Comparison of linear normal deviations

 

C o m p a r is o n o f lin e a r a n d n o r m a l d e v i a tio n s o f f o r m s o f engineering surfaces M. S. Shunmugam F o r m e r r o r s o f e n g i n e e r i n g c o m p o n e n t s a r e u s u a l ly ly e v a l u a t e d u s i n g l in in e a r d e v i a t io io n s , a l t h o u g h s o m e t im i m e s , n o r m a l d e v i a t i o n s a re r e a l s o c o n s id id e r e d . T h is is p a p e r c o m p a r e s l in i n e a r a n d n o r m a l d e v ia i a t o n s , u s i n g l e a s t s q u a re r e s a n d m i n i m u m d e v i a tit i o n techniques.

K e y w o r d s : f o r m m e a s u r e m e n t , s t a t i s tit i c a l a n a l y s is is

T h e m e a s u r e m e n t o f f o rm rm d e v i a t i o n s o f e n g i n e e r i n g c o m p o n e n t s i s c a r r i e d o u t wi t h r e s p e c t t o a r e f e r e n c e datum or trajectory. For example, straightness m e a s u r e m e n t s a r e c a r r i e d o u t wi t h r e f e r e n c e t o a straight edge and roundness measurement with reference to a straight edge and roundness measurem e n t wi t h r e f e r e n c e t o a c i r c u l a r t r a j e c t o r y . T h e deviation s have to be measured norm al to the surface of the geom etric elements. This is achieved by a ligning

n o t n o r m a l t o t h e f e a t u re re u n d e r c o n s i d e r a t i o n . A c l o s e r exam ination of the relation used to express the d e v i a t i o n w o u l d r e v e a l t h i s fa fa c t . So m e d o u b t t h e r e f o r e a r is is e s in in t h e m e a s u r e m e n t a n d a n a l y s i s o f f o r m e r r o r a s t o t h e e x t e n t o f e rr rr o r i n v o l v e d i n a p p r o x i m a t i n g t h e n o r m a l e v i a t i o n b y a l i n e a r o n e . Co n f u s i o n m a y a l s o a r i s e i f f o r m e r r o r s a r e c o m p a r e d wi t h o u t c o n s i d e r i n g t h e n a t u r e o f t h e d e v i a t i o n . So m e t i m e s t h e n o r m a l l e a s t squares technique is applied on values obtained on a

t h e g e o m e t r i c f e a t u r e wi t h t h e r e f e r e n c e f e a t u r e . I n o t h e r wo r d s , t h e r e f e r e n c e s t r a i g h t e d g e s h o u l d b e m a d e p a r a l le le l to to t h e s t r a i g h t f e a t u r e a n d t h e c i r c u l a r t r a je je c t o r y s h o u l d b e c o n c e n t r i c w i t h t h e c i r c u la la r f e a t u r e . Ho we v e r , d u e t o p r a c t i c a l d i f f i c u l t i e s , t h e a l i g n m e n t c a n n e v e r b e p e r fe fe c t . F o r p r a c t i c a l measurements, such misalignm ents should be kept to a minimum . This procedure shou ld ensure that d e v i a t i o n s ar a r e w e l l w i t h i n t h e r a n g e o f t h e m e a s u r in in g d e v i ce c e s w h i c h u s u a l l y e m p l o y h i g h e r m a g n i f i c a titi o n s . For evaluating form error, the International St a n d a r d s Or g a n i s a t i o n s p e c i f i e s t h a t a n i d e a l g e o m e t r i c a l f e a t u re re m u s t b e e s t a b l i s h e d f r o m t h e a c t u a l m e a s u r e m e n ts ts s u c h t h a t t h e m a x i m u m d e v i a t io io n b e t we e n i t a n d t h e a c t u a l f e a t u r e c o n c e r n e d i s t h e l e a s t p o s s i b l e v a l u e 1. T h e m a x i m u m d e v i a t i o n f r o m t h e i d e a l geom etric feature thus established is taken to represent

c i r c u la l a r i t y tr tr a c e w i t h o u t c o n s i d e r i n g t h e a c t u a l d i m e n s i o n o f t h e c o m p o n e n t . T h i s p r a c titi c e w o u l d a ls ls o lead to confusion. I n t h i s p a p e r , t h e l e a st st s q u a r e s a n d m i n i m u m d e v i a t i o n f e a t u r e s a re re e s t a b l i s h e d u s i n g l i n e a r a n d normal deviations. The simplex search method is used f o r m i n i m u m d e v i a t i o n a n d n o r m a l l e a st st s q u a r e s t e c h n i q u e s w h e r e i n s i m p l e r s o l u t io io n s c a n n o t b e o b t a i n e d . T h e d i f f e r e n c e b e t we e n n o r m a l a n d l i n e a r d e v i a t i o n s i s d e m o n s t r a t e d wi t h n u m e r i c a l e x a m p l e s f o r s i m p l e g e o m e t r i c f e a tu tu r e s .

ti dh ee aflofrema t ue rr er o cr .a nT hbee ot raikeennt attoi o nr e ap nr eds/ oe rn tl ot hcea t imo ins -o f t h e a l i g n m e n t e r r o r in in s e t t i n g t h e w o r k p i e c e d u r i n g t h e m e a s u r e m e n t . Ho w e v e r , t h e s ta ta n d a r d d o e s n o t s p e c i f y a n y m e t h o d b y wh i c h t o e s t a b l i s h s u c h a n i d e a l geom etric feature. T h e l e a st st s q u a re re s t e c h n i q u e i s u se se d t o e s t a b l i s h t h e i d e a l g e o m e t r i c f e a tu tu r e . T h i s a p p r o a c h i s b a s e d o n t h e l i n e a r d e v i a t i o n s a n d t h e p a r a m e t e rs rs o f t h e i d e a l g e o m e t r i c f e a t u r e a r e o b t a i n e d i n a s t r a i g h t f o r wa r d wa y 2 7 . Ho we v e r , t h e d e v i a t i o n s o b t a i n e d b y t h i s p r o c e d u r e a re re n o t t h e m i n i m u m . A t t e m p t s h a v e therefore been made to arrive at the minimum d e v i a t i o n ; t h e s e m e t h o d s a l wa y s r e q u i r e a s e a r c h p r o c e d u r e4 e 4 '5 '5 . M o r e o v e r , t h e d e v i a t i o n s c o m p u t e d a re re

List of symbols ei

f ht hs K L

m0, l0 N R0 ri, 0i rw xi, Yi, zi

x0, Y0, z0

Ax, Ay, Az * D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r in in g , I n d i a n I n s t i t u te te o f Technology, Madras 600 036, India

96

Linear deviation at ith point N o r m a l d e v i a t i o n a t itit h p o i n t F u n c t i o n t o b e m i n i m i se se d Peak-to-valley height Root-mean-square value Nu m b e r o f p o i n t s i n a s e c t i o n N u m b e r o f s e c t io io n s Es t im im a t e d s l o p e v a l u e s Total number of points Es t i m a t e d r a d i u s o f a c i r c l e / c y l i n d e r / sphere Po l a r c o o r d i n a t e s o f i t h p o i n t N o m i n a l r a d iu iu s o f w o r k p i e c e coordinates of ith p oint Es t im im a t e d c o o r d i n a t e v a l u e s Av e r a g e v a l u e s Latitudinal direction of ith point on a sphere I n t e r v a l b e t we e n v a l u e s

01 41 6359 /87/0 20 0 96 0 7/$03 .00 ,~j~,1987 Butt Butterwor erworth th & Co (Publ (Publis ishers hers)) Ltd

AP RI L 198 7 VO L 9 NO 2

 

S hun mu gam ~L inea r

a n d n o r m a l d e vi v i a tit i o ns n s o f f o r m s o f e n g in i n e e r in i n g s u rf r f ac ac e s

a n d ci rc u la ri ty

Straightness

A m i n i m u m d e v i a t i o n p r i n c ip ip l e c a n a l so so b e u s ed ed t o o bt ain t he es t im at es of Y 0 and / 0 by m inim is in g

The s t r aight line, c ir c le and ot her plane pr of iles ar e t w o - d i m e n s i o n a l g e o m e t r i c f e at at u re re s . S t ra ra i g h t a n d circular features. Straight and circular features are the m os t c om m on f eat ur es in engineer ing applic at ions . The s t r aight f eat ur e has no s iz e, w her eas t he c ir c ular feature, being a closed form, has a size. Straightness

f = l emaxl + t~ t~m~nl m~nl

E qn ( 4) w ill not y ield a uniq ue v alue of Y o. o. W hile us ing a s ear ear c h pr oc edur e, dif f er ent v a lues of Y 0 giv in g ident ic al v alues of f or m er r or m ay be ex pec t ed. S o m e t i m e s s u c h c o m p u t a t i o n a l i n s t a b i l i ty ty i s a v o i d e d by f ix ing t he v a lue of Y 0 as the the leas leastt s quar es es t im at e 4. A logic al appr oac h w o uld be t o f ix t he line line s uc h t h a t th th e m a x i m u m d e v i a t i o n a b o v e a n d b e l o w i s e q u a l.l. H e n c e th th e m o d i f i e d f u n c t i o n g i v e n b e l o w i s u s e d :

error

The m eas ur em ent dat a ar e giv en by {xi, Yi}. If the as s es s m ent line is r epr es ent ed b y Yo + Ioxi as in Fig 1, t hen t he nor m al dev iat ion ei is ex pr es s ed as 1/22 i = [ Y i - ( Yo Yo + / o X i ) ] [ 1/ ( 1 + / 2) ] 1/

f = I ~ m . x I + l emin I + 1 /I ~max l a m ~ o I

( 1)

h, = l a m ~ x I + I ~ m , o I

(2) (3)

hs =

Eqn (3) requires a search procedure to arrive at the

[ T . ~ N ] 1 /2 /2

(7)

emaxx and l ~ r n inin a r e s uit ably r eplaced ema eplaced f or linear dev iat ions . Table 1 inc ludes t he m eas ur em ent dat a. The es t im at es o f Y 0 and / 0 bas ed linear and nor m al dev iat ions ar e als o s how n in t he t able f or t he leas t squares and minimum deviation tecniques.

estimates. Forsquares a linearestima deviation, ei and is replaced by e~ and the least te of Y0 / 0 are evaluated directly. Y

Circularity error

Pi(xi,vi) Pi(xi, vi)

C ir c ular it y m eas ur em e nt s ar e repr repr es ent ed by {ri , 8i}. As the feature has a size, the rad ius, R0, of the assessm ent c ir c le le has t o be det er m ined along w it h t he c ent re re ( x 0, Y 0) as in Fig 2. The nor m al dev iat ion is giv en by

yo+ IoXi

ai = [(x ;-

xi

(6 )

T h e r o o t - m e a n - s q u a r e v a l u e c an an a l s o be be c o m p u t e d f ro r o m t h e f o l l o w i n g r e l a titi o n :

The leas t s quar es t ec hnique m inim is es f = T.#2 T.#2

(5 )

This function, when minimised, yields a minimum valu e o f I er erna naxx + I ~r ~rni ninn I and a spec ific valu e o f Y0 suc h that l emaxl = l emin eminl.l. For linear de viatio n, e i is used in place of ei. Eqn (5) also requires a search proc edure t o f in d t he v alues of Y 0 and / 0. For t he pr es ent w or k , a w ell es t ablis hed s im plex s ear ch ch m et hod is us ed s . The f or m er r or is is t hen c o m p ut ed as a m ax im um pea k - t o- v a lley height , h t , w he r e

w her e x is deno t e t he m eas ur em ent point s and y is ar aree t he dev iat ions m eas ur ed. I n pr ac t ic e, t he dis t anc e bet w een t he m eas ur em e n t p o i n t s w o u l d u s u a l ly ly b e 2 5 m m o r m o r e a n d t h e dev ia t ions w o uld be of t he or der of m ic rons rons . I f t he f eat ure ure is w ell a ligned w it h t he x - ax is , the the dev ia t ion c an be ex pr es s ed in t he linear f or m : ei = Y i- (Yo + Ioxi)

(4 )

X o ) 2 + ( y ; - y o ) 2 ] 1 /2 /2 _ R o

(8)

The nom inal s iz e of the the w o r k piec e m ay be in in m illim illim et r es w it h t he r adial dev iat ions in m ic r ons . For a w ell c ent er ed t r ac e, t he dev iat ion c an be appr ox im at ed by a linear relation:

0

ei = ri - (R o + Xo cos 8i + Yo sin 8i)

F i g 1 D e v i a t i o n s f r o m a s t r a ig ig h t l i n e

(9)

Table 1 Straightness me asurement

Data:

N=5 xi, m m

y;, /~ /~m m

-50 3

V alues #m

-25

0

25

50

5

2

1

2

Least squares

Minimum deviation

N or m al

Linear

N or m al

Linear

)/0

2.59954 -0.59967

2.60000 -0.60000

2.99880 -0.66666

2.99908 -0.66666

hhst

21 . 80 50 08 63 40

21 . 0850 80 30 0

21 .. 61 36 46 86 76

21 .. 61 36 46 96 78

IoAx

P R E C I S IO IO N E N G I N E E R I N G

97

 

S hun mu gam --L inea r

a n d n o r m a l d e v ia i a t io i o n s o f f o r m s o f e n g i n e e r i n g s u rf rf a ce ce s Z

Y P i r i,i , S i )

ri~R

-o

o

J

paneyzo+l laneyzo+loXi oXi+mo +mo

(Xo,Y o) 0

y

X

Assessment circle

Fig 3 Deviations from a plane

Table 3 give s the the data an d the results results from the flatness ev al uati on. Fig 2 Deviations from a circle

Cylindricity

C y l i ndri c i ty meas urements are denoted by {ri, 8i, zi}. The radius, Ro, of the assessment cylinde r and the ax is represented by Xo + Iozi and Y o + m o z i have to be evaluated as in Fig 4. The normal deviation ei is given by

The measurement data, the values o f Xo, Yo Yo and R o and the circularity error are given in Table 2. F l a t n e ss ss , c y l i n d r i c i t y a n d s p h e r i c i t y Three-dimen sional features are are planes, cylinders, spheres etc. Features such as planes planes an d cylinde rs are c ommonl y us ed i n engi neeri ng appl i c ati ons . S ome of the three-di mens i onal features , namel y the c y l i nder and sphere, have size.

a~ =

{ [ ( x ; - X o ) - I o z i ] 2 + [(~'i- ~o) - m o z i ] 2 XiXo) - / o ( y ~ - Y o) ] 2}1 2}1/2 /2 + [mo ( XiXo) x {1/(1 + l ~ + m 2 ) } l / 2 - R o

(12)

For a c y l indri indri c al feature feature al i gned properl y w i th the z-axis, the deviation can be expressed in the linear form:

Flatness error

The measurements of flatness are given by {xi, Yi, zi} as sh ow n in Fig 3. The assessment plane is represent represented ed b y z o + Ioxi + m oY i and the normal dev i ati on i s expressed as ~ i = [ z i - ( zo z o + lo lo x o + m o Y i ) ] [ 1 / ( 1

error

ei = ri - [ (Ro + (Xo + Iozi ) cos el) + ( Y o + m o z i ) sin el]

(13)

The estimates of R o, x o, Yo, Io and m o for for the gi v en data are included in Table 4.

+ 1 2 + m 2 ) ] 1/ 1/2

(10)

For a surf surface ace aligned paral parallel lel to the x - y plane, the linear

Sphericity

dev i ati on i s gi v en by :

The data {r/, 8i,/~} represent the sphericity measurements. The centre of the assessment sphere

(11 )

~i = zi - (Zo + Ioxi + moy i)

error

Table 2 Circularity measurement

D at ata :

rw = 1 0 m m ,

N= 8

~i, deg

0

45

90

135

180

225

( r i - r w ),), p m

4

4

3

5

2

3

270 1

31 5 2

t

V alues alues ( Ro Ro - r w ) Xo Yo ht hs

98

Leas t s quares quares

M i ni m um

dev i ati on

Normal

Linear

Normal

Linear

2.99997 0.14640 1.20 698 2.45708 0.87220

3.00 0 0 0 0.14644 1.20711 2.45710 0.87219

3 .0 .0 0 0 3 1 -0.12131 1.11 212 6 1. 2.24264 0.89457

3.00 05 7 -0.12 131 1.11 2131 1. 2.24264 0.89457

APR IL 1987 VOL 9 NO 2

 

S h u n m u g a m - - L i n e a r a n d n o r m a l d e v i a titi o n s o f f o rm r m s o f e n g i n ee e e r in i n g s u r fa fa ce ce s Table 3 Flatness measurement

ata:

K=5,

L=3,

N=15

z , #m X/, mm

-50 Yi, mm

-25

0

25

50

+ 25 0

5 4

4 3

1 3

2 2

2 2

-2 5

3

4

2

1

2

Values

Least squares

#m z0 ~Ax mo&y h hs

Z

Minimum deviation

Norm al

Linear

Norm al

Linear

2.66666 - 0 . 59 9 9 7 0 .2 . 2 0 0 12 12 2 .8 .8 0 0 2 6 0.73635

2.66667 -0.6000 0 0 .2 .2 0 0 0 0 2 . 80 80 0 0 0 0.73636

2.25136 - 0 . 7 50 50 0 0 -0.00 154 2 . 49 49 9 9 9 0.8870 0

2.2502 9 - 0 . 7 50 0 0 -0.00 047 2 .5 .5 0 0 0 0 0.88735

A x i s o f a s s e ss ss m e n t cylinder Xo+loZi

Yo+moZi

For the l eas t s quares and mi ni mum dev i ati on tec hni ques , the func ti ons gi v en by E qs (3) and (5) are minimised. Table 5 gives the sphericity data and the results obtai ned w i th l i near and normal dev i ati ons . Computational

e

0

P ( r , S ,z )

(Xo,Yo) X

Assessment cylinder

R~

,,

F i g 4 D e v i a t io io n s f r o m a c y l in in d e r

(Xo, Yo, Zo Zo)) and its radius can be de termined from the normal or l i near dev i ati ons gi v en bel ow : ei = [ ( x ;- Xo) 2 + (Y i - Y o) o) 2 + (z i - Zo)2 Zo)2]] 1/ 1/22 - R o (14) ei = r; - R0 + x0 cos/~; cos ei + Y0 cos/Y; sin ~i + z0 sin/~i) (1 5) PRECISION ENG INEERING

details

To fac l i tate c omputati on w hi l e us i ng l i near dev i ati on, t h e xi, Yi and z ; v al ues are trans formed formed us i ng the relations ( x i - 7c ) /& /& x , ( Y i - ;') /& Y a n d ( z i - 2 ) / ~ . res pec ti v el y6. y6. The de viation s from the reference reference feature are meas ured ured i n mi c rons . The v alues alues c orres pondi ng to radial radi al distanc es are are sim plified by referri referring ng to a nomina l di mens i on. To c ompute the normal dev i ati ons , the ac tual di mens i on of the c omponent s houl d be c ons i dered. Thi s may l ead to c om putati ons i nv ol v i ng v ery larg largee and extremely small values. Therefore, a double precision mode hasresults. to be us ed i n the thess e c om putati ons to ob tai n accurate For approaches requiring a search procedure, the simplex search method is used. The procedure is ex pl ai ned i n the appendi x . Thi s method requi res an initial simplex. For example, three sets of values form the i ni ti al s i mpl ex i n the c as e of a s trai trai ght l ine, ine, w here the values o f Y0 and I 0 are are to be evaluated. evaluated. The search is carried out in three different modes, namely refl ec ti on, ex pans i on and c ontrac ti on, dependi ng on the closeness to the final value. The coefficients used for refl ec ti on, ex pans i on and c ontrac ti on are 1, 2 and 0 .5 res pec ti v ely ely . A c onv ergenc e c ri terion terion of 1 x 10 -7 has been used in the present work. The form errors are calculated from the respective equations. For example, if the parameters of the geometric feature are evaluated from the linear dev i ati on, the form error i s al s o c omputed from the l inear inear dev i ati on. Tabl Tables es 1 -5 s how the v al ues c omp uted for different geometric features. For comparison, the 99

 

S h u n m u g a m - - L i n e a r a n d n o r m a l d e v ia ia t io i o n s o f f o r m s o f e n g i n e e r in i n g s u r fa fa c e s

T a b le le 4 C y l i n d r i c i t y m e a s u r e m e n t ata: rw=10mm,

K=8,

L=3,

N=24

( r i - rw), m

0i, deg

+2 5 0 -25

zi, mm

Values

0

45

90

135

5 4 3

3 4 2

4 3 4

3 3 3

180

225

270

315

1 3 2

2 2 3

2 2 1

3 2 2

Least squares ormal

/~m /~ m (Ro (R o - r~) Xo Yo /o Az mo &z h hs

2.79168 0.55897 0.67 680 0.64004 - 0.03 694 2.24834 0.6266 3

Min imum deviati on

Linear Line ar

Normal

Linearr Linea

2.79167 0.55892 0.67 678 0.64016 - 0.03661 2.24791 2.24 791 0.62653

2.59878 0.53834 0.73441 0.86295 - 0.1 5358 2.00027 0.67295

2.60803 0.58917 0.65671 0.80914 - 0.11801 2.00009 0.66240

parameters for normal deviations are presented after

or zo in the case of a straight line or plane and t he

suitable transformation. The computer programs were written in FORTRAN IV and run on an IBM 370.

radius Ro of a circle, cylinder or sphere are not determined u niquely. However, the modifi ed functio n determined suggested sugges ted in this paper does does yield a unique value of these parameters. The minimum deviation technique gives values of form error whi ch are less than tho se obtained obtained wit h the least squares technique. The difference between the t he values is quite appreciable. In both these these techniques techniques,, the normal deviation

Conclusions The mini mum deviat ion approach suggested by other research workers does not fix the position or the size of the geometri c feature. For exampl example, e, th e intercept Y0

Table 5 Sphericity m easurem ent D a t a : rw r w = 1 0 m m, m, K = 8 ,

L=4,

N=26

(ri - rw) , I~m I~m Oi, deg

deg

+90 + 45 0 -45 -90

0

45

90

135

180

225

270

5 5 4 3 3

3 4 2

4 3 4

3 3 3

1 3 2

2 2 3

2 2 1

Values

100

Least squares

315 3 3 2

Minimum deviation

m

Normal

Linear

Normal

Linear

(Ro- rw) x0 Yo z0

2.88468 0.68123 0.80618 0.41 213

2.88461 0.68121 0.80622 0.41213

2.96336 1.38382 0.40219 0.61625

3.01090 1.41129 0.24230 0.58449

ht hs

3.39757 0.78551

3.39760 0.78552

2.84099 0.91724

2.83087 0.95197

APRIL 1987 VOL 9 NO 2

 

S h u n m u g a m --L in e a r a n d n o r ma l d e via tio tio n s o f fo fo r ms o f e n g in e e rin rin g su r fa ce ce s

R

~

p;(r~,e;,~.

/ 0oxo \

Assessment

/

movement of the simplex is achieved by reflection, ex pans i on and c ontrac ti on. For the s ak e of s i mpl i c ity ity , c ons i der a tw o dimensional space (ul, u2) and the initial simplex formed by U 1, U 2 and U 3 as s how n i n Fi g A -I. The v al ues of the the obj ec ti v e fun c ti on are obtai ned at the vertices of the initial simplex. If Urn (say U2) is the v ertex c orres pondi ng to the hi ghes t v al ue and U o i s the c entroi d of the remai ni ng poi nts , then a poi nt on the l ine ine j oi ni ng Urn and U o may be expected to have the smallest value. First, a reflected point Ur is obtained from the relation U,=(1 +~)Uo-~Urn

X

Fig 5 De via tio tio n s fr o m a sp h e r e

wh ere ¢ > 0 is the reflection coefficient. If the refl ec ti on produc es a new mi ni mum (ie f ( U r ) < f ( U 1 ) where U1 represents the vertex c orres pondi ng to the mi ni mum func ti on v al ue of the initial simplex), one can generally expect to decrease the func ti on v al ue by mov i ng further aw ay . The ex panded poi nt U e i s giv giv en by : U e =

app roach g enerally results in in different form error values from thos e obtai ned w i th the l i near dev i ati ons . H ow ev e r, the di fferenc e be tw een thes e v al ues is is qui te te i ns i gni fi c ant for prac ti c al meas urement probl ems . C omputati on of the form error w i th normal dev i ati on s h oul d be carri carried ed out i n doub l e prec i s i on as the equa ti on c ompri s es v ery l arge and ex tremel tremel y s mal l values. It is also found that the normal deviation approac h requi res l onger c omputati on ti me, w hi c h i s not justifiable in view of the marginal difference in the values. If the expressions for linear and normal deviations are analysed, the expression for normal deviation is s een to c ontai n the ac tual di mens i on of the c ompo nent. While using linear deviation, the transformation of xi, y;,, z; and r~ values ca n be co nve niently carried ou t y; w i thout affec ti ng the bas i c equati on. In fac t, meas urement from a c i rc ul ari ty c hart s houl d be fol l ow ed by c omputati ons bas ed on l i near dev i ati ons onl y . The i niti nitial al radi us and the the m agni fi c ation ation empl oy ed i n trac i ng w i l l not affec t the res ul ts w hen s uc h an approach is used. In some cases, the result is based on a smaller numbe r of data poi nts and therefore may hav e l i mi ted value. However, the results serve to demonstrate the pri nc i pl es c onv eni entl y . The pri nc i pl es outl i ned here may be ex tended to any geometri c feature.

(A-l)

~ U r - ~~- ( 1 -

~') U 0

(A- 2)

wh ere 7 is the ex pan sion c oefficient. 7 > 1. If the expansion is successful, ie f ( U e ) > f ( U l ) , then a ne w s i mpl ex is formed repl ac ing ing Um by Ue. Otherw ise, Ur replaces Urn, form ing a new simplex. If the reflection process results in a value greater than the highest value of the initial simplex, ie, f ( U r ) > f ( U r n ) , then contraction is carried out using U c =/~ =/~U U rn + (1 -/ ~ ) U 0

(A -3 )

w here /~ is the c ontrac ti on c oeffi ci ci ent, 0 </~ < 1. For c ondi ti ons f ( U r ) < f ( U r n ) a n d f ( U r ) > f (U i )i)i ~ rn rn the c ontrac ti on i s ac c om pl i s hed by repl ac i ng t h e p o i n t Urn by U r and us i ng E qn (A -3). The contraction is successful if the value at the contracted point Uc is less than the minimum of f ( U m ) a n d f ( U r ) and U c replaces Urn. Otherw ise, all U is are modified as (U i + U e )/2 and a new s i mpl ex i s formed.

u2 U 1

~,j

Appendix

JJ

J

J

A l g o r i t h m f o r s i m p l e x s e a rc rc h The geometri c fi gure formed by a s et of n + 1 poi nts i n n-di mens i onal s pac e i s c al l ed a s i mpl ex . W hen the points are equidistan t, the simp lex is said to be regular. For ex ampl e, a regul ar s i mpl ex i n tw o di mens i ons i s an equilateral triangle and in three dimensions it is a tetrahedron. In general , the s i mpl ex method c ompares the vofalaues of ex theand obj mov ec ti ves e func tgrad he n ual + l1y vtow ertic erti c es s i mpl thi s tis impl ion mplat ex the ar ds the mi ni mum poi nt duri ng the i terati v e proc es s . The P R E C I S IO IO N E N G I N E E R I N G

U2

J

-

Um )

y

Fig A - 1 P r in cip cip le o f simp le x se a r ch

uI

101

 

S h u n m u g a m L i n e ar ar a n d n o r m al a l d e v i at ati o n s o f f o r m s o f e n g i n e e r i n g s u r f ac ac e s S

i n i t iaia l i z e t h e s i m p l e x

U1, U2 , ........ ........ Un+ 1 Specify

r|

Find

Um

~,~,-~,{

V

and

U

such that

f(Um)=max

i=1 to n+ l [f ui)]

Y [ f l U l l = m/i=nI tO . + 1 l f l L l i } l

,x - R e f l e c t io io n c o e f f ic ic i e n t a > 0

1 3- C o n t r a c t i o n c o e f f i c i e n t 0 < . ~ < 1

*/ - E x p a n s i o n c o e f f i c ie ie n t ~ > 1

e - Value for convergence

Find the centroid UO = n+l

_

~m

R e f l e c t u s i ng ng

No

I

~-I set I

set

U ° °U~

~=~ ,

co~ oto

~<r,

A -4 )

References

I

1 Tec hnical drawings: toleranc e of form and o f position, position, Part I:I: Generalities. I S O / D I S 1 1 0 1 , 1 9 7 2 2 Assessment of departure from roundness. B S 3 7 3 0 , 1 9 6 4 3 G o t a M . a n d L i z u k a K . A n an an a ly ly s is is o f t h e r e la la ti ti on on s h ip ip b e t w e e n minimum zone deviation and least squares deviation in circularity and cylindric cylindricity. ity. P r o c . I n t.t. C o n f . P r o d. d. E n g g . , N e w D e l h L I n d i a ,

I

~

I

I

t heet hsoi m lexxp lsaein is g i vTehne ifnl o Fwig i gd iAa-g2r.a m T hoef m d pele i an recdh apbroo vc ee dcuarne be easily extended to n-dimensional space.

E x p a n d u s i ng ng

Ue=~Ur+(I~)Uo

n + - ; I-I-

T h e c e n t r o i d o f t h e f i n a l s im im p l e x i s t a k e n t o r e p r e s e n t the minimum point.

U = (I+~)Uo-eU m

I

With the new simplex, again the reflection process i s st art ed and al l ot h er st eps are f ol l owed t i l l t h e c o n v e r g e n c e c r i te te r i o n i s s a t is is f ie ie d . T h e m e t h o d i s assumed to have converged whenever the standard deviation of the function at the n + 1 vertices of the curren t si m pl ex i s sm al l er t h an t h e prescri bed val u e

I

Q = ( N [flui)- flUol] 2 in+l) V=

1 9 7 7, 7, x 6 1 - x 7 0

Is~t um~u, I

1

c = ~Um + (1-~] )U 0

,

I

4 M u r t h y T . S . R . a n d A b d i n S . Z . M i n i m u m z o ne ne e va va lu lu at at io io n of surfaces. I nt nt. J . M a c h . T o o l D e s . R e s . , 1 9 8 0 , 2 0 2 ) , 1 2 3 - 1 3 6 5 F u k u d a M . a n d S h i m o k o h b e A . A l g o ri ri th th m s f o r f o rm rm er er ro ro r e v a lua t ion - me t h ods of m inimum zone a nd t h e le a s t s qua res res .

P r o c . I n t.t . S y r n p . M e t r o l o g y a n d Q u a l i ty ty C o n t r o l i n P r o d u c t i o n , Tokyo, 1984, 197-202 6 S h u n m u g a m i . S . O n a s se se s sm sm e n t o f g e o m e t r ic ic e rr rr o rs rs . I n t . J.

I

~ r : l ~+ U ~

F i g A 2 F l o w d i ag ag r am a m o f s i m p le le x s e ar ar c h m e t h o d

102

J

R eusg. , a 1m9 8i6. , 2S4. C 2 )r,itit er 7 PSrho ud n. m e4r1ia ia3 f- o4 r2 c5o m p u t e r - a i d e d f or or m ev ev a lu lu a ti ti on on .

1 2 th t h D e s i g n a n d A u t o m a titi o n C o n f e r e n c e - A S M E , O h i o , O c t 1 9 8 6

8 Ra o, S . S. O pt imis a t ion t h e ory a nd a pplic pplic a tions tions . W i l e y E a s t e r n

Ltd, India, 1984

APR IL 1987 VOL 9 NO 2