Dual-Quaternions. From Classical Mechanics to Computer Graphics and Beyond. Ben Kenwright

Transcription

1 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon Dual-Quaenions Fom Classical Mechanics o Compue Gaphics an eyon en Kenwigh bkenwigh@xbev.ne bsac This pape pesens an oveview of he analyical avanages of ual-uaenions an hei poenial in he aeas of oboics, gaphics, an animaion. While uaenions have poven hemselves as poviing an unambiguous, un-cumbesome, compuaionally efficien meho of epesening oaional infomaion, we hope afe eaing his pape he eae will ake a paallel view on ual-uaenions. Despie he fac ha he mos popula meho of escibing igi ansfoms is wih homogeneous ansfomaion maices hey can suffe fom seveal ownsies in compaison o ual-uaenions. Fo example, ual-uaenions offe incease compuaional efficiency, euce ovehea, an cooinae invaiance. We also emonsae an explain how, ual-uaenions can be use o geneae consan smooh inepolaion beween ansfoms. Hence, his pape aims o povie a compehensive sep-by-sep explanaion of ual-uaenions, an i compising pas (i.e., uaenions an ual-numbes in a saighfowa appoach using pacical eal-wol examples an uncomplicae implemenaion infomaion. While hee is a lage amoun of lieaue on he heoeical aspecs of ual-uaenions hee is lile on he pacical eails. So, while giving a clea no-nonsense inoucion o he heoy, his pape also explains an emonsaes numeous wokable aspec using eal-wol examples wih saisical esuls ha illusae he powe an poenial of ual-uaenions. Keywos: ual-uaenion, ansfomaion, blening, inepolaion, uaenion, ual-numbe Inoucion (Why shoul we use ual-uaenions? Dual-uaenions ae a nea mahemaical ool ha beaks away fom he nom. Pobably one of hei mos impoan popeies is in classical mechanics since hey can epesen complex poblems in a unifie compac way. ual-uaenion combines he linea an oaional componens ogehe ino a single vaiable ha can be inepolae, concaenae an ansfome using a single se of algebaic ules. While i has been emonsae ha uaenions ae he bes geneal soluion fo oaions [] hey can only epesen half he igi ansfomaion. Since, a full 3D igi ansfomaion is compose of a anslaional an oaional componen, which is aiionally manage as a 4x4 homogenous maix. Howeve, he maix conains a gea eal of ovehea an is ifficul o inepolae beween ansfoms. lenaively, he ansfomaions can be manage using wo inepenen componens (e.g., anslaion veco an a uaenion. Theefoe, ual-uaenions ake us in a iffeen iecion an pesen us wih a unifie componen ha pesens us wih a huge numbe of avanages. In a nushell: They combine oaion an anslaion ino a unifie sae vaiable They ae a compac epesenaion (8 scalas They ae easily convee o ohe foms (e.g., maices They can be inepolae easily wihou ambiguiy o gimbals' lock They ae compuaionally efficien (compaable wih maices an uaenions [][3] They can be inegae ino a cuen sysem wih lile isupion (i.e., maix alenaive They pesen a single invaian cooinae fame o epesenaion igi ansfoms [4] Dual-uaenions ae an algoihmically simple an compuaionally efficien appoach of epesening igi ansfoms (i.e., oaion an anslaion. They ae use in he same way as uaenions bu povie he ae avanage of encapsulaing boh anslaion an oaion ino a unifie sae ha can be concaenae an inepolae effolessly. In fac, we believe ha he eae afe eaing his pape will be sufficienly familia wih how ual-uaenion algeba woks, an how i can be use in pacical siuaions, o begin o appeciae he enomous poenial ual-uaenions can offe, boh fo he gaphical communiy bu also in ohe aeas of eseach. Oveview (Wha we nee o know Dual-uaenions ae a combinaion of ual-numbe heoy an uaenion mahemaics. Wheeby, o have a goo unesaning of how we can exploi ual-uaenions o ou avanage, we nee o unesan he basics of uaenions an ual-numbe heoy. Hence, his pape begins by explaining he funamenal componens of ualuaenions o help esablish a common goun fo eaes, afe which, we hen focus on ual-uaenions an he applicabiliy fo epesening ansfomaions boh compuaionally an ynamically (e.g., calculaing iffeences an inepolaing. en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp -

2 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon asically, a ual-uaenion is he concaenaion of uaenion an ual-numbe heoy (see Figue. Quaenion Mahemaics Dual-Numbe Theoy Figue : Dual-Quaenions Componens. To avoi confusion an enable he eae o easily isinguish a uaenion fom a ual-uaenion we use wo iscenible symbols o ienify hem (see Euaion. Quaenion( Dual Quaenion( Dual-Quaenions i is essenial o have a goo unesan of is unepinne pas wok (i.e., uaenions an ual-numbes. Fuhemoe, once he eae unesans how uaenions wok, i shoul be ouble-fee an saighfowa o see how ual-uaenions opeae ue o hei likeness. uaenion is epesene by wo funamenal pas, a scala eal pa (w an an imaginay veco pa ( v x, y, z. In pacice we ae only concene wih a uniuaenion since hey offe he mos benefis an epesen he oaion on a 4D uni-hypesphee. While he majoiy of people ae familia wih he ecomposiion an pinciples of uaenions, hee can, howeve, be a eficiency in he pacical consieaions. Quaenion Real Complex = w + ix + jy + kz Imaginay 4 scala vaiables Dual-Quaenion Real Dual-Pa = + Dual-Opeao 8 scala vaiables Figue : Visual Oveview of Quaenion an Dual-Quaenion Componens. n oveview of boh he uaenion an ual-uaenion componens is shown in Figue. While a uaenion consiss of fou scala values, a ual-uaenion consiss of eigh scala values. Howeve, a uaenion can only epesen oaion, while a ual-uaenion can epesen boh oaion an anslaion. Dual-uaenions ae a valuable ool ha can be ae o an iniviual's libay o achieve a paicula ask, e.g., igi hieachy concaenaion, inepolaion, chaace skinning. They opeae simila o exising mehos (i.e., maices an can be ansfome o an fom ohe foms easily (i.e., uaenions, maices which enables hem o be inegae o exchange wih lile isupion ino a sysem o gain hei ewas. Fo a beginnes inoucion o ual-uaenions wih an emphasis on compaison beween ivese mehos (e.g., maices an Eule angles an how o go abou implemening a saighfowa libay I efe he eae o he pape by Kenwigh [3]. Quaenion lgeba While walking wih his wife in 843, Si William Hamilon [5] gave bih o a evoluionay new concep ha lae became known as Quaenions. While i ook some ime fo uaenions o be accepe, hey evenually emonsae hemselves as being he mos compeen, memoy efficien, ambiguiy-fee meho of epesening oaions. Fuhemoe, since uaenions ae he founaion upon which ual-uaenions ae buil i comes as no shock, an is uie unesanable, ha hese popeies ae inheie. Neveheless, o ensue he eae is uly able o unesan he poenial of ual-uaenions =(w,x,y,z=(w, v The funamenal mahemaical opeaions ae efine fo uaenions (i.e., aiion an muliplicaion of uaenions an he muliplicaion of a uaenion by a scala. Quaenion fom xis-ngle Given an angle an axis of oaion, we can consuc a uaenion using Euaion 3. ˆ cos, ˆ nsin o w cos, x nx sin, y ny sin, z nz sin whee is he angle an ˆn is a uni-veco epesening he axis of oaion. While i is ecommene ha you consisenly use uaenions fo oaion, we can, howeve, ewie Euaion 3 o give us he axis-angle fom he uaenion o ai in visualizing angle-axis iffeences as shown in Euaion 4. cos ( w x y z nx, ny, nz sin sin sin 3 4 en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp -

3 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon In pacice, if you o ecie o conve o he axis-angle epesenaion, you shoul ensue he uaenion is always a uni-uaenion an be awae of he ivie by zeo causaliy ha may occu (i.e., sin is zeo. Quaenion Veco Tansfomaion The uaenions ansfomaion can be applie o a 3D veco cooinae by means of muliplicaion. Wheeby, o ansfom a veco posiion by a uaenion we simply conve he veco o a uaenion (i.e., he imaginay pa is he veco posiion, an he scala eal pa zeo an muliply i by he uaenion ansfom an is conjugae, as shown in Euaion 5. Opionally, we can conve he uaenion ansfom o a maix wih lile o no exa wok fo sysems ha opeae wih maices (e.g., ansfoms ae one on he GPU using maices. whee p' ˆ ˆ p 5 ˆ is a uni-uaenion epesening he oaion ansfom ˆ is a uni-uaenion ha epesens he invese of he oaion uaenion p is he 3D veco poin in uaenion fom (i.e., p (0, v wih v ( v, v, v x y z p ' is he 3D ansfome veco poin in uaenion fom (i.e., p' (0, v Howeve, i is exemely impoan o noe ha fo a uniuaenion he invese is he same as he conjugae. This is ue o he mahemaical an compuaional efficiency by which he conjugae is calculae. The conjugae of a uaenion is simply he negaion of he veco componen (shown in Euaion 6. * w (, v 6 Quaenion o Maix Due o he populaiy of maices, i is vial o be able o ansfom a uaenion o maix fom an vice-vesa. uaenion can be ansfome o a maix using lile moe han muliplicaions an aiions as shown in Euaion 7. -(y +z (xy+zw (xz+yw M (xy+zw -(x +z (yz+xw 7 (xy+yw (yz+xw -(x +y whee M is a maix euivalen of he uaenion, an x, y, z an w epesen he elemens of he uaenion. Quaenion iion ing wo uaenions ogehe is accomplishe by simply summing he iniviual componens ogehe as shown in Euaion 8. (,,, 8 0 0w w 0x x 0y y 0z z Quaenion Muliplicaion Quaenion muliplicaion is analogous o maix muliplicaion; wheeby, muliplying uaenions ogehe is euivalen o combining hei ansfoms. Fo example, when wo uaenions ae muliplie ogehe i is euivalen o he fis uaenion being oae by he axis an angle of he secon uaenion. Howeve, uaenion muliplicaion is non-commuaive (i.e., oe of muliplicaion maes bu can be simplifie by being epesene using he o an coss pouc (shown in Euaion 9. ( ( 0 w0 v0 w v ( ( w0 w v0 v w0 v w v0 v0 v whee w0 an w epesen he eal scala componens of each uaenion an, v0 an v epesen he veco componen of each uaenion. Quaenion Diffeence Since each uaenion epesens an axis-angle, hen muliplying wo uaenions ogehe is euivalen o ansfoming one uaenion by anohe. Hence, i shoul be obvious, ha we can use his o eemine iffeences beween uaenions. If boh uaenions ae he same, an we muliply one by he invese of iself, i will cancel ou (see Euaion 0 an give us an ieniy uaenion. ˆˆ 0 So if we have wo uaenions, we simply muliply one by he invese o ge he iffeence beween hem (Euaion. I is vial o emembe ha he invese of a uniuaenion is he same as he conjugae. ˆ ˆ ˆ iff Fo example, a simplifie numeical example of he iffeence beween wo uaenions is shown in Euaion. : 0, nˆ 0, 0, :, nˆ 0,0, 0 0 ˆ cos, 0,0, sin,0,0,0 ˆ cos, 0, 0, sin 0, 0, 0, * ˆ ˆ 0,0,0, ˆ ˆ ˆ iff ((0 (0,0,0 (0,0,, ˆ iff ((0, 0, (0(0, 0, 0 (0, 0, 0 (0, 0, 0,0,0, :, nˆ 0,0, To help visualize he esul fo he example in Euaion, imagine compaing he iffeence beween wo scala numbes an (e.g., 0 an n. Then he iffeence, - 9 en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp 3-

4 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon = (0 - n = -n, which is analogous o wha we calculae in he example. Quaenion Spheical Linea Inepolaion (SLERP Quaenion spheical linea inepolaion is he ansfomaion along he suface of he 4D unihypesphee. Saing wih he well known exponenial funcion fom complex numbes i can be shown ha in Euaion 3. e i cos isin 3 Then we can euae ou uaenion an epesen i as an exponenial given by Euaion 4. e v cos v sin 4 whee an v is a uni veco (noing ha v. We can hen wie he uaenion in he fom (Euaion 5: cos( v ˆ sin( 5 Then he Slep expession is given by Euaion 6. SLERP(, : ( Fo example, le us consie wo vey simple cases when =0 an = an 0 cos( vsin( 0 0 cos(0 v sin(0 ( ( ( cos( vsin( cos( vsin( ( ( ( ( n alenaive, an moe popula, epesenaion of Euaion 6 can be calculae using a geomeic appoach an is shown in Euaion 7 (fo a moe eaile escipion see Shoemake []. sin ( sin( SLERP( 0, : 0 7 sin( sin( Dual-Numbe Theoy Cliffo [6] publishe his iniguing wok on ual-numbes in 873, an povie us wih a poweful ool fo faciliaing he analysis of complex sysems (e.g., mechanical, geomeic. In fac, i was no long befoe hey foun a place in he movemen of igi boies [7][8] an lae in geomey [9]. The elevan fomalism ha was evelope an wha we pimaily make use of in his pape is he scew calculus ha allows he unificaion of anslaion an oaion. The efiniion an popeies of a ual-numbe ae given in Euaion 8. Dual-numbes ae akin o complex numbes. Howeve, wheeas complex numbes have a eal-pa an an imaginay-pa an ual-numbes have a eal-pa an a ual-pa. z wih bu whee is known as he ual-opeao, is he eal-pa an, he ual-pa. Dual-Numbe iion ( ( ( ( 9 Dual-Numbe Muliplicaion ( ( ( ( emembe 0 0 Dual-Numbe Division ( ( ( ( ( ( ( ( Dual-Numbe Diffeeniaion Fom elemenay calculus pinciples shown in Euaion. s( x x s( x s ( x lim x x0 x We use Taylo seies o fin he iffeeniable (Euaion 3. f ( f ( f ( f f!! 3! f( f as! f ( f ( 3 ( ( ( (... ( (, 0 Remakably, ue o he coniion 0, we en up wih an exemely elegan soluion. Fo a moe in-eph explanaion of he aionale behin ual-numbe heoy see Kele [0] o Pennes e al []. Dual-Quaenion lgeba The ual-uaenion is an exension of ual-numbe heoy wheeby he numbes fo he ual-numbe euaion ae epesene by uaenions. Remakably, he ualuaenion algeba ha esuls is vey saighfowa an elegan an povies us an algebaically compac an 3 en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp 4-

5 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon efficien sysem fo solving ohewise complex poblems. Fo example, we can epesen a igi ansfoms wih eigh scala vaiables; we can combine ansfoms effolessly hough concaenaion, an we ae able o pouce smooh consan inepolaion beween igi ansfomaions. s shown in Figue, he ualuaenion is ecompose ino wo pas he eal pa an he ual-pa. Dual-Quaenion Ieniy The ieniy of a ual-uaenion is shown in Euaion 4 an is analogous o a uaenion ieniy. Theefoe, any ual-uaenion ha is muliplie wih an ieniy ualuaenion emains unchange. To efine an ieniy ualuaenion we se he fis scala value o an he ohe seven scala values ae all 0. [,0,0,0][0,0,0,0] 4 Dual-Quaenion fom Posiion an Roaion To consuc a uni-ual uaenion fom a oaion an a anslaion we use Euaion 5. We consuc he ualuaenion fom a pai of uaenions ha epesen he oaion an anslaion. whee is a uni uaenion epesening he oaion an is a uaenion escibing he anslaion. The iniviual elemens of he wo uaenions fom Euaion 5 ae shown in Euaion 6. [cos(, nx sin(, n y sin(, nz sin( ] [0,,, ] x y z whee n is he axis of oaion, is he angle of oaion, an x, y, z is he posiion in Caesian cooinaes. Fo example, if we wan o consuc a ual-uaenion ha only has a oaion we have: [cos(, nx sin(, n y sin(, nz sin( ][0,0,0,0] an, if we wan o consuc a ual-uaenion ha only has a anslaion we have: x y z [,0,0,0][0,,, ] Compaable o maices an uaenions we can concaenae ual-uaenion ansfomaions using muliplicaion. Hence, you can ceae a pue oaion ualuaenion an a pue anslaion ual-uaenion an muliply hem ogehe o fom a combine ualuaenion ha possesses boh he anslaion an oaion componens; howeve, be awae ha he muliplicaion oe is impoan. Dual-Quaenion o Posiion an Roaion We can exac he posiion an oaion fom a ualuaenion. In evese o Euaion 5 ha ceae a ualuaenion fom a posiion an oaion, we convesely exac he posiion an oaion using Euaion 9. Dual-Quaenion iion The aiion of ual-uaenions is one of he simples opeaions since we only nee o a each iniviual componen ogehe (see Euaion 30. ( a a i a j a k ( a a i a j a k * ( b b i b j b k ( b b i b j b k (( a b ( a b ( a b ( a b (( a b ( a b ( a b ( a b Dual-Quaenion Muliplicaion Due o ual-numbes euiing 0 esuls in he muliplicaion of ual-uaenions being a vey nea an iy opeaion (see Euaion 3. Hence, he esuling ual-uaenion muliplicaion can be boken own ino hee uaenion muliplicaions an a uaenion aiion opeaion. 0 3 ( ( 0 3 ( Dual-Quaenion Conjugae The ual-uaenion conjugae is essenially an exension of he uaenion conjugae, an is given by Euaion * * * 3 Dual-Quaenion Magniue ual-uaenion muliplie by is conjugae gives he magniue suae an hence he suae oo of his is he scala magniue lengh (see Euaion 33. * 33 I is cucial o noe ha a uni ual-uaenion has a magniue of. Hence, we can say ha he magniue of a uni ual-uaenion muliplie by is conjugae mus eual. en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp 5-

6 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon ˆ ˆˆ* 34 Dual-Quaenion Veco Tansfomaion Euivalen o a uaenion a ual-uaenion can ansfom a 3D veco cooinae as shown in Euaion 34. Noe ha fo a uni-uaenion he invese is he same as he conjugae. whee p' ˆ ˆ p 35 is a ual-uaenion epesening he ansfom is a ual-uaenion ha is he invese of he ualuaenion p is a ual-uaenion epesening he igi ansfom (e.g., 3D veco poin p (,0,0,0 (0, v, v v x y z p ' is a ual-uaenion wih he esuling ansfom. Plücke Cooinaes Plücke cooinaes [] ae use o ceae Scew cooinaes which ae an essenial echniue of epesening lines. We nee he Scew cooinaes so ha we can e-wie ual-uaenions in a moe elegan fom o ai us in fomulaing a neae an less complex inepolaion meho ha is compaable wih spheical linea inepolaion fo classical uaenions. The Definiion of Plücke Cooinaes:. p is a poin anywhee on a given line. l is he iecion veco 3. m = p l is he momen veco 4. ( lmae, he six Plücke cooinae We can conve he eigh ual-uaenions paamees o an euivalen se of eigh scew cooinaes an vice-vesa. The efiniion of he paamees ae given in Euaion 36. scew paamees (,, lm, ual uaenion ( w v ( w v whee in aiion o l epesening he veco line iecion an m he line momen, we also have epesening he anslaion along he axis (i.e., pich an he angle of oaion. Conveing o an fom a ual-uaenion an is scew paamees is shown in Euaion 37 an Euaion 38 (see Daniiliis [3] fo eails. 36 an ual uaenion scew paamees cos ( w w v l v v v w m v l w cos v lsin w sin v sin m cos l v v scew paamees ual uaenion Dual-Quaenion Powe We can wie he ual-uaenion epesenaion in he fom given in Euaion 39 (see Daniiliis [4] fo eails. whee ˆ cos ( l msin ˆ ˆ cos ˆ sin v ˆ is a uni ual-uaenion ˆv is a uni ual-veco vˆ l m ˆ is a ual-angle ˆ The ual-uaenion in his fom is excepionally ineesing an valuable as i allows us o calculae a ualuaenion o a powe. Calculaing a ual-uaenion o a powe is essenial fo us o be able o easily calculae spheical linea inepolaion. Howeve, insea of puely oaion as wih classical uaenions, we ae insea now able o inepolae full igi ansfomaions (i.e., oaion an anslaion by using ual-uaenions. v ˆ ˆ ˆ cos ˆ sin v 40 Dual-Quaenion Scew Linea Inepolaion (ScLERP ScLERP is an exension of he uaenion SLERP echniue, an allows us o ceae consan smooh inepolaion beween ual-uaenions. Simila o en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp 6-

7 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon uaenion SLERP we use he powe funcion o calculae he inepolaion values fo ScLERP shown in Euaion 4. Weigh ˆ ˆ ˆ ˆ ˆ ScLERP (, : ( 4 whee ˆ an ˆ ae he sa an en uni ual-uaenion an is he inepolaion amoun fom 0.0 o.0. The implemenaion of ScLERP involves fis using Euaion 37 o conve he ual-uaenion paamees o scew paamees, so we can calculae he powe funcion wih Euaion 40. fewas, we use Euaion 38 o conve back o a ual-uaenion o complee he calculaion an give he esuling inepolae esul. scew paamees (,, lm, ual uaenion ( w v ( w v asic Un-Opimize Implemenaion Seps of ScLERP (fo 4 Weighing (Fo example, see Lising fo a pacical implemenaion example. lenaively, a fas appoximae alenaive o ScLERP was pesene by Kavan e al. [5] calle Dual-Quaenion Linea lening (DL. Fuhemoe, ual-uaenions have gaine a gea eal of aenion in he aea of chaace-base skinning. Since, a skinne suface appoximaion using a weighe ual-uaenion appoach pouces less kinking an euce visual anomalies compae o linea mehos by ensuing he suface keeps is volume (fo example, see Figue 3. Dual-uaenions eliminae skin collapsing aefacs an while hey ae slighly slowe han he linea blene skinning meho hey ae, howeve, gaphical pocesso uni (GPU fienly. Fuhemoe, hey ae simple o inegae ino a 3D engine an cause vey lile isupion since he same igging as sana linea blening skinning can be use Weigh Tansfom Tansfom Tansfom Tansfom Tansfom Tansfom Linea Dual-Quaenion Figue 3: Visual compaison beween linea an ual-uaenion weighing fo veex skinning. Euaion 4:. Calculae Invese of (i.e., Conjugae of. Muliply Inv( an 3. Calculae Scew Paamees fo esul Inv( 4. Calculae o he powe of 5. Conve scew paamees fom back o he classical ual-uaenion fom 6. Muliply wih o ge he answe Inepolaion In geneal, one of he geaes avanages of using uaenions an ual-uaenions ove any ohe meho is hei abiliy o inepolae smoohly beween ansfoms. Naively, wo values can epesen he sa an en, an a scala consan epesens he inepolaion amoun (scala aio is fom 0.0 o.0. Fo a saigh-line veco we can ea each componen sepaaely an use a paameic euaion shown in Euaion 43. This has he ae avanage of being compuaionally fas an simple. en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp 7-

8 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon LERP( a, b : b ( a b 43 whee a an b epesen he sa an en value an he in-beween aio. In fac, fo small changes we can use Euaion 43 o inepolae beween uaenions an ual-uaenions. Howeve, as he uaenion an ual-uaenion become moe issimila hee is a geae eo an he inemeiae seps become less smooh an less coec. The inemeiae seps beween he sa an en o no epesen a uni-uaenion oaion o ual-uaenion oaion. Hence, we nee o e-nomalize he value a each sep o ensue i falls on he uni-hypesphee. Mos impoanly, hough, is ha he inepolaion ae is no consan. We can euce he eo an make he linea inepolaion appoximaion moe oleable by nomalizing he values beween seps. This is known as Nomalize Linea Inepolaion (NLERP an has he ae avanage of ensuing ha he inemeiae values ae always of unilengh (see Euaion 44. gain, i shoul be sesse ha he linea inepolaion appoximaion is only suiable fo small changes. + ( - NLERP(, : + ( - 44 whee a is he sa, b is he en an is he inepolaion amoun (i.e., 0.0 o.0. While i has numeous poblems fo boh uaenions an ual-uaenions, i is compuaionally fas an easy o implemen an can, howeve, give easonably goo appoximaions fo small inepolaions. The ouble is, uaenions an ual-uaenions o no avel along saigh-line ajecoies. Howeve, we can use an alenaive inepolaion meho ha follows he unihypesphee sphee. This is accomplishe by inepolaion along he uni-hypesphee o pouce a consan an smooh ae of change. Dual-uaenions can use he exponenial epesenaion simila o uaenions o geneae an inepolaion scheme o pouce consan smooh inepolaion. Shoes o Longes Inepolaion Pah Conay o popula belief, a uaenion an ualuaenion by efaul will no ake he shoes pah beween poins when inepolae. This is because a uaenion can epesen he same oienaion using wo iffeen epesenaions, an conseuenly a ualuaenion. This means ha boh uaenions an ualuaenions o no offe a uniue epesenaion of an oienaion o ansfomaion (i.e., hee ae wo. The iffeence beween he wo epesenaion becomes appaen uing inepolaion an povie a meho fo eemining he shoes o longes pah o be aken uing inepolaion. The inepolaion iecion can be calculae by examining he angle beween he wo ansfoms. If he angle beween he wo uaenions (o ual-uaenions is geae han hen he inepolaion will ake he "longes pah". We can eece easily in pacice by aking he o pouce of he wo uaenions (fo a ual-uaenion we use he uaenion fo he oaion. If he o pouc is less han zeo hen he longes pah will be aken. Howeve, if we wan o peven he longes pah fom being aken we simply negae all he elemens fo he uaenion o ual-uaenion befoe inepolaing. Likewise, if we esie he longes pah we can check ha he o pouc is geae han zeo befoe negaing he uaenion o ual-uaenion. Camull-Rom Spline-ase Inepolaion Fo iegula space key-fame aa, we can exploi he Camull-Rom spline-base veco inepolaion funcion an ual-uaenions algeba as a meho fo geneaing a unifie, smooh, coninuous ajecoy pah. Eaicaion of he Suae Roo We can opimize some opeaions by eaicaing he suae oo ovehea. Since boh uaenions an ualuaenions ae nomalize he same way as vecos (see Euaion 45, we can ienify cases wheeby an elemen is muliplicaion wih anohe elemen o cancel ou he suae oo. ˆ Howeve, he muliplicaion of wo uaenion elemens esuls in he suae oo being eunan. Fo example, when we consuc a maix fom a uaenion (as shown in Euaion 7 we muliply pais of elemens. This can be use o cancel ou he necessiy o nomalize he esul as shown in Euaion 46. ˆ ˆ x y x y x y Pefomance Compaison I can be shown wihou ifficuly ha in geneal a ualuaenion akes less opeaions o compue a geneal ansfom concaenaion compae o a maix (see Table. Maix4x4 Maix4x3 : 64mul + 48as : 48mul + 3as DualQuaenion : 4mul + 38as Table : Compuaional cos of combining maices an ual-uaenions. Fuhemoe, fo igi skeleal animaions, he compuaion of wol space ansfoms in aiion o he ovehea cos of ansfeing he aa o he gaphics pocessing uni (GPU can be noiceably bee. Fo example, o ansfe he ansfoms o he GPU each fame a ual-uaenions euies only eigh floas compae o a 3x4 maix ha euies welve pe join. en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp 8-

9 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon Invese Kinemaics The convenional meho fo epesening an concaenaing links ogehe in hieachical sysems is he Denavi-Haenbeg [6] maix convenion, an while Wang an Ravani [7] popose an alenaive moe efficien fowa ecusion meho fo kinemaic euaions, we popose using ual-uaenions, since hey offe an analogous alenaive ha is numeically sable an compuaionally efficien. Dual-uaenions have shown pomising esuls fo poviing singulaiy-fee soluions fo invese kinemaic (IK poblems wih nonlineaiies [8]. I is clealy an avanage o use ualuaenions fo igi hieachies since each ualuaenion can be concaenae easily, inepolae smoohly an povie igi ansfom compaisons effolessly. Poing o Dual-Quaenion Conveing an exiing maix scheme o a ual-uaenion sysem is saighfowa since much of he opeaions (i.e., concaenaion of ansfoms ae one he same way. Fo example, he concaenaion of ansfoms wih a maices an ual-uaenions: Maix Dual-Quaenion M M M M M whee he subscip epesens he ansfom, while maix ansfom M 0 coespons he ual-uaenion ansfom 0. Howeve, unlike maices, ual-uaenions povie an aiional epeoie of valuable funcions o easily compae an inepolae beween ansfoms. Conclusion an Final Thoughs This pape has aempe o inoucion he eae o he pacical poenial of ual-uaenions an hei avanages in solving kinemaic poblems (i.e., sysems wih oaional an anslaional popeies. The funamenal feaues an wokings of ual-uaenions have been ouline. I has also been shown, ha in geneal, hey povie a compac an efficien ool fo epesening igi ansfomaion (i.e., simulaneously oaion an anslaion. In pacicaliy, a ual-uaenion is a ool like any ohe ool o be use o solve a poblem. I is a novel an fesh alenaive o he e-faco meho of maices wih numeous benefis ha can be inegae ino a sysem wih lile isupion o complicaion. I is hope ha he eae afe eaing his pape will go fowas an implemen a saighfowa ual-uaenion class o enable hem o exploe he poenial an ecie fo hemselves if hey ae he igh ool fo he job. Refeences [] K. Shoemake, nimaing oaion wih uaenion cuves, In Poceeings of he h annual confeence on Compue gaphics an ineacive echniues. CM Pess, pp , 985. [] M. Schilling, Univesally manipulable boy moels ual uaenion epesenaions in layee an ynamic MMCs, uonomous Robos, vol. 30, no. 4, pp , 0. [3]. Kenwigh, eginnes Guie o Dual-Quaenions: Wha They e, How They Wok, an How o Use Them fo 3D Chaace Hieachies, The 0h Inenaional Confeence on Compue Gaphics, Visualizaion an Compue Vision, no. June 6 8, pp. 0, 0. [4] Q. Ge,. Vashney, J. P. Menon, an C. F. Chang, Double uaenions fo moion inepolaion, in Poceeings of he SME Design Engineeing Technical Confeence, 998. [5] S. W. R. Hamilon, On uaenions; o on a new sysem of imaginaies in algeba, Philosophical Magazine an Jounal of Science, no. July, pp. 0 3, 844. [6] W. Cliffo, Mahemaical Papes. Lonon, Macmillan, 88. [7]. P. Koelnikov, Scew calculus an some of is applicaions in geomey an mechanics, Kazan (in Russia, 895. [8] Leipzig, Geomeie e Dynamen, E. Suy, 903. [9] I. M. Yaglom, simple non-eucliean geomey an is physical basis, Spinge Velag, vol. New Yok, 979. [0] M. L. Kele, On he heoy of scews an he ual meho, In Poceeings of Symposium Commemoaing he Legacy, Woks, an Life of Si Robe Sawell all Upon he 00h nnivesay of Teaise on he Theoy of Scews, vol. July 9, 000. [] E. Pennes an R. Sefanelli, Linea lgeba an Numeical lgoihms Using Dual, Muliboy Sysem Dynamics, vol. 8, no. 3, pp , 007. [] J. Plùcke, On a new geomey of space, Philosophical Tansacions of he Royal Sociey of Lonon, vol. 55, no. 865, pp , 865. [3] K. Daniiliis, Han-Eye Calibaion Using Dual Quaenions, The Inenaional Jounal of Roboics Reseach, vol. 8, no. 3, pp , Ma [4] K. Daniiliis an.-c. Euao, The ual uaenion appoach o han-eye calibaion, Poceeings of he 3h Inenaional Confeence on Paen Recogniion, vol., pp. 38 3, 996. [5] L. Kavan, S. Collins, J. Žáa, an C. O Sullivan, Skinning wih ual uaenions, In 007 CM SIGGRPH symposium on ineacive 3D gaphics an games, vol. CM Pess, no. pil/may, pp , 007. [6] J. Denavi an R. S. Haenbeg, Kinemaic Noaion fo Lowe-Pai Mechanisms ase on Maices, Jounal of pplie Mechanics, vol., no. June, pp. 5, 955. [7] L. T. Wang an. Ravani, Recusive compuaions of kinemaic an ynamic euaions fo mechanical manipulaos, IEEE Jounal of Roboics an uomaion, vol. Sepembe, no. 3, pp. 4 3, 985. [8] Y. yın an S. Kucuk, Quaenion ase Invese Kinemaics fo Inusial Robo Manipulaos wih Eule Wis, IEEE Inenaional Confeence on Mechaonics, vol. July 3 5, pp , 006. en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp 9-

10 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon ppenix Sample Dual-Quaenion Class Implemenaion public class DualQuaenion_c public Quaenion m_eal; public Quaenion m_ual; public saic eaonly DualQuaenion_c Ieniy = new DualQuaenion_c(; public DualQuaenion_c( m_eal = new Quaenion(0,0,0,; m_ual = new Quaenion(0,0,0,0; public DualQuaenion_c( Quaenion, Quaenion m_eal = Quaenion.Nomalize( ; m_ual = ; public DualQuaenion_c( Quaenion, Veco3 m_eal = Quaenion.Nomalize( ; m_ual = ( new Quaenion(, 0 * m_eal * 0.5f; public saic floa Do( DualQuaenion_c a, DualQuaenion_c b eun Quaenion.Do( a.m_eal, b.m_eal ; public saic DualQuaenion_c opeao* (DualQuaenion_c, floa scale DualQuaenion_c e = ; e.m_eal *= scale; e.m_ual *= scale; eun e; public saic DualQuaenion_c Nomalize( DualQuaenion_c floa mag = Quaenion.Do(.m_eal,.m_eal ; Debug_c.sse( mag > f ; DualQuaenion_c e = ; e.m_eal *=.0f / mag; e.m_ual *=.0f / mag; eun e; public saic DualQuaenion_c opeao +(DualQuaenion_c lhs, DualQuaenion_c hs eun new DualQuaenion_c(lhs.m_eal + hs.m_eal, lhs.m_ual + hs.m_ual; // Muliplicaion oe - lef o igh public saic DualQuaenion_c opeao *(DualQuaenion_c lhs, DualQuaenion_c hs lhs = DualQuaenion_c.Nomalize( lhs ; hs = DualQuaenion_c.Nomalize( hs ; eun new DualQuaenion_c( hs.m_eal * lhs.m_eal, hs.m_ual * lhs.m_eal + hs.m_eal * lhs.m_ual; public saic DualQuaenion_c Conjugae( DualQuaenion_c eun new DualQuaenion_c( Quaenion.Conjugae(.m_eal, Quaenion.Conjugae(.m_ual ; public saic Quaenion GeRoaion( DualQuaenion_c eun.m_eal; public saic Veco3 GeTanslaion( DualQuaenion_c Quaenion = (.m_ual *.0f * Quaenion.Conjugae(.m_eal ; eun new Veco3(.X,.Y,.Z ; public saic Maix DualQuaenionToMaix( DualQuaenion_c = DualQuaenion_c.Nomalize( ; Maix M = Maix.Ieniy; floa w =.m_eal.w; floa x =.m_eal.x; floa y =.m_eal.y; floa z =.m_eal.z; // Exac oaional infomaion M.M = w*w + x*x - y*y - z*z; en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp 0-

11 Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon M.M = *x*y + *w*z; M.M3 = *x*z - *w*y; M.M = *x*y - *w*z; M.M = w*w + y*y - x*x - z*z; M.M3 = *y*z + *w*x; M.M3 = *x*z + *w*y; M.M3 = *y*z - *w*x; M.M33 = w*w + z*z - x*x - y*y; // Exac anslaion infomaion Quaenion = (.m_ual * Quaenion.Conjugae(.m_eal *.0f; M.M4 =.X; M.M4 =.Y; M.M43 =.Z; eun M; public saic DualQuaenion_c ScLERP( DualQuaenion_c fom, DualQuaenion_c o, floa // Shoes pah floa o = Quaenion.Do(fom.m_eal, o.m_eal; if ( o < 0 o = o * -.0f; // ScLERP = a(a^- b^ DualQuaenion_c iff = DualQuaenion_c.Conjugae(fom * o; Veco3 v = new Veco3(iff.m_eal.X, iff.m_eal.y, iff.m_eal.z; Veco3 v = new Veco3(iff.m_ual.X, iff.m_ual.y, iff.m_ual.z; floa inv = / (floamah.s( Veco3.Do(v, v ; // Scew paamees floa angle = * (floamah.cos( iff.m_eal.w ; floa pich = - * iff.m_ual.w * inv; Veco3 iecion = v * inv; Veco3 momen = (v - iecion*pich*iff.m_eal.w*0.5f*inv; // Exponenial powe angle *= ; pich *= ; // Conve back o ual-uaenion floa sinngle = Sin(0.5f*angle; floa cosngle = Cos(0.5f*angle; Quaenion eal = new Quaenion( iecion* sinngle, cosngle ; Quaenion ual = new Quaenion( sinngle*momen+pich*0.5f* cosngle *iecion, -pich*0.5f*sinngle ; // Complee he muliplicaion an eun he inepolae value eun fom * new DualQuaenion_c( eal, ual ; #if false public saic voi SimpleTes( DualQuaenion_c 0 = new DualQuaenion_c( Quaenion.CeaeFomYawPichRoll(,,3, new Veco3(0,30,90 ; DualQuaenion_c = new DualQuaenion_c( Quaenion.CeaeFomYawPichRoll(-,3,, new Veco3(30,40,90 ; DualQuaenion_c = new DualQuaenion_c( Quaenion.CeaeFomYawPichRoll(,3,.5f, new Veco3(5,0,66 ; DualQuaenion_c = 0 * * ; Maix ToMaix = DualQuaenion_c.DualQuaenionToMaix( ; Maix m0 = Maix.CeaeFomYawPichRoll(,,3 * Maix.CeaeTanslaion(0,30,90; Maix m = Maix.CeaeFomYawPichRoll(-,3, * Maix.CeaeTanslaion(30,40,90; Maix m = Maix.CeaeFomYawPichRoll(,3,.5f * Maix.CeaeTanslaion(5,0,66; Maix m = m0 * m * m; #enif // En DualQuaenion_c Lising : Dual-Quaenion Implemenaion Class (noe, his vesion of he class was wien fo claiy a poucion eay vesion coul be opimise an mae moe compac. en Kenwigh (bkenwigh@xbev.ne (Ocobe 0 pp -