Trajectory Planning for Spacecraft Rendezvous with On / Off Thrusters

T ajec o y Planning o Spacec a Rendez ous wi h On/O Th us e s ? Ra ael Vazquez ∗F ancisco Ga ilan ∗Edua do F. Camacho ∗∗ ∗Depa amen o de Ingenie ia Ae ospacial ∗∗ Depa amen o de Ingenie ia de Sis emas y Au oma ica Uni e sidad de Se illa, Camino de los Descub imien os s/n, 41092, Se illa, Spain (e-mails: [email p o ec ed], [email p o ec ed], edua [email p o ec ed]) Abs ac : The objec i e o his wo k is o p esen a ajec o y planning algo i hm o spacec a endez ous ha is able o inco po a e Pulse-Wid h Modula ed (PWM) con ol signals. The algo i hm is based on linea iza ion a ound a p e iously compu ed solu ion. To ini ialize he algo i hm, a i s solu ion needs o be ob ained. To do so, he ajec o y planning p oblem is sol ed using Pulse-Ampli ude Modula ed (PAM) con ol signals; hese a e hen con e ed o PWM signals, which a e used as an ini ial guess. I e a ing, he solu ion is e ined un il an op imal alue is eached. Simula ions show ha his me hod con e ges a e a ew i e a ions. The algo i hm is simple and as , hence i could be implemen ed online o used oge he wi h a Model P edic i e Con olle . Keywo ds: Spacec a au onomy, Space obo ics, Pulse-wid h modula ion, T ajec o y planning, Op imal ajec o y, Linea iza ion. 1. INTRODUCTION Technology enabling simple au onomous spacec a en- dez ous and docking is becoming a g owing necessi y as access o space con inues inc easing. A e decades o de elopmen , many app oaches ha e been p oposed; see Wo inden and Gelle (2008) o an his o ical accoun o Fehse (2003) o he basics. Classicaly (Wie (1998)), he p oblem is modeled using impulsi e maneu e s, compu ing an ini ial and inal im- pulse (∆V) o achie e endez ous. Impulsi e changes o speed a e also used in ecen wo ks (Tong e al. (2007); Gelle (2006)). La ely, app oaches based on ajec o y planning and op imiza ion (B ege and How (2008)) and p edic i e con ol (Richa ds and How (2003); Rossi and Lo e a (2002); Asawa e al. (2006)) a e eme ging. Howe e , ypically hese me hods allow he con ol signal ( h us ) o ake any alue in an allowable ange. This ype o con ol signal is usually e e ed o as Pulse-Ampli ude Modula ed (PAM). A mo e ealis ic modeling o spacec a h us e s would ake in o accoun ha ypically, h us e s a e ON-OFF ac ua o s, i.e., he h us e s a e no able o p oduce any alue o o ce, bu can be only swi ched on (p oducing he maximum amoun o o ce) o o (p oducing no o ce). Thus only he imes whe e he h us e s a e u ned ON o OFF ( he swi ching imes) can be con olled. This ype o con ol signal is usually e e ed o as Pulse-Wid h Modula ed (PWM). ?The au ho s would like o acknowledge inan ial suppo o he Spanish Minis y o Science and Inno a ion and o he Eu opean Commission o unding pa o his wo k unde g an s DPI2008- 05818 and p ojec FPT ICT HD-MPC. The objec i e o his wo k is o p esen a ajec o y planning algo i hm o spacec a endez ous ha is able o inco po a e PWM con ol signals. The algo i hm is based on he ac ha , e en hough in he endez ous model he PWM signals appea nonlinea ly, his depen- dence on he swi ching imes is only weakly nonlinea . Thus, linea iza ion a ound e e ence alues can ai h ully cap u e he dependence o he sys em on he swi ching imes, a leas close o he e e ence alues. To ob ain a i s solu ion a ound which linea ize he sys ems, he ajec o y planning p oblem is sol ed using PAM con ol signals; hese a e hen con e ed o PWM signals, which a e used as ini ializa ion o ou algo i hm. I e a ing, he solu ion is e ined un il an op imal alue is eached. The esul ing algo i hm is simple and as and could be imple- men ed online, o wi h a Model P edic i e Con olle (such as Ga ilan e al. (2009)), allowing he MPC o include PWM con ol signals. While he idea o linea iza ion o compu e op imal PWM con ol signals is, as a as we know, o iginal, we no e ha local linea iza ion echniques ha e been used o op imal ajec o y p oblems in o he con ex s (see e.g. Kim e al. (2002)). The s uc u e o his pape is as ollows. In Sec ion 2 we in oduce he ma hema ical model o endez ous space- c a used o ajec o y planning, bo h in he PAM and PWM case. We ollow in Sec ion 3 whe e we o mula e he planning p oblem, de ailing he equali y and inequali y cons ain s, and he cos unc ion. Sec ion 4 desc ibes he me hod we p opose o sol e he planning p oblem. In Sec ion 5 we show simula ions o he p oposed me hod. We close he pape wi h some ema ks in Sec ion 6. 2. MODEL OF SPACECRAFT RENDEZVOUS In his wo k we use he he linea Hill-Clohessy-Wil shi e (HCW) equa ions, as in oduced in Hill (1878) and Clo- P oceedings o he 18 h Wo ld Cong ess The In e na ional Fede a ion o Au oma ic Con ol Milano (I aly) Augus 28 - Sep embe 2, 2011 978-3-902661-93-7/11/$20.00 © 2011 IFAC 8473 10.3182/20110828-6-IT-1002.02445 hessy and Wil shi e (1960). Those desc ibe he ela i e posi ion o he spacec a i he a ge is o bi ing in a ci cula keple ian o bi and app oaching ehicle is close enough o he a ge . The HCW model also assumes ha he a ge ehicle is passi e and mo ing along a ci cula o bi o adius R. Thus he angula speed o he a ge h ough i s o bi is n=pµ R3, whe e µis he g a i a ion pa ame e o he Ea h, µ= 398600.4 km3/s2. Deno e by k he ime ins an = 0+kT , whe e Tis an adequa ely chosen sampling ime ( he sampling ime chosen in his wo k is 60 s). We o mula e wo e sions o he HCW equa ions in disc e e ime. In he i s e sion, he con ol inpu s a e cons an du ing a whole sampling ime. This is e e ed o as he PAM disc e e model. In he second e sion, i is assumed ha h us e s a e no able o p oduce any o ce alue, bu can be only swi ched on (p oducing he maximum o ce) o o (p oducing no o ce), and only once du ing each sampling ime. This is e e ed o as he PWM disc e e model. 2.1 PAM disc e e model Assuming he con ol signal cons an h ough he sampling ime, i can be shown ha x(k+ 1) = ATx(k) + BTu(k).(1) In (1), x(k), u(k) deno e espec i ely he s a e a ime k, and he inpu ampli ude ( om ime k o k+ 1). Also, x= [x y z ˙x˙y˙z]T,u= [uxuyuz]T.(2) In hese de ini ions, x,y, and zdeno e he posi ion o he chase in a local– e ical/local–ho izon al (LVLH) ame o e e ence ixed on he cen e o g a i y o he a ge ehicle. In he LVLH ame, x e e s o he adial posi ion, y o he in- ack posi ion, and z o he c oss- ack posi ion. The eloci y o he chase in he LVLH ame is gi en by ˙x, ˙y, and ˙z. The a iables ux,uy, and uza e he inpu s ( h us ac ua ion) ac ing on he chase ehicle, e e ed o he LVLH axes as indica ed by hei espec i e subsc ip s. The ma ices ATand BTappea ing in (1) a e gi en by AT=             4−3C0 0 S n 2(1 −C) n0 6(S−nT) 1 0 −2(1 −C) n 4S−3nT n0 0 0 C0 0 S n 3nS 0 0 C2S0 −6n(1 −C) 0 0 −2S4C−3 0 0 0 −nS 0 0 C             ,(3) BT=                 1−C n2 2nT −2S n20 2(S−nT) n2−3T2 2+ 41−C n20 0 0 1−C n2 S n21−C n0 2(C−1) n−3T+ 4S n0 0 0 S n                 ,(4) whe e S= sin nT and C= cos nT. Compac o mula ion Nex we de elop a compac o mula ion ha simpli ies he no a ion o he p oblem. The s a e a ime k+ 1, gi en he ini ial s a e a ime 0 (which is deno ed as x0) and he inpu signals om ime 0 o ime k, is compu ed by applying ecu si ely Equa ion (1): x(k) = Akx0+ k−1 X j=0 Ak−1−jBu(j).(5) De ine now xSand uSas a s ack o Nps a es and inpu signals, espec i ely, spanning om ime 1 o ime Np, whe e Npis he planning ho izon: xS=    x(1) x(2) . . . x(Np)     ,uS=    u(0) u(1) . . . u(Np−1)     . Then, xS=             ATx0+BTu(0) A2 Tx0+ 1 X j=0 A1−j TBTu(j) . . . ANp Tx0+ Np−1 X j=0 ANp−1−j TBTu(j)             ,(6) which can be w i en as xS=Fx0+GuS,(7) whe e Gis a block lowe iangula ma ix wi h i s non- null elemen s de ined by (G)ij =Ai−j TBTand he ma ix Fis de ined as: F=     AT A2 T . . . ANp T      .(8) 2.2 PWM disc e e o mula ion Conside ing now he case o ON-OFF h us e s, i is assumed ha he h us e s a e no able o p oduce any alue o o ce, bu can be only swi ched on o o . Also, i is assumed ha he e is an aligned pai o h us e s o each di ec ion i= 1,2,3 wi h opposing o ien a ion. To dis inguish be ween he posi i e and nega i e hey a e deno ed as u+ iand u− i, whe eas he espec i e maximum h us is e e ed o as u+ imax and u− imax , espec i ely. Finally, du ing each sample ime each h us e is allowed o i e only once. Thus, he PWM ou pu o each ime in e al kis com- ple ely desc ibed by wo new con ol a iables o each pai o h us e s: he pulse wid h κ+ i(k) and he pulse s a ime τ+ i(k) ( o he posi i ely o ien ed h us e in he di ec ion i) and simila ly κ− i(k) and τ− i(k) o he nega i ely o ien ed h us e in he di ec ion i. Then, o ∈[kT, (k+ 1)T], we ha e: u+ i( ) =    0, ∈kT, kT +τ+ i(k), u+ imax , ∈kT +τ+ i(k), kT +τ+ i(k) + κ+ i(k), 0, ∈kT +τ+ i(k) + κ+ i(k),(k+ 1)T, (9) 18 h IFAC Wo ld Cong ess (IFAC'11) Milano (I aly) Augus 28 - Sep embe 2, 2011 8474 and simila ly o he nega i ely o ien ed h us e s. The new a iables con ol a iables e i y κ+ i(k)>0, τ+ i(k)> 0 and τ+ i(k) + κ+ i(k)< T, and simila ly o he nega i ely o ien ed h us e s. The las cons ain p e en he PWM signal o spill o e o he nex ime in e al. Call he PWM con ol a iables as uP(k): uP(k) =                    τ+ 1(k) κ+ 1(k) τ− 1(k) κ− 1(k) τ+ 2(k) κ+ 2(k) τ− 2(k) κ− 2(k) τ+ 3(k) κ+ 3(k) τ− 3(k) κ− 3(k)                    .(10) To ind he HCW equa ions o PWM inpu s, de ine b1 =              1−C n2 2(S−n ) n2 0 S n 2(C−1) n 0              , b2 =              2n −2S n2 −3 2 2+ 41−C n2 0 21−C n −3 + 4S n 0              , b3 =           0 0 1−C n2 0 0 S n           ,(11) which a e he h ee columns o he ma ix B in (4), bu de ined o di e en . Then he sys em e olu ion equa ion (1) is eplaced in he PWM case by x(k+ 1) = Ax(k) + BPWM(uP(k))umax,(12) whe e BPWM(uP(k)) =            AT−τ+ 1(k)−κ+ 1(k)b1 κ+ 1(k) AT−τ− 1(k)−κ− 1(k)b1 κ− 1(k) AT−τ+ 2(k)−κ+ 2(k)b2 κ+ 2(k) AT−τ− 2(k)−κ− 2(k)b2 κ− 2(k) AT−τ+ 3(k)−κ+ 3(k)b3 κ+ 3(k) AT−τ− 3(k)−κ− 3(k)b3 κ− 3(k)            T ,(13) umax =         u+ 1max −u− 1max u+ 2max −u− 2max u+ 3max −u− 3max         .(14) No ice ha sys em (12) is nonlinea in he PWM a iables uP(k). Compac o mula ion The compac o mula ion de eloped be o e can be eadily adap ed o PWM inpu s. Equa ion (7) is now w i en as xS=Fx0+GPWM(uPS)umaxS,(15) whe e uPSis a s ack ec o wi h all he PWM sig- nals, GPWM is a block lowe iangula ma ix wi h i s non-null elemen s de ined by (GPWM(uPS))ij = Ai−jBPWM(uP(j−1)), and he cons an ec o umaxS is de ined as: umaxS=    umax umax . . . umax            N imes.(16) 3. FORMULATION OF THE PLANNING PROBLEM Nex we o mula e ou planning p oblem, in oducing he cons ain s (bo h equali y- and inequali y- ype con- s ain s) and he objec i e unc ion ha has o be mini- mized. The o mula ion is done o bo h PAM and PWM con ol signals. 3.1 Cons ain s on he p oblem Inequali y cons ain s on he s a e Fo sensing pu poses (see B ege and How (2008)), du ing endez ous i is equi ed ha he chase ehicle emains inside a line o sigh (LOS) a ea. To simpli y he cons ain , we conside a 2-D LOS a ea as shown in Figu e 1. This LOS egion is he in e sec ion o a cone, gi en by he equa ions y≥cLOS(x−x0) and y≥ −cLOS(x+x0), and he egion y≥0. Fig. 1. Line o Sigh egion. The LOS cons ain is ALOS x(k)≤bLOS, whe e ALOS ="0−10000 cLOS −10000 −cLOS −10000#, bLOS ="0 cLOSx0 cLOSx0#. (17) Using he compac o mula ion ha was de eloped in Sec ion 2.1, he cons ain s equa ions o he s a e can be ew i en as: AcxS≤bc,(18) 18 h IFAC Wo ld Cong ess (IFAC'11) Milano (I aly) Augus 28 - Sep embe 2, 2011 8475 whe e Acand bca e gi en by: Ac=    ALOS ALOS ... ALOS     ,bc=    bLOS bLOS . . . bLOS     .(19) Then, o he case o PAM con ol and using equa ion (7), one can e o mula e he LOS cons ain s as cons ain s o he con ol signals in he ollowing way: AcGuS≤bc−AcFx0,(20) and simila ly o he case o PWM con ol. Equali y cons ain s on he s a e Equali y cons ain s a e o mula ed o ensu e ha he chase spacec a a i es a he o igin wi h ze o eloci y a he end o he planning ho izon. Thus, hese cons ain s can be w i en as x(Np) = 0. De ining Aeq as Aeq =    0 ... 0 Id6×6     ,(21) he equali y cons ain can be w i en as: AeqxS=0.(22) Then, o he case o PAM con ol and using equa ion (7), one can e o mula e he a i al condi ions as equali y cons ain s o he con ol signals in he ollowing way: AeqGuS=−AeqFx0,(23) and simila ly o he case o PWM con ol. Inpu cons ain s Fo he case o PAM con ol, one would ha e limi a ions on he magni ude o he con ol, gi en by umin ≤uS≤umax.(24) Fo he case o PWM con ol, he cons ain s a e gi en by κ± i(k)>0, τ± i(k)>0 and τ± i(k) + κ± i(k)< T. 3.2 Objec i e unc ion The objec i e unc ion o be minimized in he planning p oblem is he 1-no m o he con ol signal, which is p opo ional o uel consump ion. PAM con ol inpu s Fo he case o PAM con ol inpu s, he selec ed con ol unc ion is gi en by: JP AM = Np−1 X k=0 kuT(k)k1=kuSk1.(25) PWM con ol inpu s Fo he case o PAM con ol inpu s, using (9) i can be seen ha he selec ed objec i e unc ion is gi en by: JPWM = Np−1 X k=0 3 X i=1 u+ imax κ+ i(k) + u− imax κ− i(k).(26) 4. COMPUTATION OF THE OPTIMAL CONTROL INPUT As shown in Sec ion 2.2, he disc e e HWC equa ions in he PWM case a e nonlinea in he swi ching imes. Thus, i is nei he easy no as o sol e an op imiza ion p oblem using he swi ching imes as con ol a iables. To ind he op imal con ol inpu wi hou needing o sol e a nonlinea planning p oblem, he ollowing scheme is p oposed: S ep 1. A PAM linea op imiza ion p oblem is sol ed. S ep 2. A PAM/PWM il e is used o con e he PAM inpu signals o PWM signals (i.e., swi ching imes) ha p oduce a e y simila sys em ou pu . S ep 3. The plan equa ions, in he PWM o mula ion, a e linea ized a ound he p e ious s ep solu ion, hus ob aining a linea plan wi h espec o he swi ching ime. A linea op imiza ion p oblem is hen posed and sol ed. The esul ing solu ion is aken as a be e app oxima ion owa ds he eal solu ion. S ep 4. Repea he linea iza ion p ocess o S ep 3 a ound he new solu ion. Op imize again o ind a be e e inemen . The p ocess is i e a ed un il he solu ion con e ges. Nex , we desc ibe all he s ep in ou scheme. 4.1 Compu a ion o PAM con ol inpu To compu e he op imal con ol plan (wi h PAM con ol signals), one seeks he con ol signal ha minimizes he cos unc ion o e he planning ho izon, sa is ying a he same ime he PAM cons ain s: min uS JP AM (uS) (27) subjec o: AcGuS≤bc−AcFx0 AeqGuS=−AeqFx0 umin ≤uS≤umax. Since he cos unc ion and he cons ain s a e linea , hen (27) can be eadily sol ed. 4.2 Ini ial PWM solu ion: A PAM/PWM il e The PAM solu ion ound when sol ing (27) is ans o med o an equi alen PWM solu ion using a PAM/PWM il e . This il e is o mula ed in he li e a u e ( o ins ance in Shieh e al. (1996); Ieko e al. (1999)) whe e se e al me hods a e p oposed. These me hods allow o, gi en he PAM inpu s o a sys em, compu e equi alen PWM inpu s ha p oduce a sys em ou pu op imally app oxima ing he ou pu o he sys em when d i en by he PAM signals. Following hese e e ences, a PAM con ol signal can be op imally app oxima ed by a PWM con ol signal by using he ollowing ules o each ime ins an kand di ec ion i: (1) Use he posi i e o nega i e h us e acco ding o he sign o he PAM signal ui(k). (2) The pulse wid h mus be compu ed using he P in- ciple o Equi alen A eas: κ± i(k) = T|ui(k)| u± imax , whe e u± imax is he maximum le el o he (posi i e o nega- i e) h us e i. 18 h IFAC Wo ld Cong ess (IFAC'11) Milano (I aly) Augus 28 - Sep embe 2, 2011 8476 (3) Since only one impulse pe sample ime is conside ed, i mus be alloca ed a he cen e o he sampling in e al, ha is τ± i(k) = 1 2(T−κ± i(k)). The PWM signals uP(k) cons uc ed by his me hod p oduce an almos iden ical ou pu o he sys em d i en by PAM signals. Howe e , he PWM a e no necessa ily op imal since hei cons ain s a e qui e di e en ; in ac hey migh e en no e i y he cons ain s. Thus, his so- lu ion is only used as an ini ializa ion o he op imiza ion algo i hm p oposed nex . 4.3 Re ined PWM solu ion: An op imiza ion algo i hm No ing ha mos nonlinea i ies appea ing in he sys em equa ions (12) a e o he o m cos n o sin n , whe e n is he o bi al angula eloci y, and no ing n 1, we app oxima e (12) by linea izing BPWM(uP(k)) a ound uP(k). Then, he sys em equa ions a e: x(k+1) = Ax(k)+BPWM(uP(k))umax+B∆(uP(k))∆(k), (28) whe e B∆=                              −A0 T−τ+ 1−κ+ 1 b1 κ+ 1 u+ 1max −A0 T−τ+ 1−κ+ 1 b1 κ+ 1 +AT−τ+ 1−κ+ 1b10 κ+ 1u+ 1max A0 T−τ− 1−κ− 1 b1 κ− 1 u− 1max A0 T−τ− 1−κ− 1 b1 κ− 1 −AT−τ− 1−κ− 1b10 κ− 1u− 1max −A0 T−τ+ 2−κ+ 2 b2 κ+ 2 u+ 2max −A0 T−τ+ 2−κ+ 2 b2 κ+ 2 +AT−τ+ 2−κ+ 2b20 κ+ 2u+ 2max A0 T−τ− 2−κ− 2 b2 κ− 2 u− 2max A0 T−τ− 2−κ− 2 b2 κ− 2 −AT−τ− 2−κ− 2b20 κ− 2u− 2max −A0 T−τ+ 3−κ+ 3 b3 κ+ 3 u+ 3max −A0 T−τ+ 3−κ+ 3 b3 κ+ 3 +AT−τ+ 3−κ+ 3b30 κ+ 3u+ 3max A0 T−τ− 3−κ− 3 b3 κ− 3 u− 3max A0 T−τ− 3−κ− 3 b3 κ− 3 −AT−τ− 3−κ− 3b30 κ− 3u− 3max                              T ,(29) and whe e he alue o κand τ a iables a ime kis used. The a iable ∆(k) ep esen s he inc emen s o dec emen s wi h espec o uP(k): ∆(k)=                    ∆τ+ 1(k) ∆κ+ 1(k) ∆τ− 1(k) ∆κ− 1(k) ∆τ+ 2(k) ∆κ+ 2(k) ∆τ− 2(k) ∆κ− 2(k) ∆τ+ 3(k) ∆κ+ 3(k) ∆τ− 3(k) ∆κ− 3(k)                    .(30) The ma ices used in (29) a e de ined as: A0 =       3nS 0 0 C2S0 6n(C−1) 0 0 −2S4C−3 0 0 0 −nS 0 0 C 3n2C0 0 −nS 2nC 0 −6n2S0 0 −2nC −4nS 0 0 0 −n2C0 0 −nS        ,(31) and b10 =           S n 2(C−1) n 0 C −2S 0           , b20 =           2−2C n −3 + 4S n 0 2S −3+4C 0           , b30 =         0 0 S n 0 0 C         .(32) Equa ion (33) is now compac ly w i en as xS=Fx0+G∆(uPS)∆S+GPWM(uPS)umaxS,(33) whe e G∆(uPS) is a block lowe iangula ma ix wi h i s non-null elemen s de ined by (G∆(uPS))ij = Ai−jB∆(uP(Np−j−1)) and ∆Sis a s ack ec o o he inc emen in he PWM a iables ∆(k). The LOS inequal- i y cons ain s (20) can be e o mula ed as cons ain s o he PWM a iables in he ollowing way: AcG∆∆S≤bc−AcFx0−AcGPWMumaxS,(34) whe e he dependence o GPWM(uPS) and G∆(uPS) on uPShas been omi ed o simplici y. Simila ly, he equali y cons ain s become: AeqG∆∆S=−AeqFx0−AeqGPWMumaxS.(35) The cons ain s on he ∆(k) a e as ollows: −∆κ± i(k)≤κ± i(k),−∆τ± i(k)≤τ± i(k) (36) ∆τ± i(k)+∆κ± i(k)≤T−τ± i(k)−κ± i(k),(37) |∆(k)| ≤ ∆MAX ,(38) whe e (38) is used o a oid oo la ge a ia ions in each i e a ion s ep. These cons ain s can be summa ized as A∆(k)∆S(k)≤b∆(k).(39) Finally, he objec i e unc ion can be ew i en as a unc ion o he PWM a iables and hei inc emen s as J(uPS,∆S) = JPWM(uPS) + J∆(∆S), whe e J∆(∆S) = Np−1 X k=0 3 X i=1 u+ imax ∆κ+ i(k) + u− imax ∆κ− i(k).(40) Thus, a linea p og amming p oblem wi h PWM ou pu s can be posed as ollows: min ∆S J∆(∆S) (41) s. .: AcG∆∆S≤bc−AcFx0−AcGPWMumaxS, AeqG∆∆S=−AeqFx0−AeqGPWMumaxS, A∆∆S≤b∆. The solu ion o (41), ∆S, can be used o ecompu e new PWM a iables: uNEW PS=uPS+∆S.(42) Then, he alues o uNEW PSis used o ecompu e he a ious ma ices appea ing in (41), and he op imiza ion p oblem is sol ed again. I e a ing, i is expec ed ha he solu ion keeps imp o ing un il a solu ion close o he global op imum is eached. Nex , his is shown in simula ions. 5. SIMULATION RESULTS We nex show simula ions o he ajec o y planning p oblem o spacec a endez ous using ou algo i hm. 18 h IFAC Wo ld Cong ess (IFAC'11) Milano (I aly) Augus 28 - Sep embe 2, 2011 8477 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x (km) y (km) PWM PAM Fig. 2. Solu ion ajec o y, showing he ini ial ajec o y wi h PAM inpu s (dashed) and he inal ajec o y wi h PWM inpu s (solid). ! " # $ % & %'!( %'" %'"" %'"$ %'"& %'"( %'# %'#" %'#$)*!+!# ,-./01*23*45016542,7 !"#$%&'( Fig. 3. Value o objec i e unc ion depending on i e a ion numbe . ! "!! #!!! #"!! $!!! $"!! %!!! !& !$ ! $'(#!!" )*+, -'*./+$, ! "!! #!!! #"!! $!!! $"!! %!!! !" ! "'(#!!" )*+, -0*./+$, Fig. 4. Ini ial PAM con ol signal. The planning ho izon Npwas chosen as 50. The alue o u± imax was chosen as 10−4km/s2. Ini ials condi ions we e 0= [0.2 0.6 0.2]Tkm, 0= [0.0015 0.002 −0.001]Tkm/s. The solu ion ajec o y is shown in Fig. 2 (solid). The PAM solu ion ha was used o ini ialize he algo i hm is shown in he same igu e (dashed). In his and o he igu es only he xand ycoo dina es a e shown o he sake o b e i y. As i can be seen in Fig. 3, he algo i hm con e ged a e 6 i e a ions o a solu ion wi h a cos unc ion app oxima ely 2.5% smalle . Each i e a ion ook less han a second on a con en ional compu e , using MATLAB’s linp og o calcula e he solu ion. Compa ing he ini ial PAM (Fig. 4) and PWM (Fig. 5) con ol signals, i can be seen ha o bo h he con ol e o is concen a ed a he beginning, wi h some mid-cou se and inal maneu e s ha a e di e en o bo h cases. ! "!! #!!! #"!! $!!! $"!! %!!! !$ !# ! # $&'#!!( )*+, -&*./+$, ! "!! #!!! #"!! $!!! $"!! %!!! !$ ! $&'#!!( )*+, -0*./+$, Fig. 5. Final PWM con ol signals. 6. CONCLUDING REMARKS We ha e p esen ed an algo i hm o compu e he op imal PWM con ol signals applied o endez ous o spacec a . The algo i hm uses he HCW model wi h LOS and equal- i y cons ain s, bu can be gene alized o mo e complica ed models and cons ain s. Since he algo i hm is e y as , i could be used online o implemen an MPC scheme (e.g. Ga ilan e al. (2009)) including PWM con ol signals. REFERENCES Asawa, S., Nagashio, T., and Kida, T. (2006). Fo ma ion ligh o spacec a in ea h o bi ia MPC. In SICE- ICASE In e na ional Join Con e ence. B ege , L. and How, J.P. (2008). Sa e ajec o ies o au onomous endez ous o spacec a . J Guid Con Dynam., 31(5), 1478–1489. Clohessy, W.H. and Wil shi e, R.S. (1960). Te minal guidance sys ems o sa elli e endez ous. Jou nal o he Ae ospace Sciences, 27(9), 653–658. Fehse, W. (2003). Au oma ed Rendez ous and Docking o Spacec a . Camb idge Uni e si y P ess. 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In P o- ceedings o ACC 2002. Shieh, L.S., Wang, W.M., and Sunkel, J. (1996). Design o PAM and PWM con olle s o sampled-da a in e al sys ems. J Dyn Sys Meas Con ., 118(4), 673–681. Tong, C., Shijie, X., and Songxia, W. (2007). Rela i e mo ion con ol o au onomous endez ous based on classical o bi elemen di e ences. J Guid Con Dy- nam., 30(4), 1003–1014. Wie, B. (1998). Space ehicle dynamics and con ol. AIAA. Wo inden, D.C. and Gelle , D.K. (2008). Na iga ing he oad o au onomous o bi al endez ous. Jou nal o Spacec a and Rocke s, 44(4), 898–909. 18 h IFAC Wo ld Cong ess (IFAC'11) Milano (I aly) Augus 28 - Sep embe 2, 2011 8478