Measuring Worst-Case Errors in a Robot Workcell

Transcription

1 SANDA REPORT SAND UC-76 Unlimitd Rlas Printd Octobr 1997 Masuring Worst-Cas Errors in a Robot Workcll Ronald W Simon, Randy C Brost, Dpsh K Kholwadwala

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4 SAND Unlimitd Rlas Printd Octobr 1997 Distribution Catgory UC-76 Masuring Worst-cas Errors in a Robot Workcll Ronald W Simon Randy C Brost Dpsh K Kholwadwala ntllignt Systms and Robotics Cntr Sandia National Laboratoris PO Box 58 Albuqurqu, KM Abstract Errors in modl paramtrs, snsing, and control ar invitably prsnt in ral robot s y tms Ths rrors must b considrd in ordr t o automatically plan robust solutions t o many manipulation tasks Lozano-Pirz, Mason, and Taylor proposd a formal mthod f o r synthsizing robust actions in th prsnc of uncrtainty [7]; this mthod has bn xtndd by svral subsqunt rsarchrs All of ths rsults prsum th xistnc of worst-cas rror bounds that dscrib th maximum possibl dviation btwn th robot's modl of th world and rality This papr xamins th problm of masuring ths rror bounds f o r a ral robot workcll Ths masurmnts ar dificult, bcaus of th dsir to compltly contain all possibl dviations whil avoiding bounds that ar ovrly consrvativ W prsnt a dtaild dscription of a sris of xprimnts that charactriz and quantify th possibl rrors in visual snsing and motion control f o r a robot workcll quippd with standard industrial robot hardwar n addition to providing a mans for masuring ths spcific rrors, our xprimnts shd light o n th gnral problm of masuring worst-cas rrors

5 Contnts 1 ntroduction 1 2 Prvious Work 3 3 Masuring vkion Dot Location 4 Error Modl 5 Random Nois 6 Camra Pixl Eifcts 6 Lns Distortion 8 Lns Error Corrction 11 Tsting Error Charactrization 12 Rsults 13 4 Arm Calibration 13 5 Joint 4 Orintation Error Masuring motion 42 Absolut/Rpatabl Positioning 43 Charactrizating Emotion 44 Corrcting motion 45 Rsults Error Charactrization Rsults Summary APPENDX A1 Normalization A2 A3 2-d ntrpolation mag to Objct Fram Mapping Bl Motion Masurmnt Rquirmnts Rfrncs

6 1 ntroduction n No ral robot systm oprats with prfct snsory information or control capabilitis consquntly, ral robot snsory data is an imprfct masurmnt of th world, and ral robot motions only approximat th intndd motions Ths factors may caus robot actions that will succd in an idal world to fail during xcution in th ral world This rality is on of th ky rasons that automatic planning and xcution systms ar not currntly mployd to solv tasks with high prcision rquirmnts For xampl, assmbly tasks oftn rquir insrtions that hav tolrancs that ar much tightr than th snsing and motion control capabilitis of currnt industrial robots Thus, an automatic task planning systm cannot simply command a robot motion to mov th parts into thir dsird positions; such a motion will invitably lad to collisions that do not occur in a prfctly xcutd motion Compliant motion stratgis ar oftn usd to succssfully accomplish insrtion tasks Part faturs such as chamfrs guid th parts into th dsird locations, ffctivly rmoving rrors Similar stratgis may b mployd to improv th rliability of part acquisition tasks To dat ths stratgis ar dsignd manually, sinc automatic planning systms do not xist that can automatically synthsiz compliant motion stratgis for ral robot tasks Lozano-Pirz, Mason, and Taylor [7] hav proposd a mthod for synthsizing robust compliant motion stratgis n brif, this mthod assums a worst-cas modl of robot rrors, and prforms an analysis of th task gomtry and mchanics to idntify compliant motions that will accomplish th task goal in th prsnc of rrors Lozano-Pirz, Mason, and Taylor xplain thir approach using an abstract modl of robot action; if w can implmnt a plannr that prforms this analysis for ral tasks and provid th ncssary input data, thn automatic planning of robust compliant motions can bcom a rality Th dvlopmnt of such a plannr has bn a primary focus of th Sandia Manipulation Laboratory for svral yars Our work toward an automatic planning systm will b dscribd in a forthcoming papr n this papr, w focus on th input part of th problm: How c a n w obtain th rquird worst-cas rror bounds? Figur 1 shows a simpl xampl that motivats this problm n this scnario, a robot is attmpting to insrt a pg into a hol Th robot has a modl of th hol s location, xprssd as (2, y) coordinats Th robot also has a modl of th pg s position rlativ to its grippr Th robot insrts th pg by calculating th arm position that will plac th pg dirctly abov th hol, moving thr, and lowring th pg into th hol Figur 1: (a) Th robot must locat th hol in vision coordinats (b) Th robot must insrt th pg into th hol n an idal world, th tru positions of th pg and hol will xactly corrspond to thir modld positions, and th robot s motion will xactly track th commandd motion Undr ths conditions, th insrtion will always succd n th ral world, rrors will aris that caus th tru positions to dviat from th modld positions, and th robot motion to dviat from th intndd motion Ths rrors may caus th in1

7 srtion to fail W can dtrmin whthr failur is possibl by prforming a worst-cas rror analysis W bgin by obsrving that th allowabl rror in th pg s position bfor insrtion is max = Thol - rpg, which is th pg/hol claranc f th distanc btwn th (z, y) coordinats of th cntr of th pg and th cntr of th hol is lss than max, thn th pg will slip into th hol Th maximum possibl dviation is obtaind by total = holc grasp motion, whr hol, grasp, and Emotion dscrib th worst-cas rrors in th hol position, th pg position rlativ to th grippr, and th robot s motion accuracy f total < Emax, thn th insrtion will b rliabl This xampl shows how worst-cas rror bounds may b usd to valuat th rliability of a proposd robot action Th Lozano-PCrz, Mason, and Taylor mthod gnralizs this tchniqu to problms with mor complicatd task gomtry and mchanics, considring additional sourcs of rror Thir work has bn xtndd and rfind by subsqunt authors, rsulting in svral prototyp automatic planning systms [5, 4, 6, 13 On assumption undrlying all of ths rsarch rsults is that worst-cas rror bounds such as hol, pg, and motion may b obtaind and providd as input to th plannr How can ths rror bounds b masurd? This papr dscribs a sris of xprimnts prformd in th Sandia Manipulation Laboratory to charactriz th worst-cas rrors in visual snsing and motion accuracy for th laboratory workcll Ths xprimnts srvd to masur th inhrnt rror charactristics of th workcll for latr us in task analysis xprimnts, as wll as providing insight into th gnral problm of stimating worst-cas rror bounds Th workcll is shown in Figur 2 An Adpt On manipulator is mountd on a rigid stl bas which is wldd to th worktabl This worktabl is roughly two mtrs squar, and is prcision-ground so that th work surfac is flat and lvl to within fo1mm Tooling balls ar mountd on th tabl to facilitat rpatabl placmnt of xprimntal moduls, attachd to standard mounting plats Surrounding th worktabl is a truss structur dsignd to support lights, lxan safty barrirs, and coars-viw ovrhad camras Th lights ar dsignd and locatd to provid vn illumination of th tabl surfac; masurmnts indicat that light intnsity varis by only 5% ovr th tabl work ara Th robot is quippd with an nd-ffctor which includs a grippr, compliant wrist, forc snsor, and clos-viw camra Visual procssing is prformd using standard vision analysis procdurs includd in th Adpt controllr (s Sction 31) n th xprimnts w dscrib hr, th forc snsor and compliant wrist ar not usd This workcll was dsignd to prform a sris of x- + + Figur 2: (a) Th workcll (b) Clos-up of th robot nd-ffctor primnts masuring th robustnss of automatically plannd manipulation actions Ths xprimnts all hav th sam basic format: Givn a dsird action to 2

8 Masuring ths valus is tricky To support th goals of compliant motion planning using worst-cas analysis mthods, w must assur that no rror can vr xcd its associatd E bound, whil also avoiding E valus that ar ovrly consrvativ Ths compting rquirmnts prvnt th blind application of convntional RMS or 6u rror modling tchniqus, at last for th cass w studid Th rmaindr of this papr will xplain how w masurd and vrifid ths rror bounds for th Sandia Manipulation Laboratory workcll Sction 2 will compar our rsults to past work in robot calibration Sction 3 will addrss and Sction 4 will addrss motio-, and motion Sction 6 will discuss som of th lssons larnd during this xrcis n brif, w viw our rsults as positiv bcaus w succssfully masurd th rquird worst-cas rror bounds for a complx robot systm, but worrisom bcaus of th ffort rquird to obtain ths masurmnts Our final rsults for th rror bounds of th workcll ar: tst, th robot visually masurs th currnt location of an objct to b manipulatd, calculats and movs to th dsird starting position rlativ to th objct, xcuts th action, and chcks th rsult using vision and othr snsors This procss is rpatd autonomously ovr thousands of trials, thus obtaining action rliability masurmnts ovr a rang of varying initial conditions A common thm in all of ths xprimnts is a basic stratgy for initiating th action: Visually masur th objct s location in th world, comput th robot position to achiv th dsird rlativ position, and mov thr Ths xprimnts rquir worst cas rror bounds and Emotion dscribing th maximum possibl rror in th masurd objct location, and th arm s position accuracy Our work currntly focuss on planar modls of action, whr points in spac ar dscribd by (t,y, 6 ) coordinats Thus to charactriz th rror proprtis of th basic masur-calculat-mov action stup stratgy, w nd four rror bounds: Evision,, Th maximum possibl distanc btwn th masurd ( a, y ) objct position and its tru position, rlativ to th camra = 119mm i i %ision, Th maximum angular dviation btwn th masurd objct orintation 6 and its tru orintation, rlativ to th camra motioh, motio-,, Th maximum possibl distanc btwn th tru ( a, y ) position of th robot s quill cntr at th nd of a motion, and th intndd position Th goal position may b an arbitrary input ( a, y ) point, and is not taught Thus, Emotiob, is a masur of th robot s accuracy, not its rpatability Emotion, = 13mm = 2 whr d is th distanc btwn th dots on th objct bing masurd Ths valus of vision,, rflct th rsults of applying corrction trms to liminat substantial lns distortion ffcts; without ths corrctions, i would i b6mm, Emotion, Th maximum angular dviation btwn th tru 8 orintation of th robot s tool flang at th nd of a motion, and th intndd orintation Again, motio- is a masur of th robot s accuracy, not its rpatability 2 Prvious Work Much work has bn don to valuat th capability of vision systms to locat objcts with a dfind prcision Th fild of photogrammtry was an arly drivr [Z, 31, as incrasingly highr lvls of accuracy wr sought Oftn a modl was dvlopd to rlat camra paramtrs to rsultant objct displacmnts from idal locations within a scn Th complxity of th modls did not always corrlat to improvd camra charactrization, and limitd th rang of applications addrssd Simulation was oftn usd to provid tstimonial of th improvmnts in prformanc, and rsults may or may not account for nois or lns distortion Svral computationally intns algorithms hav bn dvlopd to provid subpixl rgistration within a scn (g, 1111) Tim constraints limitd th applicability of many of ths routins n 1985 Rogr Tsai [13] proposd a mthod for a camra calibration using standard TV camras and lnss, and in 1987 n ordr to obtain consistnt visual position masurmnts ovr a broad class of task objcts, w mploy a uniform objct position masurmnt procdur W paint two whit circular dots on an objct, and paint th objct and background flat black to liminat othr faturs and shadows f th dots hav diffrnt diamtrs, thn th objct position and orintation ar obtaind dirctly from th masurd dot positions Thus, i i %is,,qual to th maximum possibl rror in locating a dot, and v i o n = tan-l(zvision,,/d), whr d is th distanc btwn dot cntrs Th motion rrors motiorg, and Emotion ar intrinsic proprtis of th robot arm and its controllr Thrfor w nd to masur, ; i,,, m o t i o,, and ti^^ for our robot systm 3

9 Ptr Sitz [8] proposd an algorithm to provid bttr accuracy for ths camras Tsai s procdur has th advantag of only nding to b prformd onc for a givn camra, but rquird a strong modl of lns distortion and ignord th ffct of nois Sitz s work rquird symmtrical distortion of th lns, th us of symmtrical filtring, and also ignord th contribution of systm nois Th prvious rsults faild to addrss all contributors to vido systm distortion Our work dviss a mthod to charactriz worst-cas rrors that accounts for all of th rror contributions W chos to analyz th rlationship btwn th camra and th imag plan as an arbitrary mapping of smooth distortion valus This allowd us to us a vry wak modl of th camra paramtrs at th xpns of systm mmory usd for storing a lookup tabl Onc th camra is corrctd, it is usd to provid a mapping of displacmnts of th robotic arm in th workcll W also chos to us lookup tabls (with inhrnt mmory rquirmnts) instad of charactrizing and corrcting a strong modl of th robot kinmatics n 1994 Shah and Aggaral[9] publishd a papr documnting a similiar procss in thir work for corrcting a fish-y lns Th approach takn by Smagt, Gron, and Kros [lo] in thir work at th Univrsity of Amstrdam in 1993 rlid upon vision systms and a nural control algorithm to corrct th kinmatics of th robot in an adaptiv control procss A drawback to this approach is th computational complxity which rquirs spcializd hardwar to provid ral tim procssing Adpt Tchnology has a commrcial product that is ssntially th sam as our position tabl lookup algorithm Thir product may b usd to map an ara as larg as ours (by ovrlapping smallr ara maps) with som incrasd complxity, howvr it dos dlivr bttr accuracy Th Adpt product rquirs nd of arm tooling (a light point sourc) attachd to th robot Points on a glass grid ar thn visitd and compard to th commandd point Errors ar rcordd in a lookup tabl and movmnts to ths points ar corrctd as thy ar mad Our mthod dos not rquir additional hardwar if th camra is rquird during xprimnts Anothr advantag of our mthod is that all masurmnts ar mad with th xprimnt tooling in plac, and drift du to tmpratur variations and war on th manipulator may b corrctd on-lin Ths corrctions ar not currntly implmntd 3 Masuring 31 systm, and not includ possibl rrors in th robot s positioning of th camra Ths rrors will b addrssd sparatly in Sction 4 Thr ar svral possibl mthods for locating circular dots in an imag; th Adpt vision systm provids at last thr: gnral blob analysis, circular arc fitting, and linar rulrs Gnral blob analysis is prformd by classifying pixls into groups by intnsity lvls and finding th cntroid of th pixls classifid as mmbrs of a dot Th circular arc fitting mthod idntifis pixls corrsponding to th boundary of th circular dot, and fitting a circl to ths points Th rulr-basd mthod is prformd by using linar dgdtcting rulrs placd vrtically and horizontally, to masur th diamtr of th dot and thus its cntr point S Figur 3 Ths mthods vary in thir robustnss Rcall that in our workcll, th lights wr dsignd to provid vn illumination of th work ara This vn illumination is corruptd by th prsnc of th robot, which obstructs a fw of th light sourcs Bcaus light coms from svral diffrnt dirctions, th rsulting shadows hav fairly low contrast; howvr, this dos caus variations in th ovrall scn intnsity rlatd to th robot s configuration Ths variations can advrsly affct th gnral blob analysis mthods, bcaus th pixl classification schms thy mploy ar snsitiv to intnsity changs n contrast, th linar rulr mthod rmains robust in th fac of smooth intnsity changs, sinc th dot boundary still producs a high-contrast dg vn whn th total illumination varis Th dot-finding mthods also vary in thir snsitivity to dot siz As th siz of th dot incrass, th numbr of pixls covrd by th dot also incrass Th rsulting incras in information can improv th ffctiv rsolution of both th blob analysis and arc fitting mthods, sinc mor pixls contribut to th statistically avragd stimat of th dot cntr or boundary Convrsly, rsolution dgrads as th dot siz shrinks Sinc w xpct to manipulat small objcts in som of our xprimnts, w would prfr to us a mthod whos accuracy dos not dgrad with dot siz For ths rasons, w chos to mploy th rulr-basd mthod of finding dots This is accomplishd by starting with a nominal ( t, y ) dot location that is known ithr from th xpctd dot position, or prliminary blob analysis A horizontal rulr is usd to dtct dgs on ithr sid of this cntr point; th rsulting dgcrossing x-valus ar usd to rfin th stimat of th dot cntr 2: valu A vrtical rulr is thn placd at this rfind x-valu, and th dg-crossing y-valus ar usd to dtrmin a rfind dot cntr y valu Sinc th original horizontal rulr was likly placd abov or blow th dot cntrlin, th crossing angl of th dot dgs was likly not prpndicular To avoid a loss of accuracy in th dot s masurd x position, a scond horizontal rulr is placd, now using th rfind x and,+ion Dot Location As xplaind abov, w ar intrstd in charactrizing th rror in th robot s ability to masur th (x,y) position of a high-contrast circular dot in its workspac, using th clos-viw camra attachd to its nd-ffctor To focus on vision-rlatd ffcts, w will xprss this rror in trms of th camra coordinat 4

10 Cntr 1=(X1 +X2)/2 assumption, how much rror can thr b btwn th masurd and tru dot positions? On possibility would b to dvlop som mans of comparing masurd valus against ground truth, tak a larg numbr of such masurmnts, and thn calculat th standard dviation Q of th rsulting rror valus W could thn tak i =67, i and assrt, that rror valus outsid this rang ar practically impossibl This approach is somwhat unsatisfying Th mthod is prhaps rasonabl whn rror distributions ar Gaussian, but so far w hav no justification for assuming that our vision rrors ar Gaussian in natur f th rrors ar not Gaussian, thn our rsulting rror bound cvision,,, may b incorrct f w tak a larg nough sampl of masurmnts to captur all of th rror phnomna that may aris, thn i will i gnrally b too larg if th rror distribution is not truly Gaussian As w shall s latr, blindly applying a Gaussian modl to raw vision rror data will yild an i that isi vry consrvativ = A bttr approach is to considr th physical procss mployd by th snsors, and idntify particular sourcs of rrors This allows us to masur ach rror componnt individually, through xprimnts dsignd to isolat th sourc of rror n visual snsing, th basic procss is that light rflcts off th scn, passs through a lns and crats an imag on a camra CCD array Th imag is convrtd to an lctronic signal and snt to th vision procssor Sampling th signal at intrvals which corrlat to th siz of th fram stor buffr provids a nw rprsntation of th camra scn Svral imag procssing computations may b prformd on th fram buffr to analyz this rprsntation This viw of th visual snsing procss illuminats svral possibl sourcs for rror: Th transmission of th light through th lns, th captur of th light by th CCD array, th convrsion and transmission of th rsulting CCD imagintnsitis to th vision procssor, th sampling routin usd to fill th fram buffr, and th imag analysis computation W assum that thr will b rror contributions from ach of ths sourcs and group thm as follows: th lns, th CCD array/vido procssing, and th lctronic transmission of th imag f ach of ths procsss has an associatd rror bound E, thn v 4 Cntr 2=(Y1+Y2)/2 Y3 4 Cntr 3=(>(3+X4)/2 Cntr 4=(Y3+Y4)/2 Figur 3: Th dot location mthod This procdur was tstd with up to twlv lins; convrgnc aftr four lins mad furthr itrations rdundant y coordinats obtaind thus far This yilds a furthr rfind z' valu This procss is illustratd in Figur 3 Furthr rfinmnts ar crtainly possibl, yilding a sris of incrmntally improvd y', z", y", valus Tsting rvald that valus rliably convrgd by z", so w adoptd a canonical stratgy of stopping rfinmnt aftr x" was calculatd Th rsulting dot-finding stratgy thus rquirs four rulr dg-dtction oprations pr dot, and som simpl arithmtic 32 Error Modl Aftr slcting a concrt dot-finding procdur, w can procd to charactriz its rror proprtis This immdiatly raiss a fundamntal qustion: How do w masur worst-cas rrors? This qustion raiss a numbr of issus, not th last of which is what w man by "worst-cas" On could argu, givn an advrsary allowd to impos any catastrophic vnt, that it is impossibl to bound th st of possibl bhaviors of a systm This includs turning off all th lights, unplugging th camra, painting dfcts on th dot, tc n th worst cas, vry larg rrors will rsult if crtain critical logic circuits fail in just th right way, rturning valus such as 345 x 167mm n ordr to covr all of ths scnarios, w nd to slct an qual to th largst numbr that can b rprsntd by th Adpt procssor This dosn't sm to b vry usful nstad, w will assum that such catastrophs ar avoidd, and th rturnd (z, y) coordinats rprsnt th rsult of a normal analysis of a propr dot imag Undr this EvisiolOy = Elns + pixl + Enoir Elns is th fixd distortion du to imprfctions in th lns, pkl is th fixd rror rsulting from th CCD charactristics, sampling rat variations, and vido procssing limitations, and,, i, rprsnts th lumpd ffct of nois in th lctronic imag transmission as wll as any nois that may b prsnt du to lns and CCD proprtis n th sctions that follow, w will dvlop a modl of th rrors giving ris to lns, cpkl, and dsign an xprimnt to charactriz ach rror, and prsnt 5,,

11 CCD Lin Camra Signal 6 Objct adpt 9 Q a Lin Monitor Figur 4: Pictoral rprsntation of systm contributrs to optical/lctronic nois in vido proccssing Nois variations &ct th imag plan location of dot cntrs Figur 5: 3 by 6 grid usd to charactriz nois contributions of vido systm discussd in Sction 31 and rcordd for ach pictur With no camra, plat or robot movmnt and no light intnsity changs, any variability in th dot location can b attributd to nois from th camra pramplifir, vido transmission lins and th vido procssing systm Th raw data wr stord to disk and transfrrd to a work station for procssing using MathmaticaTM For ach pictur th (z,y) cntr location was computd for ach dot Th (2,y) cntr locations wr also computd aftr avraging conscutiv frams in groups of 3, 5, and 7 xposurs For ach data st avrag locations along with minimum, maximum, and standard dviations wr computd Comparison of th data sts (s Figurs 6 and 7) showd that th minimum, maximum and standard dviations ar invrsly proportional to th numbr of xposurs usd for avraging Avraging also rinforcs th data s Gaussian natur Aftr studying th improvmnt in standard dviation attaind vrsus th amount of tim rquird to gathr additional camra frams, w dcidd to avrag fiv frams for all subsqunt static dot locating trials This yildd a standard dviation of 15mm for ach dot cntr location W chos a 6c rror bound to rprsnt th worst cas location rror du to nois in th vido systm, rsulting in = f9mm th rsults of ths xprimnts applid to our robot workcll Th masurmnts will produc bounds on th individual componnts of th rror, which w will thn s u m to obtain i ifinally,, w, will vrify this aggrgat rror bound by comparing Evision,y against masurd rror valus obsrvd ovr a larg numbr of random trials 33 Random Nois Th first part of th proposd rror modl to b analyzd is dot cntr location uncrtainty causd by th prsnc of nois in th CCD, transmission mdium and procssor lctronics Nois may b dfind as any unwantd signal prsnt in an optical or lctronic systm Th primary nois contributions in th camra CCD ar thrmal currnts, photon shot nois, and pramplifir lctronic nois Furthr nois contributions ar mad by transmission of th signal along th vido coaxial cabl, and rpatd signal convrsions from analog to digital and back to analog t is difecult to accuratly isolat and charactriz ach of ths sourcs of nois rror indpndntly, but som infrncs can b drawn W xpct that th nois will b ssntially Gaussian in natur, and that it will contribut a small amount of rror to th masurmnt of th dot cntr in th imag fram To charactriz th nois contribution to dot location rror th following xprimnt was prformd: A black anodizd aluminum plat (s Figur 5) had ightn hols bord to a dpth of 16 inch Th hols wr paintd whit to provid high contrast dots against th black plat Th plat was placd in th camra fild of viw and on hundrd picturs wr takn ovr an intrval of approximatly twnty sconds Th dot (2,y) locations wr found using th dot location mthod 34 Camra Pixl Effcts An rror trm rlatd to th dtrministic physical and lctronic proprtis of th camra CCD and vido procssing will b calld pixl quantization A camra CCD dvlops a charg on ach pixl rlatd to th numbr of photons rflctd by an objct which strik that pixl A pixl fully illuminatd by rflctd photons will dvlop som maximum charg, and a pixl with no illumination will dvlop som minimum 6

12 35 Max 1322 Min Sigma Max Min Sigma ; > 15 3?? S LL (u 3 g 1 t Error (mm) Error (mm) Figur 6: Histogram of raw data shows magnitud of hol location rror attributabl to lctronic nois Not th quantization ffcts, which w attribut to th fixd numbr of bits usd to rprsnt floating-point numbrs in th Adpt controllr charg Th pixls on th priphry of an objct will b partially illuminatd (s Figur 8), and dvlop a charg somwhr btwn ths limits Analog to digital convrsion will quantiz this charg to th narst digital rprsntation availabl With infinit rsolution th digital valu will xactly rprsnt th charg on th pixl Givn that infinit rsolution is not possibl (th vido quantizr uss svn bits to dscrib th charg of ach pixl) th digital rprsntation is only an approximation of th illumination of th pixl This approximation shifts th apparnt location of th dg of an objct at ach point along th priphry of th objct Givn that th dots hav a diamtr of 125in (3175mm) and th pixl siz is 4mm thr will b approximatly 5 pixls illuminatd by ach dot Dpnding on th location of th dot in th imag, thr may b as many as 14 pixls which ar only partially illuminatd Th actual numbr of partially illuminatd pixls is unknown, as is th dgr of illumination of ach of ths pixls With a singl axis translation of th dot locations, and uniform sizd and spacd pixls, rcurrnt pixl illuminations should occur for ach tabl movmnt qual to a pixl priod Ths assumptions ld us to xpct that th rror in Figur 7: Histogram of avragd data shows magnitud of hol location rror attributabl to lctronic nois Each data point rflcts th avrag of 5 hol cntr masurmnts Th improvmnt du to avraging suggsts that th nois contribution is Gaussian dot cntr locating ability of th vido systm would b priodic, with th priod dirctly proportional to th pixl siz Anothr contribution to pixl quantization is th inability of th vido procssing to xactly rsolv th pixl into infinit divisions for th purpos of dg dtction Adpt grayscal procssing convolvs a 3 x 3 matrix of pixl illuminations to dtrmin imag intnsity gradints Th first partial drivativ is thn usd to idntify th location of th dg within th pixl Tian and Huhns [12] rport th rror associatd with this typ of sarch as varying from 4 to 3 pixl, dpnding on scn paramtrs t is difficult to valuat ach of ths sourcs and thir contributions to camra rror as individual quantitis W chos to charactriz th ffct of ths camra rrors as an nvlop and valuat thir ffct on th problm of rgistring imag cntr locations accuratly This is analogous to th approach w usd in charactrizing nois rrors Th following xprimnt was run to valuat th prsnc and ffct of quantization rror on dot cntr location accuracy Th grid usd to study th ffcts of nois rrors (Figur 5) was mountd on two sts of micromtr controlld tabls at right angls to ach othr W placd th grid in th imag fram with its axs alignd to th vision fram axs

13 Pixl Grid Pixllation, a gross ffct Pixl siz is xaggratd, Pix& incrmnts How dos cntr location chang with rspct to pixl location? Figur 1: Th solid lin dpicts th idal rspons of masurd dot locations with rspct to plat movmnt Th dottd lins dpict th xpctd nvlop of variations du to quantization of pixl valus in th imag plan Ths variations ar xpctd to hav a nar sinusoidal distribution whr th priod is qual to on pixl siz (FZ 4mm), and an amplitud qual to th worstcas quantization rror Figur 8: llustration dpicts th various placs a pixl may li with rspct to th dot boundary Micromtr Tabl placmnt along th a: or y axis of th vision fram Th absnc of quantization would yild a straight lin plot whr ach dot cntr in th imag fram is displacd xactly ( f,, i ) th sam distanc as th grid is physically translatd (Figur 1) Rfr to Figur 11for th following discussion of th tst rsults Plots of th data vrsus th grid translation for ach stp yild rsponss typifid by that shown in Figur 11 Th sinusoidal rspons of th data has a priod qual to th siz of a pixl, as xpctd To slct an rror nvlop w drw lins paralll to th idal rspons, through th maximumamplitud points of th sinusoid Th maximum amplitud obsrvd in th plots is f6mq or about f pixl This valu is th maximum rror that may rsult from th dot's particular placmnt rlativ to th grid of pixls Thus Pkl = 6mm Fiv pictur avrag at vry 2 mm Figur 9: Th mthod usd to vary th grid location for pixl quantization masurmnt 35 A baslin st of picturs was takn (5 xposurs avragd pr pictur) and th dot cntr locations wr found and stord to mmory W monotonically translatd th grid's physical location in incrmnts of 2mm in th 2: dirction (s Figur 9) for a total movmnt of 2mm ( x 5 pixls) Aftr ach stp th imag fram locations of th dot cntr ar found and stord to mmory This procss is thn rpatd for displacmnts in th y axis A plot was mad of th location of ach dot cntr with rspct to th grid dis- Lns Distortion Lns distortion is th failur of a point in th objct plan to proprly map into its corrct location in th imag plan Th major sourcs of lns distortion ar drivn by lns proprtis and physical imprfctions of th camra Thr is a distortion proportional to th angular diffrnc btwn a rflctd light ray and th optical axis of th camra lns This distortion causs objct points to map into th imag plan with som displac8

14 r Figur 12: Som fild of viw altrations causd by various typs of lns distortion ?,? S? r O U ) b a,, m lngth lnss which would nd to b mountd furthr away from th objct plan to provid th sam fild of viw To provid th rquird fild of viw, th camra lns must focus rflctd light rays with significant angular displacmnt from th optical a x i s onto th camra CCD ncrasd distortion of th imag plan is a byproduct of th rquirmnt to focus rays with significant angular displacmnt onto th CCD Th lns manufacturr s data sht listd th lns distortion as approximatly 45% of th distanc from th optical a x i s to th point of intrst At a distanc of loomm from th imag cntr, barrl distortion should b approximatly 45mm Th magnitud of this rror would drastically limit our ability to conduct xprimnts unlss som sort of corrction is applid to apparnt imag plan locations W charactrizd th imag plan distortion using th following procdur: W placd a calibration grid with a 25 x 25 array of dots in th robot workcll Each dot was 3175mm in diamtr, and milld to a dpth of 46mm Th dots wr spacd lomm apart (s Figur 13) This plat rprsnts th objct plan and is locatd at a focal distanc of 28mq which producs th dsird 2mm x 2mm fild of viw Th camra (imag fram) is cntrd ovr th grid (objct fram) cntr point with a known location and orintation offst This provids a known transform btwn th cntr of th imag fram and th cntr of th objct fram Prfct physical camra lmnt alignmnt and quill mounting would also produc an optical axis xactly prpndicular to th objct plan This would liminat th nd to tak tangntial distortion into considration as a contributor to lns distortion Th inabil- 5 rnm incrmnts Figur 11: Graph showing actual dviations from idal cntr location causd by pixl quantization ffcts Th worst-cas rror is koo6rnm,as shown by th arrows mnt from thir tru location in th objct plan Ngativ valus of th angl caus shrinkag of th imag spac (barrl distortion), and positiv valus caus xpansion of th imag spac (pin cushion distortion) This distortion is primarily drivn by th lns shap (wid angl, tlscopic tc) Tangntial lns distortion displacs imag points in a dirction normal to radial lins from th cntr of th imag rsulting in a skwd imag plan Causs of tangntial distortion includ imprfct physical alignmnt of th camra componnts, imprfct camra mount to th robot quill, or a non-prpndicular quill Figur 12 shows a graphical rprsntation of th chang in imag shap with rspct to svral typs of distortion Onc again it is vry difficult to indpndntly isolat and charactriz th individual sourcs of distortion rror within a givn systm Our charactrization of rror componnts distorting th imag (lns imprfctions, physical mounting problms, tc) will b as a composit rror, rfrrd to as lns distortion for th body of this work Our workcll mploys a wid angl lns for our camra to provid th rquird fild of viw (2mm x 2mm) in th prsnc of svr physical occlusions prsntd by th quill mount location Th forc snsor and th grippr both provid constraints to long focal 9

15 Figur 13: Th 25 x 25 grid usd to charactriz lns distortion proprtis of th camra/lns Not th fiducial mark usd for angular alignmnt Figur 14: Error magnituds of ach hol cntr location bfor corrction Th rror magnitud is d(xrror)2 + (Krror)*Locations ar rfrncd to th imig fram (z, y) origin ity to prfctly align and orint th two frams rquirs masuring and saving th dx, dy, and d offsts as a transformation matrix to dscrib th rlativ position of th frams A sris of fiv picturs wr takn and th dot cntr locations with rspct to th grid cntr wr computd and stord using th prviously mntiond dot locating procdur Th dot cntr locations ar transfrrd to a workstation for additional procssing using MathmaticaTM Th rsulting data showd worst-cas location rrors of 498mm to 578mm (s Figur 14) Th bowl shapd rror surfac clarly shows how th rror magnituds incras with distanc from th imag cntr Th non-symmtrical rror distribution indicats that additional tangntial rror sourcs ar combind with ordinary barrl distortion rror W rjctd us of th approximat barrl distortion curv providd by th lns manufacturr basd on ths tst rsults W bliv lookup tabls basd on th raw rror charactrization will nabl a mor accurat and comprhnsiv corrction of all sourcs of lns distortion W collctd th raw rror valus into two 21 x 21 matrics of rror stimats, on for,,,,x and on for yyrrors On way to dscrib ths tabls of rror valus is that ach row dscribs th rror as a function of z for a spcific valu of y, for ach y E [, 211 Likwis ach column dscribs th rror as a function of y for ach x E [l,211 Svral typs of fitting routins wr tstd for accu- racy in duplicating th rror charactristics of th lns A 3-d surfac fit allows rror stimat gnration using a singl quation 3-d surfac fits using 1st through 8th ordr quations for x and y rrors yildd worst cas rror diffrncs of 9mm at th gridpoints n an attmpt to obtain a mor accurat rprsntation, w valuatd othr mthods for fitting th data Fitting th data row by row and column by column providd th bst rror rplication Bcaus of fairly wll bhavd data for any givn row or column, a vry accurat fit was found Fits for 3rd, 5th,?th, and 9th ordr quations wr valuatd to dtrmin worst cas rrors Error stimats wr vry good for all curvfits, gtting bttr as th ordr incrasd (as xpctd) Using numrous 2-d individual curvfits is mor complx than using a singl 3-d surfac fit quation, but th incrasd accuracy maks this a worthwhil tradoff Onc th curvfits hav bn found, ach gridpoint is procssd using th curvs to gnrat rror stimats Th rror stimats ar compard to th original rror magnituds to assss th quality of th curvfit (s Figur 16) W dcidd to us th 5th ordr fit bcaus it rprsnts a 5% improvmnt ovr th 3rd ordr, and highr ordrs providd asymptotically improvd prformanc not worth th incrasd complxity Thr ar two sts of curvfit cofficints gnratd for ach variabl (x,y) On dscribs th rror with 1

16 mag to Objct 5 Fit Dgr Objct to mag L 9 # Figur 16: Error magnituds of ach hol cntr location aftr corrction Hr th rror magnitud is Emcasurd fit Locations ar rfrncd to th imag fram ( r, y ) origin 3rd 5th Th first stp in corrcting a location in ithr dirction is th normalization of th location to th appropriat lookup tabl coordinat systm (s Appndix A) This procss assigns ach point an indx valu signifying a rlativ (2, y) location in th currnt fram f th normalization rsults in a fractional numbr, th point lis btwn charactrizd gridpoints, and intrpolation will b rquird Th intgr valu of th assignd indx (in both x and y), and th intgr valu plus on ar usd to slct th st of curvs rquird to calculat an rror stimat W now hav th point location rlativ to th grid curv numbrs of th rror lookup tabl For points which li xactly on a gridpoint, th rror can b dirctly calculatd using th dsignatd curvfits Th calculatd rror is addd to th non-normalizd (z,y) location to yild th corrctd point location Points which do not li dirctly on a gridpoint ar adjustd for rrors using an additional st of curvs and a 2-d intrpolation to calculat th rror magnituds Th intrpolation routin is discussd in Appndix A Aftr implmnting all of ths intrpolation mthods, th worst-cas rsidual distortion rror can b stimatd basd on th dviation of obsrvd data from th fit curvs, and th additional rror that may b introducd by th intrpolation This yildd a valu of lns= 5mm 7th Fit Dgr Figur 15: Lin graphs show ffct of incrasing ordr of curvfits in attmpt to modl rror bhavior Thr is on lin for ach curvfit function Notic asymptotic improvmnt past 5th ordr for both graphs rspct to a point's x location, and th othr dscribs th rror with rspct to it's y location W gnratd a st of datafils containing th cofficints of th 5th ordr curvfit (a 21 x 6 array), and transfrrd ths to th Adpt controllr Bcaus th curvfits ar basd on data takn at spcific gridpoints, an intrpolation mthod is rquird to assss rror valus at points of intrst that do not li on th grid 36 Lns Error Corrction Corrcting lns distortion rrors in th vido systm is accomplishd using lookup tabls gnratd by th dsignatd curvfits Thr ar two scnarios to b addrssd: 1 For a point at a known location in th camra (imag) fram, whr is th point in th robot coordinat (objct) fram? 2 For a known point in th robot coordinat fram, whr should it b found in th camra imag fram? 11

17 1 1 * = ' ::: - Enois + Epixi Error (mm) Figur 18: This histogram dpicts th rror in masurd lg lngths, without distortion corrction Ths data rflct th masurmnt of 57 triangls, or 171 data points th quill camra A st of locations was dvisd to provid imag fram viwing of triangls dfind by slctd points on th plat grid Th triangls w slctd rprsnt a wid rang of lngths and sizs, and wr chosn to li in various aras of th imag fram as th manipulator movd among svral tst positions A cardboard ovrlay was mad with on inch diamtr hols punchd out at spcific locations, allowing only th dots of th dsird triangl to show through Whn th cardboard is ovrlaid on th grid, a sris of triangls ar visibl to th camra from th slctd locations Th robot visits th st of locations and rcords th imag fram locations of th visibl dots Sparations btwn th dots ar calculatd twic, onc bfor distortion corrction and onc aftr Th calculatd lngths ar compard to th known lg lngths and th rrors ar stord in two fils S Figur 18 for prcorrction rrors, and Figur 19 for post-corrction rrors Not that for th raw data (s Figur 18) all rror valus ar ngativ (imag distancs ar shortr) with rspct to th actual distanc btwn th two points This is a dirct rsult of barrl distortion Th imag fild is comprssd, and as an objct movs furthr from th cntr of th imag plan, th clustring ffct bcoms mor pronouncd From th rsults of th prvious sctions w hav - - Tsting Error Charactrization Evision,, 1 Figur 17: Grid usd for chcking vision rror modl Triangls of various siz and shap ar producd by an ovrlay with hols cut in spcific locations to allow dots to show through Max Min Sigma Elns = OOOQmm+ 6mm + 5mm = 119mm t should b notd that th lns distortion rror contribution aftr corrction is lss than th combind nois and pixl rrors componnt This is du to th quality of th curvfits mployd to rplicat lns distortion rrors W dvisd an xprimnt to tst th quality of th rror charactrization A black anodizd plat of dimnsions 9144mm x 8848mm with a grid pattrn of dots paintd whit is mountd in th robot workcll (s Figur 17) Th grid is positivly locatd by fixtur balls mountd to th work surfac allowing vry accurat position rpatability of th grid in any of th four quadrants of th workcll Th grid spacing is 5mm in both th x and y dirctions, and th 875in dots occur vry 2mm Th diffrnt dot sizs allow gross position idntification basd on dot siz and facilitat xtnsion of th calibration routins to th ovrhad camras usd in th workcll Both dot sizs ar usd in th vision calibration and tst routins of 12

18 25 2 r n a, 2 15 i Max Min Sigma T Max Min Sigma _ LL Figur 2: Error in masurd lg lngths, with distortion corrction Ths data rflct th masurmnt of 347 triangls, or 1,41 data points Error (mm) Figur 19: Th rror in masurd lg lngths, with distortion corrction Ths masurmnts wr mad with th sam imags usd in Figur tors Th rsolution of th ncodrs usd by ach joint to dciphr location information is on such factor Diffrncs btwn th kinmatic modl of th robot and th actual dimnsions of th robot linkags and lngths, misalignmnt of th robot coordinat fram with an arbitrary world coordinat fram imposd by th usr, friction proprtis of th robot joints, and variations in th robot oprating nvironmnt ar additional factors limiting th ability of th robot to b positiond with absolut accuracy t is important to distinguish btwn th absolut positioning capability and th position rpatability of a robot arm Absolut positioning capability is th ability of th robot arm to go to a spcifid non-taught point in th workcll t is dirctly rlatd to variations btwn th kinmatic modl of th robot and actual physical dimnsions of th individual robot For xampl, th kinmatic modl of a robot dpicts linkags as crtain spcific lngths vn though linkags and link lngths vary from robot to robot du to manufacturing limitations Most robot manufacturrs do not spcify absolut positioning accuracy du to th rquirmnt to gathr and analyz a larg amount of data to crtify a prformanc lvl for ach robot manufacturd Typical robot systms dpnd on taught points or xtrnal snsing and fdback routins to minimiz th rquirmnt for this typ of positioning accuracy t is critical to th typical robot usr to b abl to visit a spcifid point (taught or non-taught) with som known dgr of consistncy This is calld rpata- Rsults Figur 2 shows a histogram of 1,41 masurmnts usd to tst th validity of our worst-cas rror modl Th largst rror obsrvd is lss than two tims i Bcaus i this xprimnt, masurs two dot locations and calculats th distanc btwn thm, w would xpct to s rrors of up to but not xcding 2 v i s i o n, Thus ths data support our stimat of th worst-cas vision rror 4 41 Arm Calibration Masuring Emotion W would lik to apply th sam approach to charactrizing and corrcting of robot kinmatic rrors that w usd in th vision cas: undrstand th procss, idntify constitunt sourcs of rror, and masur thm in isolation Unfortunatly, w wrn t abl to clarly idntify all such sourcs, and tim did not prmit an invstigation that was as dtaild as our vision invstigation Thus w applid a statistical mthod of masuring rrors and post procssing of data to quantify rrors and apply corrctions 42 Absolut/Rpatabl Positioning Th ability of a robot arm to position itslf at som spcifid point in th workcll is affctd by svral fac13

19 4 T Max Min Sigma > LL r o? k? q q? k o? p p p p 7 Figur 21: Grid usd for chcking absolut positioning rror of robot Ara to b charactrizd is outlind by whit box T- Figur 22: This histogram dpicts th arm location rrors compild in thr sparat visits to th spcifid locations (in ach quadrant) Ths data wr usd to dvlop th curvfits for ach of th points visitd in th workspac W will program th robot to visit points on th grid in thr quadrants of th workcll using a softwar mapping routin with accss to a matrix of stord world coordinat locations Th robot arm movs to a location and taks a pictur of th grid By locating multipl hols on th grid, th rsulting location of th quill cntr aftr ach mov can b calculatd (within th vision modl uncrtainty) Th diffrnc btwn th spcifid location and th rsultant location is rcordd as an offst xmotion,,o,,ymotioll,,, Th arm thn visits ach of th othr grid points and rpats this procss Th (a, y) location rrors masurd by th vision systm at ach point ar mappd as a grid of rror valus Compltion of th routin rsults in th storing of th rror grid to disk so it can b transfrrd to MathmaticaTM for post procssing (s Figur 22) Th sam gnral curvfit routins from th lns distortion work wr usd to dvlop curvfits for th arm rror data Th siz of th rctangular arrays varid from thos for th lns work, and varid with th workcll quadrant For quadrants 1and 3 th data was groupd into a 17 x 5 matrix, and for quadrant 2 data was groupd into a 5 x 17 matrix Th curvfit coficints ar stord in a st of data fils and transfrrd back to th Adpt controllr Charactrizating Emotion Th grid usd to gnrat triangls for chcking th vision systm rrors will b usd in this xprimnt A rctangular subst (s Figur 21, ara in rctangl) of all th points on th grid will b usd to charactriz a subst of th workcll Th grid may b positiond in ach quadrant of th workcll facilitating arm position rror charactrization in a portion of ach quadrant Th siz of th corrctd ara is limitd by svral factors To simplify curvfit routins th shap of th ara must b squar or rctangular, and th arm must b abl to rach all points on th boundary of th ara An undrlying assumption for this xprimnt is that th corrctd camra rror is smallr than th absolut positioning rror of th robot That is, w assum Evisio-, Error (mm) bility Rpatability is affctd by factors which vary lss than thos govrning absolut positioning Evn though th physical paramtrs of a robot vary from th kinmatic modl, thy ar constant (xcluding tmpratur ffcts) for a givn robot A manufacturr can calculat rpatability basd on worst cas manufacturing capabilitis, lctrical oprating paramtrs tc for a givn class of robot Th rsultant valu dtrmins th nvlop of positions to which a robot, whn commandd to visit a spcifid point, will go to vry tim n th cas of a robot commandd to visit a nontaught point, th rsultant position may not b within th rpatability nvlop of th commandd position 43 < Emotion 14

20 44 To tst th quality of th curvfits, w visitd ach point again Bfor ach mov, th point to b visitd was corrctd using th curvfits as follows Th world coordinats of th dsird point wr analyzd to slct th appropriat quadrant Th point was thn normalizd to th plat grid coordinats, and th rror associatd with that point was calculatd using th curvfit quations Th rror was thn subtractd from th slctd absolut point to provid an rror corrctd absolut location W thn commandd th robot to mov to th corrctd location Aftr th mov th vision systm was usd to masur th arm's location Th offst from th dsird absolut location was thn calculatd and compard to th pr-corrction rror magnitud for th sam point Th ability to corrct arm position rrors is quantifid in Figurs 22 and 23) Moving to th corrctd location rsultd in th robot achiving th commandd absolut position within an nvlop of 113mm, yilding Emotion=, = 13mm 45 Max = Min = Sigma = 364 Corrcting Emotion C 2 12 p 1 t: 8 2 L q - m Co 9 Rsults From ths data w conclud that our robot location rror charactrization, whil not as dtaild as th vision systm charactrization, dos nabl accurat corrctions of th systm rrors Not that th rror corrctd movmnts to absolut locations ar accomplishd with an rror nvlop only slightly largr than th corrctd vision rror Furthr tsting was don on th quadrant 1 data bcaus it displayd what appard to b systmatic rror in th rotation offst calculatd btwn th robot coordinat systm and th usr dfind coordinat systm (s Figur 24) This offst is rquird to stablish a map from th robot coordinat systm to th grid coordinat systm prior to calculating absolut raw dot locations Th (z,y) rror offst rquird to corrct ach data point to a zro rotation was calculatd All offsts wr summd and avragd Th avrag was thn subtractd from ach raw data point, ffctivly minimizing th rotation offst Th nw data was usd to dvlop anothr st of curvfits for comparison with th original curvfits Th tst rsults trackd th original curvfit rrors almost xactly On conclusion w draw from this comparison concrns th quality of th curvfit and corrction routins Th curvfit is vry good at prdicting systmatic rrors, and th routins ar robust in producing appropriat corrctions to th rrors n addition to running rpatd corrction trials, tsts wr also run vs tmpratur from a rang of approximatly 6 F,to 85OF, which rprsnts th xtrm rang of tmpraturs sn in our laboratory Th corrction sprad maintaind a dirct corrlation to th distribution sn in th baslin tsts, howvr m 9 o m o m o! Error (mm) Figur 23: Arm motion rrors, aftr corrction Err M Figur 24: Graphical dpiction of th arm position rror magnituds bfor corrction This quadrant is shown bcaus it had th largst rrors th mdian valu shiftd with tmpratur Ths data could b usd to dvlop a tmpratur-corrctd lookup tabl, but this rmains for futur work 15

21 M a = 1333 Min = Sigma = 5723 Figur 25: Graphical dpiction of th arm position rror magnituds aftr corrction 5 51 Joint 4 Orintation Error Error Charactrization To this point th rror modl of th robot arm consistd of rror contributions of joints 1 and 2 only W now turn our attntion to th orintation of joint 4 To charactriz th motion accuracy of joint 4, a st of ight locations was visitd by th arm Each location was dividd into 24 diffrnt orintations, on vry 15 dgrs At ach location and orintation th grid dots wr masurd visually to dtrmin th actual orintation of joint 4 Th diffrnc btwn th intndd orintation and th actual orintation was rcordd as a joint 4 rotation rror for that point All locations wr world locations, corrctd via th lookup tabls Aftr ach masurmnt, th arm rturnd to a hom location bfor procding to th nxt location and angl This nsurd that all points hav approximatly th sam rror contribution attributabl to initial location rror, and avoidd stiction problms associatd with motion through a small angl Th array of rror valus wr thn stord to disk and transfrrd to MathmaticaTM for furthr procssing 52 Error (dgrs) Figur 26: This histogram shows th rsults of visiting ach of th ight hols at 24 diffrnt orintations with a stp siz of 15 dgrs Th valus dpictd in th histogram ar th rror at ach tst point 6 Summary n this papr w succssfully found rror numbrs that appar to captur worst-cas rror charactristics, and dvlopd systmatic mthods for masuring and vrifying worst-cas rror bounds t is intrsting to not that simply applying 6 would hav yildd poor rsults To s why, considr th data shown in Figur 14 for ths data, Q = 11798mm Th rsulting 6 bound would b 57788mm, which is x 54 tims th bound obtaind through our xprimnts xploiting th natur of th physical snsing procss On of th strong points of this work is th us of a common mthodology for gnration/corrction of rrors associatd with widly variant procsss Th sam rror gnration/corrction schms ar usd for lns distortion and arm position corrction Ths mthods may also b usd to corrct forc snsing and fingr positioning rrors in futur applications Th only variations on th schm involv th diffrnt siz Rsults Figurs 26 and 27 show th rsults of this xprimnt Evaluation of th rror data from th ight hols shows that whil th rror magnituds ar consistant, th individual locations of th paks and vallys ar not This dnots th nd for individual curvfits basd on valuations of all individual aras that may b visitd during th cours of an xprimnt W dcidd that givn th magnitud and complxity of this rquirmnt, th prformanc improvmnt attainabl did not warrant th additional ffort Worst cas rror prformanc du to orintation rrors will ncompass th worst cas rror documntd by th charactrization of 16

22 Grid Lins = 15 dgrs Y min = -25 Ymax = 25 Ystp =O xtrmly high rliability is rquird (g, assmbly of inrtial navigation instrumnts [14])n th sam vin, rlatd work in our laboratory has concludd that th complxity of implmnting plannrs that utiliz ths worst-cas bounds is also onrous Continuing work is xamining th us of ordinary statistical rror modls for both planning and rror charactrization; ths sm much mor promising v=25 1 l J r Figur 27: This circular plot compars th variation from ach of fiv diffrnt sts of data for on hol Th bold lin plots th avrag of th data sts This hol is an xampl of th typical variations sn in plots of all th data points data fils rquird to handl th cofficints of th curvfit quations as thy vary by th siz of th row column matrics On th down sid, this calibration xrcis rquird significant thought and work, vn though th problms studid ar ultimatly prtty simpl This suggsts that th work rquird to produc ths charactrizations for th myriad of paramtrs rquird to dscrib gnral manipulation tasks will b too costly to systmatically apply in industrial practic Anothr problm is that as th spac dimnsionality incrass, th rquird tabl siz also incrass For this and othr rasons, our mthods bcom incrasingly impractical as th task dimnsionality incrass Thus it sms likly that prior arm kinmatic idntification schms will provid bttr prformanc for full 3-d rror stimation Whn using ths mthods, how should w stimat worst-cas rror bounds? Furthr, our initial ida of braking rrors down into thir constitunt componnts workd wll in th vision cas, but brok down in th arm cas, laving us with wak histogram-basd mthods This is somwhat unsatisfying Sinc this is rquird for corrct application of LMT worst-cas analysis mthods, w ar ld to conclud that this approach is impractical xcpt in cass whr 17

23 Appndix Appndix A 1: Normalization Th grids dscribing rror magnituds ar arrangd so that th intrsction of curv 11 in th x and y dirctions rprsnts th grid origin ( ) This point corrsponds to th cntr of th imag fram locatd at (Xcntr, Ycntr) TOcorrct th position of a point, th point s (2, y) imag location must b normalizd with rspct to th grid of intrpolation curvs W accomplish this with th following quations, basd on a lomm grid spacing: Ynorm - Ycoordinat -?/cntr \ -+Y7 \ \ Appndix A2: 2-d ntrpolation ntrpolation will b usd to corrct rrors in objct and imag fram locations which do not li prcisly (within +OOlmm in both x and y) on a charactrizd gridpoint W tstd svral intrpolation mthods to idntify th most accurat routin W slctd a bidirctional linar intrpolation schm to calculat th rror magnituds of points off th grid Bidirctional corrction maks us of data dscribing an rror magnitud for ach point, basd on th point s (2, y) location W bliv this routin producs an accurat physical dscription of location rrors causd by various sourcs of lns distortion Th first intrpolation is a wightd linar intrpolation to provid an initial rror stimat A scond intrpolation will adjust this rror stimat by adding a curvatur corrction Th curvatur corrction is significantly smallr in magnitud than th position rror at all points of th imag fram Th rason for this is that curvatur ovr any tn millimtr sgmnt conncting adjacnt gridpoints is quit small on th ordr of 2mm to 5mmmm Whil an argumnt could b mad for ignoring this rror contribution cu,tur << Eposition, our goal of high prcision rquirs it s inclusion Th following xampl dmonstrats th proccss usd for finding a point s imag plan location givn an arbitrary objct plan location Four curvfits ar rquird to fully dscrib th rror associatd with any nongrid point Thy ar as follows: 1 x location rror x,, Figur 28: Graphical rprsntation of th intrpolation mthod usd to gnrat rror stimats for points of intrst not locatd on spcific gridpoints Th following paragraphs will xplain how this intrpolation is prformd for an xampl (x,y) point (37,64) Rfr to Figur 28 for a graphical rprsntation of th points that ar gnratd Qustion: Givn a dot locatd at point (37,64) in th imag plan, whr is th dot in th objct plan? Solution: W first normaliz th point (37,64) to dtrmin which st of curvs to us for intrpolation Using quations (1) and (2) with valus of (Zcntr, ycntr) = (14833mq 17814mm), w obtain = and ynom = This shows that th point (37,64) lis btwn x curvs 4 and 5, and btwn y curvs 7 and 8 Nxt w dtrmin th corrction valus x,,,, and ynor through intrpolation W us th sam intrpolation mthod to dtrmin both of ths valus, using th appropriat tabls of Xmor and y,, curvfit functions n th discussion that follows, w will us th trm z to rfr to th x,,, or y,, trm that is bing dtrmind Th first stp in dtrmining th valu of z at (x,,,, yno-) is to intrpolat btwn th adjacnt x curvs W will dnot ths curvs Xcurv4(Y) and Xcurv5(y);both of ths curvs ar functions of y Evaluating ths functions at y = yn/norm yilds 24 = Xcurv4(norm) and 25 = X c u r v ( n o r m )corr, sponding to th points lablld 4 and 5 in Figur 28 W thn obtain our first stimat of th dsird corrc- as a function of a 2 x location rror xmoras a function of y, 3 y location rror yy,,,, as a function of x 4 y location rror yyaor as a function of y Th gnration of ths curvs from obsrvd data is dscribd in Sction 36 18

24 Max 63 Min -586 Sigma 2722 T 9 -- loo > 3 Lt 7 x 2 -- al l o t ( ll -- t 8,,,,,,, : ;;! *-- Error (mm) c O ( D d N O ( U d ( D g ) 9 9 9? q q Q Error (mm) Figur 29: Dot location rrors aftr singl axis intrpolation using a matrix of points that li at worst cas locations halfway btwn th curvs Figur 3: Dot location rrors aftr doubl axis intrpolation using matrix of points projctd to li at worst cas locations Not major diffrnc in span from Figur 29 dz7 = Ycurv7 (=norm) tion valu z by wightd linar intrpolation: - 27 dz7 rflcts th surfac dviation from linar at curv Ycurv7W us a similar calculation to dtrmin dzg, th curvatur corrction at Ycurp8 w thn stimat th curvatur corrction at ( a n o m, ynorm) as Th rsulting rror stimat is rasonably accurat givn th small amount of curvatur prsnt across th span of th boundary curvs Figur 29 shows th rror magnituds at worst-cas points of intrst using this singl axis intrpolation W can improv our stimat of z by using th y curvs to stimat th surfac curvatur btwn Xcurv4 and Xcurv5 Our goal is to comput a curvatur corrction dz which stimats th surfac dviation from a straight lin conncting points 4 and 5 W accomplish this by computing dz for y curvs 7 and 8, and linarly intrpolating btwn th rsulting dz7 and dz8 valus To obtain th valu of dz7, w first comput 274 = Ycurv7( [ZnormJ) and 275 = Ycurv7( rzn?rml), corrsponding to th points lablld 74 and 75 in Figur 28 W now calculat th linarly intrpolatd valu W thn obtain our final rror stimat z' = z + dz This mthod is usd to find th corrction trms xrr, and gmor,using th appropriat curvfit lookup tabls Ths trms ar thn addd to th point coordinats (37,64) to dtrmin th corrctd point in th imag fram Figur 3 shows th rror magnituds at worst-cas points using this bidirctional intrpolation Th narrowr rang compard to Figur 29 corrsponds to th improvmnt providd by curvatur corrction, which dcrasd th worst-cas rror by a factor of two Not that th final worst cas rror stimat is wll within th limits of th hol location rror attributabl to rsidual lns distortion rrors This valu corrsponds to a pur linar intrpolation btwn points 74 and 75 W can now dtrmin th curvatur rror for Ycurv7as 19

25 Max 98 Min -49 Sigma 27 Appndix A3: mag To Objct Fram Mapping All prvious work dalt with th problm of mapping objct points to th corrct imag fram location This hlps us answr th qustion: Givn a point on an objct whr should it show up in th imag fram? Anothr qustion w would lik to b abl to answr is: Givn a point in th imag fram, whr is th corrsponding point on th objct? To answr this qustion w will rquir anothr st of rror tabls to map-points from th imag fram to th objct fram Th rror data to dscrib th offsts in this dirction will b gnratd using th prvious corrction tabls, a grid slctd to provid convnint indx locations, and an itrativ routin for finding offsts btwn th two frams Whn slcting th gridpoints and spacing to b usd for th nw rror tabl, a coupl of issus should b considrd: First, for convninc th grid should b a 21 x 21 matrix of imag locations, which allows us to r-us our curvfit and intrpolation routins Scond, all valus of th imag-to-objct rror grid must li within th boundaris of th prviously dfind objct-to-imag rror grid This rquirmnt is drivn by th us th objct-to-imag grid in an itrativ mannr to assign rror valus to imag locations on th nw grid Locations outsid of th prviously found grid will caus th rror stimation program to crash W will construct our imag-to-objct mapping by building x-curvs and y-curvs analogous to th objctto-imag mapping This is accomplishd by visiting ach imag grid point (zi,yi), and numrically sarching for th corrsponding objct point (zo, yo) which yilds corrction valus (X-r, y,,,) such that xi Yi = = 8 > Q) 9? L Error (mm) Figur 31: Histogram rprsnts th diffrnc btwn th actual imag points (for all points on th calibration grid), and th rror corrctd imag points - to-objct corrction valus for (zi,yi) Othrwis, st (zo,yo) (zo, yo) (dz, dy) and go to Stp 2 + Onc corrction valus hav bn found for ach imag grid point (zi,yi), thn 5th-ordr curv fitting and bidirctional intrpolation may procdas in th objct-toimag cas Figur 31 shows th rsidual rrors that rmain at th nd of this numrical sarch All rrors ar lss than th loop xit thrshold of 1mm;most ar significantly smallr W ran th following xprimnt to tst both th intrpolation routin and th nwly dvlopd rror grid Slctd points in th objct fram wr corrctd to find th imag fram coordinats of th objct location Aftr convrsion, rturnd locations wr usd as objct locations and input to th objct to imag corrction routin Th locations rturnd from this routin wr compard to th original imag points for rror offsts Th doubl convrsion usd points chosn to rprsnt what w blivd would rprsnt worst cas conditions for th intrpolation schms Th prsumd worst cas occurrnc is a point halfway btwn two curvs on both th x, and y axs, th point furthst from th curvs usd to charactriz th worst cas rror offst W collctd data using an array of locations which rprsnt th midpoints of all 21 curvs W also collctd data using a st of 11 curvs (vry othr curv) Th location array for this data st consistd + Xrror Yo + Yrror zo Th corrction trms associatd with th imag grid point (zi,yi) ar thn -2rror and -ynor For a givn imag grid point (xi, yi), this numrical sarch is accomplishd using- th following - -procdur: 1 St our initial stimat of th objct fram location to th givn imag grid point: (zo, yo) +- (zi,y;) 2 Lookup th rror corrction for th currnt ( d o,yo) using th xisting objct-to-imag corrction tabls This yilds corrction trms =rror and yrror for th currnt objct point stimat (xo,yo) 3 Comput th imag point corrsponding to th currnt objct point: (xi,yi) +- (x,,yo) (zrror, yrror) 4 Comput th discrpancy in th imag points dx = x; - xi and dy = yi f dz and dy ar both lss than zkooolmm, xit th loop and rturn -xrror and -?/rror as th imag- + 2