Spinor Geometry Based Robots Spatial Rotations Terminal Control

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1 Mthemtcl Models nd Methods n Moden Scence Spno Geomet Bsed Robots Sptl Rottons Temnl Contol A Mlnkov T. Ntshvl Deptment o Robotcs Insttute o Mechncs o Mchnes Mndel st. Tbls 86 Geog lende.mlnkov@gm.com t_ntshvl@hoo.com Abstct: In the tcle usng spno epesentton o othogonl tnsomtons the epessons between second ode comple unt tnsomtons mtes nd el othogonl mtes o sptl ottons n thee dmensonl Euclden spce L e eceved tht llows esl clcultng o coespondng Eule s ngles. The obtned esults hve enbled educng the ctull thee-dmensonl poblem o sptl moton contol to the onedmensonl poblem; contol knemtcl unctons o Eule s ngles nd contol spno mt o otton wee constucted b mens o whch contol pocess o sptl ottons s completel detemned. Kewods: contol spno mt hemtn unctonls; sptl ottons; temnl contol. Poblem omulton Methods o epesentton o thee-dmensonl ottons used n solvng o vous engneeng poblems e usull conned to the descpton o ndvdul concete ottons centeed t the ogn zeo cente. Among these methods s n ptcul the well known method o othogonl el mtces whose elements e unctons o Eule ngles [ ]. At the sme tme t should be sd tht the poblem o descbng so-clled genelzed ottons [] evokes much gete nteest both om the theoetcl stndpont nd om the stndpont o pplctons n the st plce we men n pplcton n obotcs nd n ptcul n the plnnng o tjectoes n the cse o obstcles. Unde genelzed ottons we men the set o ll possble ottons wth both zeo nd nonzeo centes whch tnsom the ntl theedmensonl pont to the nte one. The bsc poblem sng n ths contet cn be omulted s ollows: Gven two thee-dmensonl ponts nd t s equed to dene the set o ll possble tnsomtons nd centes o ottons whch bng bout the tnsomton o the pont to the pont. It s obvous tht ths poblem cn be esl etended to the cse whee nsted o two ponts we consde two nte sets o ponts { } nd { } m whch coesponds to the cse o ottons o sold.. Poblem soluton.. Spnol Repesentton o sptl ottons Let L be lne Euclden spce wth othonomlzed bss vectos e e e. To ech vecto e e e o the spce L we ssgn tceless Hemtn mt X whose elements e the so-clled spno components o the vecto. When we pss om the usul Euclden components o the vecto to the spno ones we theeb dent the vecto wth Hemtn unctonls on the two-dmensonl lne spce С ove the eld o comple numbes С [4]. Denote b LС the set o ll Hemtn unctonls on С nd consde the ollowng decomposton X σ σ σ whee σ σ σ e Pul mtces [4]. ISBN:

2 Mthemtcl Models nd Methods n Moden Scence Fom decomposton t ollows tht the set LС s lne thee-dmensonl spce ove the eld o el numbes nd thus t cn be dented wth L. Note tht to ech bss vecto o the two-dmensonl spce C we cn ssgn the bss elements σ σ σ o the spce LС nd lso the othonomlzed bss vecto e e e due to the dentcton o L nd LС : ech o the mtces σ s epesented s some lne combnton o tenso poducts o bss vectos o the spce C [4]. The oegong esonng mples tht o n mt С С whch s mt o tnsomton between two bss vectos o the spce C thee lso ests tnsomton mt o the coespondng othonomlzed bss vectos n the spce L. Poposton The mt o tnsomton o the bss elements n C s unt. Poo. I on the spce C we consde Hemtn unctonls o the om X o then the wll coespond to the ou-dmensonl Fom 5 we hve vectos o pseudo-euclden spce wth sgntue nd wth bss vectos Re 6 σ ; σ σ σ. nd Now tnsomtons o the bss vectos o the twodmensonl spce C led to tnsomtons o the bss vectos whle the tnsomton mtces emn the sme s n the cse o unctonls o om. The othogonl complement σ o the st bss vecto σ s n nt-euclden spce becuse o the pseudo-euclden popet o the spce dened b vectos nd te chngng the sgns o the scl poducts thee-dmensonl Euclden spce tht concdes wth LС. The nowng o the cton o mtces o bss vecto tnsomton n С to the subspce σ mens tht these mtces sts the condton C T T σ C σ.e. C C Q.E.D. The poblem posed n Subsecton cn be now eomulted n tems o the spno spce С : Gven two tceless mtces o Hemtn unctonls X и Y t s equed to dene: set o unt mtces C whch sts the eqult T Y C XC ; 4 one-dmensonl subspces whch e nvnt wth espect to tnsomtons epesented b mtces C.e. set o espectve otton centes. Note tht snce the tnsomton C s unt the vecto noms dened b the detemnnts o mtces o the Hemtn unctonls X nd Y concde nd theeoe 4 denes otton. Fom eqult 4 we cn obtn the ollowng sstem o lne homogeneous equtons wth espect to the unknown vbles nd :. 5 Fo bt soluton o 5 s gven b.5 Im. 7 Usng the one o the popetes o untt o the mt С det C we cn dene ethe Re o Im. Note tht one o these pmetes emns bt. Thus 6 denes otton o nd. The nvnce o the otton cenete zz z z wth espect to the tnsomton C s wtten s condton C T ZC Z whence we obtn z z ; z z ; z z. It s not dcult to ve tht the detemnnt o ths sstem consdeed o the unknown vlues z z nd z s dentcll zeo nd theeoe o gven nd nd thee lws est nontvl solutons wtten n the om ISBN:

3 Mthemtcl Models nd Methods n Moden Scence ; z z whee z s bt. 8 z z Thus 7 togethe wth the nomlzton popet dene genelzed otton tnsomng to wth espect to the set o centes whch s dened b 8... Reltons Between Tnsomtons n C nd L We cn estblsh the coespondence between the elements o the tnsomton mt C n C nd the elements o the othogonl el mt o otton A n L. ; ;. Epessons enble to clculte the elements o the mt A though the gven coodntes o thee ponts ntl temnl nd the cente whch dene otton. On the othe hnd tkng nto ccount tht the mt A o bsc epesentton cn be wtten n the om [] A ϕ ψ θ snϕ snψ ϕ snψ θ snϕ snψ snϕ ψ θ ϕ snψ snϕ snψ θ ϕ ψ ϕ snθ snψ snθ ψ snθ snϕ snθ θ whee π<ϕ π θ π и π<ψ π e Eule ngles t esl ollows tht epessons enble to dene Eule ngles s well θ ; snϕ snθ nd snψ snθ The mt A s b denton the mt o tnsomton between two othonomlzed bss.. Contol o Sptl Rottons Hvng epessons 7 nd t s es to vectos o the spce L nd ts ows e clculte the Eule ngles whch ensue otton o decompostons o the new bss vectos n tems o the ntl bss vectos. Hence due to the the pont to the pont. I t s ssumed tht to the ntl pont thee dentcton o the spces LС nd L we hve coespond the zeo Eule ngles θ φ ψ T C σ C σ 9 then the contol o otton conssts n mkng 9 tmedependent chnge o the Eule ngles om the ntl whee σ e the Pul mtces coespondng to the vlues θ ; φ; ψ to the temnl vlues ntl bss σ e the Pul mtces o the new θ ; φ ; ψ clculted b omuls. bss nd e the elements o the mt A -. Fomul cn be wtten eplctl n the om o In genel om the contol pocess cn be epesented s chnge unctons o the Eule ollowng thee ngles θ t ; φ t ; ψ t whch must sts the equltes condtons * * θ t ; φ t ; ψ t θ t θ ; φ t φ ; ψ t ψ * *. whee t nd t e the ntl nd temnl * * moments o tme. The bove-sd ntull mples the poblem on whch edl eld the ollowng epessons o denng the contol unctons θ t ; φ t ; ψ t. clcultng the elements o the mt A b the It should be emphszed tht dependences θ t ; elements o the mt C: φ t ; ψ t hve knemtcs chcte snce the ; tke nto ccount nethe moments no elstctes ; ; no n othe dnmc chctestcs o the pocess ; nd theeoe te denng them thee ses ; ; poblem o sntheszng on the bss o these ISBN:

4 unctons the dnmc dptve contol. Ths ssue s dscussed n [4]. Wth ntl nd nl stte vectos nd coespondngl thee s lso the ntemedte ottng vecto whch t the ntl moment o tme t t concdes wth the ntl otton vecto nd t the temnl moment o tme t t wth the temnl vecto. The movng ngle between the vectos nd s equl t the ntl moment o tme t t to zeo nd t the moment o tme t t to whee * nd s the dot poduct o the vectos nd. It s obvous tht the movng ngle between the vectos nd s equl to. Let us dene the coodntes o the vecto ssumng tht t oms the ngles nd wth the vectos nd nd s locted n the plne. To ths end we ntoduce the vecto whch s the s poduct o the vectos nd. Then the bove condtons cn be wtten n the om o the ollowng sstem o lne equtons: ; ;. It s not dcult to see tht the vecto dened om sstem stses the ollowng condtons:. o whch ollows om the second equton o sstem snce n ths cse * snce dung otton. whch s possble onl povded tht ;. o whch ollows om the thd condton o sstem snce n ths cse whch s possble onl povded tht ;. whch ollows om the second nd thd equtons o sstem. Theeoe the vecto dened om sstem coesponds to Fg..e. t cn ctull be egded s the vecto ottng condton om the vecto condton to the vecto condton. Note tht n ths cse the ngle chnges n wthn. The equtons o sstem cn be wtten n the coodnte om s ollows:. It s not dcult to see tht ts detemnnt s equl to. Othe detemnnts o Kme's omuls o sstem wll be equl to ; ;. Mthemtcl Models nd Methods n Moden Scence ISBN:

5 Mthemtcl Models nd Methods n Moden Scence I we ntoduce the new vectos ; ; nd ; ; whch equl to the vecto poducts [ ] nd [ ] espectvel then the coodntes o the ottng vecto wll be pesented n the ollowng om: ω ω t t t In these epessons the ngle s n ndependent vble nd cn be teted s tme uncton whch mens tht the coodntes o the vecto e lso tme unctons. We would lke to emphsze tht the poblem o snthess o sptl moton contol thus educes to denng uncton t o the concete om whch s connected wth the otton pocess dnmcs nd ws dscussed n [4]. Hee we ssume tht t s sucentl smooth nd stses the condtons t t и t t. Fo denteness we ssume tht t ωt whee ω π s the constnt ngul veloct. As hs led been noted the vecto s ottng vecto nd theeoe t ech moment o tme t cn be consdeed s temnl vecto o the cuent moment o the otton pocess. I n omuls 7 we eplce the coodntes o the pont b epessons 4 then we obtn epesenttons o the pmetes o the spno mt С the othogonl mt А omuls nd Eule ngles n tems o tme unctons. Thus we obtn tme-dependent knemtcs epesentton o the otton o the pont to the pont. Howeve st we should pedetemne the mt С so tht t the ntl moment o tme the spno equton o otton 4 would hve the om T X C XC whch s evdentl possble onl C s unt mt. Ths cn be done b n ppopte choce o the pmetes nd. Indeed settng ; we obtn Re ; Im. B substtutng the lst omuls nd 4 o coespondng elements o o C t t t t t t t 4 whee t t. t Note tht detct o n t. It s obvous tht t the ntl moment o tme t the mtc t t o snce n tht cse t nd ; ;. Fo t t we hve t ; ;. Fom the bove-sd t ollows tht the obtned spno mt o otton 4 s dened coectl. But n tht cse the Eule ngles too e dened coectl. The lso tun out to be the unctons o tme t t t θ t c ; t t ϕ t csn : tsnθ t t ψ t csn. 5 tsnθ t Epessons 5 solve the poblem we hve omulted on denng the knemtcs unctons θ t ; φ t ; ψ t.. On the othe hnd t should be noted tht the poposed theo llows one to educe n ctull thee-dmensonl poblem o sptl moton contol to one-dmensonl poblem. Indeed o ths t s sucent to snthesze n one w o nothe uncton t stsng the coespondng bound condtons [5]. Then t s obvous tht the contol pocess s completel dened b the spno mt o otton 4 nd the Eule ngle unctons 5.. Concluson Let us consde numecl emple llusttng nd concludng the bove esonng. Assume tht the ntl vecto 45 nd the temnl vecto 5.5 e gven btl. The ngle between them s equl to ISBN:

6 Mthemtcl Models nd Methods n Moden Scence Assumng o the ske o smplct tht ω wth step equl to lets us clculte n two deent ws thee ntemedte postons o the ottng vecto ; ;. Usng omuls 4 we obtn the ollowng coodntes o the ottng vecto o thee ngle vlues: nd ngle 5.88 ; nd ngle 5.77 nd ngle The pocedue o veng whethe the Eule ngles hve been clculted conssts n the ollowng: usng the obtned coodntes o the ntemedte postons o the vecto ; ; o ech o thee ngle vlues gven bove we should clculte the Eule ngles b omuls 6 nd the thee-dmensonl othogonl mt A o the bsc epesentton nd then gn the ntemedte coodntes o the ottng vecto b the omul A whee s the ntl otton vecto. The obtned vlues should concde n both cses. The esults o the coespondng clcultons e pesented n Tble. The mt A ws clculted o the Eule thee ngle vlues b omuls 5. The coodnte vlues o the ottng vecto ; ; wee clculted b multplng mt A b the ntl otton vecto 45: A. Fom Tble we see tht the coodntes o the ottng vecto concde wth the coodntes clculted b omuls. Angle Coodntes o ottng vecto omuls Tble No m o Reeences: [] K. S. Fu. R.C. Gonzlez nd K Le. Robotcs: Contol Sensng Vson nd Intellgence McGw- Hll Book Compn 987 [] Y.Tokung T.Hkukw T.Inoue Algothm nd Desgn o n Robot J.o Robotcs nd MechtoncsVol.No. 999 pp.7-8. [] Mlnkov A.A. Pngshvl A.I. Rodon I.D. A Spno Model o -Dmensonl Genelzed Rottons Automton nd Remote Contol 5 6 p. -7 [4] Mlnkov А.А.. New Pocedue o Robots Sptl Rottons Temnl Contol. th Intentonl Coneence on Appled Mechncs nd Mechncl Engneeng. Co Egpt 8 [5] I. M. Gelund R.A. Mnlos nd Z.Y. Shpo Repesenttons o the Goup o Rottons nd the Loenz Goup. Fzmtgz Moscow 958 n Russn. ISBN: