Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION ORBIAL MANEUVERS USING LOW-HRUS VIVIAN MARINS GOMES, ANONIO F. B. A. PRADO, HÉLIO KOII KUGA Ntionl Institute for Spce Reserch INPE - DMC Av. Dos Astronuts 1758 São José dos Cmpos SP 12227-010 BRAZIL vivin.gomes@uol.com.br; prdo@dem.inpe.br; hkk@dem.inpe.br Abstrct: - o perform the orbitl mneuvers, softwre tht clcultes n optiml mneuver is developed. his method will be used s reference for comprison nd nlises of the suboptiml methods to be used on bord. his method id bsed on n nlyticl development tht generte equtions tht cn be computed in shorter time, llowing rel time pplictions. Key-Words: - Orbitl mneuvers, low thrust, strodynmics, rtificil stellites, orbitl dynmics. 1 Introduction he problem of clculting orbitl mneuvers is very importnt topic in Orbitl Mechnics. hus, the problem of trnsferring spcecrft from n orbit to nother hs grown in importnce in recent yers. Applictions of this study cn be found in vrious spce ctivities, such s plcing stellite in geosttionry orbit, the mneuvers of spce sttion, orbit mintennce of stellite, mong others. In ctul pplictions, there my be need to mke n dditionl mneuver, both for trnsfer orbit or only for periodic corrections of lesser mgnitude. his issue of trnsfer is to chnge the position, velocity nd mss of the stellite from its current sttus to new stte pre-determined. he trnsfer my be completely constrined or prtilly free (free time, free finl velocity, etc.). In the most generl cse, the choice of direction, sense nd mgnitude of the thrust to be pplied should be mde, respecting the limits of the vilble equipment. o crry out this trnsfer, it is intended to use optiml or suboptiml continuous mneuvers [1], [2]. So, to fulfill tht tsk, two methods for clculting mneuvers were developed. he first of them will get n optimiztion without worrying bout the processing time. he second method is sub-optiml nd it will pproch the directions of ppliction of the thrust to llow fster clcultion of the control. he optimum method will be used to compre the consumption obtined by the sub-optiml method, which involves simplifictions for ech specil sitution, in order to obtin high processing speed, fvoring the possibility of using it in rel-time. In both methods, it will be ssumed tht the mgnitude of the thrust to be pplied is constnt nd smll nd the serch will be to find its direction. his direction cn be free (optiml method), [] or with some kind of constrints (sub-optiml method). 2 Sub-optiml Method he gol of this topic is to develop sub-optiml method with high-speed computing for the clcultion of orbitl mneuvers bsed on continuous thrust nd smll mgnitude. he ide is to hve method tht genertes quick result nd, if possible, with result in terms of cost of fuel not much different from the optiml method described bove. his method should be used in cses of trnsfers with smll mgnitude, which usully re more frequent in the steps tht follow the insertion of the spcecrft in its nominl orbit. o solve this problem, it ws chosen in the literture bse method to mke expnsions nd djustments to the needs of this work. he method is described below. A ner optiml method for clculting orbitl trnsfer nd with minimum time (so, minimum consumption, s the mgnitude of the thrust is constnt, wht implies tht the time of ppliction of thrust nd consumption re directly proportionl) round the Erth for spcecrft with electric solr propellnt ws developed by [4]. It used technique of direct optimiztion to solve the problem of optiml control, with pproches towrd ppliction of thrust. he optiml trjectories clculted by the direct pproch present very close results to the optiml trjectories obtined from vritionl clcultion. he equtions of motion for the vehicle when the thrust is cting re shown below. he equtions re written in terms of non-singulr equinotil elements to cover both circulr nd plnr orbits (i = 0, 180). he reltion between the equinotil elements (, h, k, p, q, F) nd the clssicl orbitl elements (, e, i, W, W, E) is given by: h = esen( ω Ω) (1) ISSN: 1790-5117 120 ISBN: 978-960-474-054-
Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION k = e cos( ω Ω) (2) i p = tn sen Ω 2 () i q = tn cos Ω 2 (4) F = Ω ω E (5) where: = semi-mjor xis, e = eccentricity, i = inclintion, W = longitude of the scending node, w = rgument of perigee, E = eccentric nomly nd F = eccentric longitude. For spcecrft moving in the grvittionl field nd subject to the propulsive force, the equtions of motion re s follows: ' x = Mαˆ (6) In eqution 6 the stte vector is x = [, h, k, p, q] nd the sign ( ') indictes the derivtive with respect to time. he vector ( x 1) is unit vector long the direction of thrust ppliction. he vlue is the mgnitude of the thrust ccelertion, given by: 2ηP0 = (7) mgisp where h is the efficiency of the propulsion system, P0 is the initil power given propulsion system, m is the mss of the vehicle, g is the grvity ccelertion t se level nd Isp is the specific impulse. he eqution of stte for F is not included becuse it hs been used the verge of orbitl elements nd thus only elements which vry slowly re considered. he processing time is significntly reduced when using orbitl verges. As ll orbitl elements used re vribles tht vry slowly, due to the fct tht the force of thrust hs little mgnitude, it cn be used mjor steps of integrtion, in the order of dys. he eqution of motion of the spcecrft cn be pproximted by clculting the increment of ech orbitl element in period nd dividing by such time. herefore, the vrition in time of the equinotil elements by complete orbit with the propeller cting cn be obtined from the eqution: π ' 1 dt x = ˆ α df M (8) df π where is the pproximtion of the stte nd is the orbitl period. he br on top of vribles mens tht they were evluted using the verge stte vector. he integrtion represents the chnge in orbitl elements in revolution with the orbitl elements kept constnt, unless the eccentric longitude F, which is vried between -p e p. Setting the direction for the thrust ppliction by the components (id, jd, kd), the product shown inside the integrl symbol cn be obtined. As the ccelertion is held constnt, it mens tht this vlue cn be plced outside of the integrl symbol. hus, the nlyticl equtions used for the terms corresponding to ech of the elements re shown below, where id, jd nd kd represent the three components of the direction vector of the thrust ppliction. For the semi-mjor xis (): 2jdA1 2idA 2 & = (9) µ µ ( 1 k cosf hsen F) ( 1 k cosf hsen F) For the element h: ( B1 B2 ) jd 1 ( B ( B4 B5 ) sen F) id 1 h & = kkd( p B6 q B7 ) (10) 1 For the element k: ISSN: 1790-5117 121 ISBN: 978-960-474-054-
Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION ( C1 C2 ) id 1 ( ( C cosf C4 ) C5 ) jd 1 k & = hkd( p*c6 q C7 ) (11) 1 For the element p: 2 hk cosf k kd(1 p q ) h 1 sen F 1 1 h k 1 1 h k p & = (12) µ 2 1 For the element q: 2 h hksen F kd(1 p q ) k 1 cosf 1 1 h k 1 1 h k q & = (1) µ 2 1 hese equtions re written in terms of equinotil orbitl elements. It is lso possible to write them ccording to the trditionl keplerin elements. hey cn be found in [5]. he clculus of integrtion shown in Eqution 8 genertes equtions tht re too long for rel pplictions, especilly tking into ccount the need to implement them in rel time for mneuver. hus, lthough these equtions, in their complete form shown here, generte new method for clculting orbitl mneuvers, their use will be focused on individul cses. his implies to define reference orbit, which my be the finl desired orbit, the initil orbit of the spcecrft or even n verge of those two orbits. As it will be considered only mneuvers with smll mplitudes, this restriction will not bring gret losses in terms of ccurcy. hen, with these pproches mde, numericl vlues cn be used, so tht, these functions re only functions of F nd numericl constnts. From there, the integrl used in eqution 8 cn be clculted nd it is obtined simple nlyticl equtions for the vrition of ech orbitl element considered s function of direction nd mgnitude of the thrust pplied. hus, the problem of obtining the lowest fuel consumption in mneuver cn be defined s to find the optiml direction for thrust ppliction tht minimizes: J = t f (14) Subject to the men equtions of motion nd the initil condition: x (0) = x 0 (15) nd lso subject to the ties in the finl stte: ψ x(t ),t = x(t ) x (16) [ ] 0 f f f f = Results Severl mneuvers were simulted to test the methods developed [5]. he first two mneuvers involve the sme initil orbit, but the directions of thrust ppliction nd time of the opertion re different, with the gol of reching finl orbit frther. he third mneuver involves greter rnge of vrition in semi-mjor xis nd it is good to demonstrte the pplicbility of the method in situtions like this..1 Mneuver 1 In this specific cse, the semi-mjor xis ws chnged nd eccentricity nd the rgument of perigee of the ISSN: 1790-5117 122 ISBN: 978-960-474-054-
Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION orbit were kept constnts. he chnges were smll in mgnitude (bout 47 meters in semi-mjor xis, the min objective of the mneuver) to be comptible with the method developed. he rgument of perigee ws kept constnt s 90 degrees, but it could be ny vlue. ble 1 shows the elements of the initil orbit nd the components of the orbit to be chieved fter mneuver 1. ble 2 shows the input dt required for the first mneuver simultion with the optiml method: orbitl elements of the initil orbit, the vehicle chrcteristics (initil mss, the mgnitude of thrust, initil position of the vehicle, true nomly), the condition imposed to the finl orbit nd estimtion of the solution to strt the process of itertion (beginning nd end of propulsion, ngles of pitch nd yw nd their rtes of initil chnge nd n estimtion of fuel consumption). ble 1: Elements of initil nd finl orbit for mneuver 1. Initil orbit Condition in the finl orbit Semi-mjor xis Semi-mjor xis 7259,650 km 7259,697 km 0,0629 0,0629 Inclintion 66,52º Long. of scending node 110º Optiml cse ble 2: Dt for the mneuver1 using the optiml method. otl mss (vehicle fuel) 2500 kg Avilble thrust = 1 N Spcecrft Initil position = 0 initil dt rue nomly = 0º Strt of the engime = 0º Stop of the engine = 5º Initil estimte of solution Sub-optiml cse Initil pitch ngle = 0º Initil yw ngle = 0º Initil rte of vrition in pitch = 0 Initil rte of vrition in yw = 0 Fuel needed to mneuver = 2 kg urning the initil keplerin elements to non singulr elements, ccording equtions 1 to 5: = 7,25965x106 m h = -0,0220906 k = -0,0588826 p = 0,614085 q = -0,208 herefore, now the numericl vlues of the orbit cn be used, which will be used s the reference orbit, crrying out the integrtion shown in eqution 8 nd, in this wy, it is possible to obtin set of equtions tht provide the vrition of ech of the elements used to describe the orbit by orbitl revolution with the propellers cting ll the time. herefore, they will become the equtions of motion of the spcecrft with the ssumptions dopted. hese equtions, lredy tking into ccount the fct tht it ws plnr mneuver, so, kd = 0, s function of the components of the vector tht defines the direction of thrust pplied, re: d= cel*(1987180*id5296820*jd) (17) dh= cel*(-0,822*id0,0081686*jd) (18) dk= cel*(-0,0081686*id0,851282*jd) (19) dp= =0 (20) dq= =0 (21) where ccel is the ccelertion imposed by the stellite propellnt. o show in detil the usefulness of these equtions, figures 1 to show the vrition of the elements by orbit s function of the direction of the thrust pplied. It is possible to get mny informtions bout the effect of the direction of the thrust pplied in orbitl elements. Figure 1, mde for the sitution where the direction of the thrust pplied is constnt, shows tht there is vlue of the component x for which the semi-mjor xis shows mximum vrition. his vlue is round 0.5. ht figure my be used for prior ssessment of the direction of thrust pplied depending on the objectives of the mission. d(km) 5.5 5 4.5 4.5 2.5 id 0.2 0.4 0.6 0.8 1 Fig. 1: Vrition of the semi-mjor xis s function of the direction of the thrust pplied. ISSN: 1790-5117 12 ISBN: 978-960-474-054-
Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION -2 10-7 -4 10-7 -6 10-7 -8 10-7 dh 0.2 0.4 0.6 0.8 1 id ble : Finl keplerin elements obtined for mneuver 1. Finl elements Optiml Sub-optiml method method Semi-mjor xis 7259,697 7259,697 (km) 0,062887 0,062887 Inclintion 66,52 66,52 Long. of scending 110 110 node 90 90 Consumption (kg) 0,1652 0,2808 Fig. 2: Vrition of the orbitl element h s function of the direction of the thrust pplied. Durtion of the mneuver (min) 02 512 dk 8 10-7 6 10-7 4 10-7 2 10-7 id 0.2 0.4 0.6 0.8 1.2 Mneuver 2 For mneuver 2, the semi-mjor xis ws lso chnged nd the eccentricity nd the rgument of perigee of the orbit were kept constnt. he chnges were of mgnitude slightly higher thn in the previous cse, bout 120 meters in semi-mjor xis, the min objective of the mneuver, lso iming to be comptible with the method developed. he rgument of perigee ws kept constnt in vlue 90 degrees, but could be ny vlue. ble 4 shows the elements of the initil orbit nd the conditions imposed for mneuver 2. Fig. : Vrition of the orbitl element k s function of the direction of the thrust pplied. As p nd q re constnts, grphic is not shown. With these equtions, the softwre Mthemtic is used to solve the optimiztion problem nd to obtin the optiml solution. Severl ssumptions cn be mde bout the direction of the thrust pplied. he simplest of them is ssuming constnt direction. So, the problem becomes to find the vlue of id tht genertes the minimum fuel consumption, becuse kd = 0 (plnr mneuver) nd jd is obtined by the condition tht the vector tht defines the direction of the thrust pplied is unit. he solution found is id = 0.8. Considering liner or prbolic reltions cn reduce the consumption so much nd the time of mneuver obtined, but it is not studied in this prt of the work. ble shows the finl orbit chieved by the spcecrft fter the mneuver, for the optiml nd sub-optiml method, s well s the fuel consumed nd time of mke the mneuver. ble 4: Elements of initil nd finl orbit for mneuver 2. Initil orbit Condition in the finl orbit Semi-mjor xis Semi-mjor xis 7259,650 km 7259,770 km 0,062900 0,062900 Inclintion 66,52º Long. of scending node 110º Optiml cse ble 2 lso shows the prmeters used for mneuver 2, with the optimum method. ISSN: 1790-5117 124 ISBN: 978-960-474-054-
Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION Sub-optiml cse As the initil orbit (which is lso used s the reference orbit) is the sme s the previous exmple, both the initil orbitl elements nd the pproximte equtions of motion re the sme. So, the softwre Mthemtic is gin used to solve the problem of optimiztion nd to get the optiml solution. Once more it will be ssumed constnt direction. In this wy, the problem becomes to find the vlue of id tht generte the minimum fuel consumption, becuse kd = 0 (plnr mneuver) nd jd is obtined by the condition tht the vector tht defines the direction of the thrust pplied is unit. he solution found is id = 0.41. ble 6 shows the finl orbit chieved by the spcecrft fter the mneuver, for the optiml nd sub-optiml methods, s well s the fuel consumed nd the time of the mneuver. References: [1] sien, H. S. ke-off from stellite orbit. Journl of the Americn Rocket Society: 2(4), 2-26, Jul-go, 195. [2] Lwden, D. F. Optiml progrmming of rocket thrust direction. Astronutic Act: 1(1), 41-56, Jn-fev, 1955. [] Prdo, A. F. B. A.; Neto, A. R. Um Estudo Bibliográfico Sobre o Problem de rnsferêncis de Órbits. Revist Brsileir de Ciêncis Mecânics, 15(1):65-78, 199. [4] Kluever, C. A.; Oleson, S. R. A direct pproch for computing ner optiml low-thrust trnsfers. Advnces in the Astronuticl Sciences, v. 97, n. 2, p. 178-1800, 1997. [5] Gomes, V. M. Determinção de órbit e mnobrs utilizndo gps e motor com bixo empuxo. Phd hesis. São José dos Cmpos, INPE, 2008. bel 6 - Finl keplerin elements obtined for mneuver 2. Finl elements Optiml method Sub-optiml method Semi-mjor xis 7259,77 7259,77 (km) 0,0628876 0,0628876 Inclintion Long. of scending node Consumption (kg) Durtion of the mneuver (min) 66,52 66,52 110 110 90 90 0,2520 kg 0,900 kg 601,8 1020,2 4 Conclusion It ws studied nd developed method for the cse of sub-optiml continuous mneuvers. his method is bsed on n nlyticl development, which genertes equtions tht cn be used for fst processing time, llowing its use in rel time. he gol is to find the direction of the thrust pplied to perform the orbitl mneuvers, with the ppliction of the liner direction of the pplied thrust. he time nd consumption re bout 20% higher when compred to the ones obtined from the optiml method, so the suboptiml method cn be used s first estimte. ISSN: 1790-5117 125 ISBN: 978-960-474-054-