EXPRESSION OF DUAL EULER PARAMETERS USING THE DUAL RODRIGUES PARAMETERS AND THEIR APPLICATION TO THE SCREW TRANSFORMATION

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1 Mathematal and Computatonal Applatons, Vol. 6, No., pp ,. Assoaton for Sentf Researh EXPRESSION OF DUAL EULER PARAMETERS USING THE DUAL RODRIGUES PARAMETERS AND THEIR APPLICATION TO THE SCREW TRANSFORMATION Ayşın Erkan Gürsoy and İlhan Karakılıç Department of Mathemats, Istanul Tehnal Unersty, Maslak 4469, Istanul,Turkey Department of Mathemats, Faulty of Senes, Unersty of Dokuz Eylül, 54 Bua, İzmr, TURKEY Astrat- Dual numers and dual etors are dely used n spatal knemats [,5-5,8]. Plüker lne oordnates of a straght lne an e represented y a dual unt etor loated at the dual unt sphere DUS. By ths ay, the trajetory of the sre axs of a rgd ody n R the real three spae orresponds to a dual ure on the DUS. Ths orrespondene s done through Study Mappng [8,9]. Conersely a dual ure on DUS otaned from the rotatons of the DUS represents a rgd ody moton n R [8]. The dual Euler parameters are used n defnng the sre transformaton n R [8], ut orgnally n ths paper these parameters are onstruted from the Rodrgues and the dual Rodrgues parameters [5]. Key Words- Knemats, Study Mappng, Dual Euler Parameters, Sre Transformaton.. INTRODUCTION The dual representaton of a lne s smply the Plüker etor rtten as a dual unt etor [9]. For any operaton defned on a real etor spae, there s a dual erson of t th smlar nterpretaton [5]. Olnde Rodrgues, the Frenh mathematan, rote a paper on rgd ody knemats n 84. Ths paper s ell knon for ts ontrutons to spheral knemats [7]. Rodrgues reealed that eery translaton an e represented n an nfnte numer of ays y omposton of to rotatons of equal ut opposte angle aout parallel axes [6]. Smlarly Euler shoed that eery dsplaement an e desred y a rotaton folloed y a translaton. There s a detaled surey rangng from Chasles moton to the Rodrgues parametrzaton and also from the theoretal deelopments of the rgd ody dsplaements to the fnte tst n Da []. Regardng the hstoral deelopments of the rgd ody dsplaement, the studes n ths feld are assoated th the fnte tst n the 99s. The fnte tst representaton and transformaton and ts ordered omnaton for seeral manpulators hh s ased on the Le group operaton are nestgated y Da, Holland and Kerr n 995[9].

2 A.E. Gürsoy and İ. Karakılıç 68 In our paper, the dual Euler parameters are used for defnng the transformaton of sres n R. The dual Euler parameters are onstruted from the Rodrgues and the dual Rodrgues parameters see [5] hh are otaned from the rotatons of the DUS. When a dual etor x ˆ s rotated to the dual etor x ˆ n DUS, ths moement orresponds to a sre transformaton n R. Ths transformaton an e gen y the dual Euler parameters. In other ords, ths paper dsusses the usage of dual Euler parameters for the transformatons of sres n R and these parameters are defned n terms of the Rodrgues and the dual Rodrgues parameters. Quaternons and dual numers ere omned and generalzed to form hat s referred as Clfford Algera as frst dsussed y Clfford n 88. Applaton to knemat analyss s dsussed y [],[]. A omprehense ntroduton to dual Quaternons an e found n [8]. Assemlng the Euler parameters,,, of a rotaton nto the quaternon Z j k j k, jk kj, k k j, j j k, rotatons n real spae an e dentfed. If a etor x x, x, x R s defned as the etor quaternon x x x j xk, then the rotaton from x to x s gen y the quaternon equaton x ZxZ, here the onjugate s defned as j k [8]. If the dual quaternon ˆ ˆ ˆ ˆ j k s gen y Z Z ˆ the dual Euler parameters ˆ, ˆ ˆ ˆ,, and the orrespondng spatal dsplaement s gen y the dual etor ˆ here and defne the angular and lnear elotes of the spatal dsplaement respetely then the transformed sre ˆ s otaned y ˆ Z ˆ ˆ Z ˆ [8]. Sne the transformed sre has the oordnates produed from the dual Rodrgues parameters, t has nformate oordnates aout the rotatons of the DUS. In seton, e ntrodue the dual numers and the Study mappng. The theoretal akground of the dual Euler parameters s deeloped n seton and the applaton of dual Euler parameters on the sre transformaton s dsussed y an example n seton... Dual numers A dual numer s a formal sum aˆ a a, here a and a are real numers. Smlar to the omplex unt, e hae here. Addton and multplaton are gen y a a a a a a. a a a For a gen real analyt funton f e an extend ts defnton to dual numers y lettng f x x k a x x k x k

3 68 Expresson of Dual Euler Parameters For nstane, k a x x k k + k kak x x x = f x x f x k Snxˆ Snx x Snx x Cosx.. Dual Vetors Cosx ˆ Cos x x Cosx x Snx e xˆ e x x A dual etor ˆ n three dmensonal dual spae here, R. The norm of ˆ, denoted y ˆ : D D s; ˆ ˆ ˆ [ ] e x D s defned y ˆ,, The dual etor th the norm, s alled a dual unt etor. Therefore a dual unt etor ˆ s the etor th and. The set of dual unt etors defnes the dual unt sphere DUS, hh s also alled the Study Sphere For detaled algera propertes of dual numers see also [8]... Study Mappng A pont p l p an e rtten as a etor, p, from orgn to l and a unt dreton etor g of l determne the equaton of the straght lne l n R. A unt fore th respet to the orgn atng to l ges the moment etor g p g. The norm of the moment etor s the smallest dstane from lne to the orgn [9]. 6 The ompenents of g, g g, g, g, g, g, g R are alled the Plüker oordnates of l. Sne g g and g g, the dual etor gˆ g g defnes a pont on DUS. The mappng hh assgns to an orented lne of Euldean spae the dual etor gˆ g g s alled the Study mappng..4. The Cayley Formula Performng the Cayley formula [8] for the dual spheral moton th the dual rotaton matrx  t s lear that  s orthogonal, e otan the ske symmetr dual matrx Bˆ and the dual Rodrgues etor ˆ see seton. In these omputatons, smlar to the real ase that s tan [8], the norm of dual

4 A.E. Gürsoy and İ. Karakılıç 68 ˆ Rodrgues etor s the tangent of the half of the rotaton angle ˆ, that s ˆ tan. Usng the algera of dual numers one an smply otan ˆ ˆ tan tan tan See[5]. THE DUAL EULER PARAMETERS AND THE SCREW TRANSFORMATION The dual Rodrgues etor ˆ s the axs of rotaton of DUS. Let us defne the dual unt etor s ˆ ˆ y sˆ ˆ s sˆ sˆ,, s s, s s, s s. Usng the dual rotaton ˆ angle ˆ and the dual unt etor s ˆ e get the dual parameters ˆ ˆ ˆ ˆ ˆ os, ˆ ˆ sn s, ˆ ˆ sn s, ˆ ˆ sn s, hh are knon as the dual Euler parameters [8]. Reeng the method of transformaton of etors gen n real spae, the smlar method for the dual ase an e proposed. As t s dsussed, the rotaton from x to x n R s gen y the quaternon equaton x ZxZ. A spatal dsplaement an e dentfed y a oordnate transformaton [T ] n terms of a rotaton matrx [A] and a dstane d, [ T] [ A, d]. Ths oordnate transformaton an e represented y a dual quaternon ˆ ˆ ˆ ˆ Zˆ os sˆ sn sˆ sn j sˆ sn k. The dual quaternon Ẑ s the sum of real Z and Z omponents. Z s the quaternon otaned from the rotaton matrx [A]. j k, here Z os, sn s, sn s, sn s are the Euler parameters of [A]. Z s the quaternon produed from Z DZ, here D s the quaternon, D d d j dk, relates to translaton etor d d, d, d. The ompenents of Ẑ are knon as the dual Euler parameters of the spatal dsplaement. Usng the dual Euler parameters, the dual orthogonal dual rotaton matrx [A] s generated y ˆ ˆ ˆ [ ˆ] sn os [ ˆ] sn [ ˆ A I S S ], see [8]. On the other hand from a gen dual rotaton matrx [A ˆ], the dual Euler parameters hene the dual quaternon Ẑ an e otaned.

5 684 Expresson of Dual Euler Parameters Orgnally n ths paper the dual Euler parameters and the dual quaternon Ẑ results from the Rodrgues stan, stan, stan and the dual Rodrgues parameters stan s Tan, stan s Tan, stan s Tan [see 5]. If a sre, here,, s the angular eloty and,, s the translaton eloty s defned y the dual quaternon ˆ j k then the fnal sre the transformed sre, ˆ j k s otaned y ˆ ZZ ˆ ˆ ˆ That s ˆ ˆ ˆ ˆ j ˆ k ˆˆ ˆ ˆ j ˆ k T A, d R O Let us expand the dual Euler parameters ĉ gen y as follos; ˆ ˆ ˆ ˆ sn sˆ sn sn os sn os Fgure. Sre transformaton sn

6 A.E. Gürsoy and İ. Karakılıç 685 sne tan, ˆ os os ot ot,,,. 4 From e get tan tan 5 Susttutng the equaton 5 nto 4 yelds ˆ os os sn,,,. 6 On the other hand, the expanson of ges, ˆ ˆ j ˆ k j k ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ { ˆ ˆ } { ˆ ˆ ˆ } j { ˆ ˆ } k 7 ˆ l l x ˆ s ˆ O ˆ x ˆ Study Mappng R DUS Fgure. The relaton eteen the rotaton of DUS and the Sre Transformaton O.. The Transformed Velotes Susttutng the dual Euler parameters ĉ n and ˆ ˆ, ˆ, n 6 nto 7 ges the transformed or the fnal angular,, and lnear,, elotes as,

7 Expresson of Dual Euler Parameters 686. os, os, os 8 os sn os sn os sn os, os sn os sn os sn os, 9 os sn os sn os sn os, here sn os,,,. It s seen from 8 and 9 that the transformed sre s omputed dretly from the dual rotaton angle ˆ, the Rodrgues parameters and the dual Rodrgues parameters.. APPLICATION OF DUAL EULER PARAMETERS TO THE SCREW TRANSFORMATION Theoretally the formulas 8 and 9 are otaned from the rotatons of the DUS. Let us examne 8 and 9 on an example y takng a dual rotaton matrx, that s an orthogonal dual matrx ˆ A on DUS and asre axs l n R.

8 A.E. Gürsoy and İ. Karakılıç 687 Let l e the sre axs passng through p,, th the dreton x,, and let, e the sre th angular eloty,, rad se and the translaton eloty,, m se at that moment. The dual etor ˆ,,, defnes ths sre. l has the moment etor x p x of a unt fore on l th respet to the orgn. Hene x,,. Therefore the Plüker oordnates of l, that s x, x,,,,,, defnes the pont xˆ x x,, on DUS. The effet of rotaton  on DUS auses the transformed sre axs l and the transformed sre ˆ, n R. xˆ T Ax ˆ ˆ  takes x ˆ to x ˆ hh orresponds to l n T R. Hene,. Then the Plüker oordnates of l are x, x,,,,,, here xˆ x x. Hene x,, s the unt dreton etor to l and x,, s moment etor of l determned for a unt fore on l th respet to the orgn. Let p denotes any pont on l. Sne x p x, y the x x etoral dson nerse operaton for the etor produt p x, here x s a real parameter. If e take then p,, on l. It s found that l s passng through p,, th the dreton etor x,,.on the other hand, y the Cayley mappng,

9 688 Expresson of Dual Euler Parameters ˆ ˆ ˆ B A I A I B ˆ, ˆ so the dual Rodrgues etor s ˆ,, [5]. Sne ˆ tan, e fnd the dual rotaton angle as, ˆ. Usng the formulas 8 and 9 the fnal angular eloty,, rad se and the translaton eloty 4 8 4,, m se are easly otaned. 8 4 l l ˆ,, s ˆ =,,,, =,, 8 4 x ˆ O ˆ x ˆ Study Mappng R =,, =,, DUS O Fgure. The applaton of a gen rotaton  to the Sre Transformaton. CONCLUDING REMARKS In the lassal method, the real part Z of the dual quaternon Zˆ Z Z s defned y Z os s sn s sn j s sn k, here s the rotaton angle and s s, s, s the rotaton axs of the spatal moton. The dual part Z s gen s y the formula Z DZ, here D d d j d k s the dual quaternon formed from the translaton etor d d, d, d. Therefore the transformed sre ˆ z ˆ ˆ zˆ s proposed from the translaton and the rotaron of the rgd ody n real spae. But n ths paper nstead of orkng on the real enttes of the spatal moton, the quantty Ẑ s estalshed usng the Rodrgues and the dual Rodrgues parameters of the one parameter moton on DUS hh orresponds to the gen spatal moton. The formulas 8 and 9 of the sre hae nformate oordnates aout the rotatons of the DUS. So for a

10 A.E. Gürsoy and İ. Karakılıç 689 gen orthogonal dual matrx  and a sre l one an easly ompute the fnal the transformed sre l y dretly usng Â. 4. ACKNOWLEDGEMENTS We ould lke to express our grattute to the reeer for hs aluale omments and gudane. 5. REFERENCES. W. Blashke, Knemats and Quaternon, VEB Verlag, Bel. Mat. Mon. Bd. 4, 96.. O. Bottema and B. Roth, Theoretal Knemats, North Holl. Pu. Co., Ams., D.P. Chealler, Le Algeras, modules, dual quaternons and algera methods n knemats, Mehansm and Mahne Theory, 6 6, 6-67, W.K. Clfford, Prelmnary Sketh of -quaternons, London Math. So., 4, F.M. Dmenterg, The Sre Calulus and ts Applatons n Mehans, Izdat Nauka, Moso, USSR, Eng. Trans.: AD6899, Clearnghouse for Federal and Sentf Informaton, Vrgna, A.P. Kotelnko, Sre Calulus and Some of ts Applatons to Geometry and Mehans, Annals of the Imperal Unersty of Kazan, Russa, J.B. Kupers, Quaternons and Rotaton Sequenes, Prneton Un. Pres, J.M. MCarthy, An Introduton to Theoretal Knemats, Cam., MIT Press, H. Pottmann and J. Wallner, Computatonal Lne Geometry, Sprnger,.. J.M. Selg, Geometr Fundamentals of Roots, Sprnger Sene, nd ed., 5.. E. Study, Geometry der Dynamen, Lezg, 9.. G.R. Veldkamp, On the Use of Dual Numers, Vetors and Matres n Instantaneous Spatal Knemats, Meh. and Mah. Theory, 4-56, A.T. Yang, Applaton of Quaternon Algera and Dual Numers to the Analyss of Spatal Mehansms, Columa Un., Ne York, No. 64-8, A.T. Yang and F. Freudensten, Applaton of Dual Numer Quaternon Algera to the Analyss of Spatal Mehansms, Jour. of App. Mehansms, pp., -8, İ. Karakılıç, The Dual Rodrgues Parameters, Internatonal Journal of Engneerng and Appled Senes,, -,. 6 T.R. Wllams and K.R. Fyfe, Rodrgues Spatal Knemats, Mehansm and Mahne Theory, 45, 5-,. 7 O. Rodrgues, Des los geometrque qu regssent les deplaements d un systeme solde dans l epae, Journal de Math. Pures et Applquees de Loulle 5, I.S. Fsher, Dual-Numer Methods n Knemats, Stats and Dynams, CRC Press LLC, J.S. Da, N. Holland, D.R. Kerr, Fnte tst mappng and ts applaton to planar seral manpulators th reolute jonts, Journal of Mehanal Engneerng Sene, Pro. Of IMehE, Part C 9C, 6-7, 995. J.S. Da, An hstoral ree of the theoretal deelopment of rgd ody dsplaements from Rodrgues parameters to the fnte tst, Mehansm and Mahne Theory, 4, 4-5, 6.