Orbit dynamics and kinematics with full quaternions

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1 bt dynamcs and knematcs wth full quatenons Davde Andes and Enco S. Canuto, Membe, IEEE Abstact Full quatenons consttute a compact notaton fo descbng the genec moton of a body n the space. ne of the most mpotant esults about full quatenons s that they can be pattoned nto a unt quatenon (whch descbes the oentaton wth espect to a sutable efeence, and a modulus (whch epesents the tanslatonal moton along the decton ndcated by the unt quatenon. Snce vectos and scalas ae also full quatenons, the equatons of moton of the body can be ewtten n quatenon fom. In ths pape the obt dynamcs and knematcs of a pont mass movng n the space ae tansfomed n quatenon fom. Smple applcaton examples ae pesented. I. INRDUCIN When dealng wth satellte atttude and obt contol, one of the fst desgn ssue s the fomulaton of spacecaft dynamcs. Accodng to classcal appoach, gd body moton can be decomposed nto two pats: 1. obtal moton, dependng on poston and velocty of the satellte Cente of Mass (CM ;. atttude knematcs and dynamcs, descbed by Eule paametes (.e.: unt quatenons o Eule angles. hs methodology s vey well known, has been wdely teated n lteatue (see [1] and [], and s commonly used n applcatons: fo example t has been employed n the desgn of a dag-fee contolle fo the Euopean satellte GCE [3]. In ths case, satellte atttude coesponds to the oentaton of the body efeence fame wth espect to a local obtal fame, unvocally defned by obt poston and velocty. Assumng that the oentaton of the body fame wth espect to an netal fame s known, t becomes necessay to paameteze the oentaton of the obtal fame wth espect to the netal efeence. he poblem, appaently staghtfowad, s tansfomng the netal coodnates of the thee unt vectos consttutng the obtal fame nto a set of fou Eule paametes. wo altenatves have been consdeed: 1. to buld the otaton matx and then explot the well E. S. Canuto s wth Poltecnco d ono, Dpatmento d Automatca e Infomatca, Coso Duca degl Abuzz 4, 119 ono, Italy (phone: ; fax: ; e-mal: enco.canuto@ polto.t. D. Andes s wth Poltecnco d ono, Dpatmento d Automatca e Infomatca, Coso Duca degl Abuzz 4, 119 ono, Italy (e-mal: davde.andes@ polto.t. known conveson ules (see [1] allowng to pass to quatenon paametezaton;. to assocate a full quatenon notaton (.e.: non-untay quatenon to obtal fame. he fome soluton has been employed n atttude detemnaton of the GCE satellte [3]. he latte one, has been developed to fnd a dect way to expess the moton of the local obtal fame entaned by the CM moton. A full quatenon can descbe the modulus and the oentaton of a vecto wth espect to a gven efeence fame. hs mples, consdeng the satellte obt, that poston and velocty can be altenatvely denoted wth a vecto o wth the assocated full quatenon. Snce that, obtal dynamcs and knematcs can be ewtten substtutng vecto notaton wth full quatenons. hs esults n hamonzaton of moton equatons: both obtal dynamcs/knematcs and atttude dynamcs/knematcs can be ewtten n quatenon fom. hen the oentaton of the obtal fame can be dectly extacted fom the elated full quatenon at any tme. hs pape s devoted to lay down the foundatons of ths technque wth the help of smple applcatons. Fst of all, defnton and elementay algeba of full quatenons wll be ntoduced n Secton II. Next, how full quatenons can epesent vecto magnfcatons and fnte otatons wll be shown. Fst and second devatves of full quatenons ae then deved n ode to ewte obtal moton equatons n quatenon fom. hs wll be explaned n Secton III, whee quatenon knematcs and dynamcs wll be deved. In Secton I, quatenon knematc and dynamc equatons wll be appled to a pa of typcal obtal efeences: the Local btal Refeence Fame and the Local etcal Local Hozontal fame. In both cases the assocated full quatenon wll be defned, as well as obtal knematcs and dynamcs. Fnally, the smple case of unfom ccula moton wll enlghten the smlates between classcal vecto fom and quatenon expesson of obtal moton. II. FULL QUAERNINS A. Defnton A quatenon A s defned as a complex numbe: A a a a j a k= a a. (1 1 3

2 Quatenons can be also expessed n column vecto fom wth espect to the bass (1,, j,k: [ a a a a ] [ a ] A = = a. ( 1 3 Remak. o allevate notaton, the scpt A wll denote: 1. quatenons n complex numbe epesentaton (1;. and quatenons n column vecto fom (. A vecto quatenon s a thee-dmensonal vecto b epesented n quatenon notatons,.e. [ ] B = b. (3 In ths case the notatons B and b wll have the same meanng. See [4] fo futhe detals. B. Algeba A bef summay of the full quatenon algeba s povded, leavng the detals to the Appendx and [4]. he nom of a quatenon A, denoted by A, s a scala quatenon and s defned n the same way as the Eucldean nom (o l nom of a geneal spatal vecto: 1 3 [ 1 3] A = a a a a = a a a a. (4 If A =1, A s called a unt quatenon, and deseves ts own notaton A. If A has non-untay nom, t s called a full quatenon. Remak. Snce scalas and vectos ae quatenons, scala and vecto algeba apples. Let A = a a, B = b b and C = c c be thee quatenons. 1 Multplcaton Accodng to [4], quatenon multplcaton s defned as: A B = a a b b = a b a b b aab a b, (5 whee the symbols and stand fo dot poduct and coss poduct. An altenatve expesson of the nom n (4 can be obtaned though quatenon multplcaton, namely: A = A A, (6 whee A = a a denotes quatenon conjugate. he same poduct n (5 can be expessed n matx fom. Fst, ewte the poduct quatenon C n vecto fom:, ab ab C= A B C = = c. (7 ab ba a b c hen, fom matx expessons fo dot and coss poducts: a3 a ab = a b, a b= a3 a1 b= C( ab, (8 a a1 equaton (7 can be wtten as: c a a b b b = = a c a ai C( a b b bi C( b a. (9 Quatenon multplcaton s assocatve and dstbutve, but not commutatve. Commutatve popety Although commutatve law does not hold n geneal, the matx expesson (9 shows A and B to commute though sgn change. heefoe, the followng matx epesentatons of quatenons can be ntoduced: a a b b A =, ai C B = a a b bi C( b, (1 whee supescpts and denote the sgn of the coss poduct matx C( and I denotes the dentty matx. Usng notatons defned n (1, the commutatve popety whch s hdden n (9, can be expessed n the compact fom: C= A B= B A, (11 whee one must pay attenton that A and B ae meant to be n column vecto fom. 3 Invese Each nonzeo quatenon A admts an nvese A -1 such 1 that A A = 1. It s smple to poof that the nvese quatenon A -1 of A holds: 1 A = A A. (1 Equaton (1 states that f A s a unt quatenon, the nvese equals the conjugate. Instead, f A s a full quatenon, ts nom has to be taken nto account. C. Magnfcaton and fnte otatons As t wll be shown below, full quatenons allow to descbe at the same tme vecto otaton as unt quatenons and vecto magnfcaton. Consde a unt quatenon R and a quatenon B. A well known method to epesent a otaton of B nto B by an angle θ aound an axs u s: B = R B R. (13 Snce evey unt quatenon admts the Eule paametes epesentaton, t s possble to expess R n tems of θ and the unt vecto u: [ ] ( θ ( θ R = 1 3 = cos sn u. (14 By applyng (11 and (A., the matx fom n (13 ensues: B = R B R = R R B. (15 Employng matces E and E defned n (A.3 yelds: 1 B = 1 = = E ( E B B RB. (16 R R R he matx R epesents a 4 4 quatenon tansfomaton n a fou-dmenson space. Snce R R = 1, matces R and ( R ae othonomal and R s a lnea opeato wth the popety of leavng nvaant quatenon noms. Fom (16 t s possble to sepaate a 3 3 otaton matx: ( ( ( ( R= E R E R = I C. (17

3 Note that the above defnton of R s consstent 1 wth the defnton of decton cosne matx gven n [1]. Now, one can apply the same concepts to a full quatenon R nstead of the unt R. In ths case equaton (13 becomes: B = R B R. (18 Moeove, any full quatenon R can be factozed nto the poduct of the nom and of the unt quatenon: R= R R. (19 hen, usng factozaton (19, one can ewte (18 by sepaatng nom and otatonal tem as follows B = R R B R. ( Expessng ( n matx fom makes explct two opeatons, nom amplfcaton and otaton as n (15: B = R R B = R R R B. (1 Employng matces E and E defned n (A.3 yelds: 1 B = R B= R RB. ( R It s clea fom pevous equaton, that whle B s otated as n (16, an amplfcaton of the quatenon nom appeas. heefoe, n case of full quatenons, poduct (18 apples two dffeent tansfomatons: 3. a magnfcaton, by the facto R, of the B nom; 4. a otaton of B by an angle θ aound the axs u (as stated by Eule heoem. In the case B s a vecto quatenon, the factozaton ( educes to: 1 = R b' = ρrb, ρ= R. (3 b' R b he use of full quatenons allows to genealze the descpton of the moton of an object n the theedmensonal space: not only otatons but also tanslatons can be paametezed. III. QUAERNIN KINEMAICS AND DYNAMICS As stated n Secton I, the goal of ths pape s to ewte the obtal dynamc and knematc equatons usng full quatenons. o ths end, fst and second devatves of a quatenon wll be detemned. Let and o be nonzeo vectos whch, accodng to Secton II.A, can be consdeed as vecto quatenons. hen, as n (18, t s possble to defne a full quatenon P elatng the vecto o to the efeence vecto though a otaton and a magnfcaton: 1 1 o= = o P P P P. (4 1 Actually, the 3 3 matx R n (17 s the tanspose of the decton cosne matx n [1], because the opposte otaton decton has been used. A. Knematcs Dffeentatng (4 yelds: P P P P P P.(5 1 1 o= o o q, q= hen, by takng the devatve of the poduct 1 = 1 P P, one can defne the quatenon W as shown below: 1 1 P P =P P = W. (6 Fom the above defnton the quatenon knematc equaton follows: P= W P. (7 By factozng P as n (19 and by emembeng the defnton (1, the pevous equaton develops nto: W= P P P P. (8 It s possble to poof that P P s a vecto quatenon (see the Appendx. heefoe, one can ewte W as: W= P P P P = w w= w w w, (9 whee the decomposton of w nto nomal and paallel components w and w wth espect to o has been exploted. Substtutng (9 nto (5 enlghtens that the devatve of o s unaffected by the paallel component w : o= W o o W q= wo w o q=. (3 = W o q, W= w w Snce o s a vecto quatenon, equaton (3 apples: 1 = P, = o= ρprp ρp P, (31 o RP whee R P s a otaton matx. Compang (3 wth the fst devatve of (31: { ( } = I ρ ρ RR ρ R=Ω ρ R, (3 o P P P P o P P o P P yelds the followng equaltes: w = ρ ρ, w = C w = R R R = C w R.(33 ( P P P P P P ne can ecognze that w epesents the angula velocty n the thee dmensonal space (see knematc equatons of moton n [1] and w o epesents the tanslaton velocty along the o decton. he ensemble (w o w=w foms a full quatenon efeed to as genealzed angula velocty. hs tem has been chosen because n the tadtonal atttude epesentaton though unt quatenons the tem w o vanshes, and W becomes a pue angula velocty. Rewtng (7 and (3 n matx notatons yelds: whee P = W P o= W o P ( P w w = w w, (34 W W / = w Ω w Ω/ Ω equals Ω unde the constant w= w.

4 Remak. Quatenon knematcs (7 s moe geneal than vecto knematcs (3. Snce the angula ate w s unconstaned, the fome equaton has fou degees of feedom (d.o.f.. In (3, the paallel component w dsappeas, then fo descbng the o otaton, only the nomal component w needs. hs s equvalent to state that, n (3, an othogonalty constant apples to w. hen, d.o.f. educe to thee n ageement wth classcal mechancs. heefoe, equatons (3 and (31 can be vewed as output equatons of the state equaton (7. B. Dynamcs Fst defne the genealzed angula acceleaton A as the devatve of the genealzed angula ate W: A= W a a= a a a = w w. (35 In (35, the decomposton of a nto nomal and paallel components wth espect to o has been exploted. Remak. Be awae that w a and w a. Quatenon dynamcs follows by takng the devatve of quatenon knematcs (7: P= W P W P = A W W P= D P, (36 [ ] whee the quatenon D gathes the effect of angula ate and acceleaton. Scala and vecto pats of D ae elated to the components of W and A though: ( D = d d= a w w a w w. (37 he second devatve of o can be obtaned by explotng (3 and (36: o= D o o D W o W q= = ( a o ( w o ( w( w o ( a o. (38 ( w ( w o q, q= ( w q ( w q q hs expesson has a clea smlaty wth the odnay equaton of the elatve moton (see [] o [5]. heefoe, a physcal meanng can be assgned to each tem n (38: 1 q epesents the acceleaton of the efeence vecto ; ( a o ( a o s the appaent acceleaton of o wth espect to. In patcula: ( ( a o s the appaent acceleaton along o ; ( ( a o s the appaent acceleaton along a nomal decton to o ; 3 ( w ( w o s the Cools acceleaton; 4 ( ( ( w w w o s the centfugal tem. o Futhe, developng (38 shows that the acceleaton of o does not depend on the paallel component a : o= D o W o W q. (39 D = A W W, A = a a Remak. Quatenon dynamcs (36 s moe geneal than vecto dynamcs (39. Snce the angula acceleaton a s unconstaned, the fome equaton has fou d.o.f.. In (39, the paallel component a dsappeas, showng an othogonalty constant on a, whch coesponds to a d.o.f. educton. hen, (39 has only thee d.o.f., n ageement to wth classcal mechancs. I. APPLICAINS nce obtaned the geneal knematc and dynamc equatons of full quatenons, a step to be done s applyng them to obtal moton. Consde a pont P wth mass m movng n the space, subject to a foce F. wo knds of local efeence fames, can be attached to the patcle: 1 a Local btal Refeence Fame (LRF, fxed to the velocty vecto v; a Local etcal Local Hozontal fame (LLH, fxed to the poston vecto. Both fames and the oentaton wth espect to an netal efeence ae shown n Fg. 1. he netal fame {,,,} R = jks a Catesan efeence wth ogn n and unt vectos coespondng to, j and k aleady ntoduced n (1. Fo each of the two obtal fames, the followng poblems wll be solved: 1. complete defnton of the fame axes;. assgnment of a full quatenon to the fame; 3. fomulaton of the dffeental equaton of the full quatenon,.e.: obtal equatons n quatenon fom. v Fg. 1. j k P k j v j P LRF and LLH wth espect to the netal fame. A. LRF Refeence Fame he LRF R = {,, j, k} s a Catesan efeence fame defned as follows: 1. the ogn concdes wth P;. les along the velocty decton; 3. j s nomal to the nstantaneous obt plane (defned by poston and velocty; 4. k completes the fame. = v v, j = v v, k = j. (4 he velocty vecto and the oentaton of the LRF tple can be expessed though the LRF quatenon R. he defnton of R s abtay: fo example the axs otates nto and the axs j otates nto j : vr R, j R j R. (41 Because thee exsts an nfnte numbe of otatons satsfyng (41, a futhe constant must be ntoduced: the ght equaton specfes that the j-axs of the netal fame must be otated nto the obtal plane nomal decton. Factozng the left equaton n (41 as n ( enlghtens the nom R of the LRF quatenon to be equal to the k k j

5 squae oot of the velocty modulus, and the untay pat R to epesent the oentaton of the velocty unt vecto wth espect to the netal fame: v= vnv= R ( R R, nv = 1. (4 R = v, R R= nv Now, one can apply fomula (3 of quatenon knematcs to compute the acceleaton v : v = w v w v= W v, W = w w, (43,, whee the devatve of, beng zeo by defnton, dsappeas and w has been decomposed nto the nomal and paallel components w and w wth espect to v. he acceleaton of the pont mass s elated to the foce F though Newton s Law and emembeng that w v : = F m= v w, = 1 mv v F, w = 1mv v F. (44 Snce v and F ae vecto quatenons, a moe compact expesson fo the LRF angula ate can be used: ( 1m 1 W = W w = v v F v w. (45 heefoe, the obtal equatons fo LRF quatenon can be wtten n quatenon fom: F R = W R = v w R, R ( = R m v = R R, ( =, (46 o n matx fom (explotng (11 and (A. as follows: ( 1 m R = R W = R v v F w, R( = R.(47 = R( R, = Remak. As stated n Secton III.A, the paallel component w does not gve contbuton to (43. hs confms the exstence of an othogonalty constant to w, meanng that the fou d.o.f. moton of the LRF quatenon s constaned n ageement wth the classcal mechancs thee d.o.f. hs s confmed by Newton s Law, showng w to be completely ndependent on F. he angula ate w affects only (46, and epesents an angula ate of the unt vectos j and k aound the axs. But f such vectos undewent a otaton, the LRF fame would be lost. heefoe the quatenon constant follows. B. LLH efeence fame he LLH fame R = {,, j, k} s a Catesan efeence defned as follows: 1. the ogn concdes wth P;. les along the poston decton; 3. j s nomal to the nstantaneous obt; 4. k completes the fame. =, j = v v, k = j. (48 he poston vecto and the oentaton of the LLH tple can be expessed though the LLH quatenon R. In accodance wth (41 t can be defned as: R R, j R j R. (49 Factozng the left equaton n (49 enlghtens the nom R of the LLH quatenon to be equal to the squae oot of the poston modulus, and the untay pat R to be the oentaton of wth espect to the netal fame: = n= R ( R R, n = 1. (5 R =, R R = n Now, one can apply the fomula (3 of quatenon knematcs to compute the velocty v: = v= w w = W, (51, whee the decomposton of w nto nomal and paallel components w and w w..t. has been exploted. he quatenon knematcs of the LLH follows by (7: R = W R, (5 and the LLH dynamcs follows fom (36: R = A W W R = D R. (53 [ ] hen, ecallng (38 and (39, the acceleaton can be detemned as: = v = ( a, ( w, ( w, ( w ( a ( w ( w =. (54 = D W W, D = A W W As done fo LRF knematcs, one can elate acceleaton expesson to foce F though Newton s Law. akng the dot poduct between poston and foce yelds: { } a, = F m w, w. (55 he coss poduct between poston and foce bngs to: 1 F a= ( w,( w w w. (56 m Expessons (55 and (56 can be compacted nto: A = A a= a, a a= ( 1 Y( F a. (57 F Y( F = W W ( W ( W m Fnally, the obtal equatons fo LLH quatenon can be wtten n quatenon fom: R = W R, R( = R. (58 W = ( 1 Y( F a, W = W Remak. As stated n Secton III.B, the paallel component a does not gve contbuton to (54. hs

6 confms the exstence of an othogonalty constant to a, meanng that the fou d.o.f. moton of the LLH quatenon s constaned n ageement wth the classcal mechancs thee d.o.f. Moeove, fom Newton s Law, t follows that a s unfoced by F. he angula acceleaton a affects only (58, and epesents an angula acceleaton of the unt vectos j and k aound the axs. But f such vectos undewent a otaton, the LLH fame would be lost. heefoe the constant a = follows. C. Unfom Ccula Moton hs secton ends wth a smple example: the unfom ccula moton of P aound, sketched n Fg.. Fg.. k j k j j P v k Unfom ccula moton aound LLH and LRF quatenon defntons ae the same as n (49 and (41. Fst, quatenon knematcs s appled, statng fom LLH case. he genealzed angula velocty of the LLH quatenon s: W = w, =. (59 w he genealzed angula velocty s concdent wth the angula ate of P aound, denoted wth ω= ( v. hs leads to the next esult showng quatenon knematc equaton to be smla to classcal vecto fom: = ω R = w R { ( = R( = R, (6 whee w = ω /. By applyng (36, quatenon dynamc equaton can be obtaned. Quatenon dynamcs, lke knematcs, looks smla to classcal vecto fom: v = ω R = [ a w w] R =w R, (61 v( = v R( = R whee a = by defnton of unfom moton. By usng LRF, dynamcs s epesented by quatenon knematcs, because the quatenon descbes pont velocty, nstead of poston. LRF dynamcs s: v = ω v R = w R { v( = v R( = R. (6 Knematcs follows fom defnton (41: = v = R R { ( {. (63 = ( = R( R( he last fou equatons show that the angula veloctes of the LRF and LLH quatenons ae the same, namely ω = w = w. hs follows fom the fact that, fo unfom / ccula moton, poston and velocty ae always othogonal, then otatng wth the same angula ate.. CNCLUSINS AND FUURE DEELPMENS he obt dynamcs and knematcs fo the pont mass moton has been tansfomed fom the classcal vecto notaton nto a new quatenon fom. he LRF equatons have been tested though MALAB mplementaton. Among futue developments, the desgn of quatenon obseve and contol wll cove the most mpotant ole. APPENDIX A. Algeba - Conjugate Multplcaton popetes When quatenon multplcaton nvolves conjugates, commutatve popety (11 stll hold. hen the poduct C= A B can be wtten n matx notaton though: C= A B = B A = A B = B A. (A.1 Fom pevous equaton t follows: = and = A A B B. (A. B. Algeba - Some Inteestng Matces It s useful to ntoduce the followng matx notatons: E E ( X = x ( xi C( x ( X = x ( xic( x. (A.3 By explotng the new notaton, the matx expesson of quatenons ntoduced n (1 can be ewtten as: ( E ( E A = A A = A A. (A.4 B = B ( E ( B = B ( E ( B C. Knematcs - Genealzed angula velocty Rewtng the tem P P n (6 by usng the matx notaton (A.4, yelds: w = = P P P P P= = E P P E ( PP. (A.5 REFERENCES [1] J.R.Wetz, Spacecaft Atttude Detemnaton and Contol, D.Redel Publshng Company, [] M.H. Kaplan, Moden Spacecaft Dynamcs & Contol, John Wley & Sons, 1976 [3] E.Canuto, B.Bona, G.Calafoe, M.Ind, Dag Fee Contol fo the Euopean Satellte GCE. Pat I: Modellng In: Poceedngs of the 41st IEEE Confeence on Decson and Contol, Las egas, Nevada USA, Decembe. [4] J.C.K.Chou, Quatenon Knematc and Dynamc Dffeental Equatons In: IEEE ansactons on Robotcs and Automaton, vol.8, no.1, Febuay 199 [5] D.. Geenwood, W.M. Geenfeld, Pncples of Dynamcs Pentce Hall, 1965