NBSIR 88-3759

A Simple Method for Measuring

Straightness of Coordinate

Measuring Machines

A. K. Elshennawy

Fang-sheng Jing

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Gaithersburg, MD 20899

Final Report

May 1988

75 Years Stimulating America's Progress

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 88-3759

A SIMPLE METHOD FOR MEASURING

STRAIGHTNESS OF COORDINATE

MEASURING MACHINES

A. K. Elshennawy

Fang-sheng Jing

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Gaithersburg, MD 20899

Final Report

May 1988

U.S. DEPARTMENT OF COMMERCE, C. William Verity, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

A SIMPLE METHOD FOR MEASURING

STRAIGHTNESS OF COORDINATE MEASURING MACHINES

Ahmad K. Elshennawy

University of Central Florida

and

Fang-Sheng Jing

Tianjin University, Peoples Republic of China
. '

-
A SIMPLE METHOD FOR MEASURING

STRAIGHTNESS OF COORDINATE MEASURING MACHINES

Introduction

Straightness errors contribute significantly to the total error

budget of coordinate measuring machines. There are two straightness

error parameters for each axis; horizontal and vertical straightness

.

According to Bryan [1] , straightness error is "a non—linear movement

of the machine axis that an indicator sees when it is either

stationary reading against a perfect straight edge supported on a

moving slide or moved by the slide along a perfect straight edge

vhich is stationary.” The MITF definition for straightness error is

that "Carriage straightness error is the motion of a carriage,

designed for linear translation, perpendicular to the intended motion

axis. Two types of straightness error are defined: Type M

straightness, vhich is the movement one would measure with a

stationary indicator against a perfect straightedge supported on the

itioving carriage and aligned with its motion axis; and Type F

straightness, vhich is that movement measured by an indicator

attached to the carriage reading against a fixed straightedge

similarly aligned" [2]. Thus, straightness error can be determined

as the deviation of measurement data from a straight line.

For the purpose of calibrating coordinate measuring machines, a

laser interferometer system equipped with straightness optics is

often used to measure straightness as is a mechanical straightedge

and indicator, in addition to the relative high cost of a laser

interferometer system, it needs and experienced operator to perform

the measurements.
2

A simple and rapid method for measuring the straightness of

coordinate measuring machines was developed using the ball bar.

Theoretical Approach

As shown in Figure 1, if point D moves at a constant distance

from another fixed point 0 forming a circular path in the X-Y plane

for example, then the difference between the exact Y- location of

this point from the Y- location as measured by the machine represent

the Y component of motion error at that point.

In using the ball bar, the fixed end was clamped at point 0 in

the X-Y plane of the coordinate measuring machine and the free ball

moved along the arc A^_

If point D is a general position of the free ball, the true

distance (Yt) between point D and line AB can be obtained if the true

length, L of the ball bar is known. Thus, by obtaining the actual

distance (Y) between point D and line AB, the deviation of Y from Yjrepresent the motion error at point D. Line AB is the reference

standard line. Fran the mathematical model of the machine, the Y

motion can be expressed by

*-*t-*yOO - S

y

(X) - X.£Z

(Y)

where

Y — actual machine reading

Yt = 'true' value for Y- coordinate

4 y00 = Scale error of Y- axis

s y(x) = Y-straightness of x-axis motion ~ Yaw of Y— axis
3

Thus,

« y

(x) = Yt - Y - 6

y

(Y) - X.EZ

(Y) ( 2 )

From Figure 1,

( 3 )

Ihere was one assumption for this analysis, that is the machine

scale errors have negligible values. If this assumption were not

valid, then a step gage would be used to measure the scale errors of

the machine following the procedure outlined in the B89.1.12 standard

[ 3 ]

.

Within a small range, the rotation of the Y-axis slide about the

Z-axis can be expressed as

By aligning the data such that straightness error is zero at both

ends, then at point Bx (Xe,Ye) ; 6 y(Xe) = 0

Using equations (2) , (3) , and (4) and substituting X^Xq, Y=Ye, ai

X=Xe, then.

e z

(Y) = K . Y

where K is a constant.

substituting equations (3) and (4) into equation ( 1 ) , then

Therefore,
4

L -Vl2-(R-Xg) 2 - Ye

- 6

y

(Ye)

Substituting equations (4) , (5) , and (7) into equation (2) , we

get the general expression for ^(x) at any point,

<5 y

(x) = Yt - Y - 6

y

(Y) - K.X.Y (8)

If the assumption of zero scale errors is valid, then ^y(Y) = o,

and for any point (X,Y) , the Y—straightness of X is given by:

VL

2-(R-X) 2-Ye\

). X.Y (9)

Xe . Ye J

liqxsrimerrts & Results

A ball bar length of about 502.00 ram was used for the

experiments. The fixed end of the ball bar was clanped at a height

of 380 mm above the machine table and the x and y coordinates for the

fixed ball were 400 and 30 mm respectively. The measuring range for

the experiments, that is the distance between points A, and B in

Figure 1, was 600 mm. Straightness errors were obtained from

equation (9)

.

Coordinates of points Ax and ^ (^,Ye) were determined from the

CMM display. The x-coordinate from point A^ was set to zero at the

beginning of the measurements and point Bx was located such that Ax and Bi have the same y-coordinate (Ye) . Distance R was obtained by

dividing Xg by 2. A computer program using HP Basic was written to

compute values for <5 y

(X) at intervals of 30.00 mm starting from point

A!

^

endin? at Bx

. The ball bar length L was accurately
measured. The ball bar length L, Xq, and Ye were input to the

computer program, and x and y coordinates were entered to the peogram

at each measurement point. The program then performed calculations

for ^y(X) and printed out straightness values after aligning the data

such that 6y(X) = 0 at both ends. This procedure was repeated ten

times.

Figures 2 through 11 show the obtained results for the ten runs,

all at the same position and under the same conditions. Measuring

the ball bar length, L is very critical and should be measured with

special care.

Accuracy of the Method

In order to check the accuracy or applicability of using the

ball bar for the measurement of straightness errors of coordinate

measuring machines a straight edge was used to measure straightness

of the same machine at the same positions where ball bar measurements

were made. This process also was run for 10 times.

Results of straight edge measurements are shown in Figures 12

through 21 which are similar to those obtained using the ball bar.

Ihis proves the applicability of the ball bar for on straightness

measurement.

The average straightness curves for both the ball bar and

straightedge methods are shown in Figures 22 and 23 respectively.

Tables 1 and 2 show the average straightness values (of 10 runs) , the

standard deviation of each point, and the process standard deviation

for the ball bar and straight edge respectively.
Further Statistical Analysis

When two population means are to be compared, it is usually

their difference that is important, rather than their absolute

values. A statistically acceptable estimate of the difference in

population means is the difference in sample means.

Hie distribution of the difference in average straightness

values between the ball bar results and straight edge results should

follow a distribution with a mean x and a standard deviation cr . x ~ ¥av(x)bb “ ^Yav(x)se

where

^av(x)hb is the average of the average straightness

values in table 1 for the ball bar.

6

YavWse is the average of the average straightness values

in table 2 for the straightedge.

1

(nl+n2

-2) ( 11)

nl - n2 = number of data points = £22 22fi?

n for theballbar neth«a - ^ = *******

Ihe composite curve for the differences in the average

straightness Values between the two methods is shown in Figure 24 . ®e average straightness values and standard deviation for that curve

are -0.7 /im and 0.9 /zm respectively.

A highly useful means of examining the applicability of the

proposed method for measuring straightness of coordinate measuring

nachines is residual analysis, a residual is the difference between
the measured straightness value and the average value. Average

straightness values are in agreement with those obtained using a

straightedge. A residual , e^, is given by: - ei= ^y(xi) “

^Yav(xi) (12)

fR __ ^diere Yav(xi) i-s the average of ten measured

straightness values at point x^.

The histograms of residuals are shown in Figure 25. Residuals

should have a distribution with a mean of about zero and the

distribution should be reasonably normal. The two distributions for

the straightedge method and ball bar method satisfy this requirement

except, as shewn in Figure 25, the histogram of residuals for the

ball bar method has a mean of l.o pim. This shift is ranswj by the

assumption that the machine has zero scale errors. The mean of the

histogram will be shifted back to zero if the scale errors are in the

range of -1.0 or -2.0 ^im. This is true for the machine used for

taking the measurements.

Sunmary

Ihe ball is a simple, easy, and rapid method that can be applied

to the calibration of coordinate measuring machines. Using the ball

bar for anm straightness measurement was proved to be an

and simple method.
Table 1

AVERAGE STRAIGHTNESS VAHJES FOR THE BALL BAR METHOD

# Average -

(mm)

Std. Deviation

(mm)

0 .0000 .0000

1 .0009 .0002

2 .0010 .0004

3 .0014 .0005

4 .0014 .0006

5 .0018 .0006

6 -.0011 .0007

7 -.0006 .0006

8 -.0018 .0006

9 -.0014 .0006

10 -.0018 .0005

11 -.0012 .0005

12 -.0025 .0013

13 -.0022 .0003

14 -.0038 .0003

15 -.0028 .0002

16 -.0022 .0002

17 -.0015 .0002

18 -.0024 .0001

19 -.0022 .0014

20 .0000 .0000

Standard Deviation = 0028 mm
TABLE 2

AVERAGE STRAIGHTNESS VALUES FOR STRAIGHT-EDGE METHOD

Pt. #

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Average std. Deviation

(mm) (mm)

.0000

.0010

.0014

.0015

.0013

-.0007

-.0005

-.0004

-.0013

-.0008

-.0005

-.0001

-.0005

-.0011

-.0013

-.0008

-.0012

-.0009

-.0009

-.0004

.0000

.0000

.0010

.0010

.0010

.0010

.0009

.0010

.0010

.0009

.0006

.0005

.0008

.0006

.0012

.0008

.0011

.0009

.0011

.0009

.0008

.0000

Standard Deviation = 0042
References

1. Bryan, J. , Precision Engineering, 1(3), 129, July 1979.

2 . Hocken, R. J. (MTTF Working Group Chairman) , "Technology of

Machine Tools - Volume 5: Machine Tool Accuracy," UCRL-52960-5

University of California, 1980.

3. ASME B89.1.12 Committee, Proposed Standard for Performance

Evaluation of Coordinate Measuring Machines, March 1983.

Wonnacott, T. and Wonnacott, R. J. , Introductory Statistics,

John Wiley & Sons, 3rd edition, 1977.

4 .
ill.

Figure 1 Principle of Using the Ball Bar for CMM

Straightness Measurement
ERROR (MM)

12!

» 21

i ZT

; O

«t.Q

• LO^-.

! C3

1 in

Q —CD

\r

Q —OJ

1

/

: cd

——CD

cn >

r^'

+• <

‘ CD \ —£3

CO

a —

« CVJ

; /

i Q \ —CD \

a

nj

—CD

; co

1—

+

\

- 1 —t— f-~1 1

1 —1- « t— -)—\

H—1-

CD CD rvj OJ CD CD

(3 CS £3 (3 -L CD (3 CD <3

CS <3 (3 (3 C3 CS CD (3

4 —

1

Figure

2

Y

-

Straightness

of

X

—

Rx

i

s

Us

i

nq

The

Ball

Bar
ERROR (MM)

13

I Q

• CD1?

i in

Q

CD

Q —OJ

'tf' Q

CDro /

Q

X

" ® ) H7

; © \ — ^ >

! w

' \S

S

i CD

j-GO

W

X

<r

i t.<a XD3

M

0 —

rc» (0 PQ

W

0} 0

c _c

•p f—

JC

0)05 — c

ffj ~

t- w +0

CO

1 >- ^ -

7-

GD

r\J

0)

S-i 3

SO

cs

CD

1 -L

l-—

l

H i- —I- "~4 1- f00 CD TT rvj Q Q Q Q CD CD

9 0 • •

\ i ..X

; is -T i

\ —I —-I —I—I

i—I1

i OJ CD CD

-E

i CD S) C3 Q

i S) Q (3 (3

i ±

#

1

•

I

«

1

•

1
ERROR (MM)

! 21

i r

14

1 a

X.Q

! CD

Q

in — <

Q —CD

d“

a

~oj

t

' CD —CD

: ro

i

CD

C3

CO

OJ

. a

—03

\

/ <

V)X

C

E

I ^

fd Xffi

<4o —

id

V) PQ

W

0 0

C X

P h“

X

05 05 — C

fd •«*

5- PD

(/)

f

X—

! (S —OJ

0) U;3

00

\\ J a

! CO

h- -1—1 —1 —f—1—j-

i-^—i- -1—

« 1 CD ' GO

a

a

CD

a

a

-I- 1

\ —1—-I—!—-h- ..u d*

a

a

rua

a

OJ

a

a

"d* a

a

CO

a

a

GO

a

a

i 1 i
ERROR (MM)

15 -

-i^CD

» CD

. <3

)

i—i- —.j —-f. CD

CD

a

H—I —f H

CD

CDQ

-c- S)

CD

CD —CD \ - \ —U

oj \

I CD

V\

\}

j

/

\

• Q — \ nj

; CD- —nj

i

v

OJ

QQ

i

-l-hH-H

OJ Tj- a cs Q Q

0

X

CE

! L.

rd x m

<4- —

0 —

rd wm

w

0 0 Cl

4-> f—

SL

D) CD — C

fd —

i- M

+>D

10

1 >*

LO

a)u

3

30 •H

U*

{- 1 f- )

j CD CD

CD Q

CD CD

I
ERROR (MM)

z:z:

16

! G3 —CD

CD

Q

• %r

LH

Q

CD

\

Q

OJ

! Q

; CD

CDQ

CO

\s w

X

CC

I L. X 0

—

V) CP

w

U <D C X

P H“

GD

4-^r

; ru

O)

•** c

fl? C- w

-PZD

01

I CD

-~>0D

/ s

/ —(\l

sO

CDUi3

£0 rQ! CO

^ I h- 4 —H— } -f--~ f

~“4

- "f~ -4-\L

qo

CD

CD

CO

o

CD

CDQ

nj

CD

CD

! CD

| j-~

_L

ru

CD

CD

H *

I H I t—

H

CD

a

to

CD

CD

CD

CD

a

i
ERROR (MM)

?“2I

S3

• S3

CO

\

17:

1 Q

—

cn s

S3

CD

S3

nj

^r

1 CD

-—CD

' CD

t

\

.s'

: CD — \

CD \

CD

r _L /

: o

\ • OJ

i /

S3

c£

X

cr

? jl

XPQ

4- —

0 —

<a com

CO© ©

c x: •Pj0)0) — c

flj —

4 w PD

in

1

/

S3

• OJ

<yu

3

SO

\

4-E)

; CD

I—h-HCD

S3

S3

CD

S3

S3

Q

S)

nj

CD

S3

+4-IH- S3

1 i—-1 1 H-+

OJ

S3

S3

S3

S3

CD

S3

S3

CD

S3

S3

f l
ERROR
ERROR (MM)

19 . I CD Ti

tl

CD

cd

CD ! LH \ \ \ I 1 1 asQ

CD 1" 1 | -J-- CD

CD

CD

CD

CD

CD

—OJ

wX

cr

—CD

fD : CD —CDm { I t.

tc xm

<+-

—o —a W CD W© CD C X

-PH X

03 03 • Q \ —^ \ ; c\i s'

<d -~

c.

w

PZ3

U) ' \CD

-LCD I V

a)u

SO •H j -

-| 4 ) 1 1 1 1 CD

CD CD a q • • I I

CD

CD

CD9 I
ERROR (MM)

20

3Z

r

I CD

-rtG)

i LD'-

Q

LD

CD

CD

CD

nj

' (S3 —CD

no

(D

CD

m

; q —

nj

./

\

\

<

>

—

: s \ CO i

/ I Q

( t2

\

i

>

1— ~!—H-

\ -^-CD

! LD

\T

CD

CD

CD

-l-—b- - 1~- i — 1 —I- -N - ~l —i--

I (2

1 •--+—i—-!—! —1—

t

CD

CD

CD

c

^rQ

CD

•

nj

CD

CD

e

! -r Q

1 CD

004 900 800

ligure

10

Y

-

Straightness

of

X

-

Rx

i

s

Using

The

Ball

Bar
LI

'

; r

i

i <3

-*-<S

; CD

' CD —^r

: in

\

/

CD —00 \ V \

\\

: CD

\ M

\ —OJ / X

xj* CE

/ I s-

\

ftf CD

\

\

XOD —CD

i co

y

r Cf*

i 0 —

5 CD —(3

m

\

- (

! Q

; eg /’

1 /! \

' <3 \ —00 >

)

ftf MOD

W

0 0

C-C

*P r-

.c

CD CD — C

rdS- M

+> D

CD

1 >

; (3 —rg

i ;

QJi-3

00 •H

Ui —(S

. CD

l i

\i

1 1~-+--+-- \ —\- -1—

^ —j_ -4—X f—

s -b-H—1—-\ —i

i

1- £ ~ OD CD rg rg CD 00

S 2T CD CD (3 (3 ! CD CD a (3

£ s (3 Q CD a i a CD a CD ce c; u

• 9 • • i

•

i

•

1

9

\

•

1
ERROR (MM)

22 -

raightness

of

X

-

Axis

sing

Straight

Edge
ERROR (MM)

23'
ERROR (MM)

24 I.

i z:

raightness

of

X

-

Rx

i

s

sing

Straight

Edge
ERROR (MM)

25

raightness

of

X

-

Rx

i

s

sing

Straight

Edge
ERROR (MM)

26 ; 21 I S

ViX

a: Q I ZD n

xu^ +3 c j:

CD

W

—

W <tf o s- r -p -pen

jC

CD O)c

•PD

in

>-

vO

0)J-3

£0

Ci>1 1
ERROR (MM)

27

;

; Tl

i 21
ERROR (MM)

51

28 ;
29
ERROR (MM)
31 ;

W

X

(L

0

I CD

ID

XLJ

<+- -P O X

03

(A —

0 i- C -P

-PC/) X

CD 03 — C

f0 •—

i- w

p Z3

0)

f >

<D3

iC

£fct

T

i "1
ERROR (MM)

32

7Z

S3

• S3

CO

S3

in

CD

CO

M*

: Q —OJ

• ^

CD

—•CD

cn

CD

CD

CO

CD

’•v' oj

y

S3

' —OD >

S3

•OJ

—S3

! co

-j- j

-i~ -h 1 i i 1i T -X.

.008 CO

S3

S3

004 S3 __

S3

i -

OJ

CD

CD

|

| j _j—__|

"tr S3

CD

f

CO

S3

S3

0

f

CD

S3

S3

•

I

Figure

22

Y

-

Straightness

of

X

-

Rx

i

s

Using

The

Ball

Bar

(Overage)
ERROR (MM)

I f

T2:

• r

1 o

-r-Q

' CD

1\

33

i —• in

• o « —GO \

t - {

i \

• CD \

-rTM ) — /

: ®/

-—CD

—

; sCD

n

1 \

•: CD —^ >

: nj / — /

I CD —05'

40

' CD ^ —nj

-UCD

: CD

•H *•—i +—

\

—i I 1 -S--

00

Q

CD

CO

CDQ

CD

CD

OJ

CD

CD

! CD

’+— J- 4—I f—\—

i

1

j ru CO 00

4- CD CD CD CD

i CD CD CD CD

i

0

l

•

1

•

l

•

1

Figure

23

Y

-

Straightness

of

X

-

fix i

s

Using

Straight

Edge

(Average)
ERROR (MM)

; 7Z

34

- CD

_G3

. CA I

CD —^ /

i m /'

cs . —oo v TT

\

CD —OJ

CD

-CD

40 ft — 0

X

cc

t

03

niL

.

©

>

xcr ^ c

0 ©

©

ft B

ft +> © ©

CCD

4=> X ft 03 © — o

«j c

4- ©

4-> 4. CO ©

1 <-*= >- a

x . CD

^—OJ

-L®

F

*arsi

oc •H—

09Q

CD

-f-~ 1— H —J (•

1 J

s

j

' | j -!-—j

j. COO cs

ru

CD 1 g

tJ- CD CD a

0 CD

«

V—w CD

• CD

f Q

I

CD

CD

•

CD

GD

CD

CD

'+ 1
Figure

25

HISTOGRAM

OF

RESIDUALS

STRAIGHTNESS

VALUES
N BS-1 14A (rev. 2 -ec) U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. BIBLIOGRAPHIC DATA

REPORT NO.

SHEET (See instructions) NBSIR 88-3759

3. Publication Date

May 1988

4. TITLE AND SUBTITLE

A SIMPLE METHOD FOR MEASURING STRAIGHTNESS OF COORDINATE MEASURING MACHINES

5. AUTHOR(S)

Ahmad K. Elshennawy, Fang-Sheng Jing and Robert J. Hocken

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

S. Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City , Stote, ZIP

)

10.

SUPPLEMENTARY NOTES

| 1 Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bl iography or literature survey, mention it here)

Straightness errors contribute significantly to the total error budget of

coordinate measuring machines. There are two straightness error parameters for each

axis; horizontal and vertical straightness. According to Bryan [1], straightness

error is "a non—linear movement of the machine axis that an indicator sees when it is

either stationary reading against a perfect straight edge supported on a moving slide

or moved by the slide along a perfect straight edge which is stationary." Thus,

straightness error can be determined as the deviation of measurement data from a straight line.

For the purpose of calibrating coordinate measuring machines, a laser interferometer system equipped with straightness optics is usually used to measure straightness

In addition to the relative high cost of a laser interferometer syste# it needs an

experienced operator to perform the measurements.

A simple and rapid method for measuring the straightness of coordinate measuring

machines was developed using the ball bar.

12. KEY WORDS (Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Ball bar. Coordinate Measuring Machine, Laser Interferometer, Machine Axis

Straightness errors. Vertical Straightness

13. AVAILABILITY 14. NO. OF

PRINTED PAGES m Unlimited

39

| | For Official Distribution, Do Not Release to NTIS

f 1 Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402. 15. Price

1 xi Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $11.95

U S COMM* D C «04S'P80