On Motion of Robot End-effector Using The Curvature Theory of Timelike Ruled Surfaces With Timelike Ruling

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1 O Moio of obo Ed-effecor Usig he Curvaure heory of imelike uled Surfaces Wih imelike ulig Cumali Ekici¹, Yasi Ülüürk¹, Musafa Dede¹ B. S. yuh² ¹ Eskişehir Osmagazi Uiversiy Deparme of Mahemaics, 6480-UKEY ² Divisio of Mechaical Egieerig, College of Egieerig Chobuk Naioal Uiversiy, Jeoju , Souh Korea bsrac ool rihedro was firsly defied for imelike ruled surface wih imelike rulig, rasiio relaios amog surface rihedro, ool rihedro, geeraor rihedro, aural rihedro, also Darboux vecors for each rihedro were foud. he differeial properies of robo ed-effecor's moio were obaied by usig he curvaure heory of imelike ruled surfaces wih imelike rulig. MS Mahemaics subjec Classificaio (000): 505, 57, 5B0 Key words phrases: uled surface, imelike surface, Darboux vecor curvaures, obo rajecory plaig, obo ed-effecor. Iroducio he mehods of robo rajecory corol currely used are based o PP (poi o poi) CP (coiuous pah) mehods. hese mehods are basically ierpolaio echiques, herefore, are approximaios of he real pah rajecory []. I such cases, whe a precise rajecory is eeded, or we eed o race a free formed or aalyical surface accuraely, he precisio is oly proporioal o he umber of iermediae daa pois for each-playback or offlie programmig. For accurae robo rajecory, he mos impora aspec is he coiuous represeaio of orieaio whereas he posiio represeaio is relaively easy. here are mehods such as homogeeous rasformaio, Quaerio, Euler gle represeaio o describe he orieaio of a body i a hree dimesioal space []. hese mehods are easy i cocep bu have high redudacy i parameers are discree represeaio i aure raher ha coiuous. herefore, a mehod based o he curvaure heory of a ruled surface has bee proposed as a aleraive [6]. uled surfaces were firsly ivesigaed by G. Moge who esablished he parial differeial equaio saisfied by all ruled surfaces. uled surfaces have bee widely applied i desigig cars, ships, maufacurig (e.g. CD/CM) of producs may oher areas such as moio aalysis simulaio of rigid body, model-based objec recogiio sysem. fer , Coos, Ferguso, Gordo, Bezier, ohers developed ew surface defiiios ha were ivesigaed i may areas. However, ruled surfaces are sill widely used i may areas i moder surface modellig sysems. he ruled surfaces are surfaces swep ou by a sraigh lie movig alog a curve he sudy of ruled surfaces is a ieresig research area i he heory of surfaces i Euclidea geomery. heory of ruled surfaces is developed by boh usig surface heory E. Sudy

2 map which eables oe o ivesigae he geomery of ruled surfaces by meas of oe real parameer. I is also kow ha heory of ruled surfaces is applicable o heoreical kiemaics. Mikowski space is amed for he Germa mahemaicia Herma Mikowski, who aroud 907 realized ha he heory of special relaiviy (previously developed by Eisei) could be elegaly described usig a four-dimesioal space-ime which combies he dimesio of ime wih he hree dimesios of space. Four dimesioal Euclidea space wih a Lorezia meric, which is he space of special relaiviy, is called Mikowski space-ime, a Lorez space is a hypersurface of Mikowski space-ime. here are los of papers dealig wih ruled surfaces i Lorez space (cf. [ ]). Prelimiaries Le be a -dimesioal Lorezia space wih Lorez meric ds² dx²+dx ² dx ². If <X,Y> 0, X Y are called perpedicular i he sese of Lorez, where <, > is he iduced ier produc i. he orm of X is, as usual, X < X, X>. Le X be a vecor. If < X,X> <0, he X is called imelike; if <X,X> >0 X 0, he X is called spacelike if <X,X> 0, X 0, he X is called lighlike (ull) vecor []. We ca observe ha a imelike curve correspods o he pah of a observer movig a less ha he speed of ligh while he spacelike curves faser he ull curves are equal of he speed of ligh. smooh regular curve α : I is said o be a imelike, spacelike or lighlike curve if he velociy vecor α (s) is a imelike, spacelike, or lighlike vecor, respecively. surface i he Mikowski -space is called a imelike surface if he iduced meric o he surface is a Lorez meric, i.e., he ormal o he surface is a spacelike vecor. imelike ruled surface i is obaied by a imelike sraigh lie movig alog a spacelike curve or by a spacelike sraigh lie movig alog a imelike curve. he imelike ruled surface M is give by he paramerizaio ϕ :I, ϕ (s,υ) α () s +υx() () i (cf. [-4-8-9]). he arc-legh of a spacelike curve α, measured from α ( s0), s 0 I is s() α '( u) du () 0 he parameer, s, is deermied such ha α '( s), where α '( s) ((dα)/(ds)). Le us deoe V₁ α call V₁(s) a ui age vecor of α a he poi s. We defie he curvaure by k (s) α''( s), α''( s).

3 If k₁(s) 0, he he ui pricipal ormal vecor, V (s), of a imelike curve α a he poi s is give by α (s) k₁(s)v (s). For ay X (x₁,x,x ), Y (y₁,y,y ), he Lorezia vecor produc of X Y is defied as X Y (x y x y, x y x y, x y x y ) he ui vecor V (s) V₁(s) V (s) is called a ui biormal vecor of imelike curve α a he poi s. Now, we give some properies of wihou proof: Le, B, C, i is sraighforward o see he followig. B -B ; < B, > < B, B> 0; or B is imelike B is spacelike; < B, C> <B C, >; B are spacelike B is imelike; < B, B> <,B> <,><B,B> B 0 B are liearly depede [-4]. Le α be a imelike curve i le us deoe V₁ α. he we ca ge Free formulas V ₁(s) k₁(s)v (s) V (s) k₁(s)v₁(s)+k (s)v (s) V (s) k (s)v (s) where k (s) is he orsio of a imelike curve α a poi s. surface i he -dimesioal Lorez space is called a imelike surface if he iduced meric o he surface is a Lorez meric, i.e., he ormal o he surface is a spacelike vecor []. imelike curve α α(s), parameerized by aural paramerizaio, is a frame field {e, e, e }, havig he followig properies <e₁,e₁>, < e,e > < e,e > <e₁,e > <e₁,e > < e,e > 0, e₁ e e e₁ e e e e e₁. he ifiiesimal displaceme of he frame is give as α (s) e₁, e₁ (s) κ(s) e e (s) κ(s)e₁+τ(s) e, e (s) τ(s) e where e₁(s) is he age imelike vecor field, e (s) is he pricipal ormal vecor field e (s) is he biormal vecor field. he fucios κ(s) τ(s) are he curvaure orsio of he curve, α(s), respecively. Lorezia vecor produc u v of u v is defied as e e e u v u u u v v v

4 Le u v be vecors i he Mikowski -space. (i) If u v are fuure poiig (or pas poiig) imelike vecors, he u v is a spacelike vecor. <u,v> u v coshθ u v u v sihθ where θ is he hyperbolic agle bewee u v. (ii) If u v are spacelike vecors saisfyig he iequaliy <u,v> < u v, he u v is imelike, <u,v> u v cosθ u v u v sihθ where θ is he agle bewee u v [0]. epreseaio of obo rajecory by a uled Surface Pah of a robo may be represeed by a ool ceer poi ool frame of ed-effecor. I Figure, he ool frame is represeed by hree muually perpedicular ui vecors [O,, N], defied as O; orieaio vecor (imelike), ; approach vecor (spacelike), N; ormal vecor (spacelike), as show i Figure. I his paper he ruled surface geeraed by O is chose for furher aalysis wihou loss of geeraliy. he pah of ool ceer poi is referred o as direcrix vecor O, rulig. he direcrix rulig represe five parameers for he 6 degrees of freedom spaial moio. he fial parameer is he spi agle, η, which represes he roaio from he surface ormal vecor, S, abou O. Pah of CP (Direcrix), α ulig of uled Surface, r O CP O N O η S N S η S η N Frames of eferece obo Ed-effecor Figure. uled surface geeraed by O of ool Frame Each vecor of ool frame i ed-effecor defies is ow imelike ruled surface while robo moves. he pah of ool ceer poi is direcrix O is he rulig. s α(s) is a spacelike curve (s) is imelike sraigh lie, le us ake he followig imelike ruled surface X(s,v) α(s)+v (s) () he ormalized parameer s, defied as d( ψ ) s( ψ ) dψ. dψ 4

5 α O, r Surface Frame S µ β φ S b b Specified φ N Pah r µ Geeraor rihedro k ρ Naural rihedro Figure. Frames of referece o describe he orieaio of ool frame relaive o he imelike ruled surface, we defie a surface frame a he CP as show i Figures. he surface frame, [0, S, S b ], may be deermied as follows. X v X s S (4) X X v is he ui ormal of imelike ruled surface i CP. s S b O S (5) is he ui biormal vecor of he surface Geeraor rihedro i Figure is used o sudy he posiioal agular variaio of imelike ruled surface. Le us ake r( ) imelike geeraor vecor spacelike ceral ormal vecor k r spacelike age vecor. he sricio curve of imelike ruled surface is where he parameer β(s) α(s) µ(s) (s) (6) µ(s) <α (s), (s)> (7) idicaes he posiio of he CP relaive o he sricio poi of he imelike ruled surface. Sice he rulig is o ecessarily a ui vecor, he disace from he sricio poi of ruled surface o he CP is µ i he posiive direcio of he geeraor vecor. he geeraor 5

6 rihedro he sricio curve of he ruled surface are uique i he sese ha hey do o deped o he choice of he direcrix of he ruled surface. herefore, from a sudy of moio of geeraor rihedro he sricio curve we ca obai he differeial moio of he edeffecor i a simple sysemaic maer. he firs-order agular variaio of he geeraor rihedro may be expressed i he marix form as where γ is defied as r 0 0 r r d 0 γ r ds U (8) k 0 γ 0 k k γ <, > γ Ur r k (9) is he Darboux vecor of he geeraor rihedro. Differeiaig Equaio (6) gives firs order posiioal variaio of he sricio poi of he imelike ruled surface expressed i he geeraor rihedro, differeiaig Equaios (7), (8) wih he aid of geeraor rihedro, β '( s) rγ+ k (0) where Γ α '( s), ( s) µ '( s) α '( s), ( s) '( s). () Ceral Normal Surface he aural rihedro used o sudy he agular posiioal variaio of he ormalia is defied by hree orhoormal vecors. ; (spacelike) he ceral ormal vecor, ; (spacelike) pricipal ormal vecor, b ; (imelike) biormal vecor, as show i Figure. s he geeraor rihedro moves alog he sricio curve, he ceral ormal vecor geeraes aoher ruled surface called he ormalia or ceral ormal surface. he ormalia, impora i he sudy of he higher order properies of he ruled surface, is defied as X (,) sv β () s + vs () () Le he hyperbolic agle, ρ, be bewee he imelike vecors r b. he, we have 6

7 r sihρ.+coshρ.b () k r sih ρ. b+ cosh ρ.. So Equaio () ca be wrie i marix form as r 0 sihρ coshρ 0 0. (4) k 0 coshρ sihρ b he soluio of Equaio (4) is 0 0 r sihρ 0 coshρ. (5) b coshρ 0 sihρ k Subsiuig Equaio (8) io (5) usig κ, i follows ha Hece ' ( r γ k) κ( sihρ.r+coshρ.k). γ cosh ρ κ sih ρ. κ he he geodesic curvaure may also be wrie as coh ρ γ. Subsiuig his io Equaio (8) we ge r 0 0 r d 0 coh ρ ds (6) k 0 cohρ 0 k from followig equaliy U (coh. ) r γ r k ρ r k. (7) he aural rihedro cosiss of he followig vecors 7

8 ' ' κ b (8) where κ is he curvaure. he origi of he aural rihedro is sricio poi of he ormalia. he sricio curve is defied as where β () s β () s µ ( S)() s (9) β '( s), '( s) µ () s (0) '( s), '( s) is he disace from he sricio poi of he ormalia o he sricio poi of he imelike ruled surface i he posiive direcio of he ceral ormal vecor. Subsiuig Equaios (6) (0) io Equaio (0), we obai µ ρ ρ () s cosh.( Γ+ coh ) he firs-order agular variaio of aural rihedro may be expressed i he marix form as 0 κ 0 d κ 0 τ ds U. () b 0 τ 0 b b where τ < ', b> is orsio. s i he case of he geeraor rihedro, Equaio () may also be wrie as U τ κb () is referred o as he Darboux vecor of he aural rihedro. Observe ha he Darboux vecor of he geeraor rihedro he Darboux vecor of he aural rihedro, describe he agular moio of he ruled surface he ceral ormal surface. he curvaure κ is defied by (6) as follows: κ cosh. () sih ρ + ρ Differeiaig Equaio (9) subsiuig Equaios (8) (0) io he resul, we obai where β Γ + b (4) ' Γ η ' Γ cosh ρ + sih ρ. (5) 8

9 he four fucios, give by Equaios (5) (7), characerize he ormalia i he same way as Equaios (8) (0) characerize he imelike ruled surface. elaioship bewee he Frames of eferece he orieaio of he surface frame relaive o he ool frame he geeraor rihedro is show i Figure. Le agle bewee he spacelike vecors S b be defied by ϕ, referred as spi agle, we have <S b, > cosϕ siϕ S +cosϕ S b O N siϕ S b cosϕ S. We may express he resuls i marix form as O 0 0 O 0 siϕ cosϕ S N 0 cosϕ siϕ S b (6) Equaio (6) may also be rewrie as O 0 0 O S 0 siϕ cosϕ. (7) S b 0 cosϕ siϕ N Le he agle bewee he spacelike vecors S b k be defied as φ. We have S S b si φ. + cos φ. k cos φ. si φ. k. We may express he resuls i marix form as O 0 0 r S 0 cosφ siφ. (8) S b 0 siφ cosφ k Wih he aid of Equaios (6) (8), we have O 0 0 r 0 si cos, (9) N 0 cos si k where ϕ +φ Σ. he soluio of Equaio (9) is 9

10 r 0 0 O 0 si cos (0) k 0 cos si N where Σ, referred as spi agle, describes he orieaio of he ed-effecor. Subsiuig he parial derivaives of Equaio () io (4) S x x u u x x s s xu x α ' + v '. s Because he surface ormal vecor is deermied a he CP which is o he direcrix, v is zero. xu xs α ' subsiuig α ' β ' + µ ' + µ ' wih he aid of Equaio (0), x x µ k u s x x µ u s +. Fially S S b µ k () + µ k+ µ. () + µ Comparig Equaio () wih (8), we observe ha µ cosφ siφ () + µ + µ Subsiuig Equaios (5) (5) io Equaio (7) gives Ur (coh ρ. r k ) κb, (4) which shows ha he biormal vecor plays he role of he isaaeous axis of roaio for he geeraor rihedro. Differeial Moio of he ool Frame I his secio, we obai expressios for he firs secod-order posiioal variaio of he CP. he space curve geeraed by CP from Equaio (6) is 0

11 α() s β () s + µ () s. (5) Differeiaig Equaio (5) wih respec o he arc legh, usig Equaio (0), he firs order posiioal variaio of he CP, expressed i he geeraor rihedro is α '( s) r( Γ+ µ ' ) + k+µ. (6) Subsiuig Equaio (0) io Equaio (6), i gives α '( s) ( Γ+µ' )O+(µsi Σ+ cos Σ)+( µcos Σ+ si Σ )N (7) µ µ α ''( s) ( '+µ'' + )r+( Γ Γ +µ'+ coh σ)+( ' coh σ)k. (8) Wih he aid of Equaio (0) gives µ Γ µ α''( s) ( Γ'+µ'' + )O+[si Σ( +µ'+ coh σ)+cos Σ( '- coh σ)] Γ µ +[-cos Σ( +µ'+ coh σ)+si Σ( '- coh σ)]n (9) Differeiaig Equaio (9) subsiuig Equaio (6) io he resul o deermie he firs order agular variaio of he ool frame subsiuig Equaio (0) io resul gives where hus, O 0 si cos O O si 0 o s Ω U, N cos Ω 0 N N (40) cohρ Ω + '. (4) Uo Ω. O cos. si. N (4) is he Darboux vecor of he ool frame. Subsiuig Equaio (0) io Equaio (4) gives Uo Ωr k. (4) wih he aid of Equaio (4) we have U 0 Σ' r + cohσ r k usig Equaio (7), i becomes

12 U o Σ ' r+u r Subsiuig Equaio (4) io he resul, i gives Uo Σ ' r+ κb. (44) he secod order agular variaio of he frames may ow be obaied by differeiaig he Darboux vecors. Differeiaig Equaio () gives U ' κ' τ'b. Differeiaig Equaio (4) gives U ' r κ'b κb' wih he aid of Equaio () he firs order derivaives of he geeraor rihedro may be wrie as U' r κ'b κτ. Differeiaig Equaio (4) gives U ' o Ω'r+ Ωr'b κ ' Wih he aid of Equaio (8), i is rewrie as U ' o Ω'r+( Ω cohσ ) or he firs order derivaives of he ool frame may be wrie as U o Σ' Ω'r+. (45) From he chai rule, he liear velociy he liear acceleraio of he CP, respecively, are V α S a α S+α S² (46) lso, he agular acceleraio of he ed-effecor, respecively, are Example w i U o S i ii i w U o S+ U' S For he ruled surface, show i Figure,. (47)

13 ϕ(s,v) ( cos s v.si s, si s+ vcos s, s+v) i is easy o see ha α (s) ( cos s, si s,s) is he base curve (spacelike) (s) ( si s, cos s, ) is he geeraor (imelike). he sricio curve is he base curve. his surface is a imelike ruled surface. Differeiaig α(s) gives α'(s) ( si s, cos s, ) where < α'(s), α'(s)> si²s+cos²s, so α (s) is spacelike vecor. Figure. he imelike ruled surface wih imelike rulig <,> 8si²s+8cos²s 9, herefore. Hece, geeraor rihedro is defied as r ( si s, cos s,) ' ( cos s, si s,0) where <, > si²s+cos²s. lso, k r e e e k de cos s si s 0 ( si s, cos s, ) si s cos s

14 where < kk, > > 0. So k is spacelike vecor. herefore Γ < α '(s),(s)> µ '( s), where µ < α','>0. So µ '( s) 0 usig his resul, i becomes Naural rihedro defied as where κ < ', '>, Γ < α', k> γ < ', k >. ( cos s, si s,0) ' (si s, cos s,0) ' (si s, cos s,0) κ <, > cos² s+si² s. So is a spacelike vecor. Hece b b e e e de cos s si s 0 (0,0,). si s cos s 0 he Darboux vecor of geeraor rihedro is, U r (0,0, ). sice τ < b ', >0, he Darboux vecor of aural rihedro is U(0,0, ). Subsiuig Equaio io Equaio (), we have So cos φ + µ² si φ µ a φ. Differeiaig equaio gives µ. +µ² µ' (+a φφ ) '. hus µ' φ '. +µ² 4

15 Sice spi agle is zero ϕ 0 so ϕ ' 0, Σ φ, Σ ' φ ', Ω φ ' +. he approach vecor he ormal vecor, respecively, are (µcos s si s,µsi s+ cos s,8) 6+µ² N ( cos s µsi s, si s+µcos s, µ). 6+µ² Firs order posiioal variaio of he CP may be expressed i he ool frame as 4µ+6 8µ α'(5+µ')o+ + N 6+µ² 6+µ² he Darboux vecor of he ool frame, is 4 µ U 0 ΩO+ + N Ωr k. 6+µ² 6+µ² he firs order derivaive of he Darboux vecor of he ool frame U' φ'' r+ φ. 0 his paper has preseed he sudy of he moio of a robo ed-effecor based o he curvaure heory of imelike ruled surfaces. efereces: [] B. S. yuh, G.. Peock, ccurae moio of obo Ed-Effecor usig he curvaure heory of ruled surfaces, J. Of Mechaisms, rasmissios, uomaio i Desig, Vol. 0, 8-87, (988). [] B. O'Neill, Semi-iemaia Geomery wih pplicaios o elaiviy, cademic Press Ic., Lodo, (98). []. urgu, H.H. Hacısalihoğlu, ime-like uled Surfaces i he Mikowski -Space. Far Eas J. Mah. Sci. Vol. 5 No. (997), [4]. urgu, H.H. Hacısalihoğlu, imelike uled surfaces i he Mikowski -space-ii. r. J. of Mahemaics, V., -46, (998). [5] J. Ioguchi, imelike surfaces of cosa mea curvaure i Mikowski -space, okyo J. Mah., No., 4-5 (998). [6] B. S. yuh, obo rajecory Plaig usig he curvaure heory of ruled surfaces, Docoral disserio, Purdue Uiversy, Wes Lafayee, Idiaa, 4 (989). 5

16 [7] M. Özdemir,.. Ergi, oaios wih ui imelike quaerios i Mikowski - space, J. of Geomery Physics, Volume 56, Issue, -6, (006). [8]. O. Öğremiş, H. Balgeir, M. Ergü, O he ruled surfaces i Mikowski -space, J Zhejiag Uiv. SCIENCE, Vol. 7(), 6-9, (006). [9] V. D. I. Woesije, Miimal surface of he -dimesioal Mikowski space, World Scieific Publishig Sigapore, 4-69, (990). [0] G., S. Birma, K., Nomizu, rigoomery I Lorezia Geomery, epried From he me. Mah. Vol 9, No.9 November (984). [].C. Çöke, N. yyildiz, Differeial Geomeric Coidiios Bewee Geodesic Curve imelike uled Surfaces i he Semi-Euclidea Space E₁³, Joural Naurel Scieces Mahemaics, Vol.4, No:, 7-5, (00). [] F. L. Livi X. C. Gao, "alyical epreseaio of rajecory of Maipulaors," reds Developmes i Mechaisms, Machies, oboics, he 988 SME Desig echology Cofereces-0h Bieial Mechaisms Coferece, Vol. 5-, 988, pp. 48~485. []. P. Paul, "Maipulaor Caresia Pah Corol," IEEE ras. Sysems, Ma., Cybereics, Vol. SMC-9, No., 979, pp. 70~7. 6