163thResearchSymPOSitLm）TouhokuDivisionofSocietyorInstmnentandControIEngineerinBS

DocumentNumber163．9

ThedcvelopmentofPC−bascdCADsystembrreverslngenglneerlng

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BingyanZhao●，ShigenoriOkubo．＋＝ FacultyofEngineering，YamagataUniversity●●

FacultyofE喝ineerinB，YamagataUniversity

Keywords＝COOrdinatemeasurementmachine，metrOlogy，CAD′CAM，reVerSeenglneerlng，CurVeandsurhcesmoothing

Correspondenceaddress：4−3−16Jonan，Yonezawa，992，Japan

Okuboresearchlaboratory，FacultyofEngineering，Yam喝ataUniversityBi喝yanZhao，Te）・‥023畠−26−3245，Fax二0238−24−6445E−mail＝Zhaobing＠mip3470・yZ■yamagata−u・aC．jp

itapartfromtheexistlng COmmerCiaLPC based

1．Introduction

Freeshape叫ectsarecomposedofalarge

CAD／CAM systems suchas Autocad，Personal

numberof丘・eeformsur血ces，this characteristic

Designer，Microstationetc・，thesystemproposed

distinguishesitfromothermechanicalproductsin

hereismainlyforthereverslngenglneenng，thatis

thedesignandmanufhcturlngPrOCeSS．Inmodem

tosay，theincozT）OrationofCMMw■iththePC＿basedCAD／CAMsystemwithwhichthefbllowlng
automobi）edesignandmanufhctunngprocess，theCOOrdinatemeasuTementmaChine（CMM）iso鮎n

免加11reShold：usedinthos￠Ofthemanufhcturlngandmeasure− 1）Tofu11ydigoutthepotentialfunctionsofCMM．
mentofdlel：5daymodel，themastermodel，the GeneralIy，the measurement softwarefor CMM
PuShdie andthe最知ures．Thisiswhatwecon−CaJlmere）ydealwithsomesimpledeviationcalmー

Cernedfbrthesystemdeveloplng．

1ationsanditsrneaSurementfunctions arevery

Tbe CAD／CAMteclmologymaketheauto−

limitedcomparedwiththeCAD／CAMsystemmobi）e body desiBn and maJlufhcturmg a great

CMMincorporateswiththeCAD／CAMsystem

PrOfits，but most CAD／CAM systems such as

見nalmost pe血rm肌ymeaSurementfunctions

directly as opposed to transform it to the
ProEngineer，UGII，CADDS5，CATIA，Euclid−IS
etc．runOntheworkstations ratherthanPC com−

CAD／CAMsystemfbrfurtherprocess．

Puterbecauseofthecomplexcalculationsoffree2）Totake the CMM as ainput device ofthe

formcurvesandsurhces・TodaythePCcomputerCAD／CAMsystemcaJ）directlyconstruCtathreeisdevelopLng鮎tanditsperformancehavealready dimensionalmodelinthesystem，heretheCMMis
beenuptotherequlrementOfCAD／CAM．Setting＿1＿

reftrredasthT・ビeEh．ensionaIdigiti＝er■

C）aymodelismaJlu血cturedtoevaluatetheshape3）Tomonitorth巳meaSurementPrOCeSSOn−1inein

the computer screen弧d deletetherude pomts inrealworldandrefineit．3）WiththeCMM，theinmlediatelyasopposedtoprocessitlatter．Also

datapomtsaredi由tized丘omtheclaymodelaJ）d

CanmakenotationsormarkstocLassiqTthegrouparetrans鈷rmedlaterintotheCAD／CAMsystem

Ofthemeasuredentitiesfbrtheeasyofpostproc−

to buildthe mathematicaL model．4）The masteress．Specifically，砧rthemeasure＝mentOftheauto−modelis manu也ctured from the mathematicalmobi）ebody，thepolntSnumberissogreatthatitis

model．5）The mathematicalmodelin the

difnculttoclassiqTpOintstothisorthat surh亡e

CAD／CAMsystemmaybechangedcompatiblyto

PatChes．

themastermodelbydigitiz）ngSOmePOmtSOfdifl4）TocombinetheCMMandCAD／CAMsystem血■enCe・6）Thedesignandman甘払ctureofthepush
andNCmachineintoaintegratedsystemcaJIPer−

dieaJldotherrelativefixturesandsoon・Thesys−

formprofi1emodeIingofworkpiece，butthispro−
temproposedhereisdevelopedaccordingtothe

PrOCeduresofthedesignandmanuhctumlgOftheme modeling havemoreflexib）ethaJtthe only

COPytheworkpiece，itcoulda1lowtheusertomod−
鮎eshapeproducts，Asthesystemisonglnated加rntheincor−i阜′theshapetOtheusersrequirements．

Forexample，itcanallowusertomeasurethe

POration ofCMMwithNC machine，the CMMdatapolntS付omthemastermodelandconstruCtandNCmachinetaketheroleofinputandoutput
theteclmicalmodelbyconstruCtlngthedrawand

devicesrespectivelyasinFig．1．nemeasurement

StePPartOfthediemodelasopposedtocreateafunctionsisthesubmodulewithintheCAD／CAM

realteclmicalmodel．

SyStem．

BasedontheconsiderationofaboYereaSOnS，

仙川

anintegratedCMM／CAD／CAM systemis devel−

Plint亡rFil亡OPed．TherearefbllowlI唱PartSinthispaper，SeC−
tion2isthe占eneralstruCtureOfthesystem，SeCtion

Fil亡

仙Iuさ亡

抑州別K亡叶oardHC

D氾＝〃GESD〉肝〃GES

3describingthecompensationoftheproberadiusFig・1nebasicstruCtureOfthesysternandthemode）ingconstruCtion，SeCtion4givesout¶le generalstruCture Ofthe systemis ar−acurveandsurfhcesmoothingalgori血m．Thelast

rangedaccordingtotheproceduresof鮎eshapeistheconclusion．Tothelimitationofthepages，

thesuI血ceGCIsmoothingandtheCAMpartis

PrOductsdesignandmanufhctureasinFigr2・The systemtakesthewir丘ame and sur血ce

not血scussedhert∋．

model，andthe coruleCtion between CMM andユ．Thesystemdesign

COmPllterisbyRS−232c．¶ledatapointsorthe
The mainprocedures of丘■ee Shape objects，

measured entities are directly displayed on thefbrexample，autOmObilebody，designandmanu−
fhcturingis as fb））owing：1）A concept desiBn

SCreenandstoredintotheCAD／CAMdatabase．

TLedatabasearecomposedofoneEntitylist

accordingtothetechnicalrequirements．2）Al：5＿2＿measured entities are directly displayed on thetypesofentitiessuchasentitynumbers，Validmark，screenandstoredintotheCADICAMdatabase．

1ayer，ViewLngmark，grOuP，fotd，COlorandetc・，

rnledatabasearecomposedofoneEntityList

thelastoneisthepointerforthegeometrydatalistof16−byte records，On巳fbr each entity andthe

Oftheentitywhichincludeseveralstandardsubre−recordsholdsattributeswhich arecommontoa11

COrdsofvadab181eng血・Fig．2 −¶16

1ayoutofthesystemurlngthe su血ce which containthis poLntSand

3．The radius compensation of CMM

PrObe且ndthreedimensiona］modelcon− PrqeCtthepoLnttOtheo飽etsurhcewiththesur・
血cenormaldirection passingthepolnt．
Stru亡ting

3．2Themeasurementofspatiallinesandcircles

Thedigitizeddatapolntisthecenterposition

Thetraditionalmethodistoa鴫ustthemeas−

oftheCMMproberatherthanthecoITtaCtpOlntOn

uredlinetoparal1eIorperpendiculartooneofthe

thesur鮎eofthebeingmeasuredobiect，Inorder

toobtainthepomtsontheobject，theproberadiusmeasurementaxis，thisisinconvenientbothfbrthemanlPulation aJld fbrthe reverslng englneenng

must be compensated.

PurPOSeS．Herethemethodofmeasurementisto

3．1¶lemeaSurementOfonepolnt
Formerelyonepo）nt，1tisdi餓culttoo飽etmeaLSuretheplanesandbyintersectionofplaJleS，the）inesareobtained．

automaticallyifthemeaLSunngdirectionisunCer−

3．3ThemeasurementofcurvesaJldsuI血ces

tain．If themeasuredoqectisfreefbrmsu血ce，themethodtoobtainthecontactpomtsisbymeas−¶lemeaSurementOfcurveiscorrLmOnlytol−ユーSPhere，the cylinder，butfbrthecubeorconvex
setoneaxisconstant，l．e．aSeCtioncurve，1tlSeaSy

toof臨etacurvewiththedistanceofproberadius POlygono叫ects，thecompensationofproberadius
ismoredi餓cuLt．

haplane・If廿氾CuⅣeisnotasEdion仙Ⅳち aS

Forapolyhedralconvex，1tismoredifRcuLtto

血Fig・3，血e su血ce也at co】血血仙e cuⅣ6

m組SureeaChedgeoftheobjectdirectlythanto

mustfirstbemeasuredandprqecteach meas−

uredpoLntSOnthecurveontotheoff岳etsu血ce， measuretheplanesandhtersectthemwitheach
whereeachpointhasitsownprq［eCtiondirection，Othertoobtaintheintersectionlines，AIso，itdoes

the relative sur血ce nomlal direction passlng

nothavetoa句usttheo匝cttotheeasyofmeas−

throughthepolnt．finalLytheprqectedpolntSis

urementandtheproberadiuscanbecompensatedsmoothedtocurve．

automatically．

bt Pl（Ⅹ．，yいZJ，Pコ（Ⅹユ，yコ，Zヱ），q（Ⅹ3，y如ち）
arethethreepolntSOnaplane．
Vector両＝声高x声高directlngtOtheinsideof
the object．Then the plaれe equation is
AX＋BY＋CZ＋D＝0．

Where

Fig．3ProberadiuscompenSationofc
Forthemeasurementofthe丘・eefbrmsur血ce，

A＝（y2−y．）（z3−Z．）−（y，−yt）（z2−Z．），
B＝（xユーⅩl）（zユーZ．）−（ⅩユーⅩl）（z3−り，severalmethods Gan be usedto smooththe data

poultStOSu血ceandobtaintherealonebytheC＝（ⅩユーⅩ．）（y3−y．ト（Ⅹ3−Ⅹl）（yユーyl），normaloffietsurfhce，thispaperproposedagen・

D＝−（AX＋BY＋CZ）．

eralschemefbrthesmoothingnetgridpolntSand

Theo臨etplanewithdistBLnCedis

thescatterdistributedpolntStOSurfaceasinsec−

AX＋BY＋CZ＋d−（Ax一＋By一＋C2：l）＝0■

tion4．

nepolyhedralconvexisclosed，becauseallnor−

Ifthe above questions are solved，thenthe
CMMcanberefEreedtotheinputdevice，adimensionaldigitizer

mal＄OftheplaneSdirecttotheinsideoftheotホct．

threewithwhichthe user canmeasureandconstruCtthethreedlmensionalmodelTllefo1lowingalgoritlmistofindal1theedgesof
血epolyhedralGDnVeXl）F血dall血planesTて謹＝1，ヱ：．．皿）

intheCAD／CAMsystem，

2）FindtheintersectionlineL ofplane7Ti，7tj

3．4The automatically construCtlngthe convex（i≠j）鮎inFig．4．

polyh喝en叫edmodelwi山CMM

Basedon CSGmodelingmethod，anyCOm− 3）Findtheintersectionpoint PkbetweenLandthe
plexoqiectsCan beconstruCted bytheBool弧1Plane T【k（k＝1，2，・・・n，k≠i，j）

0PerationofthestBuldardprimitivessuchascube，

4）Bycheckingwhetherthe Pkisinsidetheobject

SPhereaJldcyLinderetc‥ItiseasiertomeasuretheOrnOttOdeteminethetwoendofL，thecheck一4−

4．1CuⅣeSmOO也hg

criteriais AmPkX＋BmPky＋CmPkヱ≧1・OeT5

ForagivingsetofpointsPj（j＝1，25）Gottostep2）untilal1theedgelineareobtained・，m），

Here，thealgoritlmfocuses on打ommears− findaparametriccurvewithco血■01poLntS

urement to modelconstruCtion，the geometricalM

Ci（i＝1，2…M），thenP（u）＝∑Bi（u）・Ci＝Pj

devi孔tionsaJldthepositiondeviationsarenotin−

i＝1

cluded．Fortheprecisionmeasurement，theproce− 0＝1，2，．．，m）（1）

duresare：Withthe variationalpnnciple，Seekthe curve f1）MeasuringthreepohTtSinclockwisewhilefacing

fromthefbllowlngfunctional

theplane，thisisto deteminethecompensation

P（q＝α・E（q＋β■8（qdirectionoftheproberadius．

（ユ）

Where E（f）isthedeviationit印l，0（f）issmooth2）Me且Suringthenetpointsontheplaneandusing

item，α，βaretheapproximationandsrn00tlmessthelea5tSquaremethodtoobtainthegeometricalⅥ喝ights．
d巳Viation．
m3）Theleast square planeis being usedas thenen p（r）＝∑αjj＝l7ti（i＝1，2，・・・n）血rtheconstruCtingofthemodell

M∑Bji（u）・CiTPj
i＝l＋宣（3）j＝1 βj‡葺Bごi（叶Ci‡ユ

Wlere mis the number ofpom短Selected to1

evaluatedthecurvatureofthecurve．
From−

塵 ＝0，thefollowingmatrixequation∂Ci

Fig．4ConstruCtlngCOnYeXObjectmodel

hold5．
BT・α・（B・C−P）＋IrT・β・B ・C＝0

4．CurYeandstJrhcesmoothing

C＝（BT・α・B＋B−T■β・B■）−■BT・α・P（5）■ThecurveaJldsurhcesmoo也血ghavebeen
studiedintensely（MJPratt（4）85，Nowackiand

Ⅶ鮎re C∈RMxl，

B∈RmrM，B■∈RmXM，■

P∈RmXl，α∈RmXm，β∈Rmxm

R巳∈S6，蕃3； Po仕mam，91；H喝餌 弧dSchultze，92；Hagen，H．and BoImeau G．−P．93；
happlication，theparameterofpoints
MIG，Bloor，M．］．Wilson and H．Hagen，95，Pj（j＝1，2，m）iscalculatedbysumofthe ac−

Schumal（er，LL76，Welch，W．andWitkin，A．92），
CumulatedchordIengths．

butLittlepayattentiontotheeasyofmeasurement，rnlis curve smoothiI唱SCheme canalso beherethesmoothingschemeha5）ittleconstraintson

usedtosmoothasetofpointson

measurement as opposed to strictly to net grid廿acks

tosur一払ce．First，SmOOtheachtrackto curvewitha

pomts・

glVen COITtrOIpoLntS KU，then smoothcontrol＿5■POlntSOfeachcurvesinanotherdirection・ヰ・Cf卜も・C・鴇＋も・C・槌＋克・C槌＝軒αP骨頂

4．2Su血∝SmOOth血g岬）

Herewetiketheboun血ryconditionintoaccount・ForagivenBridpointsPu（Ⅰ＝l，2 M，J＝l，2，…，N），justascurvesmoothing，知rsu血ce

Letthecontrolnetpointsmatrix C＝X＋Esmoothing，theVariationa）functionis

where XistheinnercontrolnetpolntSmatrix，and EistheboundarycontroIpomtsmatrix．

p（r）＝†・E（r）＋†。・8（r）．

也enweyields

Where E（f）deviationitem， 0（f）smooth

へ・Ⅹ旦＋互．・Ⅹ・観l＋A能・Ⅹ・鴇ユ＋克・Ⅹ・粘＝D

item，†approxjmationweight，†8SmOOthweight▲（9）

Accordingtotheshe11血1erenergyPreSentation，

Wも∈r6

址e8（f）＝比（D三。r＋D：

D＝B：・α・P・β・Bv−A．・E・B．＋

r＋D三，r）ds as山e

A朗・E・B別＋A8ユ・E■B自2＋A。3■E■B舶smoothitem and representitindiscrete fbrm，

where D≡。＝Bこ・C・B：，D三。＝B。・C・BこT，

Al＝Bニ・αtB。，Bl＝B：・β†Bv，

D三u＝B∴C・Bご．

A。i＝B忘i・α。■B。。i，B。i＝Bニi・β。■Bv。i（i＝1，2，3）

SoIvingthe abovelinear equation，the controI

画料儲B山（u）・B耶−Pu ‡ヱPOlntSCisobtained＋卦藁B胸（叫ー1‡2［A仇⑳B言iヰl咄＋毒 ⑳BF】＋∑n m

N M ＋∑†柑∑∑B血（叶軋（Ⅴ）
i＝lJ＝1

Ⅰ＝t J＝t∑（10）Ⅶleren m

N M 十∑・ロ（D）
l ‡You∑∑BJu（u）・BL力（叶C毎

Ⅰ＝t J＝l∑

ロ：CmⅧ＝∋CmnXl，SOthat．ロ（C）＝Ⅹ∈CⅧXl

i＝】J＝l¶le（10）canbeusedtosmoothbothnetpointsa皿d（占）inwhichN，M

arethenumberofl］Olnt50nthe Onlyfburorthreeboumdaries poLntStO Sur血ce．

Sur血ce仙如issought．

ButfbrtheconvemientofmeaLSurement，1tisoRen

There are nxmequation offbrm（6）whichis

tomeasurethe臨atureline＄aStheboundariesand

equaltothecontrolpolntSOfsurfhce・

SOmeSCatterdatapolnt5inside，theschemeshould¶lenthefbllowlngequationisholds．

havetheabilitytodealwiththecircumstance，

ユ・B：・α・（B。・C・Bニ【叶β・Bv

Foron6POintPk，finditsparameter（uk，Vk）by＋2・叱l・α・（B。。1tC・B：こ．）サBv。，

Newton−Raphsonmethod丘■Om

十2・Bニ云ヱ・α・（B。。2・C・Bエコ）・βtB，。ユ

漂＋2・Bニ，j・α・（B鵬・C・B：。3）・β・B咄＝0（11）

WhereS（u，V）istheCoonssurhceconstruCtedby（T）

In hctthe above equation hasthe fbrm of

the fbur boundaries．Thenthe scheme fbrscatter−6−

datapolntSSmOOthingis
Re俺reれ亡eS

x＝蛸］ヰ欄ヰ￣1・畠中k）
怯l）H喝en，H．andG．Schul2：e，G．（1992）Varia−
tionaldesignofsmoothB−SPlinesurfhces，1n：（12）Hagen，H・，ed・，Topicsinsur鮎emodeling，SIAM，Philadelphia，PA，PP邑5−942）Hagen，H・andBonneauG．一P．（1993），Varia−

tionaldesignofsmoothrationalBeziersur−
fhces，ComputLngSuppl・8，pP133−138
3）M・JPRATr（1985）Smoothparametricsur−
Fig．5Smootl血gscatterdatapolntStOSurhce，thefhceapproximationstodiscretedata，Com−

maximumdeviationbetweenpolnttOthesu血ceis

PuterAidedGeometric Design2 pp165−

O．02，fbrthesmootlmessweightO．01inbothuaJld

171．

VParameterSdirections．

4）MlI・G・Bloor，M・J・Wilson，H．Hangen（1995），The smoothing properties ofvari・

占Com仁1usion

tionalschemesfbrsurhcedesign，ComputerTYleSyStemPrOPOSedinthispaperrumlmgOnAidedGeometricDesign12 pp381−394．

thePCcomputerhavingtheadvantageofeasyfor5）Nowacki，H−aJldReese，D．（19g3），Design
use．WiththeCMMasathreedimensionaldigit−

izer，the regularthreedimensionalmodelcanbeand蝕血gofshipsur鮎e3，in：13ami1il），R．E．

COnStn）Cted directly丘omthe measurement．TheaJld Boelm，W．，eds，，助所cesin〔1GD，furtherinvestlgationconcerrtratesontheboundary

NorthJio11and，Amster血m，pP12l−1346）Schumaker，L■L（1976）Fitting surthces todetection ofdle meaSunng POlntS丘・Omthe舟ee
formsur払ceot函ctbythe鮎tureanalysistocoIl−

SCattereddata，in：Lorent2：，G．G．，ChuiC．Kand

StruCtthesur払ceautomatica））y．

Schumaker，L・L・，eds・，AHrOXimationmeoly
EAcademicPress，NewYork，Ppユ03−26S7）Pottmarm，H−（1991），Scatterdatainterpola−tion based upon generalizedminin1umnet−
WOrk，ConstrainedAppro・7，PP247−2568）We）ch，W．andWitkin，A．（1992），Variational
Su血ce mode）hg，Comput．Gra，Ph．26，pP
157−166．

Fig．6Acarroofhlishedwiththesystem＿7＿