of Results 12

5. Sources of Error 14

6. Graphical Display of Roundness Profile 14

7 . Example , . 16

8. Conclusion 16

References 20

The Calibration of a Roundness Standard

by

Charles P. Reeve

1. Introduction

The Dimensional Metrology Group of the National Bureau of Standards

has been making roundness measurements for many years. In some of the

measurements a high level of precision is not needed. This usually

occurs for objects such as cylinders, spheres, and ring gages where

roundness is not the parameter of primary importance. In these cases it

is routine to make a single trace with a roundness measuring instrument

whose output is a polar graph. Much literature has been devoted to the

interpretation of data taken in this form.

When calibrating objects which are to be used solely as roundness

standards, such as hemispheres, a higher level of precision is called

for. During the last decade a measurement algorithm has been used at

NBS which is unlike any thus far found in the literature. It involves

making several traces of the roundness standard (also referred to as the

"workpiece") where the standard is rotated between traces. A least

squares analysis is performed on the resulting measurements enabling the

noncircularity in the spindle to be separated from the profile of the

standard

.

The primary purpose of this paper is to present a detailed description of the mathematical model for this particular measurement process.

Certain related topics are discussed briefly. A method of graphically

displaying the roundness profile is described and an example is given.

Topics not discussed because of their availability in the literature are effects of stylus type, alignment of stylus, alignment of workpiece, and

effects of imperfect centering.

2. Preliminaries

2.1. Roundness Measuring Instruments

There are two basic types of high precision roundness measuring

instruments. One type has a high precision spindle which carries a sensitive indicator (stylus) around the circumference of the workpiece

which is held stationary. The other type has a high precision turntable

which rotates the workpiece against a stylus which is held stationary.

NBS currently has both types of instruments.

1
2.2. The Polar Graph

The output in most roundness calibrations is a polar graph. The

roundness profile of the workpiece is transferred to the graph as the

surface of the workpiece deflects the stylus. The continuous trace

represents the radial deviations of the workpiece from the center of

rotation of the spindle. Included in the trace are the deviations from

circularity of the path of the spindle. The magnification factor exaggerates the profile in the radial direction, but the angular positions

on the graph retain the same relationship as on the workpiece itself.

In some cases the output of a roundness trace consists not of a polar graph but of a set of n numbers which represent the radial deviations of the workpiece at n equally spaced points around the circumference. This is equivalent to a polar graph except that the trace consists of a discrete number of points instead of a continuous curve.

Whether discrete or continuous, the trace may be interpreted in

several ways as described in the following section.

2.3. Methods of Determining Roundness

There are several ways to assign numerical "roundness" values to the

workpiece from its polar graph. One way is to draw either the minimum

circumscribing circle or the maximum inscribed circle and state the

maximum departure from one or the other. Another way is to draw the pair

of concentric circles which enclose the graph and have the minimum

radial separation. This separation is then a measure of the "out-ofroundness". Examples of the above are illustrated in some of the

references [2,5,8,10].*

A slightly more sophisticated approach is to fit a "least squares

circle" (LSC) to the graph. The LSC is fit through an integral number

of equally spaced points on the curve rather than through the entire

curve. The maximum deviation from the LSC is then another measure of the

"out-of-roundness" of the workpiece.

The following derivation of the equations for the LSC is a summary

of the more rigorous derivations given in several of the references

[2,5,7,10,11].

Consider a polar graph given in a rectangular coordinate system as shown in figure 2.1. Let point 0 be the center of rotation of the spindle

and y^ be the distance from 0 to point P on the curve at the angular

position 0^. Let point (a,b) be the center of the LSC and R be the

radius of the LSC. Let c

2 = a

2 + b2 and tan a = b/a. Then the triangle

between points 0, P, and (a,b) has the relationship

y ± = [(R+A^)2 - c^sin^(0^-a) + c cos (0^-a) (2-1)

*Figures in brackets indicate literature references listed at the end of

this paper.

2
DETERMINATION OF LEAST SQUARES CIRCLE

Figure 2.1

3
where A_^ is the deviation of point P from the LSC.

Since c is many orders of magnitude less than R (in a well centered

trace), the term c

2 sin2 (0^-a) can be neglected, thus

y. = R + c cos(0.-a) + A. . (2-2)

’l 11

Since a = c cosa and b = c sina this expression can be expanded to give

y. = R + c cos0 . cosa + c sin0 . sina + A. , (2-3) ; i i ii or

y . R + a cos0 . + b sin0 „ + A

„

J ± l ii (2-4)

Let {0_^, i=l,n) be n equally spaced angles. Then the sum of squares of n

the deviations, ^ A

2

, is minimized by the usual method of least squares

i=l

to give the following estimates for the LSC parameters:

-i n i = i L y. n f—' y i 1=1

= — £ y .cos0 . n f—L J i i i=l

and (2-5)

n

= V y . sm0 . n t-*

J i i i=l

The deviations A^ are given by A-j_ = y^ - R - a cos0^ - b sin0^

where i=l,n.

( 2 -6)

When roundness values are assigned from a single graph, as in the

above cases, the graph contains both the workpiece profile and the spindle

profile. In a precision instrument the spindle error is usually small

and within known limits and can be ignored except when the most precise

measurements are needed.

There are currently two methods available for removing spindle error

at the cost of making multiple traces. One method requiring two traces

is given by Donaldson [4] for use on a turntable-type instrument. Between

traces the workpiece and stylus positions are rotated 180° while the

shaft and housing positions remain the same. If both graphs are recorded

on the same chart the true workpiece profile is obtained by drawing a third graph halfway between the two. Though well suited for a turntabletype instrument, this method is not easily adapted to a spindle-type

instrument

.

4
The other method of removing spindle error is the main topic of

this paper. It requires several polar graphs and works equally well on either type of instrument. A detailed discussion of this method as applied to a spindle-type instrument is given in the following sections.

3. Procedure for Taking Data

The workpiece should be centered as well as possible under the

spindle. The mark on the workpiece which donotes the zero angular position is aligned with the zero position of the spindle as shown in figure

3.1. A trace is made with the workpiece in this position. The workpiece

is then rotated clockwise by 360/n degrees as shown in figure 3.2 and

another trace is made. This process is continued until n traces have

been made. The value of n is arbitrary but is usually at least 4. Note

in the above figures that the zero spindle position is shown pointing

"south". This convention was arbitrarily chosen for the current measuring

instrument

.

It is important to record the angle <5 between the lever arm of the

stylus and the tangent to the workpiece at the point of contact. This

angle can usually be set to 0° for a sphere, hemisphere, or cylinder as shown in figures 3.3, 3.4, and 3.5 respectively. If an obstruction

exists, as shown in figure 3.6, then the angle must necessarily be greater

than 0°. The correction for 5 assures that the profile of the workpiece

will be measured normal to the point of contact. A sphere is normally

measured about its equator. A hemisphere is measured as nearly to the

base as possible without the stylus being obstructed. A cylinder is

measured at some specified distance above its base.

If the output of the instrument is a polar graph for each trace of

the workpiece, then the observer should measure the radial distance (in

polar graph units) from the center of each graph to the curve at each of

the n angular positions and record the values on the graphs at the proper

places. The graphs should be numbered from 1 to n and the scale factor

(K) of the instrument should be noted on at least one of them.

4. Mathematical Model

4.1. Parameters

Let the n angular positions on the circumference of the workpiece be

numbered counterclockwise from 1 to n beginning with the 0° position.

Let the angular positions of the spindle be similarly numbered without

regard to the direction in which the spindle rotates. The i^ angular

position is then denoted by

0 .x 360(i-l) n degrees

5
270

270

180 ° Workpiece in

I s

' position

\

- 90 °

Workpiece

circumference

Workpiece base

Spindle Spindle

path

Figure 3.1

180° Workpiece in

2

nc* position

0

°

Figure 3.2

6
PROBE POSITIONS FOR 4 TYPES OF STANDARDS

Sphere Hemisphere

8 = 0

°

Figure 3.3 Figure 3.4

Cylinder

8 = 0

°

1

3

1

:

Figure 3.5

Cylinder (with cap)

8 > 0

°

Figure 3.6

7
where i=l,n. Let the deviations from the LSC of the workpiece at the 0_^

position be given by a . , and let the deviation of the spindle from its

t h

LSC at the 0_^ position be given by B^. For the j graph let the three

parameters which define the LSC on the gr aph be given by R_. , a_. , and b_. (see equation 2-4) . In an idealized measurement system these parameters

would be constant for all j. In reality, though, each rotation of the

workpiece causes it to shift a small amount vertically and horizontally.

To account for this shifting, separate parameters are needed for each

trace.

Let y.. be the observed distance (in polar graph units) from the

il t h

center of the j graph to the point on the curve which corresponds to

the 0^ position of the spindle . Let K be the magnification factor of

the measuring instrument given in (micro inches/polar graph unit)*, and

let z . . be given by

1

1

z . . = (K cos6) y . . il il

for i=l,n and j=l,n. The z..’s are then the observations to be used in

the mathematical model. 1~1

4.2. Measurement Equations

The measurement equations take the form

z.. = a.,. - B . + R. + a.cos0. + b.sinO. +

il i+l-l il 1 il i il

where the e.. are independent error values from a distribution with mean

il zero and variance a

2

. (The subscript of a is reduced modulo n.) Let

z (z n ... znl z 12 a = (a, ... a )

’ , 1 n

In

z )* nn

9

B = (B

1 ... 3 n

)

f , (4-1)

<j) = (R_ a_ b_ . . . R a b )

' 111 nun

e = (e.11 nl 12

* •

’ n2

* * In

’ ‘ e ) ’ , and nn *Customary U.S. units are used in this report rather than the recognized

metric (SI) units. The well established ongoing calibration procedures

described here employ customary units exclusively. The conversion to SI

units will be made at a future time..

S
P° -I A 0 0

P

1

-I 0 A 0

X =

Pn X -I 0 0 A

(4-2)

where ' denotes matrix or vector transposition, and P is the n x n permutation matrix given by

P

1

0 10

0 0 1

0 0 0

10 0

0 0

“

0 0

0 1

0 0

(4-3)

I is the nxn identity matrix, and A is the nx3 matrix given by

1 cos6^ sin0^

1 cosQ^ sin02

(4-4)

1 cos0 sin0

L n nj

(Note that P° = pn

= I.) The measurement equations can then be written

in the matrix form

z X

A

+ e (4-5)

4.3. Restraints

2 The n by 5n system given above has no unique least squares solution

because the restraints which were arbitrarily set on a and 6 (that they

be deviations from least squares circles) have not yet been included in

the model. These are included by adding the six equations

n n n y a. =

1

V a. cos0

1 1 X

i=l

sin0 .x0

and

9

(4-6)
E

i=l

n

- E

i=l

$ . COS0

nT, 3 . sinG . = 0

i-1

1

to the model. In matrix notation the restraints take the form

A’a = A'B = 0 (4-7)

where A is defined as in equation 4-4.

4.4. Method of Solution

The model can now be solved by the method of restrained least

squares [3]

.

The least squares estimation takes the form

where

E(z)

Var(z) = a

2 I.

(4-8)

The normal equations (incorporating the six restraints) take the form

A 0

X'X o A

0 0

A' 0 0 0 0

0 A' 0 0 0

(4-9)

where X = (X^ X . .

.

V are Lagrangian multipliers entering in the minimization process. The least squares estimates of the unknowns are given

by

a A 0

-1

B 0

8 X'X 0 A X'z C 0 B X'z

V = 0 0 = D E

A' 0 0 0 0 0 B' 0 D' 0 0 0

X 0 A’ 0 0 0 0 0 B' E' 0 0 0

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

values of the observations are given by

10
z = X (4-11)

and the deviations by

d = z = z (4-12)

The estimate of a is given by

= T d / (n^ - 5n + 6 ) (4-13)

and the standard deviation of the estimates by

s~ = +lc . . s and ss = A /C ~ . . s a. \ li \ n+i,n+i (4-14)

4.5. Simplified Expression of Results

The matrix of normal equations and its inverse, both given in equation 4-10, can be written more specifically as

nl -J PnA • • • P

1

A A 0

-J nl -A • • • -A 0 A

A'P

1

-A' A'A 0 0 0

A t A’P -A’ 0 A’A 0 0

A’ 0 0 . . . 0 0 0

0' A’ 0 0 0 0

and

11
to

0 0 0 AQ

o 0

0 i(I-AQ

0

A ') 0 » • • 0 0

o co

0 0 o o

•

0 A o o

0 0 0

•

•

o o i

—

i

0

1

o o

<

o

O' 0

o 0

1 0 • • -Vi 0 0

0 QqA

' °0 • • • o o 0 0

\

= (A’P^)"1

. The expressions for the estimates can be reduced

n n

a

t - £ Z

j=l k=l

t k-i+j+n z, . 9 kJ

n n

6 x

-5=1 V=1

t k-i+l-fnz i

• > kJ

n

R.

3 k=l 'KJ

’ (4-15)

n

i. = - T J n k=i

z, . cos0, , and

kj k

n

b. = - Vz,.

3 n « k3 sin0.

k=l

where

t = m

n-3

n

if m=l, and

- 1

„ (1 + 2 cos0 ) if 2<m<n

L m

(4-16)

12
(The subscripts of t are reduced modulo n.) The standard deviation of

the profile estimates is given by

/n-3

s s (4-17)

a .1 n

for i=l,n.

5. Sources of Error

Some known sources of error in roundness measurement are:

(1) horizontal and vertical fluctuations of spindle path from one trace to the next,

(2) variability due to stylus type,

(3) nonuniformity of polar graphs on which traces are recorded, and

(4) variability in observer's interpolation of trace position on polar graph.

These errors are assumed to accumulate in a random fashion and thus are included in the estimate of the standard deviation.

In the measurement of roundness it is particularly useful to have a graph of the measured profile. The n values obtained from the previously

described method of calibration can be plotted on a polar graph to give

an indication of the shape of the workpiece. This graph can be enhanced

by connecting the points with a smooth curve. One good method of doing

this involves the use of periodic cubic spline functions. Let the set

of points {(0^,a^), i=l,n+l} represent the profile of the workpiece in

rectangular coordinates where 0 . .. = 0 + 0 O and a = a- as shown in

figure 6.1. In each of the intervals ® jq-q) > i=l,nj- a cubic polynomial is fit to the points subject to the condition that the cubic

between 0. . and 0. must agree with the cubic between 0. and 0.., at 0.

i-1 i 6 i i+l i

in their first and second derivatives. Similarly, the cubic between 0^

and 0 O must agree with the cubic between 0 and 0 ... at 0, and 0 _ . 2 6 n n+1 1 n+1

The last requirement makes the spline periodic.

6. Graphical Display of Roundness Profile

13
RECTANGULAR GRAPH OF ROUNDNESS PROFILE

ANGLE —*

Figure 6.1

POLAR GRAPH OF ROUNDNESS PROFILE

Figure 6.2

14
The resulting equations form a system of 3n equations in 3n unknowns

which is tridiagonal except for nonzero elements in the upper right and

lower left hand corners of the associated matrix. In the current computer

program the equations are formulated by extending the method given in

UNIVAC [12] for a nonperiodic cubic spline. The resulting equations are

then solved by the method given in Ahlberg [1] for a periodic spline.

Once the coefficients of the n cubics are obtained the interpolated value

at a given 0q is determined by plugging the value of 0q into the appropriate cubic. The rectangular* graph is then transformed to a golar graph

by plotting the set of points { (p+a_^, 0 ^) , i=l ,n} where p+cu is the distance

from the origin and 0_^ is the angle (see fig. 6.2). The parameter p is

an arbitrary constant. Note that 0 ^ is identical to 0^ on the polar

graph.

7 . Example

A glass hemisphere designated L9474 was measured on a spindle-type

instrument. Twelve traces were made with the hemisphere rotated 30°

between each trace. The radial distances y.. (in polar graph units)

were recorded at 30° intervals on each trace"1 as shown in figure 7.1. The

scale factor was determined to be K = 5.56 micro inches /unit and 6 was approximately 0°. The true radial distances z_^_. = (K cos6) y_ where

i=l,n and j=l,n were computed. Then using equations 4-15 and 4-16 the

estimates of the unknown parameters were computed as given in figure 7.2.

(Note also the standard deviation values.) The profile of the hemisphere

was plotted by the computer as shown in figure 7.3.^ The small signs

indicate the values of the least squares estimates a. for i=l,12 relative

to the dotted circle (outward is +) . In this example interpolated values

were computed at 720 equally spaced points. The values were connected

by straight lines, but due to their closeness they give the appearance

of a smooth continuous curve. The estimates R. , a., and b. are omitted

J 3 1 because they are normally not of interest. The total uncertainty of each

profile value was taken to be the three standard deviation limit for

random error

.

8. Conclusion

The data reduction process is now fully computerized and incorporates

the simplified expressions for the estimates given in equations 4-15 and

4-16. The output includes the usual Report of Calibration plus a plot of

the computed roundness profile.

The strong point of this method of roundness measurement is that it

allows the spindle error to be separated from the workpiece profile while

giving an estimate of the measurement precision. It should be used whenever the desired level of precision is so great that the out-of-roundness

of the spindle path becomes significant.

15
Observed Values of

i

ii 1^ 2 3 4 5 6 7 8 9 10 11 12

1 8.1 6.9 6.7 5.3 7.5 6.9 5.6 7.2 7.0 7.8 7.2 7.2

2 7.2 6.8 6.2 5.1 7.4 6.7 5.4 7.1 6.9 7.8 7.2 6.3

3 6.

6

6.8 6.0 5.1 7.2 6 .

3

5.3 6.9 6.8 7.3 7.1 6.0

4 6.1 6.7 6.0 5.1 7.0 5.9 5.4 6.3 6.8 6.9 7.0 5.7

5 5.6 6.2 6.2 5.2 6.8 5.7 5.6 6.0 6.9 6.2 7.1 5.4

6 5.5 6.0 6.8 5.3 6.5 5.5 5.8 5.8 7.0 5.7 7.1 5.5

7 5.8 5.7 7.4 5.4 6.5 5.6 6.0 5.7 7.3 5.4 7.1 5.9

8 6.3 5.6 7.9 5.6 6.7 6.0 6.2 5.8 7.6 5.3 7.0 6.2

9 6.9 5.8 8.0 5.6 6.8 6.0 6.1 5.9 7.7 5.3 7.0 6.9

10 7.7 6.0 8.0 5.7 7.0 6.4 6.0 6.2 7.3 6.0 7.0 7.0

11 8.3 6.3 7.8 5.5 7.2 6.8 5.9 6.7 7.2 6.5 7.0 7.3

12 8.5 6.8 7.3 5.3 7.3 6.9 5.7 7.0 7.0 7.1 7.0 7.2

K = 5.56 microinches/polar chart unit

5 = 0°

Figure 7.1

Least squares estimates of profiles

(micro inches)

Position

(ap

Hemisphere

«i>

Spindle

0° .14 -.48

30° -.25 .10

60° -.15 .10

90° .07 .04

120° .21 .12

150° -.06 .05

180° .02 -.15

210° -.01 -.30

240° -.20 .11

270° .09 .26

300° .00 .09

330° .13 .06

Std

.

dev. of residuals = .50 micro inch

Std

.

dev

.

of estimates = .13 micro inch

certainty of estimates = .38 micro inch

Figure 7.2

16
MICROINCHES

2.000

p

1.800 -

1.600 -

1.400 -

1.200 -

1.000 -

.800 -

.600 -

.400 -

.200 -

0 -

-.200 -

-.400 -

-.600 -

-.800 -

- 1.000 -

- 1.200 -

-1.400 -

-1 600 -

-1.800 -

- 2.000- 2.000

ROUNDNESS STANDARD NO. L9464 ZERO POSITION AT RIGHT EDGE

i i i i i i i i j i i i i i i i i -1.600 -1.200 -.800 -.400 0 .400 .800 1.200 1.600 MICROINCHES

Figure 7.3

j i 2.000

17
References

[1] Ahlberg, J. H. , Nilson, E. N. , and Walsh, J. L. , The Theory of Splines

and Their Applications, Academic Press, 1967, Section 2.1.

[2] British Standard 3730:1964, Methods for the Assessment of Departures from Roundness, British Standards Institute, 1964.

[3] Cameron, J. M. , The Use of the Method of Least Squares in Calibration, NBSIR 74-587, September 1974.

[4] Donaldson, R. R. , A Simple Method for Separating Spindle Error

from Test Ball Roundness Error, UCRL-73805 Preprint, Lawrence

Livermore Laboratory, Livermore, California, 1972.

[5] Farago, Francis T. , Handbook of Dimensional Measurement, Industrial

Press Inc., New York, 1968.

[6] Good, Charles H. and Laakso, L. H. , Frequently Ignored Sources of

Error in Roundness Measurement, SP64-42, American Society of

Tool and Manufacturing Engineers, 1964.

[7] Miller, L. , Engineering Dimensional Metrology, Edward Arnold Ltd.,

London, 1962.

[8] Moore, Wayne R. , Foundations of Mechanical Accuracy, The Moore

Special Tool Company, 1970.

[9] Reason, R. E., The Talyrond Handbook, Taylor, Taylor & Hobson, Ltd.,

1954, Part III.

[10] Reason, R. E., 1966 Report on the Measurement of Roundness, The

Rank Organization, Rank Taylor Hobson.

[11] Thomas, Geoffrey G. , Engineering Metrology, Butterworth and Company

Ltd., London, 1974.

[12] UNIVAC, MATH-PACK Programmers Reference, Sperry Rand Corporation,

1970, pp. 2-68 through 2-72.

18
NBS-114A irev. 9-7*)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 79-1758Z. fteciprent's Accession No.

4. TITLE AND SUBTITLE

The Calibration of a Roundness Standard

5. Publication Date June 1979

- MMWI

7. AUTHOR(S)

Charles P. Reeve

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

(Same as No. 9)

13. Type of Report & Period Covered

15. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200~word or less factual summary of most si ghificant information. If document includes a significant bibliography or

literature survey, mention it here.) For the past decade the National Bureau of Standards has been making high

precision roundness measurements using an algorithm which allows the spindle

out-of- roundness to be separated from the true profile of the workpiece. The

algorithm requires multiple traces on a roundness measuring instrument, and the

resulting redundancy provides an estimate of the measurement precision. The

primary purpose of this paper us to describe in detail the mathematical model used

for this measurement process. Some related topics are discussed briefly and an example is given.

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 aemicolona

)

Least squares circle; least squares estimation; polar graph; restraints;

roundness standard; spindle error; standard deviation; stylus.

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