Simulation of Spacecraft Attitude and Orbit Dynamics

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1 Smulton o Spcect Atttude nd Obt Dynmcs Ps Rhmäk, Jen-Pete Ylén Contol Engneeng Lbotoy Helsnk Unvesty o Technology PL-, TKK E-ml: ps.hmk@tkk., pete.ylen@tkk. KEYWORDS Smulton Model, Stellte, FDIR, Qutenon ABSTRACT In ths ppe, the smulton model o stellte tttude nd obt dynmcs s dscussed. The stellte tttude model hs been epesented n tem o qutenon nd odny deentl equton s used to descbe the stellte obtl moton. The deent ctutos nd sensos hve been modeled wth sutble ults nd lues. The smulton model enbles us to consde the stellte moton unde deent envonmentl petubtons (o exmple eodynmc dg, extenl celestl body etc.) nd lue n ctutos nd sensos. The smulton model s utlzed n the development o tttude nd obt contol lgothms o ult detecton, solton nd ecovey (FDIR) technologes. Smulton esults e lso gven. INTRODUCTION Dung the lst decdes modelng, smulton, nd wde computtonl scence nd engneeng hve become moe nd moe mpotnt tools n the esech nd development pojects. The desgn phse hs to be educed n tme nd cost when the use o new des nd tools becomes possble. Ths s lso the tend n spce pplcton n whch the el tests e not possble o t lest they e expensve. New demnds on the eospce nd contol engneeng hve become up nd they hve to be ble to nswe to equements. Spcect smultos o smultos n genel, e sotwe tools tht cn be used by eseches, engnees, students o eveybody to nlyze nd ssess system opetons, behvos, nd to nswe to the questons egdng phenomenon o poduct. The smultons e essentl tools n the msson nd spcect contol desgn. Fo exmple, the scentc mssons e unque nd the nstumentton o spcect s desgned only o ths specc msson. Thee e not ny edy-to-use pltoms tht cn be used. Hence, t s not possble to vey the opeton o contol lgothms nd stteges n el pocess but the smulton envonments cn be used. Thee e plenty o compnes tht oe the smulton sevces to the esech nsttutes nd spce compnes. MODEL STRUCTURE AND MATHEMATICS Model Stuctue The spcect smulton model s ognzed lke ny ctul contol loops (See Fgues.). The nteces o the components e dened nd modeled n such wy tht the smulton model would be s modul s possble. Modctons to the smulton model e esy to do nd one pt o the model cn be esly eplced wth nothe. The model s ntlzed nd contolled om the coodnton level. Ths mens tht the model pmetes nd possble ults nd lues n the FDIR smulton cse e dened. Fgues. Spcect smulton model stuctue. Coodnte Systems Thee deent coodnte systems e dened n the smulto:. Inetl Coodnte System (ICS),. Obt Coodnte System (OCS), nd 3. Body Coodnte System (BCS). The netl coodnte system s usully dened such tht the cente o mss o the Eth (cm) cts s ogn nd the decton o the xes e xed to the sol system. Ths knd o coodnte system s not exctly netl but t s enough o ll engneeng puposes (Sd 997). The Z-xs o the ICS s the otton xs o the Eth n postve decton nd the X-Y plne s the equtol plne o the Eth, whch s pependcul to the Eth s otton xs. The venl equnox vecto ϒ s selected to be the X-xs o the ICS. Fnlly, the Y-xs

2 hs been chosen n such wy tht the ICS s ghthnded othogonl coodnte system. Obt coodnte system s lso ght-hnded othogonl coodnte system wth ogn n the cente o the stellte mss. The Z-xs s pontng towds the cente o the Eth; X-xs to the decton o stellte pependcul to the Z-xs, nd Y-xs completes the coodnte system such tht t s ght-hnded nd othogonl. The thd coodnte system, whch hs been xed to the movng nd ottng spcect, denes stellte oentton. Rotton The tttude tnsomton n spce cn be executed by usng vous deent spects. In the smulton model, the qutenon technque s used. The mn etue o qutenons s tht they povde convenent poduct ule o successve ottons nd they hve smple om o knemtcs (Wetz 978, Ws newsk 996). The bsc denton o the qutenon s consequence o the popety o the decton cosne mtx A tht t hs t lest one egenvlue o unty. Ths mens tht thee s n egenvecto e (Eule xs) tht s unchnged n evey otton. The qutenon s dened s vecto () whee q R,, j nd k stsy the Hmlton s ule () nd whee the length o the qutenon s unty. (Sd 997) q = q + q+ q j+ q k () 4 3 = j = k = jk = j = j = k jk = kj = k = k = j () When the Eule xs e o the otton s known the connecton between qutenon nd the otton Eule xs s q = esn q = esn q3 = e3sn q4 = cos whee e s component o Eule xs nd α s the mgntude o the otton. The nl combned otton o two successve ottons cn be peomed s mtx-vecto multplcton (3) whee q nd q e the ndvdul ottons. q q q q q ' ' ' ' 4 3 ' ' ' ' q q3 q4 q q ' ' ' ' q q q q4 q 3 3 ' ' ' ' q q q q3 q4 4 q'' = qq ' = (3) SIMULATION MODEL The smulton model s elzed n the MATLAB/ SIMULINK-envonment. Obt Model The moton o celestl body s bsed on the qute elementy pncples o celestl mechncs. In the 7 th centuy, J. Keple povded thee bsc empcl lws tht descbe the moton o plnet n unpetubed plnety obt. The obtl dynmcs o stellte s extensvely explned n mny books, o exmple (Sd 997) nd (Wetz 978). I we consde system o two ptcles P nd P o msses m nd m nd pply Newton s second lw nd the lw o gvty to the two-body system, we cn get the undmentl equton (4) o the moton o the twobody system whee the symbol µ = G(m +m ) nd G s the unvesl constnt o gvtton. Ths equton descbes the moton o the ptcle P eltve to the second mss P. µ + = (4) 3 In genel, ptcle P moves n oce eld F, the momentum o the oce F bout ogn O s M= F whee s the poston vecto o the ptcle P. The ngul momentum bout ogn s h=m( v)= p whee p s the lne momentum o the ptcle. Thus, the tme te o the ngul momentum h s equl to the moment o the oce F. dh d = ( mv) = + F = M () dt dt The equton () sttes the undmentl ct tht the momentum ctng on ptcle s equl to the tme te o the chnge o ts ngul momentum. In spce scence t s common to descbe the stellte obt by ve numbes, known s obtl elements o clsscl obtl elements (COE). A sxth element s dded to detemne the locton o the stellte n ts obt (Wetz 978 nd Sd 997). The clsscl obtl

3 elements hve been descbed n the Tble. Becuse these elements e pooly dened e nd/o s equl to zeo, so-clled equnoctl obtl elements (EOE) hve been dened n tems o the clsscl obtl elements. The equnoctl obtl elements hve been dened n Tble. Tble. The clsscl obtl elements. (Sd 997) Symbol e Ω ω M the sem mjo xs the eccentcty the nclnton the ght scenson o the scendng node the gument o pegee the men nomly the devtve o clsscl obtl elements when the petubng oce s consevtve o non-consevtve. Knowng the ntl condton o COE the Guss equtons cn be ntegted to clculte the evoluton o the elements. Guss equton s epesented n equton (7), whee s the ngle between stellte locton vecto nd the vecto pontng towds pegee (See Fgues 3.), p = (-e ), n = sqt(µ/ 3 ), = p/(+e cos()), nd, nd z e the components o the petubng oce long the dus vecto decton, the tnsvese obt decton nd the decton o the noml to the obt plne, espectvely. To vod the sngulty due to the pooly dened pmetes, the Guss equtons cn be ewtten n the tems o the equnoctl elements s n (8), whee b= sqt(-p -P ), h=nb, p/=+ P sn(l)+p cos(l), L=ω+Ω+, nd K=ω+Ω+E. (We nd Rothmy, C.M. ) Fgues. The dentons o the elements. Tble. The dentons o the equnoctl obtl elements (EOE). (We nd Rothmy, C.M. ) EOE P esn( Ω+ ω) P ecos( Ω+ ω) Q tn sn( Ω) Q tn cos( Ω) l Ω+ ω + M In Keplen obt the devtve o the st ve obtl elements e equl to zeo. I the stellte obtl elements e known the stellte locton nd the velocty vecto v cn be clculted, nd vse ves. Algothms to do ths cn be ound n ny textbook concenng obtl dynmcs, o exmple (Sd 997), (Wetz 978). In the genel cse, n whch ny knd o petubng oce cn exst, the equton o obtl moton s µ + = 3 p wth ntl condton nd whee p s the petubng oce pe unt mss. Due to the petubng cceleton the obtl elements e not constnts. Hence, so-clled Guss om o Lgnge s plnety equtons descbes Fgues 3. The spcect obt nd uxly ccle. Atttude Model Dynmcs Fom equton () we get tht the toque ctng on the stellte body s equl to the devtve o the ngul momentum o the spcect n the netl coodnte system. Hence, n the ottng body coodnte system. T= hi = h+ h I momentum exchnge devces e used n the contol, the ngul momentum vecto h = h B +h w whee h B s the ngul momentum o stellte gd body nd h w s the ngul momentum o the momentum exchnge devces. Hence, the tme te o ngul velocty o the stellte body s lke n equton (6). = ( Is ) I s + T h w ( Is) hw (6) Knemtcs The spcect tttude hs been modeled s qutenon epesentton q=(q, q, q 3, q 4 ). Hence, the equton (9), whee ω s the stellte ngul velocty bout stellte body xs, gves the devtve o qutenon vecto.

4 = esn + + ecos µ p ( ) ( ( ) ) ( ω + ) sn() ( ω + ) cos µ psn() ( ) ( ) ( ) p e + cos e = sn( ) + cos( ) + µ ecos + z cos( ω ) + = µ p Ω= sn z sn ω = z µ p p cos + sn e µ p e M n n ne p = + cos( ) + sn( ) p = ( Psn ( L) Pcos( L) ) + h p p P = cos( L) + P+ + sn ( L) h ( cos sn) P Q L Q L z p p P = sn ( L) + P + + cos( L) h ( cos sn ) + P Q L Q L z Q = ( + Q + Q) sn( L) z h Q = ( + Q + Q) cos( L) z h p b l = n Psn ( L) Pcos( L) h b p + + ( Pcos( L) Psn ( L) ) + b ( cos sn ) + Q L Q L z q ω3 ω ωq q d ω3 ω ω q = dt q3 ω ω ω3 q3 q4 ω ω ω3 q4 (7) (8) (9) Tt = t F t () The de o ecton wheels (o momentum exchnge devces) s to tnse the ngul momentum o the whole system between deent pts o the spcect wthout chngng ts ovell ntenl ngul momentum. The cheved toque level s o the ode o. Nm. (Sd 997) The ecton-wheel s modeled s equton (). w = Tdem hw = Iww µ () In mgnetotoque, the contol toque T mg s geneted by n ntecton o the Eth s geomgnetc eld B(t) wth the mgnetc dpole momentum m(t) (See equtons () nd (3)) whee n col s the numbe o col, col (t) s the mgnetotoque cuent, A col loop e, nd ˆn s the unt noml vecto to the plne o the loop. Sensos Tmg () t = m() t B () t () m() t = n () t A n ˆ (3) col col col In the smulton model the modeled sensos e: cose Eth nd Sun Senso (CESS), st tcke, mgnetomete, gyo, nd GPS. The CESS s modeled s component tht gves the decton o Sun nd Eth n the body coodnte system. The st tcke s modeled s component tht gves the stellte tttude contmnted wth n uncetnty tht depends on the stellte ngul speed. Mgnetomete s modeled s component tht gves the mgntude nd decton o the Eth s mgnetc eld. WMM mgnetc model s used n the smulto. Actutos The ctutos e used to poduce the contol toques nd oces o the stellte tttude contol. The modeled ctutos e: thuste, ecton-wheels, nd mgnetotoque. The thuste hs been modeled s thust oce vecto F t ectng the stellte n poston t. Hence, the toque bout the cente o the mss o the spcect s the coss poduct between the poston nd the oce vectos (Equton ()). A gyoscope s modeled s n nstument tht mesues the ngul speed o the spcect. The ctul ngul speed s contmnted wth eltvely smll Gussn ndom uncetnty. A GPS s modeled s n nstument tht gves the stellte locton n the netl Eth centeed coodnte system. Fults One o the mn ms o the spcect smulton model s tht t cn be used n the FDIR-smulton. Hence, the ults nd lues hve to be tken nto ccount ledy n the desgn nd modelng phse. The ults cn occu n ny pt o the model nd ny knd o

5 ults e possble. Usully, the ults e ethe ddtve o pmetc but lso totl blckout o component s possble. Pehps, the most pevlent ult s ce buldng on the suce o ny optcl nstument ncesng the nccucy o ths element. Envonmentl Toques The mn souces o the envonmentl toques e epesented n the Tble 3. Tble 3. The mn envonmentl toques (Wetz 978). Souce Dependence Domnnt Aeodynmc e -α below ~ km Mgnetc / 3 ~ - 3 km Gvty Gdent / 3 ~ - 3 km Sol Rdton Independent Inteplnety spce bove synchonous lttude Mcometeotes ndependent Nomlly neglgble The eodynmc dg s one o the mn envonmentl toques o the spcect n low obt. The eodynmc dg model hs been explned extensvely, o exmple, n the book (Wetz 978). The oce d on the suce elements da s gven by equton (4) whee ˆN s outwd noml o the suce element da, ˆV unt vecto o the tnsltonl velocty, ρ s the densty nd C D s the dg coecent o the suce. In el tems, the dg coecent C D s uncton o the suce stuctue nd the locl ngle o ttchment nd ts vlue s usully between nd. Fo ll pctcl pplctons, the vlue C D = cn be used. (Wetz 978) d ˆ ˆ ˆ eo = CDρV N V V da (4) In the smulton model, the stellte stuctue hs been ppoxmted by collecton o smple geometcl gues. Hence, the totl eodynmc toque s the sum ove the toques ctng on ndvdul pts o the spcect. T = eo = C ρv A Nˆ Vˆ Vˆ D Any nonsymmetcl body n obt s subject to gvttonl toque becuse o the vton n the Eth s gvttonl oce ove the object. Usully n the ltetue, the gvty-gdent s only deved o the unelstc sphecl Eth model. Due to the nonsphecty nd the non-homogenous mss dstbuton o the Eth the el gvttonl eld s gnul. Fo sphecl Eth, the gvttonl oce d ctng on s/c mss element dm locted t poston R s d µ R dm =. 3 R Hence, the toque bout the stellte geometc cente due to oce d, t poston, s ( ' ) dt = d = + d whee ρ s the vecto om the geometc cente to the cm nd om cm to the mss element dm. Assumng tht the cm nd the geometc cente o the s/c le n the sme pont the gvty-gdent s 3µ T = ˆ ˆ R I R gg 3 s s s R s whee R ˆ s s unt vecto long R s nd I s s the spcect netl mtx. (Wetz 978) A SIMULATION CASE In ths secton, some smulton esults obtned by the bove-descbed smulton model e pesented. The smulton cse s smple nd nced. The obt o the smulton cse s ccul wth n lttude o 4 km nd nclnton 87. The moments o net o the stellte e I xx =36, I yy =7, I zz =6, nd I xy = I xy = I xz = I yz = kgm. The m s tht the tttude contol system shll ensue thee-xs stblzton o the stellte. The stellte tttude s mesued by GPS senso nd only thee ecton wheels e used n the contol. The ecton wheels e mounted othogonlly such tht the otton xes e long X, Y nd Z-xs o the stellte body. Thee PID-contolles e used to clculte the contol toques. The smulton esults hve been epesented n Fgues 4-8. Roll ngle, [deg] Yw ngle, [deg] Ptch ngle, [deg] Tme, [s] Fgues 4. Atttude ngles.

6 ω x x q q q 3 q 4 ω y ω z e x x x Tme, [s] Fgues. The stellte ngul tes ω Ω M - Tme, [s] Fgues 6. The clsscl obtl elements n smulton. CONCLUSION The stellte tttude nd obt smulton model wth the most common ctutos nd sensos hve been ntoduced n ths ppe. The smulton model cn be utlzed both n the contol lgothm desgns nd n the development o FDIR methods. The smulton models hve been mplemented n the MATLAB/SIMULINK envonment. 3 q Tme, [s] REFERENCES Fgues 8. The tttude qutenons. Nsz, B.J.. Clsscl Element Feedbck Contol o Spcect Obt Mneuves Thess, Mste o Scence. Vgnn Polytechnc Insttute nd Stte Unvesty, Aeospce Engneeng. 9 p. Sd, M.J Spcect Dynmcs & Contol A Pctcl Engneeng Appoch Cmbdge Unvesty Pess. 49 p. ISBN We, B. Rothmy, C.M.. Integted Obt, Atttude, nd Stuctul Contol Systems Desgn o Spce Sol Powe Stelltes NASA/TM--84. [.3.] -tm84.pd Wetz, J.R Spcect Atttude Detemnton nd Contol D. Redel Publshng Compny, Dodecht, Hollnd.88 p. ISBN Ws newsk, R Stellte Atttude Contol Usng Only Electomgnetc Actuton Ph. D. Thess. Denmk, Albog Unvesty, Deptment o Contol Engneeng. 7 p. AUTHOR BIBLIOGRAPHIES PASI RIIHIMÄKI ws bon n Ähtä, Fnlnd nd went to Helsnk Unvesty o Technology, whee he eceved mste s degee n utomton nd system scence. Nowdys, he s postgdute student n the Contol Engneeng Lbotoy n Helsnk Unvesty o Technology. ω w ω ω ω 3 JEAN-PETER YLÉN ws bon n Helsnk, Fnlnd nd went to Helsnk Unvesty o Technology, whee he eceved mste s degee n chemcl engneeng nd docto s degee n utomton nd system scence. Tme, [s] Fgues 7. The ngul tes o ecton wheels.