MCM-based Uncertainty Evaluations practical aspects and critical issues

Transcription

1 C-based Uncertanty Evalatons practcal aspects and crtcal sses H. Hatjea, B. van Dorp,. orel and P.H.J. Schellekens Endhoven Unversty of Technology

2 Contents Introdcton Standard ncertanty bdget de wthot dervatves onte-carlo ncertanty evalaton achne eleent effects Exaples of vrtal easreent systes

3 Introdcton: What s a vrtal nstrent? Conventonal easrng nstrent Vrtal easrng nstrent d =.577 d =.577 =.75 A vrtal nstrent calclates ts ncertanty tself

4 The standard ncertanty bdget accordng to EAL and GU: =,..); = ) Qantty Vale of qantty Standard ncertanty ) Senstvty coeffcent c = Contrbton to standard-ncertanty n c ) = ) c ) ) c ) : : : : : N N N ) c N N ) total: ) + ) = +.. N

5 Uncertanty bdget wthot explct partal dervatves = Qantty.. ± ),..); =...) Vale of qantty Standard ncertanty Contrbton to standard-ncertanty n ) ) ) = +r ),,,..., N ) -,,... N ) ) =, +r ),..., N ) -,... N ) : : : : : N N N ) N =,,..., N +r N )) -,... N ) total ) = N Vary each paraeter separately wth ts ncertanty? ± = + r r = ± rando nber)

6 onte Carlo Uncertanty Bdget ) ± + = = r r,,...),....); Vary all paraeter wth ts ncertanty at a te? ± = + r r = ± rando nber) Or r = rando nber wth <r>=, s r = Slaton nber k Estaton of standard-ncertanty ),..., )) ),..., N N N N r r + + = ),..., )) ),..., N N N N r r + + = : K ),..., )) ),..., N N KN N K K r r + + = K.. K =

7 Characterstcs of onte Carlo Uncertanty Bdget: dfferent dstrbtons can be cobned hgher order ters are taken nto accont nknown systeatc errors can be slated correlatons can be slated can be sed for very coplcated probles not too any slatons needed N=5? s s =%)

8 Inflence of flterng effects Calclate Paraeters e.g Ra =.86 µ Calclate standard ncertanty for any paraeter P = P nonal P + wth P nonal e.g Ra =.86 ±. µ Calclate Paraeters e.g Ra =.76 µ Exaple how to slate/evalate probe effect

9 Inflence of teperatre effects o L = f L, α, T, δa, δt ) = L + α δ T + T C) δα) L s length at C L s easred length T s average teperatre of object and scale a s average expanson coeffcent δα s dfference n expanson coeffcent δt s teperatre dfference between object and scale ethod to randose: calclate L back), randose paraeters, calclate new L

10 -Densonal Geoetrcal devatons Scale devatons are never rando: take correlatons nto accont ) ) ) ) j j j j x x j j x x x AC x x AC x x x x + = + = AC s atocorrelaton fncton dx x x L AC L + = δ δ δ δ ) ) )

11 -Densonal Geoetrcal devatons analytcal calclaton for easred length 3 devaton n µ poston n easred devatons along a easreent lne of a C 4 ncertanty n µ 3 L δ AC δ ) = x) x + δ L δ ) dx easred length n Standard ncertanty n a length easreent along ths axs

12 -Densonal Geoetrcal devatons: How to randose the: Atocorrelaton fncton st be constant 4 evdent possbltes : org nal sgnal flpped left rght error n croeter error n croeter reversed sgn flpped left rght and reversed sgn error n croeter error n croeter poston n - 3 poston n

13 -Densonal Geoetrcal devatons: How to randose the: Alternatve for perodc sgnal: Take forer transfor randose phases transfor back exaple: rondness dagra alternatve: rotate dagra over rando angle

14 -Densonal Geoetrcal devatons: How to randose the: Proble for non-contnos sgnal Dscontnty gves hgher haroncs

15 lt-densonal Geoetrcal devatons Lnear, straghtness and rotatonal errors can all be slated lkewse antanng atocorrelaton) Correlaton between axes: sqareness: slated separately

16 Exaple: Vrtal roghness tester No onte-carlo: jst ncertanty bdget Qantty vale standard ncertanty contrbton to ncertanty n Ra 8,3 n contrbton to ncertanty n RS 4 µ x-axs ) %,4 µ calbraton z-axs ) %,8 n calbraton λ c,5 %,5 n,4 µ λ s,5 µ %,43 n,3 µ F N 5%, n,3 n rads µ 5%, n, µ step sze,5 µ %, µ nose 3 n RS, n, n total s),6 n,6 µ

17 Exaple: Srface plate easreent onte-carlo: sperpose rando ncertanty on easreents, re-calclate paraeter exaple: ncertanty n easred heghts

18 Exaple: Vrtal Fzea nterferoeter Randoze reference srface: flp, rotate, negate

19 Vrtal C Calbrated/easred devatons n -D s T y y T y x y T x y x T x x

20 Vrtal C easreent strategy: easre real probed objects/dsplaceents: ths ncldes software correcton easre sall and larger wavelengths For each lnearty/straghtness: 4 µ wth µ steps.6 wth 4 µ steps 3 wth µ steps.5 wth steps)

21 Vrtal C Straghtness easreent: cerac straghtedge

22 Vrtal C Straghtness easreent: cerac straghtedge.5 error xty, easred, stepsze..5 error xty, easred, stepsze.6 error n croeter error n croeter poston n poston n Short wavelengths Long wavelengths

23 Vrtal C Lnearty easreent: laser-nterferoeter step gage

24 Vrtal C Lnearty easreent: laser-nterferoeter step gage xtx error, easred, stepsze. xtx error, easred, stepsze.6 error n croeter error n croeter poston n poston n Short wavelengths Long wavelengths

25 Vrtal C Note short-wavelength yty-devatons nflence on slghtly salgned straghtedge n easreent of xty.8 straghtness devaton n µ x posto n n

26 Vrtal C Synthesse rando sgnal fro 3 easreent ranges.8 exaple for xtx.6 error n croeter poston n

27 Vrtal C Slate and calclaton: synthetc data generaton wth own progra calclaton of geoetrcal paraeters easy ones: own progra coplcated ones: Zess Uess va backdoor, not easy) 8 9 x axs y axs

28 Vrtal C Reslts for rng gages, as a fncton of nber of easred pont.5 daeter.5 daeter 5.45 odelled easred.45 odelled easred standard ncertanty, croeters standard ncertanty, croeters nber of easreent ponts nber of easreent ponts

29 Vrtal C Unsolved probles: traceablty: what to do wth ncertanty n calbrated objects repeatablty: enters n all calbratons, bt only once n real easreents dstncton between rando, nknown systeatc, known systeatc vanshes n ths approach alas of short-wavelength errors n long-wavelenght calbratons what to do wth ncertanty-bas n for easreents