ࡱ> w5@~bjbj22XX (" B &-&-&-8^--B b.$/"F/F//u_n`tWa<Ցבבבבבב$R6! a_^u_aaF///UjUjUjaXF/: /ՑUjaՑUj2Ujjv /. `؊&-d!"20bCȖgPȖD 4 Ȗ aaUjaaaaaB B "&-9jB B &-CONSECUTIVE COLLISION ORBIT PROBLEMS TRANSFER MANEUVERS FROM ONE BODY BACK TO THE SAME BODY Denilson Paulo Souza dos Santos Antnio Fernando Bertachini de Almeida Prado Evandro Marconi Rocco Diviso de Mecnica Espacial e Controle INPE C.P. 515, 12227-310 So Jos dos Campos - SP, Brasil denilson@dem.inpe.br; prado@dem.inpe.br; evandro@dem.inpe.br ABSTRACT The objective of this work is to study optimal space maneuvers that try to reduce the fuel consumption to make interplanetary missions possible. In this type of maneuver, the spacecraft leaves and comes back to the same body. This is called "Application in Space Maneuvers of the Problem of the Multiple Encounters". For the dynamics of the problem, it is assumed that the three bodies involved are points of mass and do not suffer external disturbances. The considerations done by Prado and Broucke [1] are extended, and the departure and arrival angles of the spacecraft M3 can be non-symmetrical. The topics studied here can be applied in missions that leave and come back to the same body, as a mission that leaves the Earth, researches the interplanetary space and comes back to the Earth. Several consecutive collision orbits were simulated and solutions were shown in new variables for better analysis and visualization of the results. 1. INTRODUCTION The study and the exploration of the space environment are priorities for the countries that want to develop space technologies. To plan and to execute maneuvers with artificial satellites that are in orbit around the earth are important tasks in space activities. The literature on this theme is vast and several researchers developed alternatives and solutions for these subjects (Gobetz [2], Goddard [3], Hohmann [4], Marec [5], Prado [6], Prado and Rios-Neto [7] and Rocco [8]), however all of them introduce specific modeling for each approach of the problem. To optimize the consumption of fuel in an orbital maneuver of an artificial satellite is an important goal, because it makes possible a larger number of maneuvers, with a consequent prolongation of the life of the satellite.  2. DESCRIPTION OF THE CONSECUTIVE COLLISION ORBITS PROBLEM M1 and M2 are the two primaries with masses (1 - () and (, respectively. M2 is in a circular orbit around M1. The space vehicle M3 leaves M2 from a point P (t = (0). It follows a trajectory around M1 and meets again with M2 in a point Q (t = (f), where (0, (f ( [0, 2(] and (f > (0. The values of (0 and (f are not necessarily symmetrical. The problem will be modeled using the dynamics of the two bodies, which means that ( = 0, implying in the reduction of the problem with three bodies to a problem with two bodies. In this way, the equations derived by Kepler can be used to find the solutions. Two impulses will be used in the transfer maneuver. It is assumed that the three bodies involved are mass points and do not suffer external disturbances (fig. 1). Hnon [9, 10] studied this problem and published results with solutions for the case of circular orbits. Howell [11] published solutions for the elliptic case, where the transfers were symmetrical with respect to the periapsis. Prado e Broucke [12] also published solutions for this problem, in the same situations, using the Lambert method to solve the problem. The results were analyzed and disposed in form of tables and graphs. 3. MATHEMATICAL FORMULATION OF THE CONSECUTIVE COLLISION ORBITS PROBLEM The Hnon problem [9, 10], formulated as a Lambert problem, can be described as below. The position of M3 is known at t = (0 (point P, initial point of the transfer orbit). The position vector R1 can be specified as a function of the angle (0, where:  EMBED Equation.3  (1) It is the same value for M2 and M3, because at the initial moment (t = (0) M2 and M3 occupy the same position. The position of M3 at t = (f (point Q, final point of the orbit). The position vector R2, similar to the item above, is described by the equation:  EMBED Equation.3  (2) The total transfer time is given by EMBED Equation.3 . Remember that the angular velocity of the system (() is unit, so ( can be considered to be the time as well as the angle. The total angle ((, which the spacecraft must travel to go from P to Q, for the case where the orbits are elliptic, has several possible values. First of all, it is necessary to consider two possible choices for the transfer: the one that uses a direction of the shortest possible angle between P and Q (short way) and the one that uses the direction of the longest possible angle between these two points (long way). After considering these two choices, it is also necessary to consider the possibilities of multirevolution transfers. In this case, the spacecraft leaves P, makes one or more complete revolutions around M1, and then it goes to Q. Thus, combining those two factors, the possible values for (( are [(f - (0 + 2m(] and [2( - ((f - (0) + 2m(]. There is no upper limit for m, and this problem has an infinite number of solutions, except in the case where the orbit of M3 is parabolic or hyperbolic, where (( has a unique value. The solution of the Lambert problem is the Keplerian orbit that contains the point P and Q and that requires the given transfer time (t = (( = (f - (0 for the spacecraft to travel between these two points. In this paper, we used the Goodings Lambert routine to solve the Lambert problem [13]. Possible applications for this maneuver are interplanetary research of the Solar System, a basis for a transportation system between the Earth (M1) and the Moon (M2) where no orbit correction is required, etc, (Chobotov [14]). 4. DEVELOPMENT AND RESULTS The solution of the Lambert problem was searched for a long time. The approach to solve this problem is to establish a group of the nonlinear equations (two body problem) and to begin a process of iterations to find an orbit that satisfies all the requirements. The largest difficulty is to choose the best group of equations and parameters for the iterations, so that it guarantees that the convergence happens in all of the cases. The presented solutions are in terms of the true anomaly ((), eccentric anomaly ((), velocity variation ((V) and we also show solutions in terms of the semimajor axis (a). They are defined in the following way: If the vehicle passes in the periapsis in ( = 0:  EMBED Equation.3   EMBED Equation.3  If the vehicle passes in the apoapsis in ( = 0:  EMBED Equation.3   EMBED Equation.3  Solutions with (V < 0.01 e = 0.1True Anomaly RedefinedEccentric AnomalyVariation of Velocity( (rad)( (rad)( (rad)(V-1.0030.997530.997770.000649-1.0020.998350.998510.000432-1.0010.999180.999260.0002160.9991.99921.99910.0005641.9990.998770.998890.000915-0.0040.995090.994570.00084-0.0030.996310.995920.000629-0.0020.997540.997280.000419-0.0010.998770.998640.000209   5. CONCLUSION We studied optimal space maneuvers that reduce the fuel consumption for interplanetary missions, in particular in situations where the spacecraft leaves and comes back to the same body, considering non-symmetrical situations. We included a swing-by in a return passage as a form of gain energy. We showed many solutions, including low-cost maneuvers, that can be used for planning of real missions. 6. ACKNOWLEDGEMENTS The authors are grateful to CAPES-Brazil (Coordenao de Aperfeioamento de Pessoal de Nvel Superior) for the scholarship given to the first author, to the National Council for Scientific and Technological development (CNPq), Brazil, for the research grant receive under Contract 300828/2003-9 and to the Foundation to Support Research in the So Paulo State (FAPESP) for the research grant received under Contract 2006/00187-0. 7. REFERENCES Prado, A. F. B. A. and Broucke R. A.; Study of Hnons Orbit Transfer Problem Using the Lambert Algorithm, AIAA Journal of Guidance, Control, and Dynamics, vol 17, number 5, 1993, pages 1075-1081. Gobetz, F. W. and Doll, J. R., 1969, "A Survey of Impulsive Trajectories," AIAA Journal, Vol. 7, pp. 801-834. Goddard, R. H.; A method of reaching extreme altitudes. Smithsonian Institute Public Miscelanea Collect, v. 71, n.2, 1919. Hohmann, W., 1925, "Die Erreichbarkeit Der Himmelskorper", Oldenbourg, Munique. Marec, J.P., "Optimal Space Trajectories," New York, NY, Elsevier, 1979. Prado, A. F. B. A., Transferncias de rbitas com aplicaes a trajetrias interplanetrias, Fundamentos de Tecnologias Espaciais, Prado, A. F. B. A. and Kuga, H. K., Editores, pg. 171-197, INPE So Jos dos Campos SP, 2001. Prado, A.F.B.A. e Rios-Neto, A., "Um Estudo Bibliogrfico sobre o Problema de Transferncias de rbitas". Revista Brasileira de Cincias Mecnicas, Vol. XV, No. 1, 1993, pp. 65-78 (INPE-5601-PRE/1816). Rocco, E.M., Prado, A.F.B.A., Souza, M.L.O., Bi-Impulsive Orbital Transfers Between Non-Coplanar Orbits with Time Limit. Applied Mechanics in the Americas, Vol. 6, 1999, pp. 259-262. Editors: Djenane Pamplona, Charles Steele, Hans I. Weber , Paulo B. Gonalves, Iwona Jasiuk, Luiz Bevilacqua. ISBN: 85-900726-1-4. M. Henon, Bull. Astronom. (serie 3) (1968), 377. Hitzl, D.L. and Hnon, M.; Critical Generating Orbits for Second Species Periodic Solutions of the Restricted Problem, Celestial Mechanics, vol. 15, 1977 pag. 421-452. Howell, K. C.; Consecutive Collision Orbits in the Limiting case (=0 of the Elliptic Restricted Problem Celestial Mechanics, vol. 40, 1987 pag. 393-407. Prado, A.F.B.A. e Broucke, R.A., "The Problem of Transfer Orbits from one Body Back to the Same Body". Advances in the Astronautical Sciences, Vol. 82: Space Flight Mechanics, Part II, Editors: Robert G. Melton, Lincoln J. Wood, Roger C. Thompson, Stuart J. Kerridge, pp.1241-1260 (INPE-5599-PRE/1814). Gooding, R. H.; A Procedure for the Solution of Lamberts Orbital Boundary-Value Problem, Celestial Mechanics, vol.48, 1990, pp. 145-165. Chobotov, V.A.; Orbital Motion Second Edition. Reston: American Institute of Aeronautics and Astronautics, 1996. 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In 88-88.27827T888-8.S828.78!87888!1.T22882.882287.8 2 @  this type of .82.188.7y2 Imaneuver, the spacecraft leaves and comes back to the same body. This is dT878818!1881288172!81881821888128S821882218188127T7188811>8212t2 ` Fcalled "Application in Space Maneuvers of the Problem of the Multiple 2888P#C782887P7PC8818PS878818!2O7P78PC 887TP7O88PS8882  Encounters".C828788!1$ 2 ( 352  RFor the dynamics of the problem, it is assumed that the three bodies involved are =8!888188T227788!887T28228S8887888!88788728181888!82 poin888}2 Lts of mass and do not suffer external disturbances. The considerations done 2 7T822888 77 88 278! 818!88 828!888182>88 188278!8882 78772 Oby Prado and Broucke [1] are extended, and the departure and arrival angles of 81#C!888"878#C!78227#8#8!7#8188788#787#87#7888!8!8"887#8!!18#88782#7-#2 .the spacecraft M88188282!7S@"Arial- 2 ;33%-2 .  can be non27888788 2 .-3!2 .# symmetrical.21TS8!28 2 .T  35d2 ;The topics studied here can be applied in missions that lear>78"8822"28878"78!8"287"78"78887"8!T22882"78"88)2 N ve and come back to 18"878!28T7"7822"72 NQthe same body, as a mission that leaves the Earth, researches the interplanetary 8827T78781828T2288887818278C8!7!8278!28827888!87888!22 Mspace and comes back to the Earth. Several consecutive collision orbits were f28828,788,27T82,8812,8,88,C8!7,C818!8,288282718,28298,8!82,G8!8}2 mLsimulated and solutions were shown in new variables for better analysis and 2T8888187812887821G8!81289G818188H218!888218!1878 18881221888-2 vi142 5sualization of the results.n2881888788 8182- 2  3, 2  3--#2 1. INTRODUCTION 8H=HNHHH=NH 2 x 38 2   38-2 <NThe study and the exploration of the space environment are priorities for the >7802781088807808188!888/70780278270881!88T8708!808!7!8208!087X2 3countries that want to develop space technologies. 2888!8287G88788188827828827888782 2   362 [OTo plan and to execute maneuvers with artificial satellites that are in orbit a >78787888818288S878818!2G88!2828882878!788!882 [round !87882 Sthe earth are important tasks in space activities. The literature on this theme is k88&88!8&8!8'S88!77'8222'8'18818'82182'=88'8!88!8&88&82'87T7'2}2 {Lvast and several researchers developed alternatives and solutions for these 182888882818!88!8288!288!2878188888778!881828888818888278!888272 Psubjects (Gobetz [2], Goddard [3], Hohmann [4], Marec [5], Prado [6], Prado and 288822 !M8871 8N8878!8 8 H78S878 8 S8!82 8 C!787 8C!887 877-                    @@@ NormalCJ_HaJmHsHtH@@@ Ttulo 1$$@&a$:CJ >A@> Fonte parg. padroXi@X  Tabela normal :V 44 la ,k@, Sem lista DB@D Corpo de texto$d a$bOb Estilo Denilson 7$8$H$5CJOJQJ\aJmH sH X>@X Ttulo$<@&a$5CJ KHOJQJ\^JaJ FP@"F Corpo de texto 2 dxHQ@2H Corpo de texto 3xCJaJjOBj FR11$dh1$5$7$8$9DH$]^a$5CJ_HmHsHtHVObVSection Header$5_HmHnHsHtHuRObR Section Body$T`Ta$_HmH sH tH4 @r4 Rodap  C"8)@8 Nmero de pgina:@: Cabealho  C"!%(+/37PSk -.Kilosw{!I+UTNLJIHFEDC:87|slje\640/-+)&%$YZ!%(+/37PSk -.Kilosw{!$      !"#I+I+%\]}%bclm./GIJKLMNOPQRSTUV O P @~W*[s $6AENOW_gjkrz  #+,3;CLMT\dmnu}YZno-.c 1!z!_")#e$$A%% '' ((((#($(%(&(>(?(B(C(F(G(J(K(M(N(P(Q(T(U(X(Y(\(](u(v(x(y(((((((()) )4)B)C)D)))))))***0*1*Q*R*S*T*p*q*************************E+F+G+J+000000000000000000000000000000000000000000 000 00 0 0000000000 0 0 0 0 0 000 000 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00000000000000000000000000000000000000000000000000000 0 0 0 0 0 0 0 0 0 0  0  0  0  0 @0@0@0@0@0@00} g0000000000000000000000000000000000000000000000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  [ !W"#)G00B11120~~!"$&2579:;<>@L[!N"j""""" #+#L#m####-'#00q2~~ #%'()*+,-./013468=A~7KM[oqsI+::::::: !!.FH$:::P`a2$XiPqikZ/l` 22$tMڪn ^&b$3QF,5LjJ-b$EMY; *+9;<b$?Iz.`_8f!gR;B:<b$>Iȭ*#+oذ[,~-b$ gi,iY$?3R$2:?E;: 2$fofHV}k3,b$_ڿצ {jjjP2$֭Rz'd1Yb$<8R9匸~/@6-O`< 8,(  j!b 8 -< W #"  \ s *\  S"`? l<k X n L,& V #" 8 -z2  s * ? #" `#9z2   s * ? #" `]#6B ! 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