8/13/2019 Comparison of linear normal deviations

1/7

C o m p a r is o n o f l i n e a r a n dn o r m a l d e v i a t i o n s o f f o r m s o fe n g i n e e r i n g s u r f a c e sM . S . S h u n m u g a mF 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 e v a l u a t e d u s i n g l in e a rd e v i a t io n s , a l t h o u g h s o m e t im e s , n o r m a l d e v i a t i o n s a re a l s o c o n s id e r e d . T h is p a p e rc o m p a r e s l in e a r a n d n o r m a l d e v ia t o n s , u s i n g l e a s t s q u a re s a n d m i n i m u m d e v i a ti o nt e c h n i q u e s .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 ti c a l a n a l y s is

T h e m e a s u r e m e n t o f f o rm d e v i a t i o n s o f e n g i n e e r i n gcomponen ts i s ca r r i ed ou t w i th respec t to a re fe renceda tum or t ra jec to ry . For examp le , s t ra igh tnessmeasurements a re ca r r i ed ou t w i th re fe rence to as t r a i g h t e d g e a n d r o u n d n e s s m e a s u r e m e n t w i t hre fe rence to a s t ra igh t edge and roundness measure -ment w i th re fe rence to a c i rcu la r t ra jec to ry . Thedev ia t i on s have to be measured norm a l to the su r faceo f the geom et r i c e lements . Th is i s ach ieved by a l i gn ingthe geomet r i c fea tu re w i th the re fe rence fea tu re . Ino ther words , the re fe rence s t ra igh t edge shou ld bemade para l le l to the s t ra igh t fea tu re and the c i rcu la rt r a 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 rfea tu re . However , due to p rac t i ca l d i f f i cu l t i es , thea l i gnm ent can never be per fec t . Fo r p rac t i ca lmeasurements , such m isa l i gnm ents shou ld be kep t to am i n i m u m . T h i s p r o c e d u r e s h o u l d e n s u r e t h a td e v i a t i o n s ar e we 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 gd e v i ce s wh 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 ti o n s .For eva lua t ing fo rm e r ro r , the In te rna t iona lS tandards Organ isa t ion spec i f i es tha t an idea lgeom et r i ca l fea tu re mu s t be es tab l i shed f rom th e ac tua lm e a s u r e m e n 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 nbe twee n i t and the ac tua l fea tu re concern ed i s the leas tposs ib le va lue 1. The max imum dev ia t i on f rom the idea lgeom et r i c fea tu re thus es tab l i shed i s taken to rep resen tthe fo rm e r ro r . The o r ien ta t i on and /o r l oca t ion o f theidea l fea tu re can be taken to rep resen t the m is -a l i g n m e n t e r r o r in s e t t i n g t h e wo r k p i e c e d u r i n g t h emeasurement . How ever , the s tandard does no t spec i f yany metho d by wh ic h to es tab l i sh such an idea lgeom et r i c fea tu re .The least squares techn iq ue i s used to es tab l i shthe idea l geom et r i c fea tu re . Th is approach i s based onthe l i near dev ia t i ons and the paramete rs o f the idea lgeomet r i c fea tu re a re ob ta ined in a s t ra igh t fo rwardway2 7 . However , the dev ia t i ons ob ta ined by th i sp r o c e d u r e a 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 ethere fo re been made to a r r i ve a t the m in imumdev ia t i on ; these methods a lways requ i re a searchp r o c e d u r e4 '5 . Moreov er , the de v ia t i ons com pute d a re* 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 g , I n d i a n I n s t i t u te o fT e c h n o l o g y , M a d r a s 6 0 0 0 3 6 , I n d i a

no t n o rma l to the fea tu re under cons idera t ion . A c loserexam ina t ion o f the re la t i on used to express thedev ia t i on w ou ld revea l th i s fac t . Some dou b t the re fo rear ises in the measurem ent and an a lys i s o f fo rm e r ro ras to the ex ten t o f e rro r i nvo lved in app rox im at ing thenorma l ev ia t i on by a l i near one . Con fus ion may a l soar i se i f fo rm e r ro rs a re compared wi thou t cons ider ingthe na tu re o f the dev ia t i on . Som et imes the norma l l eas tsquares techn ique i s app l i ed on va lues ob ta ined on ac i r c u la r i t y 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 ld 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 ti c e wo u l d a ls olead to con fus ion .In th i s paper , the least squares and m in imumdev ia t i on fea tu res a re es tab l i shed us ing l i near andnorma l dev ia t i ons . The s imp lex search method i s usedfo r m in im um dev ia t i on and norm a l l east squarest e c h n i q u e s wh e r e i n s i m p l e r s o l u t io n s c a n n o t b eob ta ined . The d i f fe rence be tween norma l and l i neardev ia t i ons i s demons t ra ted w i th numer i ca l examp lesfo r s imp le g eome t r i c fea tu res .

L i s t o f sy m b o l se ifh thsKLm0, l0NR0r i , 0irwxi , Yi , z ix0, Y0, z0

Ax, Ay, Az

L inear dev ia t i on a t i t h po in tNo r m a l d e v i a t i o n a t it h p o i n tF u n c t i o n t o b e m i n i m i se dPe a k - t o - v a l l e y h e i g h tRo o t - m e a n - s q u a r e v a l u eNumber o f po in ts i n a sec t ionNu m b e r o f s e c t io n sEs t imated s lope v a luesT o t a l n u m b e r o f p o i n t sEs t imated rad ius o f a c i rc le /cy l i nder /spherePo la r coord ina tes o f i t h po in tNo m i n a l r a d iu s o f wo r k p i e c ec o o r d i n a t e s o f i t h p o i n tEs t imated coord ina te va luesAverage va luesLa t i tud ina l d i rec t i on o f i t h po in t on asphereIn te rva l be tween va lues

96 01 41 6359 /87/0 20 0 96 0 7/$03 .00 ,~j~, 1987 But terworth & Co (Publishers) Ltd AP RI L 198 7 VO L 9 NO 2

8/13/2019 Comparison of linear normal deviations

2/7

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 vi a ti o ns o f f o r m s o f e n g in e e r in g s u rf ac e sS t r a i g h t n e s s a n d c i r c u la r i t y A m in im um dev ia t i on p r i nc ip le c an a l so be us edThe s t ra igh t l ine , c i r c le and o ther p lane pro f i les aret w o - d im ens io na l geom e t r i c f eat u res . S t ra igh t andcircular features. St raight and c ircular features are themost common fea tures in eng ineer ing app l ica t ions .The s t ra igh t f ea ture has no s ize , whereas the c i r cu la rfeature, being a c losed form, has a s ize.S t r a i g h t n e s s e r r o rThe measurement da ta are g iven by {x i , Yi }. I f theassessment l ine is represented b y Yo + Ioxi as in Fig 1,then the normal dev ia t ion e i i s expressed as

i = [ Y i - ( Yo + / o X i ) ] [ 1 / (1 + /2) ] 1 /2 (1 )where x is deno te the measurement po in t s and y is a rethe dev ia t ions measured.In prac t ice , t he d is tance between the measure-m en t po in t s w ou ld us ua l ly be 25 m m o r m or e and t hedev ia t ions wo u ld be o f t he order o f m ic rons . I f t hefeature is we l l a l igned w i th the x -ax is , the dev ia t ioncan be expressed in t he l inear f o rm:

e i = Y i - (Yo + Iox i ) (2)The leas t squares techn ique min im ises

f = T.#2 (3)Eqn (3) requires a search procedure to ar r ive at theest imates. For a l inear deviat ion, e i is replaced by e~and the least squares est ima te of Y0 and / 0 are evaluateddirect ly .

YPi(xi,vi)

y o + I o X i

x i 0F i g 1 D e v i a t i o n s f r o m a s t r a ig h t l i n e

t o o b ta in the es t imates o f Y0 and / 0 by m in im is in gf = l emaxl + t~m~nl (4 )

Eqn (4) w i l l no t y ie ld a un iq ue va lue o f Yo. W hi le us inga search procedure , d i f f e rent va lues o f Y0 g iv in gident ica l va lues o f f o rm er ror may be expec ted.S om e t im es s uc h c om pu t a t i ona l i ns t ab i l i ty i s av o idedby f ix ing the va lue o f Y0 as the least squares es t imate 4 .A log ica l approach wo u ld be to f i x t he l ine sucht ha t the m ax im um dev ia t i on abov e and b e low i s equa l.H enc e the m od i f i ed f un c t i on g i v en be low i s us ed :

f = I ~ m . x I + l emin I + 1 /I ~max l a m ~ o I ( 5 )Th is f unc t i on , w hen m in im is ed , y i e l ds a m in im umvalu e o f I ernax + I ~rnin I and a spec if ic valu e o f Y0 suc hthat l emaxl = l eminl . For l inear de viat io n, e i is used inplace of e i . Eqn (5) a lso requires a search proc edureto f in d the va lues o f Y0 and / 0 . For t he present work ,a we l l es tab l ished s implex search method is used s .The fo rm er ror is t hen co mp uted as a max im umpea k- to-va l ley he igh t , h t , whe re

h, = l a m ~ x I + I ~ m , o I ( 6 )The r oo t - m ean - s qua r e v a lue c an a l s o be c om pu t edf rom t he f o l l ow in g r e la ti on :

hs = [ T . ~ N ] 1 /2 ( 7 )emax and l~ r n in a r e su i tab ly rep laced fo r l inear dev ia t ions .Tab le 1 inc ludes the measurement da ta . Thees t imates o f Y0 and / 0 based l inear and normaldev ia t ions are a lso shown in t he tab le fo r t he leas ts qua r es and m in im um dev ia t i on t ec n iques .C i r c u l a r i t y e r r o rCircu lar i t y measureme nts are represented by { r i , 8i} . Asthe feature has a s ize, the rad ius, R0, of the assessm entc i r c le has to be determined a long w i th the cent re ( x0 ,Y0) as in F ig 2 . The normal dev ia t ion is g iven by

a i = [ ( x ; - X o ) 2 + ( y ; - y o ) 2 ] 1 / 2 _ R o ( 8 )The nom ina l s ize o f the wo rkp iec e m ay be in m i l l imet resw i th the rad ia l dev ia t ions in m ic rons . For a we l lcentered t race, t he dev ia t ion can be approx imated bya l inear relat ion:

ei = r i - (R o + Xo cos 8i + Yo sin 8i) (9)

T a b l e 1 S t r a i g h t n e s s m e a s u r e m e n tData: N = 5

xi , m m - 5 0 - 2 5 0y;, /~m 3 5 2

2 51

502

Values# m

Least squaresNormal L inear

M i n i m u m d e v i a t i o nNormal L inear

) / 0I o A xh ths

2 . 5 9 9 5 4- 0 . 5 9 9 6 72 . 8 0 0 6 41 . 0 5 8 3 0

2 . 6 0 0 0 0- 0 . 6 0 0 0 02 . 8 0 0 0 01 . 0 5 8 3 0

2 . 9 9 8 8 0- 0 . 6 6 6 6 62 . 6 6 6 6 61 . 13487

2 . 9 9 9 0 8- 0 . 6 6 6 6 62 . 6 6 6 6 81 . 1 3 4 9 7

P R E C I S IO N E N G I N E E R I N G 9 7

8/13/2019 Comparison of linear normal deviations

3/7

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 t io 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 a ce sY

0

r i ~ R o

(Xo,Y o)

Assessmentcircle

P i r i , S i )Z

- o J planeyzo+loXi+mo

F i g 3 D e v i a t i o n s f r o m a p l a n eyX

Table 3 give s the data an d the resul ts f rom the f latnesseva luat ion .F i g 2 D e v i a t i o n s f r o m a c i r c l eThe measurement data, the values o f Xo, Yo and R o andthe c i rcular i ty error are given in Table 2.F l a t n e 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 yThree-dimen sional features are planes, cylinders,spheres etc. Features such as planes an d cyl inde rs arecommonly used in eng ineer ing app l i ca t ions . Some ofthe three-d imens iona l fea tures , namely the cy l inderand sphere, have s ize.F l a t n e s s e r r o rThe measurements of f latness are given by {x i , Yi, zi }as sh ow n in Fig 3. The assessment plane is representedb y z o + Iox i + moY i and the normal dev ia t ion i sexpressed as

~ i = [ z i - ( zo + lo x o + m o Y i ) ] [ 1 / ( 1 + 1 2 + m 2 ) ] 1/2(10)For a surface al igned paralle l to the x - y plane, the l ineardev ia t ion i s g iven by :

~ i = z i - (Zo + Iox i + m oy i ) (11 )T a b l e 2 C i r c u l a r i t y m e a s u r e m e n t

C y l i n d r i c i t y e r r o rCyl indr i c i t y measurements are denoted by {r i , 8 i , z i } .The radius, Ro, of the assessment cy l inde r and the ax isrepresented by Xo + Ioz i and Yo + m o z i have to beevaluated as in Fig 4. The normal deviat ion ei is givenb y

a ~ = { [ (x ; - Xo ) - I o z i ] 2 + [(~'i- ~o) - m o z i ] 2+ [mo (X iXo) - / o (y~ - Yo) ] 2 }1/2x {1 / (1 + l ~ + m 2 ) } l / 2 - R o (12)

For a cy l indr i ca l fea ture a l igned proper ly w i th thez-axis, the deviat ion can be expressed in the l inearform:ei = r i - [ (Ro + (Xo + Ioz i ) cos el)

+ (Yo + m oz i ) sin el ] (13)The est imates of R o, x o, Yo, Io and m o for the g ivendata are inc luded in Table 4.

S p h e r i c i t y e r r o rThe data {r / , 8 i , /~} represent the spheric i tymeasurements. The centre of the assessment sphere

D ata: r w = 1 0 m m , N = 8~i, deg 0 4 5 9 0( r i - rw ), pm 4 4 3

135 180 225 270 31 55 2 3 1 2

t

Values Leas t squares M in im umNormal Linear

dev ia t ionNormal Linear

(Ro - rw ) 2 .99997 3 . 0 0 0 0 0 3 .0 0 031 3 . 0 0 05 7X o 0 . 1 4 6 4 0 0 . 1 4 6 4 4 - 0 . 1 2 1 3 1 - 0 . 1 2 1 3 1Yo 1.20 698 1.20711 1.1 212 6 1.1 2131h t 2 . 4 5 7 0 8 2 . 4 5 7 1 0 2 . 2 4 2 6 4 2 . 2 4 2 6 4h s 0 . 8 7 2 2 0 0 . 8 7 2 1 9 0 . 8 9 4 5 7 0 . 8 9 4 5 7

98 A P R IL 1987 V OL 9 N O 2

8/13/2019 Comparison of linear normal deviations

4/7

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 ti o n s o f f o rm s o f e n g i n ee r in g s u r fa ce sT a b l e 3 F l a t n e s s m e a s u r e m e n t

K = 5 , L = 3 , N = 1 5ata:z , # m

Y i , m m

X / , m m- 5 0 - 2 5 0 2 5 5 0

+ 2 5 5 4 1 2 20 4 3 3 2 2- 2 5 3 4 2 1 2Values

#mLeast squares Mi n i mum dev i a t i on

Norm al Linear Norm al Linearz0 2 .66666 2 .66667 2 .25136 2 .2502 9~ A x - 0 . 59 9 9 7 - 0 . 6 0 0 0 0 - 0 . 7 50 0 0 - 0 . 7 50 0 0m o & y 0 .2 0 0 12 0 .2 0 0 0 0 - 0 . 0 0 1 5 4 - 0 . 0 0 0 4 7h 2 .8 0 0 2 6 2 . 80 0 0 0 2 . 49 9 9 9 2 .5 0 0 0 0h s 0 . 7 3 6 3 5 0 . 7 3 6 3 6 0 . 8 8 7 0 0 0 . 8 8 7 3 5

Z A x i s o f a s s e ss m e n tc y l i n d e rX o + l o Z i Y o + m o Z i

0 ( X o , Y o )P ( r , S , z )e

X

R ~ , ,A s s e s s m e n tc y l i n d e r

F i g 4 D ev i a t ions f rom a cy l inde r

(Xo, Yo, Zo) and i ts radius can be de termined f rom thenormal or l i near dev ia t ions g iven be low:e i = [ ( x ; - Xo) 2 + (Y i - Yo) 2 + (z i - Zo)2] 1 /2 - Ro(14 )ei = r ; - R0 + x0 cos/~; cos ei + Y0 cos/Y; s in ~ i

+ z0 s in/~ i ) (1 5)

For the leas t squares and min imum dev ia t iontechn iques , the func t ions g iven by Eqs (3) and (5)are minimised.Table 5 gives the spheric i ty data and the resul tsobta ined w i th l i near and normal dev ia t ions .C o m p u t a t i o n a l d e t a i l sTo fac l i ta te computa t ion whi le us ing l i near dev ia t ion ,t he xi, Yi and z ; va lues are t rans formed us ing therelat ions ( x i - 7c ) /&x , (Y i - ; ' ) /&Y and ( z i - 2 ) / ~ .respec t i ve ly6. The de viat ion s f rom the reference featureare measured in mic rons . The va lues cor respond ing toradial d is tanc es are s im pl i f ied by referr ing to a nomina ld imens ion.To compute the normal dev ia t ions , the ac tua ld imens ion o f the component shou ld be cons idered.Th is may lead to com puta t ions invo lv ing very la rge andextremely smal l values. Therefore, a double precis ionmode has to be used in these com puta t ions to ob ta inaccurate results.For approaches requir ing a search procedure, thesimplex search method is used. The procedure isexp la ined in the appendix . Th is method requ i res anini t ia l s implex. For example, three sets of values formthe in i t i a l s implex in the case o f a s t ra ight l ine , w herethe values o f Y0 and I 0 are to be evaluated. The searchis carr ied out in three di f ferent modes, namelyre f lec t ion , expans ion and cont rac t ion , depending onthe c loseness to the f inal value. The coef f ic ients usedfor re f lec t ion , expans ion and cont rac t ion are 1 , 2 and0 .5 respec t i ve ly . A convergence c r i te r ion o f 1 x 10 -7has been used in the present work.The form errors are calculated f rom the respect iveequat ions. For example, i f the parameters of thegeometr ic feature are evaluated f rom the l ineardev ia t ion , the form er ror i s a lso computed f rom thel inear dev ia t ion . Tab les 1 -5 s how the va lues comp utedfor di f ferent geometr ic features. For comparison, the

P R E C IS ION E N G IN E E R IN G 99

8/13/2019 Comparison of linear normal deviations

5/7

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 t io n s o f f o r m s o f e n g i n e e r in g s u r fa c e sTable 4 Cyl indr ic i t y m e a s u r e m e n t

K = 8 , L= 3 , N = 24ata: rw=10mm,( r i - rw), m

0i, deg0 45 90 135 180 225 270 315

+2 5 5 3 4 3 1 2 2 3zi , mm 0 4 4 3 3 3 2 2 2- 25 3 2 4 3 2 3 1 2Values Least squares Minimum deviation

/~m Linear Normal Linearormal(Ro - r~) 2.79168 2.79167 2.59878 2.60803Xo 0.55897 0.55892 0.53834 0.58917Yo 0.67680 0.67678 0.73441 0.65671/o Az 0.64004 0.64016 0.86295 0.80914mo &z - 0.03694 - 0.03661 - 0.15358 - 0.11801h 2.24834 2.24791 2.00027 2.00009hs 0.62663 0.62653 0.67295 0.66240

parameters for normal deviations are presented aftersuitable transformation.The computer programs were written inFORTRAN IV and run on an IBM 370.C o n c l u s i o n sThe minimum deviat ion approach suggested by otherresearch workers does not fix the position or the sizeof the geometric feature. For example, the intercept Y0

or z o in the case of a straight line or plane and theradius Ro of a circle, cylinder or sphere are notdetermined uniquely. However, the modified functionsuggested in this paper does yield a unique value ofthese parameters.The minimum deviation technique gives values ofform error which are less than those obtained with theleast squares technique. The difference between thevalues is quite appreciable.In both these techniques, the normal deviation

T a b l e 5 S p h e r i c i t y m e a s u r e m e n tData: rw=10mm, K=8, L=4, N=26( r i - rw) , I~m

Oi, deg0 45 90 135 180 225 270 315

deg+90 5+45 5 3 4 3 1 2 2 30 4 4 3 3 3 2 2 3- 45 3 2 4 3 2 3 1 2-90 3

Valuesm

Least squaresNormal Linear

Minimum deviationNormal Linear

(Ro- rw)x0Yoz0hths

2.884680.681230.806180.41 2133.397570.78551

2.884610.681210.806220.412133.397600.78552

2.963361.383820.402190.616252.840990.91724

3.010901.411290.242300.584492.830870.95197

100 APRIL 1987 VOL 9 NO 2

8/13/2019 Comparison of linear normal deviations

6/7

S hun m uga m --L ine ar and norma l dev ia tions o f fo rms o f eng ineering sur faces

R ~ p;(r~,e;,~./ 0 o x o\ Assessment / X

Fig 5 Devia t ions f rom a sphere

app roach g eneral ly resul ts in di f ferent form error valuesf rom those obta ined w i th the l i near dev ia t ions .Howeve r , the d i f fe rence be tween these va lues is qu i teins ign i f i cant fo r prac t i ca l measurement prob lems.Computa t ion o f the form er ror w i th normaldev ia t ion sh ou ld be car r ied out in doub le prec is ion asthe equa t ion compr ises very la rge and ex t remely smal lvalues. I t is also found that the normal deviat ionapproach requ i res longer computa t ion t ime, wh ich i snot just i f iable in v iew of the marginal di f ference in thevalues.I f the expressions for l inear and normal deviat ionsare analysed, the expression for normal deviat ion isseen to conta in the ac tua l d imens ion o f the compo nent .Whi le using l inear deviat ion, the t ransformat ion of x i ,y ;, z ; and r~ values ca n be co nve nient ly carr ied ou tw i thout a f fec t ing the bas ic equat ion . In fac t ,measurement f rom a c i rcu lar i t y char t shou ld befo l lowed by computa t ions based on l inear dev ia t ionson ly . The in it ia l rad ius and the m agni f i ca t ion employedin t rac ing w i l l no t a f fec t the resu l ts when such anapproach is used.In some cases, the resul t is based on a smal lernumbe r o f data po in ts and there fore may have l imi tedvalue. However, the resul ts serve to demonstrate thepr inc ip les conven ient l y . The pr inc ip les out l i ned heremay be ex tended to any geomet r i c fea ture .A p p e n d i xA l g o r i t h m f o r s i m p l e x s e a rc hThe geomet r i c f i gure formed by a se t o f n + 1 po in tsin n-d imens iona l space i s ca l led a s implex . When thepoints are equidistan t , the s imp lex is said to be regular.For example , a regu lar s implex in two d imens ions i san equi lateral t r iangle and in three dimensions i t is ate t rahedron.In genera l , the s implex method compares theva lues o f the ob jec t i ve func t ion a t the n + 1 vert icesof a s implex and moves th is s implex grad ua l l y towar dsthe min imum po in t dur ing the i te ra t i ve process . The

movement of the s implex is achieved by ref lect ion,expans ion and cont rac t ion .For the sake o f s impl i c it y , cons ider a two -dimensional space (ul , u2) and the in i t ia l s implexformed by U1, U2 and U3 as shown in F ig A- I .The va lues o f the ob jec t i ve fun c t ion are obta inedat the vertices of the init ial simplex. If Urn (say U2) isthe ver tex cor respond ing to the h ighes t va lue and Uois the cent ro id o f the remain ing po in ts , then a po in ton the l ine jo in ing Urn and U o may be expected to havethe smal lest value.Fi rst , a ref lected point Ur is obtained f rom therelat ion

U , = ( 1 + ~ ) U o - ~ U r n ( A - l )wh ere > 0 is the ref lect ion coef f ic ient .I f the re f lec t ion produces a new min imum(ie f (U r ) < f (U 1 ) where U1 represents the vertexcor respond ing to the min imum func t ion va lue o f theini t ia l s implex), one can general ly expect to decreasethe func t ion va lue by mov ing fur ther away. Theexpanded po in t U e i s g iven by :

U e = ~U r -~- ( 1 - ~ ' ) U 0 (A- 2 )wh ere 7 is the ex pan sion c oef f ic ient . 7 > 1.I f the expansion is successful , ie f ( U e ) > f ( U l ) ,then a ne w s implex is fo rmed rep lac ing Um by Ue.Otherw ise, Ur replaces Urn, form ing a new simplex.I f the ref lect ion process resul ts in a value greaterthan the highest value of the in i t ia l s implex, ie,f ( U r ) > f ( U r n ) , then contract ion is carr ied out using

Uc = /~Urn + (1 - / ~ ) U 0 (A -3 )where /~ is the cont rac t ion coef f i cien t , 0 < /~ < 1 .For cond i t ions f ( U r ) < f (U rn ) and f (U r ) >f (U i )i~ rn the cont rac t ion i s accom pl i shed by rep lac ingthe po i n t Urn by Ur and us ing Eqn (A -3) . Thecontract ion is successful i f the value at the contractedpoint Uc is less than the minimum of f ( U m ) and f (U r )and U c replaces Urn. Otherw ise, al l U is are modi f iedas (U i+ Ue ) /2 and a new s implex i s fo rmed.

u 2

JU 2 -U m )

U 1~ , j J J JJ

Fig A- 1 P r inc ip le o f s imp lex search

y

u I

P R E C IS ION E N G IN E E R IN G 101

8/13/2019 Comparison of linear normal deviations

7/7

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 ti 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 e sS i n i t ia l i z e t h e s i m p l e xU1, U2 , ..... . . . Un+ 1

S p e c i f y ~ ,~ , - ~ , {Vr F i n d U m a n d U s u c h t h a t| f ( U m ) = m ax ,x - R e f l e c t io n c o e f f ic i e n t a > 0i = 1 t o n + l [ f u i ) ] 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 < . ~ < 1Y [ f l U l l = m i n/ = I tO . + 1 l f l L l i } l */ - E x p a n s i o n c o e f f i c ie n t ~ > 1e - V a l u e f o r c o n v e r g e n c e

~ - I s e t ~ = ~ , I

I s ~ t u m ~ u , I

F i n d t h e c e n t r o i dU O = n + l

~ m

R e f l e c t u s i ngU = ( I + ~ ) U o - e U mN o

I E x p a n d u s i ngU e = ~ U r + ( I ~ ) U o I

I s e t U U ~ I~

I c o ~ o t o IQ = ( N [ f l u i ) - f l U o l ] 2 i n+ l ) V =

1 Ic = ~Um+ (1-~ ] )U 0 ,

I ~ r : l ~ + U ~ JF i g A 2 F l o w d i ag r a m o f s i m p l e x s e ar c h m e t h o d

Wi t h t he new s i mpl ex , aga i n t he re f l ec t i on processis started and al l other steps are fol lowed t i l l theconve rgence cr i te r i on i s sa t is f ied . The met hod i sa s s u m e d t o h a v e c o n v e r g e d w h e n e v e r t h e s t a n d a r ddev i a t i on o f t he f unct i on a t t he n + 1 ver t i ces o f t hecurren t simplex is smal ler than the prescribed valu e_ n + - ; I- I ~ < r , A - 4 )

Th e cent ro i d o f t he f i na l s impl ex is t aken t o representt he mi n i mum po i n t .Th e f l ow d i agram of t he s i mplex search procedureis g i v e n i n F ig A - 2 . T h e m e t h o d e x p l a in e d a b o v e c a nbe eas i l y ex t ended t o n - d i mensi ona l space .R e f e r e n c e s1 Tec hnical drawings: toleranc e of form and o f position, Part I:Generalities. I S O / D I S 1 1 0 1 , 1 9 7 22 Assessment of departure from roundness. B S 3 7 3 0 , 1 9 6 43 Go t a M . a nd L izuk a K . An ana lys is o f t h e re la tions h ip be t we e nminimum zone deviation and least squares deviation in circularityand cylindricity. P r o c . In t . C o n f . P ro d . E n g g . , N e w D e l h L I n d i a ,

1 9 7 7 , x 6 1 - x 7 04 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 e va lu at io nof surfaces. In t . 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 65 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 th m s f o r f o rm er ro reva luat ion - methods of m inimum zone and the least squares .P r o c . In t . S y rn p . M e t r o l o g y a n d Q u a l i ty C o n t r o l n P r o d u c t i o n ,T o k y o , 19 8 4 , 1 9 7 - 2 0 26 S h u nm ug a m i . S . On a s se s sme nt o f ge ome t r ic e rro rs . I n t . J .P r o d . R e s ., 1 9 8 6 , 2 4 2 ) , 4 1 3 - 4 2 57 Sh un m ug a m i . S . C r it er ia f o r c omput e r - a ide d f orm ev a lua tion .1 2 th D e s i g n a n d A u t o m a t io 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 68 Ra o, S . S. O pt imisat ion theory and appl ications . W i l e y E a s t er nL t d , I n d i a , 1 9 8 4

1 0 2 A P R I L 1 9 8 7 V O L 9 N O 2