CONTROL OF A SPACE ROBOT FOR CAPTURING A TUMBLING OBJECT

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1 CONROL OF A SPACE ROBO FOR CAPURING A UMBLING OBJEC Ou Ma (), Agl Flos-Abad (), Khah Pham (2) () Dpam o Mchacal ad Aospac Egg, Nw Mxco Sa Uvsy, Las Cucs, NM 88, USA Emal: oma@msu.du, a_abad@msu.du (2) U.S. A Foc Rsach Laboaoy, Spac Vhcl Dcoa, Klad A Foc Bas, NM , USA Emal: AFRL.RVSV@klad.a.ml ABSRAC hs pap pss a opmal cool sagy o a spac obo o capu a -umblg objc ud h codo o havg mmal mpac o h bas sall dug h capug opao. h da s o s pdc a ag m, locao ad spd o h umblg objc o h obo o cp wh such ha, wh h obo had physcally ouchs h objc, wll as a mmal agula mpuls o h bas sall. h, a opmal moo ajcoy ad h cospodg jo oqus wll b gad o cool h obo o ach h objc a h agd m ad locao. Jo a ad oqu lms wll b ak o accou h opmal cool soluo. Sc h cool acs bo a physcal coac happs, wll o ac ay xsg complac cool capably o h spac obo gadg boh mplmao ad opao. ho, h poposd mhod ca co-xs wh a xsg complac cool mhod h obo cool sysm. A umcal smulao xampl s psd o dmosa h cvss o h poposd mhod. A umcal smulao xampl s psd o dmosa h cvss o h poposd mhod.. GENERAL SPECIFICAIONS Spac mapulaos hav b succssully usd o may applcaos such as mauvg asoaus, bhg ad dployg lag spac sucus, cosucg ad maag h Iaoal Spac Sao (ISS), xplog ad sampl-collcg, sall o-ob svcg (chology dmos oly), c. All o hs mapulao acvs dal wh coopav payloads o ag objcs ad hus, h xsg obocs chologs ca hadl hm qu wll, hough may mpovms such as h opaoal ccy ad dxy may sll b do. Howv, a mapulao s xpcd o pom mo challgg ad sk asks, such as o capu a ukow objc, lk a pc o spac dbs, o a o-coopav objc, such as a umblg sall, h culy avalabl spac obocs chologs a sll a om bg ady. o mak hs challgg asks paccal, may ablg chologs hav o b uh advacd. hs sach dvlops ablg chology o a spac mapulao o capu a umblg sall. h opao o capug a umblg sall a complo o h dzvous pocss may b dvdd o ou phass. h s phas s h obsvg ad plag phas, whch s o acqu moo (maly oaoal moo) omao o h ag sall ad h dm wh ad wh o gasp h ag sall. h scod phas s h al appoachg phas, whch h obo s coolld o mov s dco o h plad gaspg locao o gaspg h ag sall a h plad m. h hd phas s h capug (o cpo) phas, whch h obo capus h ag sall. hs s h phas wh physcal coac happs ad hus s also h mos sky phas. h las s h pos-capu sablzao phas whch h umblg ag sall s dumbld ad sablzd by h obo ad svcg sall. h wok pod hs pap s cocd oly wh h plag pa o h s phas ad h whol scod phas. Ou ocus o h s phas s o dm a opmal m ad locao basd o h umblg moo o h ag sall o h obo s d-co o cp wh h ag sall. h ocus o h scod phas s o cool h obo o ach h opmal locao wh a mmal dsubac o h aud o h svcg sall o sa capu o h umblg ag sall. Rsach o h obsvao pa o h s phas has b do by may sachs h lds o compu vso ad ssg chologs. h hd ad ouh phass a mo sky ad challgg bcaus o h volvm o physcal coac. Som sach wok has b do bu much mo uu wok s absoluly qud od o hav guaad sa ad succssul uu mssos. W wll o dscuss hs h bcaus hy a ou o h scop o h pap. h mpac mmzao poblm o capug a ag objc has b sudd by a w sachs om d pspcvs. Yoshda al [] modld h collso dyamcs dug h capug pocss usg h xdd galzd so. hy ocusd o h moms jus bo ad a h mpac usg vlocy laos. Yoshda ad Nchv [2] oducd h cocp o aco ull-spac o aalyz h mpac ad pos-mpac moms o h capug pocss. hy oud ha choosg coguaos wh h aco

2 ull-spac o h svcg mapulao sysm ca sul a opao wh mmum mpac o h aud o h svcg sall. Papadopoulos ad Paaskvas [] poposd a mhodology basd o h pcusso po o bods o mmz h ocs sad o h momum asmd o h bas o h mapulao wh gaspg a objc. I all h pas suds h coac oc was jus a assumd mpulsv oc xd o h p o h mapulao whou akg o accou o h gomy o h coacg bods ad h umblg moo o h ag sall. I hs sach, w mov a sp owad o cosd h umblg moo o h ag sall h obo cool sagy o achvg mmal aud mpac o h svcg sall. h poblm o opmal ajcoy plag o a spac mapulao was addssd al by Duvowsky ad os [4]. hy oducd a hacd dsubac map, whch ca ad slcg a pah ha ducs h dsubacs o h bas spacca by dyg h dco o ach jo movm whch suls mmum o maxmum dsubacs. Agawal ad Xu [5] poposd a global opmum pah plag o duda spac mapulaos usg a vaaoal appoach o mmz h objcv ucoal wh cosas h la ad agula momum. Lampallo al [6] poposd a opmal moo plag mhod usg ca h jo spac. Huag al [7] poposd a opmal appoach ajcoy plag mhod o mmzg h mpac o h bas sall, h opmal ajcoy s oud basd o a gc algohm ad h dyamc couplg aco. Aghl [8] dsgd a opmal cooll o capu a umblg sall usg a objcv uco mmzg h opao m ad lav vlocy bw h obo p ad h ag.. Ok al [9] also poposd a opmal cool mhod o capu a umblg sall bu hy ocusd maly o mmzg h opaoal m o as capu. h ma dc bw ou appoach ad hos opmal cool appoachs s ha w ocus o h mmzao o h aco oqu o h svcg sall o sa capu opao. I hs pap, h ms svcg sall ad bas sall a xchagabl, so a h ms ag objc ad ag sall ad h ms mapulao ad obo. 2. DYNAMICS MODELLING 2.. Basc assumpos h dvlopm o h mhodology dscbd hs pap s basd o ollowg basc assumpos: (a) Boh h svcg sall ad h ag objc a assumd o b gd bods. h mapulao also cosss o gd lks. (b) h mass pops ad moo sa o boh h bas sall ad h ag objc a assumd kow. (c) h mauvs a clos poxmy ag ad hus h c o obal mchacs s glcd. (d) h aud o h bas sall s ully coolld ulss ohws sad. Assumpo (a) s a vy usual assumpo h obocs ld, spcally o dvlopm ad paccal mplmao o cool mhodologs bcaus a gd-body dyamcal sysm s much as o modl ad aalyz. I may applcaos such assumpo s also paccally suc. hs assumpo may b oo o aly o a log spac mapulao o capu a as umblg objc. Assumpo (b) s o ocus h sach o h obo cool poblm ad avod dalg wh h a dcao ad moo sa smao poblms, whch a wo sach aas havg b wll sudd ad a couously bg sudd by may oh sachs. Assumpo (c) s o ocus h sach o h scop o poxmy dzvous ad capu, wh h ocs/moms lad o obal mchacs a glgbl compad o h a ocs causd by h obo moo ad h coac ocs causd by h physcal cpo. Assumpo (d) has b a commo appoach o all h paccal capug opaos spac bcaus ucoolld aud ca sgcaly cas h possbly o msso alu. W a wll awa ha hs assumpos may o b alsc may applcao cass. hy a mposd o acla ou aly dvlopm o h chology. W wll b laxg hs assumpos h uu sach Dyamcs Modllg o h Svcg Sysm h mulbody sysm o h svcg sall ad h mapulao cosss o gd bods cocd by jos, as show Fg.. Body s h sall whch s also h bas o h obo ad body (,2,, ) s h -h lk o h mapulao. Jo has 6 dgs o dom whch cocs h a am o h svcg sall ad Jo ( j,2,, ) has oly o dg o dom whch aculas lks - ad. h symbols appag Fg. a dd as ollows: θ R : galzd jo coodas τ R : galzd jo oqus R : poso vco o h CM o Body R : poso vco o h mass c o h c svcg sysm R : poso vco o h mapulao d-co

3 a R :abodyvco o lk xpssd F am c R : poso vco o h CM o Lk masud om Jo z R : oaoal axs o h h jo v R : la vlocy o h mass c o Lk ω R : agula vlocy o h hlk v R : la vlocy o h svcg sall ω R : agula vlocy o h svcg sall v R : la vlocy o h d-co ω R : agula vlocy o h d-co R 6 : xal oc ad mom xd o h d-co 6 R : xal oc ad mom xd o h svcg sall R : aco ocs a h h oo o h mapulao τ R : aco oqu a h h oo o h mapulao z 2 2 Lk F2 ω a v C z τ F c Lk 2 a2 c2 ω 2 C2 a v 2 ω ω F- O- c oc c- C Lk - a- c C- F z v F c Lk a C Svc Sall Body (B ) v F ω ω Fg. Mulbody dyamc sysm o a svcg sall ad a spac mapulao I h al plag ad appoachg phass h a o xal ocs acg o h sysms ad hus, h momum o h svcg sysm wll b cosvd, om whch w ca dv h dyamcs quao o h spac obo ms o s jo vaabls θ as ollows []: wh Hθ Cθ τ () Cθ θ Hθ R θ 2 whch cluds h ola Cools ad cugal ocs acg o h sysm ad b b b R v (2) H H H H H () Moov, H s h galzd a max o h mapulao wh s aachd o a -loag bas. h oh vaabls o h sysm a dd as ollows: H J I J R : galzd a max o h mapulao as s aachd o a xd bas. 6 b m R : H J H couplg a max bw h svcg sall ad h mapulao. H b m mr mrc H svcg sall. c R 66 : a max o h I R :a max o lk wh spc o s mass c J z z2... z... R : Jacoba max o h agula vlocy o h -h body. J z ρ z ρ z ρ v c 2 c2 c R Jacoba max o h la vlocy o h -h body J mj m v R m v R H I J Z J m : oal mass o h svcg sysm m : mass o h -h body R : dy max z() z(2) Z z () z () R : max om o z(2) z() ps h coss poduc opao o ay vco. H I m RR I R ρcj R : poso vco om h jh jo o h mass c o h h body. 2.. Dyamcs Modllg o h ag Sall Sc h ag sall s assumd o b a sgl oag gd body, s dyamcs quao s ah smpl, Iω +ω Iω = τ (4) wh

4 I R : a max o h ag sall. ω R : agula vlocy o h ag sall. ω R : agula acclao o h ag sall. τ R : xal oqu appld o h ag sall. Basd o h assumpo o gog h obal mchacs, h xal oqu τ s zo bo a capu opao ad s h coac oqu dug capug.. DEERMINAION OF HE OPIMAL CAPURE IME As w hav sad aly, h ovall goal o h cool dsg s o capu h umblg sall wh mmal mpac o h aud moo o h svcg sall. h s sp o achvg such a goal s o dm a bs m o h obo o gasp. I s udsadabl ha h sula coac oc xd a h obo p (sulg om a capu aco) passs h mass c o h svcg sysm, h coac oc wll o caus ay aud dsubac o h svcg sall, as show Fg. 2. Howv, h dco o coac oc dpds o h lav vlocy, coacg spos ad coac gomy, whch mak vy dcul o pdc advac. Alhough such a pdco may o b mpossbl w hav a accua coac dyamcs modl, hs wll qu mo xsv sach wok h uu. Fo hs wok, w appoxma by assumg ha h coac oc s alog h dco o h lav vlocy bw h obo p ad h gaspg hadl o h ag sall. ho, o aud dsubac o h svcg sall ca b achvd by h o h ollowg codos: ) h lav vlocy bw h obo p ad h gaspg hadl o h ag sall s zo. 2) h lav vlocy s ozo bu s dco passs hough h mass c o h svcg sysm. h dsg o cool sags o m s codo s a commo appoach such as h wok dscbd [8]. I qus ha h obo p mus mov as as as h gaspg hadl o h umblg sall. hs s vy dcul o mpossbl wh h ag sall has a as umblg moo bcaus h p spd o a mapulao s always lmd o oly by h jo a lms bu also by h aud olac o h svcg sall. I such a cas, a sagy usg h scod codo as s cool goal bcoms mo aacv bcaus dos o qus zo lav vlocy, bu such a appoach has o b sudd h pas ho, w wll ocus ou sudy o achvg h scod codo. As show Fg., h scod codo mas ha h agl β (bw h lav vlocy ad h poso vco o h gaspg hadl o h ag sall) should b zo. I such a cas, h majo compo o h mpac oc (assumg maly alog h lav-vlocy dco) wll pass hough h mass c o h svcg sysm ad hus, caus o agula mom o h svcg sall. O cous, hs s oly a dal cas. I a gal umblg cas, h dco o h lav vlocy may v pass hough h mass c o h svcg sall. Howv, v h β agl ca v ach zo, wll always hav a mmal valu a a ca m. Hc, w wll jus ocus o h poblm o dm such a mmal agl. hs ca b omulad as: gv a s o al moo codos o h ag sall, d a uu m such ha h vlocy o h gaspg hadl, v ( ), wll hav zo o mmum momum abou h mass c o h svcg sysm. Mahmacally, hs ca b xpssd as wh v( ) ( ) max cos max v( ) ( ) v( ) v R( ωa) () Ra c Mass c o h svcg sysm (5) (6) Fg. 2 Icpo o mmal mpac o h bas sall basd o h coac oc dco. I h abov poblm do, R R s h oao max dg oao o h sall am F wh spc o h obal am F. Poso vco a pos o h gaspg hadl om h mass c o h ag sall, xpssd h sall s body-xd am F. No ha boh max R ad vco ω a ola ucos o m alhough h sall s udgog a oqu oao. R () ad ω () ca b solvd om h ollowg dyamcs quao ad kow al codos:

5 I ω + ω I ω = ()=, v() v, ω () ω (7) Soluo o ω () o ay gv m h sall local am ca b obad closd-om hough h valuao o Jacoba Ellpc ucos []. Howv, h closd-om soluo o ω () global am ad h closd-om soluo o h oao R() a vy dcul, as dscussd cly [4]. A ay a, umcal soluo o h opmzao poblm (5) s always possbl. Wh h a o xal ocs ad moms appld o h ag sall, s umblg moo should b podcal ov h m. hus, h abovdscussd opmal capu m wll b pad wh h pod o h sall s umblg moo. hs mas ha w ca hav ough m o ppa o a sa ad opmal capu bcaus h dsd capu m ad oppouy wll com padly ov h m. O cous, h aly, h ag wll ulkly b dog pcly podc oao bcaus h always xs som o-zo xal ocs ad moms as wll as dampg acos h sysm. ho, h umblg moo may o b kp h sam pod ov. Howv, such chagg s lkly b slow m ad hus, w wll sll hav m o pla ad pom a opmal capu ask, as dscbd h x sco. ω() ω R () R Gaspg hadl F z Mass c o svcg sysm h() h v() ω() Mass c o z ag sall () y y ω( ) ω? R( ) R v () v( ) F ( ) x ( )? Fg. Icpo o mmal mpac o h bas sall basd o h coac oc dco 4. OIMAL CONROL FOR HE ROBO S FINAL APPROACHING Oc a opmal m o capug s dmd, h cospodg moo sa o h ag sall ca also b calculad. hs opmal m ad moo sa wll b usd as h al m ad ag pos o h dco o dvlopg h cool o h obo o pom h capu ask. o ocus o h obocs cool, s assumd ha h svcg sall has z y F x ( ) a v ω( ) b coolld o kp a xd dsac o h ag sall such ha h ag sall s wh h ach o h oboc am. o dvlop h opmal cool, h obo s dyamcs quao () s w o a sa spac om as ollows x (x) G(x)τ (8) 2 2 wh x R s h sa vco; R s h sa 2 uco, G R s h cool max; ad τ R s h jo cool oqus. hy a dd as x θ x x2 θ x (x) (9) H(x ) C(x) x2 G(x) H(x ) Assumg ha h svcg sall s ully coolld, w ca d h mpac o h obo moo o h svcg sall by dvg h aco oc ad mom τ (s Fg. ) o h oo o h obo (a h s obo jo), amly, c τ τc ( a ) c () mv ( x, x) (, ) c ( ) m τ x x I ω a v wh c R ad τ c R a h a oc ad mom acg a h mass c o h h body, spcvly; R ad τ R a h oal oc ad mom h obo appls a h s oo, spcvly. ho, h oal aco oc ad mom causd by h obo moo a h mass c o h svcg sall a τ c τc c m v ( x, x) (, ) c m τ x x I ω Rv () Ou cool goal s h o d a m hsoy o ach jo s cool oqu such ha, wh h mapulao s p s coolld by hs s o jo oqus o mov om s al pos o s al pos,

6 wll hav mmal aud dsubac o h svcg sall. o d hs s o opmal cool oqus, w ca us h ollowg objcv uco θ θ J τ τd, x() x, x( ) x (2) θ θ Fo hs opmal cool poblm, h al sa x s kow ad h al m ad al sa x( ) a dmd by solvg h cosad opmzao poblm dd sco. h Maxmum pcpl [2] lls us ha a cssay codo o a opmal cool τ () s o maxmz h ollowg Poyag Hamloa H H( x, λ, τ) τ τ λ ( Gτ ) () 2 wh λ R s h vco o cosa vaabls. h, h cssay codos o h soluo ca b w h ollowg om λ H H, x (4) x λ wh h al ad al codos gv (2). Fo ou poblm, h al m ad al moo sa o h opmal cool a kow. I oh wods, s a xd m ad xd bouday poblm. Howv, bcaus o s complx ola au, sll has o b solvd umcally. 5. SIMULAION EXAMPLE o show h applcao o h ao-mod opmal cool sagy, w ps a xampl usg a 2-DOF plaa mapulao hs sco. h paams o h salls ad obo a dd Fg. 4 ad abl. Rlav vlocy pog o h svcg sall s COM c.5m Fully Coolld Bas Sall c 2 v() v( ) Mass c o h svcg sall sysm 4 ω ω Gaspg hadl Ed-co ajcoy wh mmu aco oqu Fg. 4 Exampl o 2-DOF spac mapulao appoachg a squa ag o capug abl. Paams o h 2-DOF spac mapulao Body Body o. (m) c (m) m (Kg) I (Kg m 2 ) Bas sall Robo lk Robo lk ag sall Dmao o h opmal m ad sa h sysm s assumd as show Fg. 4, h c o mass o h svcg sysm cludg h obo ca b oud o b: C m W also assum ha : m 2 a.5 m s.2.4 m R(), () [2.7.4 ] m, ω =.47 ad/sc, v().75 m/s I h 2-D cas, h oaoal moo s smpl ad hus, w ca asly d ha h opmal m ad oao o h ag sall o h obo o capu wh zo aud mpac a 4s v( ) R( ω a ) [ ] m/s R ( ) Opmal Cooll o h Fal Appoachg Followg h mhod oducd Sco 5, h opmal cool o h obo s dd as ollows: subjc o: 4sc dx d τ τ Mmz J d J( x,τ ) dx2 dx dx4 x ; x 4 ; x ad x 4, d d d ( ) [2 ] ; x ( ) [ ] ( ) [.47.2 ] ; x ( ) [.5.5 ] h opmal cool poblm was solvd umcally usg h omlab sowa. Fgu 5 shows h mapulao s jo dsplacms. Fg. 6 shows h appld jo oqus dvg h obo om s al

7 Raco mom (Nm) Jo oqus (Nm) Jo agls (ad) Raco mom (Nm) Jo agls (ad) coguao o h ag coguao o capug h ag sall. I ca b ocd ha boh jos ach h dsd al posos a h dsd m. I s 4sg o s ha h scod lk movs a lo (owad s ad h backwad). hs s cssay o h pupos o achvg h goal o havg a mmum accumulad aco mom o h bas sall dug h am mauvg. Fg. 7 claly dcas ha h aco oqu a h aco oqu o h svcg sall s vy small ( h lvl o mco Nwos). ho, w ca coclud ha h dsd opmal cool goal has b achvd cool. As a sul, such a obo mauv wll hav a sgca sk o dsablzg h bas sall m (sc) 2 Fg. 8 Jo ajcos o h obo om h soluo o a o-opmal cool m (sc) Fg. 5 Jo ajcos o h obo om h soluo o h opmal cool m (sc) Fg. 6 Jo cool oqus om h soluo o h opmal cool x m (s) Fg. 7 Raco oqu o h bas sall I od o s h advaag o h poposd opmal cool, w also mplmd a usual obo cooll o achvg h sam p moo ag wh h sam m lm. h sulg jo ajcos o such a oopmal cool a show Fg. 8. Obvously, h bhavo o h jo moo, as show Fg. 8, sms b ha ha om h opmal cool show Fg. 5. Howv, h aco oqu hs cas, as dpcd Fg. 9, s much lag ha ha om h opmal 2 x y z m (sc) Fg. 9 Jo cool oqus om h soluo o a o-opmal cool 6. CONCLUSIONS A opmal cool sagy o a spac mapulao o capu a umblg sall was psd. h goal o h cool sagy s o mmz h mpac o h aud o h svcg sall. hs s do wo sps: ) d a opmal m ad h cospodg moo sa o h umblg sall such ha h physcal cpo om capug opao wll hav zo o mmal aud mpac o h svcg sall; 2) cool h obo p o ach h umblg sall a h opmal m ad also caus mmal aco mom o h svcg sall dug h moo o h obo. hs appoach s maly amd a sa opao o capug a as umblg sall, whch s ohws a vy dcul ad sky opao. A 2D xampl was psd o dmosa h applcao o h poposd mhod. h xampl shows ha, wh h opmal cool, h oaoal dsubac o h bas sall s almos zo whl h ag ca sll b achd a h opmal m. Culy, w a sudyg h sags ad pomac o h cass wh h s sp us a sul havg adom os (.., h smad opmal capu m ad sa hav ucas). Fuh, h sudy o h pomac o h poposd chology o gal 6- DOF mapulaos cludg capug coac dyamcs s also h plad uu wok o h hs sach pojc. x y z

8 7. KNOWLADGMENS hs maal s basd o h sach wok sposod by h US A Foc Rsach Laboaoy (AFRL) ud agm umb FA h U.S. Govm s auhozd o poduc ad dsbu ps o Govmal puposs owhsadg ay copygh oao ho. h vws ad coclusos coad h a hos o h auhos ad should o b pd as cssaly psg h ocal polcs o dosms, h xpssd o mpld, o A Foc Rsach Laboaoy o h US Govm. 8. REFERENCES. Yoshda, K., Kuazum, R., Sashda, N., Uma, Y., (992). Modlg o Collso Dyamcs o Spac F-Floag lks wh h Exdd Galzd Ia so. IEEE Iaoal Coc o Robocs ad Auomao, Nc, Fac, pp Yoshda, K., ad Nchv, D. N. (995). Spac Robo Impac Aalyss ad Sall-Bas Impuls Mmzao Usg Raco Null-Spac. IEEE Iaoal Coc o Robocs ad Auomao, Nagoya, Japa, pp Papadopoulos, E. ad Paaskvas, I. (25). Dsg ad Coguao Cool o Spac Robos Udgog Impacs. 6h Iaoal ESA Coc o Gudac, Navgao ad Cool Sysms, Louak, Gc, pp Dubowsky, S., ad os, M. (99). Pah Plag o Spac Mapulaos o Mmz Spacca Aud Dsubacs. IEEE Iaoal Coc o Robocs ad Auomao, Sacamo, CA., pp Agawal, O. P. ad Xu, Y. (994). O h global opmum pah plag o duda spac mapulaos. IEEE asacos o Ma ad Cybcs. 24(9) Lampallo, R., Agawal, S., ad Hzg, H. (2). Opmal Moo Plag o F-Flyg Robos, IEEE Iaoal Coc o Robocs ad Auomao, ap, awa, pp Huag, P., Yua, J., Xu, Y., ad Lu, R. (26). Appoach ajcoy Plag o Spac Robo o Impac Mmzao. IEEE Iaoal Coc o Iomao ad Acquso, Shadog, Cha, pp Aghl. F. (28). Opmal Cool o Roboc Capug ad Passvao o a umblg Sall wh Ukow Dyamcs. AIAA Gudac Navgao ad Cool Coc, Hoolulu, Hawa, pp Ok,. Nakash, H ad Yoshda, K. (28). m- Opmal Mapulao Cool o a F-Floag Spac Robo wh Cosa o Raco oqu. IEEE Iaoal Coc o Illg Robos ad Sysms, Nc, Fac, pp Xu, Y., ad Kaad,., Eds. (99). Spac obocs: Dyamcs ad Cool. Kluw Acadmc Publshs.. Yoshda, K., ad Nchv, D. N. (999). Impac Aalyss ad Pos-Impac Moo Cool Issus o a F-Floag Spac Robo Subjc o a Foc Impuls. IEEE asacos o Robocs ad Auomao, 5(), Poyag, L.S., Bolyask, V.G., Gamkldz, R.V., Ad Mshchko, E.F. (964). h Mahmacal hoy o Opmal Pocsss (aslad by D. E. Bow), Macmlla Compay.. Jupp, A. H. (974). O h oao o a gd body. Clsal Mchacs, 9(), Zo, R.V. ad Schold, J. (27). Numcal mplmao o h xac dyamcs o gd bods. Joual o Compuaoal Physcs. 25 (),