FORMATION FLYING IMPULSIVE CONTROL USING MEAN ORBITAL SPEED

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1 (Pepint) AAS XX-XXX FORMATION FLYING IMPULSIVE CONTROL USING MEAN ORBITAL SPEED Sunghoon Mok, * Yoonhyuk Choi, an Hyohoong Bang INTRODUCTION In this poeeing, an impulsive ontol metho fo satellite fomation flying is intoue. Seula ift ause by mean obital spee iffeene is use to eease impulse magnitue an the ift motion is epesente by mean anomaly iffeene among six obital elements. Obital elements illustate obital motion an simulation esults show the ontol histoy. In aition, to valiate the teneny of ontol histoy of pevious eseah s metho, optimization metho is applie fo same mission example. To use optimization metho, elative ynami equations ae nomalize by sale fatos. Simulation stuies investigate the pefomanes of analytially popose metho an optimization metho. The esults show that the tenenies of obital elements eos ae simila between them. Satellite fomation flying has been eseahe sine 196s 1-1. It eplaes one lage satellite by multiple small satellites, so it makes launh an maintenane osts be eue. Vaious eseahes, inluing guiane, navigation an ontol, ae stuie in elate aea. Dynamis of elative motion between hief an eputy satellites ae essential to peit satellite state an Clohessy an Wiltshie 1 evelope elative ynami equations whih ae alle Hill s equations. Howeve, elative motion an be expesse by obital elements also, an Gauss s vaiational equation an be use as govening equation. In this pape, six obital elements ae hosen to epesent an obit of eputy satellite. In aition, in-plane motion is only ealt fo obital oetion. Out-of-plane is assume to be oete aleay. Thust-type fo obit oetion an be ivie into two majo types whih ae ontinuous-type thuste an impulsive-type thuste. Impulsive type is hosen in this eseah fo obit oetion. An impulsive ontol metho by Shaub an Alfien fomulates base metho of this eseah. Then, by inopoating seula ift effet, two altenative ontol methos ae intoue 1. Seula ift is ause by unmathe obital peio an it makes the ift of elative position. In some ases, it an be benefiial to use it by euing equie impulse. Two intoue methos using seula ift ae eive by analytial appoah. In this pape, numeial optimization tehnique is use to ompae the teneny of obit elements eo histoy. Optimization oe is obtaine by open-soue whih is alle GPOPS 1 an ontinuous thust- * Ph.D. Caniate, Division of Aeospae Engineeing, KAIST(Koea Avane Institute of Siene an Tehnology), shmok@asl.kaist.a.k. Postotoal Reseahe, NASA Ames Reseah Cente, CA, Yoonhyuk.hoi@nasa.gov. Pofesso, Division of Aeospae Engineeing, KAIST, hbang@asl.kaist.a.k. 1

2 type optimization esults ae ahieve by using optimization. To apply optimization metho in fomation flying, non-imensionalize elative ynamis in Hill fame ae use. Howeve, it shoul be note that beause the type of thuste is iffeent, ompaison of ontol aeleation ost ietly an be meaningless. In this pape, instea ompaing the ost, we tie to see the teneny of obit element histoy only to veify ontol histoy of popose ontol metho. This poeeing is oganize as following. In fist setion, oiginal impulsive ontol algoithm is biefly intoue an two ontol methos ae popose. Then, nomalize elative ynami equations ae eive fom oiginal ynami equations. Finally, simulation stuies ae illustate fo atifiial mission example. PROPOSED METHODS This setion intoues a basis metho of this pape an popose methos in pevious eseah. Fistly, basis metho esigne by Shaub an Alfien is biefly intoue in following. Impulsive feebak ontol appoah is selete as an obit oetion between uent obit an esie obit. Gauss s vaiational equations ae use as govening equations an thee impulses fomulate ontol sequene. Thee impulses an be expesse by h æ DWsin i ö è Di ø na é(1 + e) ù D v p = - ê ( D w + DW os i ) + DM ú 4 ë h û D vh = D i + DW sin i at qit = atan ç na é(1 - e) ù D v a = - ê ( D w + DW os i ) + DM ú 4 ë h û nah æ Da De ö D vq = p ç + 4 è a 1+ e ø nah æ Da De ö D vq = a ç - 4 è a 1- e ø whee ( W, i, w, a, e, M ) ae six obital elements whih ae longitue of asening noe, inlination angle, agument of peigee, semi-majo axis, eentiity an mean anomaly. D means obital elements eo between uent obit an esie obit, fo example D a = a - a whee a is a esie semi-majo axis. D vh is a fist impulse maneuve whih makes obit plane be equalize with esie one an it takes plae at tue latitue q it. Subsipts (, q ) enote impulse ietions whih ae aial ietion an tangential ietion. ( p, a ) mean impulse fiing position whih ae peigee an apogee. Finally, ( Dv, Dv, Dv, Dv ) ae two impulsive pais to oet p a qa qa obit shape an anomaly along obit tak. Eqs. (1-) epesent thee impulses to ontol eputy satellite an Eq. (1) oets the obit plane by obit-nomal ietion impulse. Howeve, in this pape we fous on in-plane motion only to simplify the poblem an easie hanling. It means that ( DW, Di) eos ae zeo an Eq. (1) is not use. Also, Eq. (-3) an be e-witten as (1) () (3) (4) () na é(1 + e) ù D v p = - ê D w + DM ú 4 ë h û (6)

3 na é(1 - e) ù D v a = - ê D w + DM ú 4 ë h û Note that equations ae slightly hange by using DW = onition. Until now, we intoue basis metho of this pape. Now, two popose methos of pevious eseah 1 ae summaize in following. Seula ift ause by mean obital spee iffeene makes mean anomaly ift as D M = D M + ( n - n) t (8) whee D M is an initial mean anomaly iffeene an ( n, n ) ae mean angula veloity of esie obit an uent obit. In this pape, we assume that satellites ae in Kepleian obit an othe obit elements iffeenes emain onstant. Hene, DM is an only vaiable whih hanges impulse magnitues of Eqs. (4-7). In pevious eseah this seula ift effet is use to eue in-plane impulse magnitue whih an be expesse by D v = D v + D v + D v + D v (9) s p a q p qa (7) whee D vs is a sum of in-plane impulse magnitues. Two impulsive ontol methos wee iven in pevious eseah. One is a elaye impulse metho whih only elays impulse fiing time an seon is an aitional impulse pai metho whih uses an aitional pai to hange D M ift ietion. Eah metho is intoue in Refeene 1, an the esult an be summaize as following. Fistly, the elaye impulse metho fies in-plane impulse afte time elay an the elay time is obtaine by whee { t } = { DM } n - DM - n { M } { M } { M } D = - a p < D < - a, a p > = - a < D < - a, a < a p p (1) (11) (1 + e) a p = Dw h (1 - e) a = Dw h whee { D M } is a mean anomaly iffeene inteval whih eues in-plane impulse magnitue D. { t } is a oesponing time elay to meet { D } inteval. Howeve, in-plane impulse v s M takes plae at peigee o apogee, so time onstaint shoul be applie as whee N t t { } (1) = ag p o a - (13) N ³ 3

4 p t p o a = t1 + N, N =,1,,... (14) n whee t po a means the time when satellite passes peigee o apogee, t 1 is an elapse time fom initial position to the neaest peigee o apogee an N is an intege exluing negative aea. Finally, N is a time elay nomalize by half-obit peio. As a seon, the aitional impulse pai metho is pesente shotly. This ontol metho exhausts pe-impulse pai befoe in-plane impulse oetion. The pe-impulses hanges D a only among six elements, an it hanges D M & magnitue whih is a seula ift ate. The ontol metho is use when obtaine time elay N is too lage o even negative. Solution ases of N an be ivie by { lim} { } { N N } ì1: N Î N N > N ï N = í : N Ï N N > Nlim, N < ï î3: N Î < whee N lim is a mission limit time. When N belongs to ase, the elaye impulse metho an be applie without aitional impulse pai. Howeve, fo ases 1 an 3, aitional impulse pai metho is suggeste. The equie semi-majo axis afte aitional impulses is obtaine in pevious wok an the esults an be expesse by whee -1 é 3p æ E öù æ 3p N ö lim aa, = a + ê ç + Nlim -1 ú ç DM - DM - ( n - n) t1 + ( a - a) ë a è øû è a ø ì1 + e, satelliteis at peigee at N = E = í î1 - e, satelliteis at apogee at N = whee a a, is a equie semi-majo axis afte pe-impulses, M inteval. (1) (16) (17) D is an element of { D } NON-DIMENSIONALIZED RELATIVE DYNAMICS FOR OPTIMIZATION This setion eives non-imensionalize elative ynamis between hief satellite an eputy satellite. The eive equations will be applie in optimization poblem. Optimization ties to fin ontol input to imize total in-plane impulse magnitue. Then, by analyzing obtaine ontol input histoy, we an see that the ontol histoy ten is simila with the popose aitional impulse pai metho. Non-imensionalization o also alle saling of ynami equations is often impotant in optimization appliation. It an affet pefomane of optimization iteation. Hene, this eseah uses sale elative ynamis fo optimization. By the way, elative motion an be expesse by two states, whih ae obital elements an elative Catesian omponents. Catesian omponents ae hosen in this pape beause it is moe intuitive an easy to sale the ynamis. Fistly, elative ynamis between satellites an be fomulate by 11 M 4

5 whee æ & ö m m && x - f& y& - y - xf& - = - + x + u è ø ç ( ) 3 x æ & ö m && y + f& x& - x - yf& = - y + u è ø m && z = - z + uz ç 3 y a (1 - e ) = 1 + e os f & = e sin f 3 f& m = (1 + e os f ) 3 3 a (1 - e ) (1 - e ) = ( + x) + y + z a m (18) (19) whee ( x, y, z ) ae Catesian position omponents of Hill fame, ( x&, y&, z& ) ae elative veloities. is a aial istane fom Eath of hief satellite, & is a veloity along aial ietion, is a aial istane of eputy satellite, f is a tue anomaly of hief satellite an f & is a ate of hange of tue anomaly of hief satellite. Also, a an e ae semi-majo axis an eentiity of hief satellite. Next, we nomalize state veto an elate paametes by using sale fatos. Sale state omponents ae efine as x = x /, y = y /, z = z / x& = x& / v, y& = y& / v, z& = z& / v && x = && x / a, && y = && y / a, && z = && z / a v = / t, a = / t u = u / u a, u = u / u a, u = u / u a x x y y z z = /, & = & / v, = /, f = f, f& = f& t ' ' ' whee ( x, y, z, x&, y&, z&, f ) ae non-imensionalize states, (&& x, && y,&& z ) ae nomalize aeleation omponents, (,,, f & ' & ) ae sale paametes use in elative ynamis, ( u, u, u ) ae x y z sale ontol aeleation an (, v, t, u ) ae sale paametes fo position, veloity, time an ontol aeleation. By using Eq. (), Eq. (18) an be nomalize by ()

6 && x f æ y & y ö x f æ t ö m æ t ö m x u u è ø è ø è ø ' ' ' ' - & ç & - ( ) ' - & - ç 3 = - ' ç 3 '3 + + x && y f æ x & x ö y f æ t ö m y u u è ø è ø ' ' ' ' ' ' + & ç & - ' - & = - ç 3 + '3 y && z æ t ö m z u u è ø ' ' ' = - ç 3 + '3 z m ' f& ' = (1 + e 3 3 os f ) t a (1 - e ) Now, by setting (, v, t, u ) popely fo speifi mission, we an use Eq. (1) as nomalize ynami equations. SIMULATION RESULTS In this setion, numeial simulations fo atifiial mission ae given. Table 1 esibes initial obit of eputy satellite, an Table shows obit element iffeene between esie obit an uent obit. Table 1. Initial obit of eputy satellite Longitue of asening noe W.8441 eg Agument of peigee w eg Semi-majo axis a 8. km Inlination angle i eg Eentiity e.3733 Mean anomaly M 1: : eg (1) Table. Obit elements iffeenes Longitue of asening noe Agument of peigee Semi-majo axis Inlination angle Eentiity Mean anomaly DW.3 eg D w. eg D a -. km D i -. eg D e -.3 D M 1: : -.6 eg In Tables 1 an, we an see that thee ae two elements in mean anomaly 6 th aw. They ae use fo ases 1 an 3 of Eq. (1). N is hosen to be 3. lim Now, six obital elements histoy with two ontol methos, popose metho an optimization metho, ae ompae. As an optimization tool, GPOPS(Geneal Pseuospetal Optimal Contol Softwae) 1 is use an ost funtion is efine by 6

7 t f J = ò u t () t 1 Fistly, esults of ase 1 ae investigate. Figues 1 an epit obit element eo histoies of both methos. In Figue 1, we an see that pe-impulse pai fies at initial time an semi-majo axis is hange. Then, at the final time fou obit elements ae oete by in-plane impulse pai. In optimization esult, simila histoy ten of semi-majo axis hange an be seen in Figue. By ontinuous fiing, semi-majo axis is hange an seula ift ate beome lage afte that buning. In aition, even ontinuous fiing is assume it looks obit elements histoies have isete hanges at speifi points. Figues 3 an 4 show esults of ase 3. In both figues, the sign of D a beome opposites an seula ift ietion is hange. Again, it is seen that obital element histoy shows isete hange behavio as an impulsive maneuve. D a[km] D W[eg] D e x D w[eg] D i[eg] time(obit numbe) D M[eg] time(obit numbe) (a) Δa,Δe,Δi (b) ΔΩ,Δω,ΔM Figue 1. Case 1: Obit Elements Eos Histoy with Aitional Impulse Pai Metho D a[km] D i[eg] D e x D W[eg] D w[eg] D M[eg] (a) Δa,Δe,Δi (b) ΔΩ,Δω,ΔM Figue. Case 1: Obit Elements Eos Histoy with Optimization 7

8 D a[km] D W[eg] D e x D w[eg] D i[eg] time(obit numbe) D M[eg] time(obit numbe) (a) Δa,Δe,Δi (b) ΔΩ,Δω,ΔM Figue 3. Case 3: Obit Elements Eos Histoy with Aitional Impulse Pai Metho D a[km] D i[eg] D e x D W[eg] D w[eg] D M[eg] CONCLUSION (a) Δa,Δe,Δi (b) ΔΩ,Δω,ΔM Figue 4. Case 3: Obit Elements Eos Histoy with Optimization Two popose ontol methos ae intoue, whih ae inopoating seula ift between satellites to eue in-plane impulse magnitue. Total impulse magnitue is efine as a sum of 1-nom of impulses an only in-plane motion is onsiee not out-of-plane motion. Optimization solution is obtaine to ompae the teneny of obit element histoy with the popose metho. GPOPS open-soue is use to emonstate optimization. Simulation stuies exae the pefomane of analytial an numeial methos an it is shown that obit elements eos histoy is simila between them. ACKNOWLEDGMENTS This eseah was suppote by NSL(National Spae Lab) pogam though the Koea Siene an Engineeing Founation fune by the Ministy of Euation, Siene an Tehnology (S A1-131). 8

9 REFERENCES 1 W. Clohessy an R. Wiltshie, "Teal Guiane System fo Satellite Renezvous." Jounal of Astonautial Siene. Vol. 7, No. 9, 196, pp H. Shaub an K. Alfien, "Impulsive Feebak Contol to Establish Speifi Mean Obit Elements of Spaeaft Fomations." Jounal of Guiane, Contol, an Dynamis, Vol. 4, No. 4, 1, pp L. S. Bege an J. P. How, "Gauss s Vaiational Equation-Base Dynamis an Contol fo Fomation Flying Spaeaft." Jounal of Guiane, Contol, an Dynamis, Vol. 3, No., 7, pp C. Sabol, R. Buns an C. MLaughline, "Satellite Fomation Flying Design an Evolution." Jounal of Spaeaft an Rokets. Vol. 1, No.,, pp P. Gufil, "Relative Motion between Ellipti Obits: Genealize Bouneness Conitions an Optimal Fomationkeeping." Jounal of Guiane, Contol, an Dynamis, Vol. 8, No. 4,, pp H. Shaub, S. R. Vaali, J. L. Junkins an K. T. Alfien, "Spaeaft Fomation Flying Contol using Mean Obit Elements." Jounal of Astonautial Sienes, Vol. 48, No. 1,, pp S. S. Vai, K. T. Alfien, S. R. Vaali an P. Sengupta, "Fomation Establishment an Reonfiguation using Impulsive Contol." Jounal of Guiane, Contol, an Dynamis, Vol. 8, No.,, pp G. Inalhan, M. Tilleson an J. P. How "Relative Dynamis an Contol of Spaeaft Fomations in Eenti Obits." Jounal of Guiane, Contol, an Dynamis, Vol., No. 1,, pp Y. Choi, S. Mok an H. Bang, "Impulsive Fomation Contol using Obital Enegy an Angula Momentum Veto." Ata Astonautia, Vol. 67, No. -6, 1, pp S. Mok, Y. Choi an H. Bang, "Impulsive Contol of Satellite Fomation Flying using Obital Peio Diffeene." 18 th IFAC Symposium on Automati Contol, Japan, Sep., H. Shaub an J. L. Junkins, AIAA Euation Seies, Analytial Mehanis of Spae Systems, 3, Chap A. V. Rao, D. A. Benson, C. L. Daby, M. A. Patteson, C. Fanolin, I. Sanes an G. T. Huntington, Algoithm 9: GPOPS, A MATLAB Softwae fo Solving Multiple-Phase Optimal Contol Poblems using the Gauss Pseuospetal Metho. ACM Tansations on Mathematial Softwae, Vol. 37, No., Apil-June, 1, pp