C-based Uncertanty Evalatons practcal aspects and crtcal sses H. Hatjea, B. van Dorp,. orel and P.H.J. Schellekens Endhoven Unversty of Technology
Contents Introdcton Standard ncertanty bdget de wthot dervatves onte-carlo ncertanty evalaton achne eleent effects Exaples of vrtal easreent systes
Introdcton: What s a vrtal nstrent? Conventonal easrng nstrent Vrtal easrng nstrent d =.577 d =.577 =.75 A vrtal nstrent calclates ts ncertanty tself
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
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)
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 + + + =
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 =%)
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
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
-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 ) ) )
-Densonal Geoetrcal devatons analytcal calclaton for easred length 3 devaton n µ - 3 4 5 6 7 8 9 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 3 4 5 6 7 8 9 easred length n Standard ncertanty n a length easreent along ths axs
-Densonal Geoetrcal devatons: How to randose the: Atocorrelaton fncton st be constant 4 evdent possbltes : org nal sgnal flpped left rght error n croeter.5 -.5 error n croeter.5 -.5-3 - 3 reversed sgn flpped left rght and reversed sgn error n croeter.5 -.5 error n croeter.5 -.5-3 poston n - 3 poston n
-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
-Densonal Geoetrcal devatons: How to randose the: Proble for non-contnos sgnal Dscontnty gves hgher haroncs
lt-densonal Geoetrcal devatons Lnear, straghtness and rotatonal errors can all be slated lkewse antanng atocorrelaton) Correlaton between axes: sqareness: slated separately
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
Exaple: Srface plate easreent onte-carlo: sperpose rando ncertanty on easreents, re-calclate paraeter exaple: ncertanty n easred heghts
Exaple: Vrtal Fzea nterferoeter Randoze reference srface: flp, rotate, negate
Vrtal C Calbrated/easred devatons n -D s T y y T y x y T x y x T x x
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)
Vrtal C Straghtness easreent: cerac straghtedge
Vrtal C Straghtness easreent: cerac straghtedge.5 error xty, easred, stepsze..5 error xty, easred, stepsze.6 error n croeter error n croeter -.5.5..5. -.5 5 5 5 3 poston n poston n Short wavelengths Long wavelengths
Vrtal C Lnearty easreent: laser-nterferoeter step gage
Vrtal C Lnearty easreent: laser-nterferoeter step gage xtx error, easred, stepsze. xtx error, easred, stepsze.6 error n croeter.5 -.5 error n croeter.5 -.5 -.5..5. - 5 5 5 3 poston n poston n Short wavelengths Long wavelengths
Vrtal C Note short-wavelength yty-devatons nflence on slghtly salgned straghtedge n easreent of xty.8 straghtness devaton n µ.6.4. -. -.4 -.6 -.8 5 5 5 3 x posto n n
Vrtal C Synthesse rando sgnal fro 3 easreent ranges.8 exaple for xtx.6 error n croeter.4. -...4.6.8.. poston n
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 8 4 6 8 y axs
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.4.35.3.5..5. standard ncertanty, croeters.4.35.3.5..5..5.5 4 6 6 3 4 6 6 3 nber of easreent ponts nber of easreent ponts
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