Poltecnco d orno Porto Insttutonal Repostory [Proceedng] rbt dynamcs and knematcs wth full quaternons rgnal Ctaton: Andres D; Canuto E. (5). rbt dynamcs and knematcs wth full quaternons. In: 16th IFAC Symposum on Automatc Control n Aerospace (ACA 4), St. Petersburg (Russa), June 14-18, 4. pp. 185-19 Avalablty: hs verson s avalable at : http://porto.polto.t/163541/ snce: June 7 Publsher: Elsever erms of use: hs artcle s made avalable under terms and condtons applcable to pen Access Polcy Artcle ("Publc - All rghts reserved"), as descrbed at http://porto.polto.t/terms_and_condtons. html Porto, the nsttutonal repostory of the Poltecnco d orno, s provded by the Unversty Lbrary and the I-Servces. he am s to enable open access to all the world. Please share wth us how ths access benefts you. Your story matters. (Artcle begns on next page)
Preprnts of the 16th IFAC Symp. on Automatc Control n Aerospace, St. Petersburg (Russa), 14-18 June 4 RBI DYNAMICS AND KINEMAICS WIH FULL QUAERNINS Davde ANDREIS, student, Enrco CANU Poltecnco d orno, Dpartmento d Automatca e Informatca,Corso Duca degl Abruzz 4, 119 orno, Italy davde.andres@polto.t, enrco.canuto@polto.t Abstract: Full quaternons consttute a compact notaton for descrbng the moton of a body n the space. An mportant result about full quaternons s that they can be parttoned nto a unt quaternon (whch descrbes the orentaton wth respect to a sutable reference), and a modulus (whch represents the translatonal moton along the drecton ndcated by the unt quaternon). Snce vectors and scalars are also full quaternons, the equatons of body moton can be rewrtten n quaternon form. In ths paper the orbt dynamcs and knematcs of a pont mass movng n the space are transformed n quaternon form. Smple applcaton examples are presented. Copyrght 4 IFAC Keywords: Aerospace raectores, System Models, Quaternons, Satellte Control Applcatons. 1. INRDUCIN When dealng wth satellte atttude and orbt control, one of the frst desgn ssue s the formulaton of spacecraft dynamcs. Accordng to classcal approach, rgd body moton can be decomposed nto: 1. orbtal moton, dependng on poston and velocty of the satellte Centre of Mass (CM) ;. atttude knematcs and dynamcs, descrbed by Euler parameters (.e.: unt quaternons) or Euler angles. hs methodology s very well known, has been wdely treated n lterature (Wertz, J.R., 1978 and Kaplan, M.H., 1976), and s commonly used n applcatons: for example t has been employed n the desgn of a drag-free controller for the European satellte GCE (Canuto, E. et al., ). In ths case, satellte atttude corresponds to the orentaton of a body-fxed reference frame w.r.t. a local orbtal frame, unvocally defned by orbt poston and velocty. Assumng that the orentaton of the body frame w.r.t. an nertal frame s known, t becomes necessary to parameterze the orentaton of the orbtal frame w.r.t. the nertal reference. he problem, apparently straghtforward, whch suggested the present work, s transformng the nertal coordnates of the three unt vectors consttutng the orbtal frame nto a set of four Euler parameters. wo alternatve solutons have been consdered: 1. to buld the rotaton matrx and then explot the well known converson rules allowng to pass to quaternon parameterzaton;. to assocate a full quaternon notaton (.e.: nonuntary quaternon) to orbtal frame. he former soluton has been employed n atttude determnaton of the GCE satellte. he latter one, whch s credted to be orgnal, has been developed wth the am of fndng a drect way to express the moton of the local orbtal frame entraned by the CM moton. A full quaternon can descrbe the modulus and the orentaton of a vector w.r.t. a gven reference frame. hs mples, consderng the satellte orbt, that poston and velocty can be alternatvely denoted wth a vector or wth the assocated full quaternon. Snce that, orbtal dynamcs and knematcs can be rewrtten substtutng vector notaton wth full quaternons. hs results n harmonzaton of moton equatons: both orbtal dynamcs/knematcs and atttude dynamcs/knematcs can be rewrtten n quaternon form. hen the orentaton of the orbtal frame can be drectly extracted from the related full quaternon at any tme. hs paper s devoted to lay down the foundatons of ths technque wth the help of smple applcatons. Frst of all, defnton and elementary algebra of full quaternons wll be ntroduced n Secton. Next, how full quaternons can represent vector magnfcatons and fnte rotatons wll be shown. Frst and second dervatves of full quaternons are then derved n order to rewrte orbtal moton equatons n quaternon form. hs wll be explaned n Secton 3, where quaternon knematcs and dynamcs wll be derved. In Secton 4, quaternon knematc and dynamc equatons wll be appled to a par of typcal orbtal references: the Local rbtal Reference Frame and the Local ertcal Local Horzontal frame. In both cases the assocated full quaternon wll be defned, as well as orbtal knematcs and dynamcs. A smple case of unform crcular moton wll enlghten the smlartes between classcal vector form and quaternon expresson of orbtal moton. Fnally, n Secton 5 some smulaton results wll be presented.
. FULL QUAERNINS.1 Defnton A quaternon s defned as a complex number: aa1a a3kaa. (1) Quaternons can be also expressed n column vector form w.r.t. the bass (1,,,k): a a a a a a. () 1 3 Remark. o allevate notaton, the scrpt wll denote at the same tme: 1. quaternons n complex number form (1),. and quaternons n column vector form (). A vector quaternon s a three-dmensonal vector b represented n quaternon notatons,.e.: b. (3) In ths case the notatons and b wll have the same meanng.. Algebra A bref summary of the full quaternon algebra s provded, leavng the detals to the Appendx and Chou, J.C.K (199). he norm of a quaternon, denoted by, s a scalar quaternon and s defned as follows: 1 3 1 3 a a a a a a a a. (4) If =1, s called a unt quaternon, and deserves ts own notaton. If has non-untary norm, t s called a full quaternon. Remark. Snce scalars and vectors are quaternons, scalar and vector algebra apples. Let a a, b b and c c be three quaternons...1 Multplcaton Accordng wth Chou, J.C.K. (199), quaternon multplcaton s defned as: ab abbaab ab, (5) where the symbols and stand for dot product and cross product. An alternatve expresson of the norm n (4) can be obtaned through quaternon multplcaton, namely:, (6) where a a denotes quaternon conugate. he product n (5) can be also expressed n matrx form as shown below: c a a b b b a c a ai Ca b b bi Cb a, (7) where matrx expressons n (8) for dot and cross product have been employed. a3 a ab a b, ab a3 a1bcab. (8) a a1.. Commutatve property Although commutatve law does not hold n general, the matrx expresson (7) shows and to commute through sgn change. herefore, the followng matrx representatons of quaternons can be ntroduced: a a b b, ai C a a b bi Cb, (9) where superscrpts + and denote the sgn of the cross product matrx C() and I denotes the dentty matrx. Usng notatons defned n (9), the commutatve property, whch s hdden n (7), can be expressed n the followng compact form:, (1) where one must pay attenton that and are meant to be n column vector form...3 Inverse Each nonzero quaternon admts an nverse -1 1 such that 1. he nverse quaternon -1 s related to by: 1. (11).3 Magnfcaton and fnte rotatons As t wll be shown below, full quaternons allow to descrbe at the same tme vector rotaton as unt quaternons and vector magnfcaton. o ths end, consder a full quaternon, a quaternon and the followng transformaton of nto :. (1) Any full quaternon can be factorzed nto the product of the norm and of the unt quaternon:. (13) hen, usng factorzaton (13), one can rewrte (1) by separatng t nto two terms, as shown below. (14) Every unt quaternon admts the Euler parameters representaton. herefore, t s possble to express n terms of a rotaton angle around the nstantaneous axs of rotaton u: r r1 r r3 cos sn u (15) By applyng (1) and (A.), t follows:. (16) Further, employng matrces E and E defned n (A.3) yelds: 1 1 E E R (17) R he matrx R s a 44 quaternon transformaton n a four-dmenson space. As remarked by Chou, J.C.K., (199), R s a lnear operator wth the property of leavng nvarant quaternon norms. From expresson (17) t s possble to separate a 33 rotaton matrx: r r rr r RE E r I r C (18) Note that the above defnton of R s consstent 1 wth the defnton of drecton cosne matrx gven by Wertz, J.R. (1978). It s clear from prevous equatons, that the product (1) apples two dfferent transformatons: 1 Actually, the 33 matrx R n (18) s the transpose of the drecton cosne matrx n Wertz, J.R., (1978), because the opposte rotaton drecton has been used.
1. a magnfcaton, by, of the norm,. and a rotaton of by an angle around the axs u (as stated by Euler heorem). In the case s a vector quaternon, the factorzaton (17) reduces to: 1 b' Rb, =. (19) b' R b he use of full quaternons allows to generalze the descrpton of the moton of an obect n the threedmensonal space: not only rotatons but also translatons can be parameterzed. 3. QUAERNIN KINEMAICS AND DYNAMICS As stated n Secton 1, the goal of ths paper s to rewrte orbtal dynamc and knematc equatons usng full quaternons. o ths end, frst and second dervatves of a quaternon wll be determned. Let r and be nonzero vectors whch, accordng to Secton.1, can be consdered as vector quaternons. hen, as n (1), t s possble to defne a full quaternon relatng the vector r o to the reference vector r through a rotaton and a magnfcaton: 1 1 o o r r r r. () 3.1 Knematcs Dfferentatng () yelds: 1 1 r o roro q, (1) q r hen, by dervng the product 1 1, one can defne the quaternon as shown below:. () From the above defnton the quaternon knematc equaton follows:. (3) After factorzaton of as n (13), t s possble to rewrte as (Chou, J.C.K, 199): 1 1 wwww w,(4) where the decomposton of w nto normal and parallel components w and w w.r.t. r o has been exploted. Substtutng (4) nto (1) enlghtens that the dervatve of r o s unaffected by the parallel component w : ro roro q w rowroq. (5) roq, w w Snce r o s a vector quaternon, equaton (19) apples: 1 r, = o PRPr P, (6) ro RP r where R P s a rotaton matrx. Comparng (5) wth the frst dervatve of (6): ro I P PR PRP roprp r roprp r, (7) yelds the followng equaltes: w P P C R R R. (8) w w C w R P P P P ne can recognze that w represents the angular velocty n the three dmensonal space (Wertz J.R., 1978) and w o represents the translaton velocty along the r o drecton, normalzed by P. he ensemble (w o +w)= forms a full quaternon referred to as generalzed angular velocty. hs term has been chosen because t s the composton of a normalzed lnear velocty and an angular velocty, both expressed n s -1 unts. Rewrtng (3) and (5) n matrx notatons yelds: ro ro r w w w w, (9) / w w / where equals under the constrant ww. Remark. As shown n (13), s composed by ts norm and unt quaternon. he latter can represent not only a smple vector rotaton n three dmensonal space, but a more general rotaton of an entre reference frame, n agreement wth the Euler heorem. herefore, the quaternon knematcs n (3) has four degrees of freedom (d.o.f.),.e.: 1. one d.o.f. related to the translaton velocty w (the dervatve of P defned n (6));. three d.o.f. related to the angular rate w of a reference frame whose one unt vector s parallel to r o (the dervatve of the rotaton matrx R P defned n (6)). Snce the vector knematcs n (5) descrbes only the moton of r o, the nformaton about the rotaton of the other two axes of the frame defned above becomes unnecessary. hs s confrmed by the vanshng of the parallel component w. hen, n agreement wth classcal mechancs, d.o.f. of vector knematcs reduce to three because of such an orthogonalty constrant. Equatons (5) and (6) can be vewed as output equatons of the state equaton (3). 3. Dynamcs Frst defne the generalzed angular acceleraton as the dervatve of the generalzed angular rate : aaaa a w w. (3) In (3), the decomposton of a nto normal and parallel components w.r.t. r o has been exploted. Be aware that w a and w a. Quaternon dynamcs follows by takng the dervatve of quaternon knematcs (3):, (31) where the quaternon gathers the effect of angular rate and acceleraton. Scalar and vector parts of are related to the components of and through: dd aww a ww. (3) he second dervatve of r o can be obtaned by explotng (5) and (31): ro roro ro q arow rowwroaro. (33) w wroq, qw qwqq 3
hs expresson has a clear smlarty wth the ordnary equaton of the relatve moton (see Kaplan, M.H., 1976 or Greenwood, D.. et al., 1965). herefore, a physcal meanng can be assgned to each term n (33): 1. q represents the acceleraton of the vector r ;. a roar o s the apparent acceleraton of r o w.r.t. r. In partcular:.1. a r o s the apparent acceleraton along r o ;.. ar o s the apparent acceleraton along a normal drecton to r o ; 3. ww r o s the Corols acceleraton; 4. w w r o w r s the centrfugal term. o Further developng (33) shows that the acceleraton of r o does not depend on the parallel component a : ro roro q. (34), a a Remark. As observed for knematcs, quaternon dynamcs (31) s more general than vector dynamcs (34). Snce the angular acceleraton a s related to the rotaton of a frame algned wth r o, t s unconstraned and then the former equaton has four d.o.f.. In (34), the parallel component a vanshes, showng an orthogonalty constrant on a, whch corresponds to a d.o.f. reducton. hen, (34) has only three d.o.f., n agreement wth classcal mechancs. 4. APPLICAINS nce obtaned the general knematc and dynamc equatons of full quaternons, the last step to be done s applyng them to orbtal moton. o ths end, consder a pont P wth mass m movng n the space, subect to a force F. Sx dfferent local reference frames can be attached to the pont mass. hey can be defned by takng all possble combnatons of two elements from the set composed by poston, velocty and acceleraton vectors. wo of the sx avalable frames are well known n Astronautcs and wll be consdered for the followng applcaton examples: 1. the Local rbtal Reference Frame (LRF);. the Local ertcal Local Horzontal frame (LLH). Both frames and ther orentaton w.r.t. an nertal reference are shown n Fgure 1. he nertal frame R, k,, s a Cartesan reference wth orgn n and unt vectors correspondng to, and k already ntroduced n (1). For each of the two orbtal frames, the followng problems wll be solved: 1. complete defnton of the frame axes;. assgnment of a full quaternon to the frame; 3. formulaton of the dfferental equaton of the full quaternon. v r k P k v r P Fgure 1 - LRF and LLH w.r.t. the nertal frame k k 4 4.1 LRF Reference Frame he LRF R,,, k s a Cartesan reference frame defned by velocty v and poston r as follows: 1. the orgn concdes wth P;. les along the velocty drecton; 3. s normal to the nstantaneous orbt plane (defned by poston and velocty); 4. k completes the frame. v v, rv rv, k. (35) he velocty vector and the orentaton of the LRF trple can be expressed through the LRF quaternon. he defnton of s arbtrary: for example the axs rotates nto and the axs rotates nto : v,. (36) Because there exst an nfnte number of rotatons satsfyng the left equaton n (36), a further constrant must be ntroduced: the rght equaton specfes that the -axs of the nertal frame must be rotated nto the orbtal plane normal drecton. Factorzng the left equaton n (36) as n (14) enlghtens the norm of the LRF quaternon to be equal to the square root of the velocty modulus, and the untary part to represent the orentaton of the velocty unt vector w.r.t. the nertal frame: vvnv, nv 1. (37) v, nv Now, one can apply formula (5) of quaternon knematcs to compute the acceleraton v : vw, vwv v, w, w, (38) where the dervatve of, beng zero by defnton, dsappears and w has been decomposed nto the normal and parallel components w and w w.r.t. v. he acceleraton of the pont mass can be related to the force F through Newton s Law and rememberng that w v : vf vf F mv w,, w. (39) mv mv Snce v and F are vector quaternons, a more compact expresson for the LRF angular rate can be used: 1 1 w m vv Fvw. (4) herefore, the orbtal equatons for LRF quaternon can be wrtten n quaternon form:,, (41) r, r r or matrx form, explotng (1) and (A.), as follows: v F, w mv. (4) r, rr Remark. As stated n Secton 3.1, the parallel component w does not gve contrbuton n (38). hs confrms the exstence of an orthogonalty constrant to w, meanng that the vector knematcs (38) has three d.o.f. he applcaton of Newton s
Law shows w to be completely ndependent on F. he angular rate w affects only (41), and represents the angular rate of the unt vectors and k around the axs. But f such vectors underwent a rotaton, the LRF frame would be lost. herefore the knematc quaternon constrant w follows. Consequently, the orbtal equaton (41) of the LRF quaternon s subect to a knematc constrant ( w ) and a statc constrant ( v ) that forces v to be a vector quaternon. hen the number of d.o.f. reduces from the eght possble to only sx, n agreement wth classcal mechancs. 4. LLH reference frame he LLH frame R,,, k s a Cartesan reference defned by poston r and velocty v as follows: 1. the orgn concdes wth P;. les along the poston drecton; 3. s normal to the nstantaneous orbt plane; 4. k completes the frame. r r, rv rv, k. (43) he poston vector and the orentaton of the LLH trple can be expressed through the LLH quaternon. In accordance wth (36) t can be defned as: r,. (44) Factorzng the left equaton n (44) enlghtens the norm of to be equal to the square root of the poston modulus, and the untary part to be the orentaton of r w.r.t. the nertal frame: rrnr, nr 1.(45) r, nr Now, one can apply the formula (5) of quaternon knematcs to compute the velocty v: rvw,rwr r, (46) where the decomposton of w nto normal and parallel components w and w w.r.t. r has been exploted. he quaternon knematcs of the LLH follows by (3):, (47) and the LLH dynamcs follows from (31):. (48) hen, recallng (33) and (34), the acceleraton can be determned as: rv a, rw, rw, w r arwwr. (49) r r, As done for LRF knematcs, one can relate acceleraton expresson to force F through Newton s Law. akng the dot product between poston and force yelds: a, rf m r w, w. (5) Cross product between poston and force brngs to: 1 rf a w, w w w.(51) mr Expressons (5) and (51) can be compacted nto: 1 aa, aa Fa r (5) F F rr rr m Fnally, the orbtal equatons for LLH quaternon can be wrtten n quaternon form:, 1 r Fa,. (53) Remark. As stated n Secton 3., the parallel component a does not gve contrbuton n (49). hs confrms the exstence of an orthogonalty constrant to a, meanng that the vector dynamcs (49) has three d.o.f. he applcaton of the Newton s Law, shows that a s unforced by F. he angular acceleraton a affects only (53), and represents an angular acceleraton of the unt vectors and k around the axs. But f such vectors underwent a rotaton, the LLH frame would be lost. herefore the dynamc and knematc quaternon constrants a and w follow. Fnally, the orbtal equaton (53) of the LLH quaternon s subect to a knematc constrant ( w ) and a dynamc constrant ( a ). herefore the number of d.o.f. reduces from the eght possble to only sx, n agreement wth classcal mechancs. 4.3 Unform Crcular Moton hs secton ends wth a smple example: the unform crcular moton of P around, sketched n Fgure. k r k P v k Fgure - Unform crcular moton around LLH and LRF quaternon defntons are the same as n (44) and (36). Frst, quaternon knematcs s appled, startng from LLH case. he generalzed angular velocty of the LLH quaternon s: w, w. (54) he generalzed angular velocty s concdent wth the angular rate of P around, denoted wth ω rv r. hs leads to the next result showng quaternon knematc equaton to be smlar to classcal vector form: r ωr w r r, (55) where w ω /. By applyng (31), quaternon dynamc equaton can be obtaned. Quaternon dynamcs, as seen for knematcs, appears smlar to classcal vector form: v ω r a w w w,(56) v v where a by defnton of unform moton. By usng LRF nstead of LLH, dynamcs s represented by quaternon knematcs, because the 5
quaternon descrbes pont velocty, nstead of poston. LRF dynamcs s: v ωv w v v. (57) Knematcs follows from defnton (36): r v r. (58) r r r he last four equatons show that the angular veloctes of the LRF and LLH quaternons are the same, namely ω/w w. hs follows from the fact that, for unform crcular moton, poston and velocty are always orthogonal, then rotatng wth the same angular rate. 5. SIMULAIN RESULS he LRF and LLH orbtal equatons n (41) and (53) have been mplemented and tested n a MALAB Smulnk orbt smulator, wth a gravty model accountng for central feld and J contrbutons. he GCE orbt has been taken as a reference (see Canuto, E. et al., ). he latter s a near-crcular ( <.5) sun-synchronous, quaspolar ( 96 ) orbt at a mean alttude h = 5Km. Fgure 3 shows the tme seres of the four components of the LRF quaternon durng a tme horzon of four orbts. he quaternon has a clear perodcty of two orbts that s equvalent to half the orbtal frequency f.19mhz. Snce the nonlnear defnton (36) of the LRF quaternon apples, the velocty v s perodc wth orbtal frequency f. he same consderatons can also be applcable on the LLH quaternon, that s llustrated n Fgure 4. Fgure 5 shows the GCE orbt around the Earth w.r.t. nertal coordnates. ne can note from the pcture that the orbt s near-crcular and quas-polar. LRF Quaternon [sqrt(m/s)] 8 6 4 - -4 LRF Quaternon for the GCE orbt -6 1 orbt = 54s -8.7.54.81 1.8 me [s] 1.35 1.6 1.89.16 x 1 4 Fgure 3 - me seres of the LRF quaternon components LLH Quaternon [sqrt(m)] 5 15 1 5-5 -1-15 - LLH Quaternon for the GCE orbt 1 orbt = 54s -5.7.54.81 1.8 1.35 1.6 1.89.16 me [s] x 1 4 Fgure 4 - me seres of the LLH quaternon components r r1 r r3 r r1 r r3 6 Fgure 5 - he GCE orbt generated by LRF/LLH smulator 6. CNCLUSINS AND FUURE DEELPMENS he orbt dynamcs and knematcs for the pont mass moton has been transformed from classcal vector notaton nto a new quaternon form. Among future developments, the desgn of quaternon observer and control wll cover the most mportant role. 7. REFERENCES [1] Wertz, J.R. (1978). Spacecraft Atttude Determnaton and Control. D.Redel Publshng Company. [] Kaplan, M.H. (1976). Modern Spacecraft Dynamcs & Control. John Wley & Sons. [3] Canuto, E., B.Bona, G.Calafore and M.Indr (). Drag Free Control for the European Satellte GCE. Part I: Modellng. In: Proc. of 41st IEEE Conference on Decson and Control, Las egas, Nevada USA [4] Chou, J.C.K. (199). Quaternon Knematc and Dynamc Dfferental Equatons. In: IEEE rans. on Robotcs and Automaton, ol.8, N.1 [5] Greenwood, D.. and W.M. Greenfeld (1965). Prncples of Dynamcs. Prentce Hall 8. APPENDIX Algebra - Conugate Multplcaton propertes When quaternon multplcaton nvolves conugates, commutatve property (1) stll hold. hen the product C can be wrtten n matrx notaton through: C. (A.1) From prevous equaton t follows: and. (A.) Algebra - Some Interestng Matrces It s useful to ntroduce the followng matrx notatons: E x x ICx. (A.3) E x x ICx By explotng the new notaton, the matrx expresson of quaternons ntroduced n (9) can be rewrtten as: E E E E. (A.4)