Estimation of uncertainty of coordinate measurements according to the B method

Transcription

1 Estimation of uncrtainty of coordinat masurmnts according to th mthod Władysław Jakubic Univrsity of ilsko-iala (TH Dpartmnt of Manufacturing Tchnology an utomation Willowa, ilsko-iała, , Poland Tl.: 48 [33] Fax: 48 [33] bstract Th papr dscribs a nw mthod for analytical stimation of th uncrtainty componnt introducd by th CMM. Th uncrtainty is stimatd sparatly for ach charactristic in th gomtrical product spcification. Th uncrtainty is calculatd dirctly, i.. no analysis of th accuracy of dtrmination of particular gomtrical lmnts is prformd. Th fundamntal condition nabling analytical stimation of th uncrtainty is assumption that uncrtainty of coordinat masurmnt dpnds on th diffrncs of coordinats of probing points usd to calculat particular charactristic ywords: Mtrology, dimnsional, coordinat masurmnt machin, uncrtainty 1. Introduction Th significanc of stimation of masurmnt uncrtainty nds no justification. So far, prcisly formulatd has bn th mthodology of stimation of masurmnt uncrtainty with th us of calibratd workpic idntical or similar to th workpic for which th uncrtainty is to b stimatd. This mthodology was publishd as ISO/TS :004 [1]. Du to high costs of calibratd workpics, it can only b implmntd for larg scal production, spcially in th automotiv industry. Othr wll known possibility of uncrtainty stimation in coordinat masurmnts is th us of computr simulation. Mthodological basis of this concption was formulatd in Physikalisch-Tchnisch undsanstalt (PT in raunschwig []. On this basis, in diffrnt rsarch cntrs (including PT, a propr softwar was dvlopd. This concption dos not rais any srious doubts, but so far it is usd only in rsarch laboratoris. Th rsarch carrid out by Prssl [3] and Hrnla [4] on th possibilitis of us of analytical mthods for stimation of uncrtainty in coordinat masurmnts has not ld to a sufficintly ffctiv solution. Thir mthod sms to b intrsting on but has not good mathmatical backgrounds. Espcially th gomtrical rrors of CMM influncing th masurmnt uncrtainty of rspctiv faturs ar chosn arbitrary by th authors of mthod. Th author of this papr was succssful in finding mthod fully compatibl with th socalld mthod prsntd in GUM [5]. This modl is also in accordanc to th gnral trnds in th rang of gomtrical product spcification [6, 7]. Th assumptions for th modl, on th D xampls, wr publishd in [8]. In th chaptr, th ky condition which th modl has to fulfil to nabl th drivation of th algorithm of analytical stimation of uncrtainty of masurmnt is prsntd. In th following chaptrs, modl of simpl masuring task for 3D masuring machins is prsntd.. Gnrality condition Th most important rason for which no ffctiv mthod of stimation of uncrtainty of coordinat masurmnt has bn found is using th masurmnt modl which dos not mt th condition of gnrality. It will b shown that if th masuring task in coordinat -113

2 masurmnt was tratd as indirct masurmnt in which th masurands ar th distancs masurd along th axs, i.. th diffrncs of individual coordinats in particular axs, thn th modl of masuring task nabling th ffctiv stimation of uncrtainty by mans of analytic tchniqus could b built [6]. It is asy to justify this statmnt. If in th masurmnt modl subtracting of valus having systmatic rrors taks plac, thn thr occurs oftn at last partial compnsation of ths rrors. If th masurand x is calculatd as th diffrnc btwn x 1 and x x = (1 x x 1 and both masurands contain systmatic rrors, rspctivly x1 and x : x = x ( x x = x (3 1 1 x1 thn th masurand x contains th systmatic rror amounting to th diffrnc of ths rrors x = x ( x x 1 (4 and thr is a chanc for at last partial compnsation of th rrors. In such cass th uncrtainty of diffrnc u x x1 should b dirctly stimatd. Finding a good stimation of masurmnt uncrtainty of diffrnc of two masurands dos not hav to b an asy task. In coordinat masurmnts, it will prov ncssary to dfin functions xprssing th maximum valus which th diffrncs of particular gomtrical rrors can assum. Mor information on th gnrality condition prsnts [9]. 3. Masurmnt of distanc btwn point and plan s a conclusion arising from th analysis of th practical masuring tasks it can b pointd out that a lot of thm can b modld as a masurmnt of distanc point-plan. On of th xampl of such a masuring task is a masurmnt of paralllism of axs (Fig. 1. Fig. 1. Masurmnt of axs prpndicularity: a spcification, b masurmnt modl with th probing points. In prsntd modl th plan is dtrmind by thr points: (x, y,, (x, y, and (x, y,. Th distanc btwn th point S(x S, y S, S and th plan C is calculatd as l = v r (4 whr v normal vctor of th plan, r vctor of any point of th plan to th point S. Th normal vctor of th plan is ( x x, y y, ( x x, y y, v = (5 ( x x, y y, ( x x, y y, -114

3 In th rmaining part of this papr th shortnd notation will b usd: instad of x x it will b usd x, instad of y y it will b usd y, tc. So aftr th transformation th formula (5 will hav a form: ( y y i ( x x j ( x y x y k v = (6 ( y y ( x x ( x y x y s vctor r can b usd vctor S, S or S. Thn th distanc l can b calculatd by following formula l = v ( x, y, (7 S For th analysis of masurmnt uncrtainty, on has to assum that th distanc l is a function of nin diffrncs of coordinats masurd dirctly (thr variants l = f ( x, x, x, y, y, y,,, (8 ccording to mthod standard masurmnt uncrtainty is calculatd as S S S S l1 l 1 l 1 l1 u l = u x u x u xs u S (9 x x xs S 4. Masurmnt uncrtainty of diffrncs of coordinats of two points Th gomtrical modl of masuring machin nabls to driv th formula for th componnts of rror vctor in point (x, y,. It has to b takn into considration that, during th masurmnt, th radings from th CMM scals x, y, dpnd also on th usd stylus tip paramtrs x t, y t, t. ppropriat rlationship is following: x = x xt, y = y yt, = t (10 Th componnts of masurmnt rror in point dpnds on gomtrical rrors of CMM and ar following: x = ( yt m xwy ( h xw xpx( x ( h xry( x ( yt m xr( x (11 ytx y yry y y m yr y tx h ry y r y ( ( ( ( ( ( ( ( = x yrx t xwy ( h yw xty( x ( h xrx( x xt xr( x ypy( y ( y x yr( y ty( ( h rx( x r( = xt xw ( yt m yw xt( x ( yt m xrx( x xt xry( x yt( y ( y m yrx( y x yry( y p( y rx( x ry( t whr h, m dsign paramtrs of CMM. To calculat th componnt standard masurmnt uncrtaintis of diffrncs of coordinats u x, u y and u on has to notic, that rspctiv masuring rrors x, y and, ar as follows: = [ xpx( x xpx( x ] [ ytx( y ytx( y ] [ tx( tx( ] [ xry( x ( h ( t xry( x ( h ( t ] [ xr( x ( yt m xr( x ( yt m] [ yry( y ( h ( t yry( y ( h ( t ] [ yr( y ( yt m yr( y ( yt m ] [ ry( ( h ( t ry( ( h ( ] [ r( y r( y ] x [ xwy ( y m xwy ( y m ] [ xw ( h ( xw ( h ( ] t t t t Th formula for y i ar analogical. ccording to gnrality condition in abov formula th particular rrors ar groupd in pairs to nabl sparat stimation of th valus achivd by th diffrncs of th t t t t t S t t (1 (13 (14-115

4 gomtrical rrors of th sam typ. It is an indirct masurmnt, in which th masuring rrors of th diffrnc of coordinats ar th sum of a fw componnts. For corrct stimation of componnt uncrtainty x th 11 componnts of rror (similarly in th rrors y and furthr componnts hav to b distinguishd: ( x xpx( x 1 = xpx ( = xw ( h ( xw ( h ( (16 t In th following, th functions dscribing th maximum valus which th diffrncs of particular gomtrical rrors can assum, in function of masurd distanc (coordinat diffrnc ar dfind (on th xampl of xpx MX : xpx ( l max xpx( x xpx( x l MX x t = (17 Th xampl chart of function dscribing th gomtrical rror and of th rspctiv function dscribing th maximum diffrnc of rrors ar shown on Fig.. a b Fig.. Th xampl function dscribing th maximal influnc of th gomtrical rror: a function of th gomtrical rror, b th rspctiv function of th maximum diffrncs of rrors Th dfining of th abov functions is on of th ssntial lmnts of th prsntd mthodology of stimation of masurmnt uncrtainty. Using th abov functions, th limiting valus for th rrors (on th xampl xpx wr stimatd: xpx ( x xpx( x xpxmx ( x x (18 Furthr, according to mthod of uncrtainty stimation, for th componnt uncrtaintis, th rspctiv componnts of th standard uncrtainty ar xprssd as products of an xtrm valus of rrors and k i cofficints arising from th probability distribution of th particular rror. u = k xpx x x (19 ( 1 1 MX 5. Exampl rsults Th gnral input data for th softwar ar th dsign paramtrs of th CMM as wll as information on th gomtrical rrors and probing systm rror. Th probing point coordinats ar givn for ach masuring task. For ach probing point th paramtrs of th usd stylus tip ar ndd. Similarly to th dscribd task (paralllism of axs in normal plan, using th pointplan distanc modl, following tasks wr solvd: paralllism of axs in common plan, paralllism of axis to plan, paralllism of plan to axis, paralllism of plans, flatnss, prpndicularity of plans, prpndicularity of axs, prpndicularity of axis to plan, prpndicularity of plan to axis, angularity, and som cass of position dviation. Th uncrtainty of masurmnt of straightnss, coaxiality, paralllism of axs (with cylindrical tolranc on, prpndicularity of axis to plan ar valuatd according to point-axis distanc modl. -116

5 6. Conclusion Th fundamntal condition nabling analytical stimation of th uncrtainty is assumption that uncrtainty dpnds on th diffrncs of coordinats of probing points usd to calculat particular charactristic. It mans for xampl, that for masurmnt uncrtainty valuation of paralllism of a hol s axis and a plan it is ncssary to us th diffrncs of coordinats of probing points both th cylindr and th plan. Th starting point for prparing th uncrtainty budgt is a formula xprssing particular charactristic as a function of diffrncs of probing points coordinats. For calculation of uncrtainty of ach diffrnc in th formula th 33 componnt rrors connctd with gomtrical rrors of CMM ar takn into account. Calculation of maximal valus that th particular componnts rrors can hav is possibl thanks to th function dfind by th author which is composd on th basis of th function dscribing th gomtrical rror and th argumnt of which is th diffrnc of coordinats. 7. cknowldgmnts Th papr draws on th findings mad within th rsarch projct financd by th Polish Ministry of Scinc and Highr Education Rfrncs 1. ISO/TS :004 Gomtrical Product Spcifications (GPS. Coordinat masuring machins (CMM: Tchniqu for dtrmining th uncrtainty of masurmnt. Part 3: Us of calibratd workpics or standards.. E. Trapt, M. Frank, F., Hartig, H. Schwnk, F. Waldl, M. Cox,. Forbs, F. Dlbrssin, P. Schllkns, M. Trnk, H. Myr, G. Morit, Th. Guth, N. Wannr. Tracability of coordinat masurmnts according to th mthod of th virtual masuring machin. PT, raunschwig, H-G. Prssl, T. Hagny. Mssunsichrhit von Prüfmrkmaln in dr oordinatnmsstchnik: Von dn Gnauigkitsangabn ds MG ur Mssunsichrhit von Prüfmrkmaln. Exprt-Vrlag GmbH, Tyskland, M. Hrnla. Mssunsichrhit bi oordinatnmssungn: bschätung dr aufgabnspifischn Mssunsichrhit mit Hilf von rchnungstablln. Exprt-Vrlag GmbH, Rnningn, Guid to th xprssion of uncrtainty in masurmnt (GUM. IMP, IEC, IFCC, ISO, IUPC, IUPP, OIML, M. Starcak. Th invstigation of masurmnt stratgy simplification influnc onto th masurmnt rsults. VIIth Intrnational Scintific Confrnc Coordinat Masuring Tchniqu, ilsko-iała, Z. Huminny. Stat of art in standardiation in GPS ara ynot papr. 10th CIRP Confrnc on Computr idd Tolrancing Spcification and Vrification for ssmblis, Shakr Vrlag, achn, W. Jakubic. Mthodology of analytical stimation of uncrtainty of coordinat masurmnts. Procdings of th 9th Intrnational Symposium on Masurmnt and Quality Control. Indian Institut of Tchnology Madras, Intrnational Masurmnt Confdration IMEO, Chnnai, W. Jakubic. dquacy and gnrality conditions in stimation of uncrtainty in masurmnts of gomtrical quatntitis. dvancs in Manufacturing Scinc and Tchnology, 007 vol. 31 nr