In this section, the optimization problem based on the PCR3BP is formulated. The following assumptions are employed:

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1 IN nd International Congress of echanical ngineering (COB 01) November -7, 01, Ribeirão reto,, Brazil Copyright 01 by ABC UN RTURBATION ON OTIAL TRAJCTORI FOR ARTH - OON FLIGHT andro da ilva Fernandes 1 Cleverson aranhão orto arinho 1 epartamento de atemática, Instituto tecnológico de Aeronáutica, ão José dos Campos Brazil BRAR.A., ão José dos Campos Brazil sandro@ita.br cleverson.marinho@embraer.com.br Abstract. In this work, the problem of transferring a space vehicle from a circular low arth orbit (LO) to a circular low oon orbit (LO) with minimum fuel consumption is presented. The optimization criterion is the total characteristic velocity. The optimization problem has been formulated using the classic planar circular restricted three-body problem (CRB) and the planar bi-circular restricted four-body problem (BR4B). In both cases, the optimization problem has been solved using a gradient algorithm in conjunction with Newton-Raphson method. Numerical results are obtained for several final altitudes of a clockwise or counterclockwise circular low oon orbit for a specified altitude of a counterclockwise circular low arth orbit Keywords: arth-oon trajectories, optimal trajectories, un perturbations 1. INTROUCTION In the last decades, new types of trajectories have been proposed to transfer a spacecraft from an orbit around arth to an orbit around oon, which reduce the cost of the traditional transfers based on the two-body dynamics (Chobotov, 005). The new trajectories are designed using more realistic models of the motion of the spacecraft such as the planar circular restricted three-body problem or the planar bi-circular restricted four-body problem (Belbruno, 004; Koon et al, 007). In these models, the motion of the spacecraft ehibits very comple dynamics that are used to design new arth-to-oon trajectories (Conley, 1968; Belbruno et al, 010). New trajectories with large time of flight (about 80 to 150 days) are calculated using the concept of weak stability boundary introduced by Belbruno (004). These new trajectories are usually referred as low energy transfers (Conley, 1968; Koon et al, 001; Belbruno, 004). Low energy arth-oon transfers can be classified into eterior or interior, according to the geometry (Topputo, 01). In the eterior transfers the spacecraft is injected into an orbit with large apogee which crosses the oon orbit. The apogee distance is approimately four times the arth-oon distance. This kind of trajectories eploits the un s gravitational attraction (Yamakawa et al, 199, 199). In the interior transfers most part of the trajectory occurs within the oon orbit. Although the new approaches reduce the cost of the mission, only few works consider the problem of minimizing the total cost (Yagasaki, 004a,b; a ilva Fernandes and arinho, 01; Topputo, 01). In this work, a preliminary analysis about the perturbation of the un on the problem of transferring a space vehicle from a circular low arth orbit (LO) to a circular low oon orbit (LO) with minimum fuel consumption is presented. It is assumed that the velocity changes are instantaneous, that is, the propulsion system is capable of delivering impulses. Trajectories with two impulses are considered in the analysis: a first accelerating velocity impulse tangential to the space vehicle velocity relative to arth is applied at a circular low arth orbit and a second braking velocity impulse tangential to the space vehicle velocity relative to oon is applied at a circular low oon orbit (iele and ancuso, 001). The minimization of fuel consumption is equivalent to the minimization of the total characteristic velocity which is defined by the arithmetic sum of velocity changes (arec, 1979). The optimization problem has been formulated using the classic planar circular restricted three-body problem (CRB) and the planar bi-circular restricted four-body problem (BR4B). Numerical results are obtained for several final altitudes of a clockwise or counterclockwise circular low oon orbit for a specified altitude of a counterclockwise circular low arth orbit. irect ascent trajectories, with time of flight of approimately 4.5 days, and multiple revolution trajectories, with time of flight of approimately 41.0 days, are considered in this study. The results for mission with multiple revolutions show that fuel can be saved if a lunar swing-by occurs.. OTIIZATION ROBL BA ON TH CRB In this section, the optimization problem based on the CRB is formulated. The following assumptions are employed: 1. arth and oon move around the center of mass of the arth-oon system;. The eccentricity of the oon orbit around arth is neglected;. The flight of the space vehicle takes place in the oon orbital plane; 4. The space vehicle is subject to only the gravitational fields of arth and oon; 5. The gravitational fields of arth and oon are central and obey the inverse square law; 6569

2 IN andro da ilva Fernandes and Cleverson aranhão orto arinho un erturbations on Optimal trajectories for arth-oon Flight 6. The class of two impulse trajectories is considered. The impulses are applied tangentially to the space vehicle velocity relative to arth (first impulse) and oon (second impulse). Consider an inertial reference frame Gy contained in the oon orbital plane: its origin is the barycenter of arth- oon system; the -ais points towards the oon position at the initial time t 0 0 and the y-ais is perpendicular to the -ais. In this reference frame, the motion of the space vehicle is described by the following set of differential equations: d du u dt dt r r where dy dt dv v y y y y dt r r is the arth gravitational parameter, is the oon gravitational parameter, r and r are, respectively, the distances of the space vehicle from arth () and oon (); that is, r y y r y y equations 1 cos (1) and. The position vectors of arth and oon are defined in the reference frame Gy by the t t y t t t t y t y t where t t, 1 sin, (), () and is the distance from the arth to the oon. The initial conditions of the system of differential equations (1) correspond to the position and velocity vectors of the space vehicle after the application of the first impulse. The initial conditions t 0 0 can be written as follows y where 0 r 0cos 0 u 0 y 0 r 0sin 0 v 0 0 v sin 0 LO r 0 0 v cos 0 LO y r 0 vlo is the velocity change at the first impulse, r h 0 a 0 and t 0 0, (4) is the angle which the position vector r forms with -ais. h 0 is the altitude of LO and a is the arth radius. It should be noted that v 0 are orthogonal (impulse is applied tangentially to LO). From qs () and (), one finds 1 0 y y 0 r and 0. (5) 1 The final conditions of the system of differential equations (1) correspond to the position and velocity vectors of the space vehicle before the application of the second impulse. The final conditions t f T can be put in the form (da ilva Fernandes and arinho, 01), T T y T y T r T, (6) u T T v T y T vlo, (7) r T 6570

3 IN nd International Congress of echanical ngineering (COB 01) November -7, 01, Ribeirão reto,, Brazil vlo. (8) r T T T v T y T y T y T u T T r T where vlo is the velocity change at the second impulse, r T a h f, h f is the altitude of LO and a is the oon radius. The upper sign refers to clockwise arrival to LO and the lower sign refers to counterclockwise arrival to LO. From q. (), one finds 1 cos T T y T T 1 1 sin T sin T y T cos T. (9) 1 The problem defined by qs (1) (9) involves four unknowns vlo, vlo 0 that must be determined in order to satisfy the three final conditions. ince this problem has one of freedom, an optimization problem can be formulated as follows: etermine vlo, vlo, T and 0 which satisfy the final constraints (6) (8) and minimize the total characteristic velocity vtotal vlo vlo. This problem has been solved by da ilva Fernandes and arinho (01) using an algorithm based on gradient method (iele et al, 1969) in conjunction with Newton-Raphson method (toer and Bulirsch, 00).. OTIIZATION ROBL BA ON TH BR4B, T and In this section, the optimization problem based on the BR4B is formulated. The following assumptions are employed: 1. arth and oon move in circular orbits around the center of mass of the arth-oon system;. arth-oon system barycenter moves in circular orbit around the center of mass of the un-arth-oon system;. The flight of the space vehicle takes place in the oon orbital plane; 4. The space vehicle is subject to the gravitational fields of arth, oon and un; 5. The gravitational fields of arth, oon and un are central and obey the inverse square law; 6. The class of two impulse trajectories is considered. The impulses are applied tangentially to the space vehicle velocity relative to arth (first impulse) and oon (second impulse). Consider a moving reference frame Gy contained in the oon orbital plane: its origin is the barycenter of arth- oon system; the -ais points towards the oon position at the initial time t 0 0 and the y-ais is perpendicular to the -ais. In this reference frame, the motion of the space vehicle is described by the following set of differential equations: where du d u cos t 0 dt dt r r r r dv dy v y sin y y y y y t 0 dt dt r r r r,, (10), r and r are the same ones defined in the preceding section, is the un gravitational r is the distance of the space vehicle from the un r y y parameter,. The distance from G to arth, oon and un are denoted by r, r and r, respectively. o, the position vectors of arth, oon and un are defined in the reference frame Gy by the equations t r cos t y t r sin t t r cos t y t r sin t, (11), (1) 6571

4 IN andro da ilva Fernandes and Cleverson aranhão orto arinho un erturbations on Optimal trajectories for arth-oon Flight t r cos t y t r sin t where t t 0 t, and t 0 r t phase of the un, r 1 and 1 taking 0. r, (1), 0 0 is the initial phase of the oon, 0 is the initial. Note that q. (1) can be obtained directly from q. (10) by The initial conditions of the system of differential equations (10) correspond to the position and velocity vectors of the space vehicle after the application of the first impulse and are given by qs (4) and (5). The final conditions correspond to the position and velocity vectors of the space vehicle before the application of the second impulse and are given by qs (6) (9). The problem defined by qs (6) (9) and (10) involves five unknowns vlo, vlo, T, 0 and that must be determined in order to satisfy the three final conditions. This problem has two s of 0 freedom, so an optimization problem can be formulated as follows: etermine vlo, vlo, T, 0 which satisfy the final constraints (6) (9) and minimize the total characteristic velocity vtotal v optimization problem has been solved using the same algorithm described in the preceding section. 4. RULT and 0 v. This In this section, results are presented for lunar missions using the optimization problems described above. The following data are used: LO LO s s r r r (distance from the arth to the oon), a 678 (arth radius) a 178 (oon radius) h 167 h 100, 00, f 5 s Tables 1 and show the major parameters for the lunar missions involving direct ascent trajectories with time of flight of approimately 4.5 days, considering clockwise or counterclockwise arrival at the oon. Tables and 4 show similar results for trajectories with time of flight of approimately 41.0 days. Figures 1 and depict a maneuver with three revolutions for the two dynamical models, CRB and BR4B, respectively, considering counterclockwise arrival at the oon. Table 1 Lunar missions, major parameters - h 167 odel aneuver LO altitude CRB Clockwise Counterclockwise v Total /s v LO /s LO v LO /s T days From the results presented in Tables 1 and, and, Figures 1 and, the major comments are: 1. un perturbation effects are too small for direct ascent trajectories. 657

5 IN nd International Congress of echanical ngineering (COB 01) November -7, 01, Ribeirão reto,, Brazil. The differences in vlo between the two models are of order of m/s for direct ascent trajectories.. The optimum initial position of the un is almost the same for direct ascent trajectories regardless the altitude of LO. For the trajectories with clockwise arrival at the oon, is approimately For the trajectories with counterclockwise arrival at the oon, is approimately un perturbation effects are significant for trajectories with three revolutions. Fuel consumption can vary significantly according the initial position of the un. 5. A swing-by maneuver with the oon is made in the trajectories with three revolutions, for the both dynamical models. 6. The velocity increment at the second impulse - vlo - is significantly affected by the presence of the un for trajectories with three revolutions. (a) arth-oon trajectory (b) LO departure (c) wing-by (d) LO arrival Figure 1 Trajectory with three revolutions CRB 657

6 IN andro da ilva Fernandes and Cleverson aranhão orto arinho un erturbations on Optimal trajectories for arth-oon Flight Table Lunar missions, major parameters - h 167 odel aneuver LO altitude BR4B Clockwise Counterclockwise v Total /s v LO /s LO v LO /s T days Table Lunar missions, major parameters - h 167 LO odel aneuver LO altitude v Total /s v LO /s v LO /s T days 0 Clockwise CRB 100 Counterclockwise Table 4 Lunar missions, major parameters - h 167 LO odel aneuver LO altitude BR4B Clockwise Counterclockwise 100 v Total /s v LO /s v LO /s T days CONCLUION In this work, a preliminary study about the perturbation of the un on optimal trajectories for arth-oon flight of a space vehicle is presented. The optimization problem has been formulated using the classic planar circular restricted three-body problem (CRB) and the planar bi-circular restricted four-body problem (BR4B). Results presented for some lunar missions with time of flight of approimately 4.5 days show that the presence of the un causes small perturbations in the main parameters defining the optimal solutions and some fuel can be saved if the duration of the transfer becomes larger (approimately 41.0 days). 6574

7 IN nd International Congress of echanical ngineering (COB 01) November -7, 01, Ribeirão reto,, Brazil (c) arth-oon trajectory (d) LO departure (c) wing-by (d) LO arrival Figure Trajectory with three revolutions BR4B ACKNOWLGNT This research has been supported by CNq under contract 0949/009-7 and by FA under contracts 08/ and 01/ RFRNC Belbruno,. A., 004. Capture ynamics and Chaotic otions in Celestial echanics, rinceton University ress, rinceton, N.J., UA. Belbruno,., Gidea,. and Topputo, F., 010, Weak tability Boundary and Invariant anifolds, IA J. Appl. yn. yst. 9 (), pp Chobotov, V.A., 00. Orbital echanics, Third dition, AIAA ducation eries, Reston, Virginia. Conley,C.C., 1968, Low energy transit orbits in the restricted three-body problem, IA J. Appl. ath. 16, pp da ilva Fernandes,. and arinho, C..., 01, Optimal Two-Impulse Trajectories with oderate Flight Time for arth-oon issions, athematical roblems in ngineering, doi: /01/ Koon, W.., Lo,.W., arsden, J.. and Ross,.., 001, Low nergy Transfer to the oon, Celestial echanics and ynamical Astronomy 81, pp

8 IN andro da ilva Fernandes and Cleverson aranhão orto arinho un erturbations on Optimal trajectories for arth-oon Flight Koon, W.., Lo,.W., arsden, J.. and Ross,.., 007. ynamical ystems, the Three-Body roblem and pace ission esign, pringer. arec, J.., Optimal pace Trajectories, lsevier, New York. iele, A. and ancuso,., 001, Optimal Trajectories for arth-oon-arth Flight, Acta Astronautica, Vol 49, pp iele, A., Huang H.Y. e Heideman, J.C., 1969, equential Gradient-Restoration Algorithm for the inimization of Constrained Functions: Ordinary and Conjugate Gradient Versions, Journal of Optimization Theory and Applications, Vol 4, No 4, pp toer, J. and Bulirsch, R., 00. Introduction to Numerical Analysis, pringer, New York, Third dition. Topputo, F., 01, On Optimal Two-Impulse arth-oon Transfers in a Four-Body odel, Celestial echanics and ynamical Astronomy, doi: /s Yagasaki, K., 004a, Computation of Low nergy arth-to-oon Transfers with oderate Flight Time, hysica, 197, pp Yagasaki, K., 004b, un-erturbed arth-to-oon Transfers with Low nergy and oderate Flight Time, Celestial echanics and ynamical Astronomy 90, pp Yamakawa, H., Kawaguchi, J., Ishii, N. and atsuo, H., 199, A Numerical tudy of Gravitational Capture in the arth-oon ystem. In: paceflight echanics 199, roceedings of the nd AA/AIAA eeting, Colorado prings, CO, pp Yamakawa, H., Kawaguchi, J., Ishii, N. and atsuo, H., 199, On arth-oon Transfer Trajectory with Gravitational Capture, Advances Astronautical ciences 85, pp RONIBILITY NOTIC The authors are the only responsible for the printed material included in this paper. 6576