Nemaa D. Zorc Research Assstat Uversty of Belgrade Faculty of Mechacal Egeerg Mhalo P. Lazarevc Full Professor Uversty of Belgrade Faculty of Mechacal Egeerg Alesadar M. Smoovc Assstat Professor Uversty of Belgrade Faculty of Mechacal Egeerg Mult-Body Kematcs ad Dyamcs Terms of Quateros: Lagrage Formulato Covarat Form Rodrguez Approach Ths paper suggests a uatero approach for the modellg ematcs ad dyamcs of rgd mult-body systems. Istead of the regular Newto-Euler ad Lagrage method used the tradtoal way, Lagrage s euatos of secod d the covarat form are used by applyg Rodrguez approach ad uatero algebra. A model of mult-body system of rgd bodes terms of uateros s obtaed, whch s useful for studyg ematcs, dyamcs as well as for research of cotrol system desgs. Keywords: uateros, ato, rgd bodes system.. INTRODUCTION I geeral, modellg of ematcs ad dyamcs of rgd bodes systems has bee mostly based o the Euler agles represetato of ato. It s well ow that three agles ca t afford a regular represetato of the ato, sce there are sgulartes. Euler proposed a soluto to crcumvet ths problem by troducg a set of four uattes, the so-called Euler parameters, based o relatos amog Euler agles. Later o, Hamlto (844 veted the uateros, a exteso of complex umbers, ad soo afterwards, t was dscovered that atos may be represeted by uateros []. I [] ad [3], Lagrage s euatos of secod d of rgd bodes system covarat form were developed usg Rodrguez matrx for the represetato of oretato of rgd body wth respect to the ertal frame. Our goal s to develop the same form of euato, but wth the help of uateros. Further research wll be based o cotrol system desgs, because uateros eable sgularty-free mathematcal represetato of oretatos [4,5].. MATHEMATICAL BACKGROUND OF THE QUATERNIONS. Defto Quateros are hyper-complex umbers of ra 4 cosstg of oe real ad three magary parts. The uateros were frst descrbed by Irsh mathematca Sr Wllam Rowa Hamlto 844 ad appled to mechacs three-dmesoal space. Crucal to ths descrpto was hs celebrated rule: -, ( deotes uatero or the Hamlto product. The uatero s defed as: Receved: December 009, Accepted: Jauary 00 Correspodece to: Nemaa Zorc, M.Sc. Faculty of Mechacal Egeerg, Kralce Mare 6, 0 Belgrade 35, Serba E-mal: zorc@mas.bg.ac.rs 0 + + + 3, ( 0 represets real part, ad, ad 3 represet magary parts of the uatero. Pure part of the uatero s defed as:. Algebrac propertes. (3 + + 3 Let p ad be two uateros. The sum of p ad ca be wrtte as: p+ p0+ 0 + p+ + p+ + p3+ 3. (4 If c s scalar, the the product of uatero ad scalar c s gve by: c c0 + c + c + c3. (5 For two uateros p ad, ther Hamlto or uatero product s determed by the product of the bass elemets ad the dstrbutve law. Ths gves the followg expresso [6]: p p00 + p0 + 0 + p p p. (6 From (6 t ca be see that uateros form ocommutatve uder multplcato. Let be the uatero. The complex cougate of uatero s defed as: 0 0 3. (7 From (6 ad (7 t ca be cocluded the followg: ( p. (8 The orm of uatero s defed as: N 0 3 + + +, (9 ad verse of uatero s:. (0 N Faculty of Mechacal Egeerg, Belgrade. All rghts reserved FME Trasactos (00 38, 9-8 9
.3 Quateroc represetato of ato of the rgd body The rgd body (V ates about axs 0τ whch s represeted by ut vector e (Fg.. Referece frame 0xyz s ertal, ad referece frame 0ξηζ s body-fxed frame. Ut vectors of axs x, y ad z are deoted by, ad, ad ut vectors of axs ξ, η ad ζ are deoted by λ, µ ad ν. Fgure. Rotato of the rgd body I tal tme these two referece frames were euvalet. Vector OM r (pot M belogs to the body (V ca be expressed both referece frames: r x+ y+ z, ( Fgure. Quateroc represetato of ato of the rgd body Vector e s varat, ad because of that property t s all the same whch the coordate frame ths vector would be expressed. The ext case s whe the rgd body ates about the movg axs. The rgd body (V ates about movg axs 0τ represeted by ut vector e. At the same tme, ths axs ates about the axs 0τ represeted by ut vector e. The referece frame 0xyz s ertal, 0ξ η ζ s body-fxed ad 0ξ η ζ s fxed o the ato axs 0τ (Fg. 3. At tal tme these three frames were euvalet. ad ( r ξλ + ηµ + ζν. ( Superscrpt ( deotes body-fxed referece frame whch vector r s expressed. Vectors ( ad ( belog to the set of vectors. I order to operate wth the uatero, a vector, whch lves R 3, eeds to be treated as a pure uatero (that s a uatero whch real part s zero whch lves R 4. The set of all pure uateros (deoted by Q 0 s the subset of Q, the set of all uateros. It ca be defed as oe-to-oe correspodece betwee the set of vectors ad the set of pure uateros, a correspodece 3 whch a vector r R correspods to pure uatero r 0 + r Q 0, that s: 3 r R r 0 + r Q0 Q. (3 The relato betwee the vector expressed ertal frame ( ad the vector expressed body-fxed frame s gve by: ( r r, (4 s ut uatero (a uatero wth orm oe whch has the followg structure: θ θ cos +e s, (5 θ represets the ato agle about axs 0τ, ad e s pure uatero whch correspods to the ut vector axs 0τ. The result of (4 s also pure uatero whch correspods to the posto vector of pot M. It ca be llustrated the followg fgure (Fg. : Fgure 3. Rotato of the rgd body about movg axs Vector OM r (pot M belogs to the body (V expressed referece frame 0ξ η ζ s: ( r r, (6 θ θ cos +e s. (7 It must be metoed that vector e s varat relato to referece frames 0ξ η ζ ad 0ξ η ζ, ad vector e s varat relato to referece frames 0xyz ad 0ξ η ζ. Vector OM r expressed ertal referece frame s: ( r r, (8 θ θ cos +e s. (9 Substtutg (6 (8, oe ca get followg expresso: 0 VOL. 38, No, 00 FME Trasactos
( r r ( r r, (0 ( represets composte uatero. 3. KINEMATICS OF OPEN CHAIN SYSTEM 3. Trasformato of coordates The ope cha system of rgd bodes (V, (V,..., (V s show Fgure 4. The rgd body (V s coected to the fxed stad. Two eghborg bodes, (V - ad (V of cha are coected together wth ot (, whch allows traslato alog the axs whch s represeted by ut vector e, or ato about the same axs body (V respect to body (V -. The values represet geeralzed coordates. Fgure 4. Ope cha of the rgd bodes system The referece frame 0xyz s ertal Cartesa frame, ad the referece frame 0ξ η ζ s local body-frame whch s assocated to the body (V at the pot C whch represets the cetre of erta of body (V. At tal tme, correspodg axs of referece frames were parallel. Ths cofgurato s called referece cofgurato ad t s deoted by (0. The symbols ξ ad ξ ca be troduced, whch are defed as: ξ, ξ 0 ( the case whe bodes (V - ad (V are coected wth prsmatc ot, ad ξ 0, ξ (3 the case whe bodes (V - ad (V are coected wth cyldrcal ot. Arbtrary vector τ, assocated wth the body (V s gve (Fg. 5. I referece cofgurato, ths vector s detcal both referece frames ( ( τ τ. (4 ( 0 ( 0 the axs e for agle, vector τ body-fxed referece frame 0ξ - η - ζ - has the followg value: ( ( τ p τ, (5 p represets ut uatero whch s defed as ( p cos +e s. (6 If rgd body (V - s coected wth (V - by cyldrcal ot, the: ( ( τ p τ, (7 ad:,, p, p p, (8 ( p cos +e s. (9 I the case of prsmatc ot, vector τ s the same both local body-fxed frames: ( ( τ τ. (30 I the geeral case, wth the help of symbol ξ, the uatero (6 has the followg form: ξ ( ξ p cos + e s. (3 Vector τ ertal referece frame 0xyz has the followg value: τ p τ p, (3 ad 0, 0, p0, p p... p, (33 p,. (34 3. Velocty of erta cetre of the rgd body (V The posto vector of erta cetre C of the rgd body (V has the followg value ertal referece frame (Fg. 4: ( ξ + + OC r ρ e ρ, (35 ρ p ρ, (36 0, 0, Fgure 5. Vector τ o rgd body (V I the case whe bodes (V - ad (V are coected by cyldrcal ot, after ato of the body (V about e p e, (37 0, 0, ( ρ p ρ. (38 0, 0, FME Trasactos VOL. 38, No, 00
Velocty of the cetre of erta s: dr r V T d t, (39 r T, (40 whch s called the uas-basc vector. Geeralzed coordate partal dervatves of the uatero are: p 0,, (4 ad p ξ s e ( ξ ξ cos + ξ Sce ξ s e ( ξ ξ + cos. (4 ( e e (43 the, multplcato of (4 wth (43 o the left sde, (43 becomes: p ( ξ ( e e the ξ s e ( ξ + cos ( e e ξ ξ ξ s cos ( e e ξ ξ ξ s + cos ( ξ e. (44 Also, tag to accout: p, (45 p p p p p p + 0, (46 ad, from the prevous expresso: p p p p p e ( ξ p. (47 Multplcato of the prevous expresso wth o the rght sde gves: p p ( ξ p e. (48 I the case, geeralzed coordate partal dervatve of vector ρ s: ( ( p 0, ρ ρ 0, 0, 0, p ( ( p ρ 0, + p0, ρ ad p0, p p......, (49 ( ξ p... e... ( ξ p0, e,, (50 p 0, p p...... ( ξ p... e... ( ξ p, e 0,. (5 Sce ( e p 0, e 0,, (5 the (50 ad (5 become: ad p0, ξe 0, p0, ξ p 0, e, (53. (54 Substtutg (53 ad (54 (49, oe ca get followg expresso: ρ ξ e 0, ρ 0, ( ξ p0, ρ 0, e ξe ρ ξρ e ξ ( e ρ + e ρ + ρ e ρ e ξ ( e ρ + e ρ + e ρ + e ρ ξ e ρ. (55 VOL. 38, No, 00 FME Trasactos
I the case >, there follows: ρ 0. (56 Smlar to the prevous expressos, t ca be wrtte the followg: e ξe e, (57 whe, e 0, (58 whe >, ρ ξe ρ. (59 Also, whe, ad ρ 0, (60 whe >. Accordg to the prevous expressos, (40 becomes: r ( ξ ξξ T e ρ + e e + + ξ e + ξ e ρ ξ e ( ρ + ξe + ρ + ξe (6 whe, ad: T 0 (6 whe >. Accordg to (6 ad (6, the expresso for velocty of the cetre erta of the rgd body (V s the followg: V T. (63 If vectors (6 are expressed local body-fxed coordate frames, the uasbasc vectors become: ξ e ( 0, T ξ p e 0, ( ( p0, ρ 0, + p0, 0, + whe. ( + p0, ρ 0, + ( + p e (64 0, 0, ξ 3.3 Accelerato of the erta cetre of the rgd body (V Accelerato of the erta cetre of the rgd body (V s tme dervatve of (39: dt ( ac a T ( +. (65 dt The secod part of the prevous expresso ca be wrtte as: dt T(. (66 dt Accordg to (40, t ca be wrtte the followg: T r T T (, (67 the (65 becomes: a T + Γ(, (68 T Γ, (69 ad Γ Γ(. (70 I the case the (see (55: T Γ ξe T, (7 ad, the case > : T Γ ξe T (. (7 The expressos (7 ad (7 ca be wrtte the uue form: Γ ξf (, ef (, Tsup (,. (73 The expresso (65 ca be wrtte the followg way: a T + Γ(. (74 The vectors (73, expressed local body-fxed coordate frames have the followg form: ef (, ( f (, p0,f (, e f (, 0,f (,, (75 ( e 0, ξ p e 0, Tsup, sup, ( ( 0, ρ 0, 0, 0, p p ξ p p + + ( + p0, ρ 0, + ( + p e. (76 0, 0, ξ FME Trasactos VOL. 38, No, 00 3
3.4 Agular velocty of the rgd body (V Agular velocty of the rgd body (V ca be obtaed from the followg expresso: or ξe ω, (77 ξ ω Ω e,, Ω 0, >. (78 If ut vector of axs (77 s expressed the local body-fxed referece frames, the (77 has the followg form: 0, ω p e p 0, ξ. (79 4. KINETIC ENERGY OF THE RIGID BODIES SYSTEM Cosder a ope cha of the rgd bodes system (V, (V,..., (V. Dfferetal of etc eergy of the body (V (Fg. 6 s: de ( dm V, (80 M C represets erta cetre, dv s ftesmal volume of the body (V whch correspods to ftesmal mass dm. E ( mvc VC + + d ω ρ ω ρ m. (84 ( V Ketc eergy of the system of the rgd bodes s eual to the sum of etc eerges of each body: E E ( mvc VC + + ( ω ρ ( ω ρ d m. (85 ( V Sce rc V C (86 the rc r C VC V C. (87 The secod part of (85, usg (78, ca be trasformed as follows: ρ ω ρ ( Ω ρ (88 the, t becomes: ( ω ρ ( ω ρ dm ( V ρ ρ dm ( V, (89 Substtutg (87 ad (89 (85, t ca be obtaed: Fgure 6. Characterstc vectors of the rgd body (V Velocty of the pot M whch belogs to the body (V s: V V V + ω ρ. (8 M C Ketc eergy of the body (V s: E ( ( VC ω ρ ( VC ω ρ d + + m. (8 V Due to (see[3]: ρdm mρc 0, (83 ( V the expresso (8 becomes: rc r C E m + ρ ρ + dm ( V a rc r C a m + ρ ρ + dm V, (90. (9 Coeffcets are called the covarat coordates of the basc metrc tesor, ad matrx [ a ] R are called basc metrc tesor. Accordg to (40, the frst part of (9 becomes: 4 VOL. 38, No, 00 FME Trasactos
rc m m ( T { T } r C mt( T (. (9 The secod part of the rght sde of (9 ca be trasformed the followg way: ρ ρ dm ( V ξξ ( e ρ ( e ρ d. (93 Sce (see[3]: V m d ( e ρ{ e ρ} ( e ρ { e }, (94 the (93 becomes: ρ ρ dm ( V d ξξ ( e ρ dm { e} ( V ξ ξ, (95 ( e J { e } C d d J C ρ m V η + ζ ξη ξζ ηξ ζ + ξ ηζ dm (96 ( V ζξ ζη η + ξ deotes erta tesor of the rgd body (V. It s most coveet for the erta tesor of the body (V to be expressed local body-fxed referece frame 0ξ η ζ, because, ths case, the erta tesor s costat. After that, covarat coordates have the followg form: or m ( T { T } + a + ξξ ( e J C e, (97 a m ( T( { T( } + sup (, tr + ξξ ( e J C { e } a + a, (98 sup (, tr a ad a deote traslatoal ad atoal compoet of covarat coordates: tr a m ( T( { T( }, (99 sup (, ad ξ ξ ( e J C e. (00 ( a sup, From (98 t ca be cocluded that coeffcets a have the followg property: (,..., (,..., a a. (0 Ut vectors (00, expressed body-fxed coordate frame have the followg form: ad ( e p, e, ( p e, (0,, ( e p, e, ( p e. (03,, 5. DIFFERENTIAL EQUATIONS OF MOTIONS OF THE RIGID BODY SYSTEM I ths secto dfferetal euatos of moto of rgd bodes system covarat form usg uatero algebra wll be derved. Cosder ope cha system of rgd bodes (V, (V,..., (V. It s assumed that costrats are holoomc, scleroomc ad deal. A system of depedet coordates (,,, ca be chose, whch allows that etc eergy to be wrtte as the fucto of these coordates ad ther tme dervatves. I ths case, dfferetal euatos of moto ca be represeted the form of Lagrage s euato expressed oly as the fucto of geeralzed coordates (,,, ad ther tme dervatves: d E E Q, (04 dt Q deotes geeralzed force of actve forces system whch act o the rgd body system, whch correspods to geeralzed coordate. Geeralzed velocty partal dervatves of etc eergy are: E a a +. (05 Accordg to (98, (05 ca be wrtte as: E a. (06 FME Trasactos VOL. 38, No, 00 5
Tme dervatve of (06 s: d E a + dt a a + +. (07 Geeralzed coordate partal dervatves of etc eergy are: E a. (08 Substtutg (07 ad (08 (04, the Lagrage euatos have the followg form [3]: a + Γ, Q, (09 a a a Γ, + (0 deotes Chrstoffel symbols of the frst d. Ths form of euatos s called covarat. Accordg to (98, t ca be wrtte: tr,,, Γ Γ +Γ, ( tr tr tr tr a a a Γ, + deotes the traslatoal compoets, ad a a a Γ, + ( (3 deotes atoal compoets of Chrstoffel symbols. Dervg (99, applyg the followg propertes: T r T (, (4 (7, (7 ad (73, ad substtutg t (, the traslatoal compoets of the Chrstoffel symbols become: T tr ( Γ, m T sup (,, mξf (, ef (, sup (,, ( Tsup (, T(, (5 ef (, ( f (, p0,f (, e p f (, 0,f (,. (6 The atoal compoets of the Chrstoffel symbols are as follows: a e ξξ JC { e } + sup (, ( e + ξξ ( e J C. (7 sup (, Accordg to (0 ad (03, ad dog the same as Secto 3., partal dervatves of ut vectors of the axs the case,, are: ad e e e ξ, (8 ( e e e ξ. (9 Substtutg (8 ad (9 to (7, t ca be obtaed: a ( e e sup (, JC { e ξξξ } + sup (, + ξξξ ( e J C { e e sup (, }. (0 sup (, Dog smlarly wth other compoets as (3, ad substtutg t to (6, the atoal compoets of the Chrstoffel symbols become: ( ( ( ( e J C { e e sup (, } ( e e sup (, J C { e } ( e JC { e e sup (, } ( e e sup (, JC { e } ( e J C { e e sup (, } Γ, ξξξ e e J e sup (, C + sup (,, + + + +. ( Accordg to the certa propertes of members the prevous expresso, ( becomes:, Γ ( ( ( e e ( J C { e } ( e e J C { e } ξξξ e e J e f (, sup (, C sup (,,.( 6 VOL. 38, No, 00 FME Trasactos
Due to the property of dual obect, the erta tesor ca be wrtte the followg way [3]: d { [ ] },(3 J C ρ d ρ ρ d m ρ I m ( V ( V ad substtutg (3 (, the Chrstoffel symbols become:, Γ ξξξ sup (,, ( e e d f (, sup (, ρ { e } + ( V ( ( ( + ( e e { ρ }( ρ { e } + ( + e e ρ ρ e m. (4 ( d The last two terms of subtegral fucto prevous expresso ca be wrtte as follows: ( e e { ρ }( ρ { e } ( + e e { ρ}( ρ { e } ( e e { ρ }( ρ f, { e sup (, } + + e e ρ ρ e sup (, f,. (5 Subtegral fucto ca be easly trasformed the followg expresso: d d ( e e f (, sup (, ρ ρ { e } + ( e e { ρ }( ρ f, { e sup (, } + e e ρ ρ e sup (, f,, (6 ad, the atoal compoets of the Chrstoffel symbols become:, Γ ξξξ sup (,, e e ρ ρ e d sup (, m f, { { } ( V ξξξ ( e e [ ] e Π,(7 sup, f (, sup (,, [ ] { ρ }( ρ Π dm (8 V deotes the plaar momet erta [3]. Ut vectors (7 ca be expressed local-body fxed coordate frame le (0, so the Chrstoffel symbols fally become: Γ, mξf (, p0,f (, sup (,, ( f (, e f (, 0,f (, T sup (, { T } + ( sup (, + ξξξ psup (,, e sup (, sup (,, ( sup (,, ( p, e, [ Π ] ( f (, pf (,, e f (, f (,,. (9 6. CONCLUSION Ths paper has show the developmet of Lagrage s euatos of the secod d of the rgd bodes system the covarat form usg the uatero algebra. It ca be cocluded that every vector whch belogs to the arbtrary body of the rgd bodes system ca be easly expressed the body-fxed referece frame of aother body mag composte uatero, whch cossts of Hamltoa product of uateros represetg the ato eghbourg bodes, avodg trgoometrc fuctos characterstc of Euler s agles. Also, t s easy to fd geeralzed coordate partal dervatves of that vector. Ule the exstg results, uateroc approach has bee appled oly for the case of ato of oe or two bodes, t s here preseted the procedure of obtag the model of mult-body system of rgd bodes terms of uateros, whch s useful for studyg ematcs, dyamcs as well as for research of cotrol system desgs. ACKNOWLEDGMENT Ths wor s partally fuded by the Mstry of Scece ad Evrometal Pecto of Republc of Serba as T.R.P. No. 05. REFERENCES [] Hamlto, W.R.: O Quateros, or a ew system of magares algebra, The Lodo, Edburgh ad Dubl Phlosophcal Magaze ad Joural of Scece, Vol. 5-36, 844-850. [] Marovc, S.: Rapd Establshmet of Dfferetal Euatos of Moto of Rgd Bodes System a Aalytcal Form, MSc thess, Faculty of Mechacal Egeerg, Uversty of Belgrade, Belgrade, 99, ( Serba. [3] Covc, V. ad Lazarevc M.: Mechacs of Robots, Faculty of Mechacal Egeerg, Uversty of Belgrade, Belgrade, 009, ( Serba. [4] Iseberg, D.R. ad Kaad, Y.P.: Computed torue cotrol of a uatero based space robot, : Proceedgs of the 9 th Iteratoal Coferece (( FME Trasactos VOL. 38, No, 00 7
o System Egeerg, 9-.08.008, Las Vegas, USA, pp. 59-64. [5] Arrbas, M., Elpe, A. ad Palacos, M.: Quateros ad the ato of a rgd body, Celestal Mechacs ad Dyamcal Astroomy, Vol. 96, No. 3-4, pp. 39-5, 006. [6] Kupers, J.B.: Quateros ad Rotato Seueces: A Prmer wth Applcatos to Orbts, Aerospace ad Vrtual Realty, Prceto Uversty Press, New Jersey, 999. КИНЕМАТИКА И ДИНАМИКА СИСТЕМА КРУТИХ ТЕЛА У КВАТЕРНИОНСКОЈ ФОРМИ: ЛАГРАНЖЕВА ФОРМУЛАЦИЈА У КОВАРИЈАНТНОМ ОБЛИКУ РОДРИГОВ ПРИСТУП Немања Д. Зорић, Михаило П. Лазаревић, Александар М. Симоновић У овом раду се предлаже кватернионски приступ за моделирање кинематике и динамике система крутих тела. Уместо регуларног Њутн-Ојлеровог и Лагранжевог метода коришћеног на традиционалан начин, употребљавају се Лагранжеве једначине друге врсте у коваријантном облику применом Родриговог приступа и кватернионске алгебре. Добијен је модел система од крутих тела у кватернионској форми који је користан за проучавање кинематике, динамике система за општи случај кретања, као и за синтезу система управљања. 8 VOL. 38, No, 00 FME Trasactos