Trajectory Optimization of Multi-Asteroids Exploration with Low Thrust

Transcription

1 Trans. Jaan Soc. Aero. Sace Sci. Vol. 52, No. 175, , 2009 Technical Note Trajectory Otimization of Multi-Asteroids Exloration with Low Thrust By Kaijian ZHU, Fanghua JIANG, Junfeng LI and Hexi BAOYIN School of Aerosace, Tsinghua University, Beijing, China (Received May 13th, 2008) Multi-asteroid tour missions require consideration of the visiting sequences and trajectory otimization for each leg, which is a tyical global otimization roblem. In this aer, the roblem is divided into a multi-level otimization roblem. Determination of the visiting sequence lays a key art for a tour mission. In this aer, the energy differences between different orbits and hase differences are used to estimate the energy required for a tour mission. First, this aer discusses the relation between fuel cost for transfer and classical orbit elements difference based on the energy relation of Kelerian orbits and the characteristics of low-thrust sacecraft. Second, the hase difference is combined with the energy difference to achieve the rendezvous energy. In addition, the lower and uer bounds of rendezvous time can be estimated by analyzing the hase difference. Very-low-thrust trajectory otimization roblems have always been considered difficult roblems due to the large time scales. In this aer, a hybrid algorithm of PSO (article swarm otimization) and DE (differential evolution) is used to achieve a solution to the energy-otimal tour mission. Based on GTOC-3 (Global Trajectory Otimization Cometition), this aer determines the exloration sequence and rovides the otimal solution. Key Words: Low-Thrust, Trajectory Otimization, Hybrid Algorithm 1. Introduction Ó 2009 The Jaan Society for Aeronautical and Sace Sciences With the develoment of sace technology and articularly dee-sace exloration technology, asteroid exlorations have come into focus. NASA has achieved the multi-urose exloration of Saturn and Titan with the Cassini sacecraft launched in The Stardust sacecraft launched in 1999 successfully made a flyby with the Wild2 comet and brought a comet samle back. In 2004, ESA launched the Rosetta sacecraft to discover the secrets of a cometary nucleus and which will take 10 years to rendezvous with the Comet 67 P/Churyumov-Gerasimenko. The HAYABUSA mission led by the Jaan Aerosace Exloration Agency is designed to exlore a small near-earth asteroid named Itokawa and to return a samle of material to Earth for further analysis. The DAWN mission was successfully launched in 2007 to exlore two asteroids with only one sacecraft, which would be a milestone in asteroid exlorations. Based on existing engine technology, the thrust required by the rocket and the fuel cost of the sacecraft would exceed human tolerances if a sacecraft were designed to exlore a celestial body directly. The trajectory must include many orbit maneuvers and wide use of a multiimulse transfer, low-thrust sacecraft has advantages of great efficiency and high recision. A great deal of recent work involves the trajectory otimization using low-thrust techniques. Many methods and theorems have been used to solve the roblem. Based on Bellman s rincile of otimality, Ross 1) develoed an anti-aliasing trajectory otimization method relating low-thrust trajectory otimization roblems to the well-know roblem of aliasing in information theory. Cui 2) used the concet of the Lyaunov feedback control law to derive semi-analytical exressions for subotimal thrust angles, and obtained a near-otimal solution that aroaches the otimal solution by using sequential quadratic rogramming (SQP) to adjust the five weights in the Lyaunov function. Ulybyshev 3) used the inner-oint algorithm to otimize the Earth ellitic orbit maneuver by rocessing the discrete legs. Betts 4) investigated orbit transfer otimization by utilizing the SQP method, reflecting the advantages and features of low thrust. Coverstone- Carroll 5) used a hybrid otimization algorithm that integrated a multi-objective genetic algorithm with a calculusof-variations-based low-thrust trajectory otimizer to identify the otimal trajectory of both Earth-Mars and Earth- Mercury missions. Olds 6) indicated that DE was a romising global method suited to trajectory otimization. However the erformance of the method is sensitive to the selection of the routine s tuning arameters. Kluever 7) alied the thrust-coast-thrust flight sequence to design the otimal trajectory for Earth-Moon transfer orbit, simulating the case from near-earth circular orbit to the lunar olar orbit. Considering the multi-revolution low-thrust trajectory otimization with the thrust-coast-thrust flight sequence. Hull 8) converted otimal control roblems into arameter otimization roblems and accomlished the method by relacing the control and state histories by control and state arameters and forming the histories by interolation. If a sacecraft is designed to exlore as many asteroids as ossible, it is imortant to determine the exloration sequence and otimize the flight trajectory so the sacecraft can rendezvous with the asteroids in roer time, which has become the focus of research. Generally, the enumeration method alied by the STOUR 9) software can search

2 48 Trans. Jaan Soc. Aero. Sace Sci. Vol. 52, No. 175 all the global otimal solutions. However, the comuting time and sace costs increase as dimensions and roblem comlexity increase, and cannot be handled by a ersonal comuter. Based on the Global Trajectory Otimization Cometition (GTOC-3) 10) organized by the Diartimento di Energetica of the Politecnico di Torino from October to December 2007, this aer rovides an algorithm for selecting candidate asteroids based on the energy and hase relation and for determining the flight sequence considering constraints on launch window and flight time. Modified equinoctial elements are used in numerical integration using non-dimensional forms. Finally, a heuristic algorithm is used for global otimization, and a attern search algorithm is used for local otimization. 2. Problem Descrition GTOC-3 chose 140 near-earth asteroids as candidates from the JPL sace objects database. A sacecraft launched from Earth must rendezvous with three selected asteroids during its tour and return to Earth. The initial mass of the sacecraft is 2000 kg which can be used as roulsion fuel. The hyerbolic velocity excess of the sacecraft relative to the Earth is 0.5 km/s and the direction can be set at will. The maximum magnitude of the engine is 0.15 N with secified direction and a secific imulse of 3000 s. The launch window is from 2016 to 2025 and the whole flight time is less than 10 years. The stay-time at each of the three asteroids must be at least 60 days. The effect of Earth and the asteroids are neglected in the whole trajectory, and Earth flybys at altitudes exceeding 6871 km can be considered. The objective of the otimization is to maximize the nondimensional quantity: J ¼ m f þ K min j¼1;3ð j Þ ð1þ m i max where m i and m f are the initial and final mass of the sacecraft, resectively. j (j ¼ 1; 2; 3) reresents the stay-time at the j-th asteroid in the rendezvous sequence and min j¼1;3 ð j Þ is the shortest asteroid stay-time. K ¼ 0:2, max ¼ 10 years is the journey time. First, we analyze the objective function. During the long journey, we must try to save the fuel and rolong the staytime at the rendezvous bodies. Only one long stay-time has no effect on the object function because the shortest stay-time will be utilized. If the stay-time of the sacecraft is longer, the flight time will be shorter, resulting in more fuel cost. Tradeoffs between stay-time, flight time and other mission arameters are a key art of the otimization. This aer divides the mission into three arts. First, we select three bodies from the 140 candidates based on the energy relation and synodic-eriod. To increase the integration recision, we divide the whole trajectory into seven arts that include four legs (connecting a celestial body with another celestial body) and three delay times. The otimization algorithm converts the otimal control roblems into arameter otimization roblems including the launch date, flight time, stay-time, magnitude and direction of thrust. 3. Dynamic Model The motion of a body is described by a system of secondorder ordinary differential equations: r þ r r ¼ a 3 ð2þ where the radius r ¼k r k reresents the magnitude of the inertial osition vector. is the gravitational constant of the central body. a is defined as erturbation acceleration. For the two-body roblem dynamic model, we rocess the low thrust as a erturbation on the small side. In the inertial coordinate, the arameters change raidly and the classical orbit elements exhibit singularities for e ¼ 0, and i ¼ 0, 90. The dynamic model in Cartesian coordinates reduces the effect of otimization. An aroriate dynamical model not only affects comuting time, but also determines the integral accuracy. Kechichian 11) analyzed the near-earth orbit transfer of low-thrust sacecraft where a dynamic model was defined using a modified set of equinoctial coordinates, which could avoid singularities in the classical elements. However, the integration stes must be increased to maintain the integration seed and lower integration recision. We alied the non-dimensional modified equinoctial elements to avoid singularities during the numerical integration. The equations of motion are described as follows: 12) _hh ¼ h n f t _f ¼ h sin L f r þ½h cos L þ n ðcos L þ f ÞŠ f t n X g f n _g ¼ h cos L f r þ½h sin L þ n ðsin L þ gþš f t þ n X f f n _h ¼ 1 2 n s2 cos L f n _k ¼ 1 2 n s2 sin L f n _L ¼ n X f n þ 1 n 2 h h where, n ¼, X ¼ h sin L k cos L, 1 þ f cos L þ g sin L s 2 ¼ 1 þ h 2 þ k 2, h ¼ ffiffiffiffiffiffi. is defined by ¼ að1 e 2 Þ. f r, f t and f n are defined as the thrust accelerations in a local radial-tangential-normal (RTN) rotating frame. Taking account of the reresentation of the asteroid orbit element rovided by the cometition organizer, we can transform the modified equinoctial elements into the classical orbital elements. h ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að1 e 2 Þ=; f ¼ e cosð! þ Þ g ¼ e sinð! þ Þ; h ¼ tanði=2þ cos ð4þ k ¼ tanði=2þ sin ; L ¼! þ þ where six classical orbital elements are defined as follows: ð3þ

3 May 2009 K. ZHU et al.: Trajectory Otimization of Multi-Asteroids Exloration with Low Thrust 49 Fig. 1. Relation of semi-axis and inclination. Fig. 3. Orbit energy. Fig. 2. Semi-major axis and RAAN. a: Semi-major axis e: Eccentricity i: Inclination : Right ascension of ascending node!: Argument of erigee : True anomaly 4. Energy Relation Analysis We analyze the inclination and energy relation of the 140 asteroids in Fig. 1 to Fig. 4 (where the bigger circle reresents Earth). The minimum semi-major axis of the asteroid is astronomical units (AU) and the maximum is AU. In the grou of asteroids, every asteroid has a number from 1 to 140 according to the incremental semi-axis of orbit from the minimum semi-axis to the maximum. The energies of asteroids is determined by the semi-major axis and is comuted by: E ¼ m ð5þ 2a where, and a are the same as above, and m is the sacecraft mass. Considering the orbit transfer, the change of angle of orbit Fig. 4. Velocity increment from Earth to asteroids. lane will lead to large fuel costs. We remove objects with inclinations bigger than 0.06 radian. For an object with large eccentricity, it is difficult to satisfy the rendezvous recision of osition and velocity as additional fuel cost is required. We remove objects with an eccentricity of more than 0.17 too. The candidate 16 asteroids are shown in Table 1. The eoch is exressed as modified Julian date (MJD) in the J2000 heliocentric eclitic frame. M is the mean anomaly of the asteroid orbit. In reliminary design of the interlanetary trajectory, we consider low thrust as low imulse and calculate the velocity increment accomlished instantaneously. At the start of velocity increment estimation, we aly the Hohmann orbit transfer with two imulses, and the velocity increment is exressed as follows: V ¼ V 1 þ V 2 V 1 ¼ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=r 1 2=ðr 1 þ r a1 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=r 1 2=ðr 1 þ r a2 Þ

4 50 Trans. Jaan Soc. Aero. Sace Sci. Vol. 52, No. 175 Table 1. Candidate asteroids. Asteroid Eoch a (AU) e i (deg)! (deg) RAAN (deg) M (deg) Note: (a, e, i,!, RAAN, M) are defined as classical orbit elements. Table 2. Candidate exloration sequences. No. A1 A2 A3 DV No. A1 A2 A3 DV Note: A1, A2, and A3 indicate the three asteroids and the number in the table reresents the serial number of the asteroids in the cometition document. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2 ¼ Vi 2 þ Vf 2 2V i V f cos i rel V i ¼ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=r a2 2=ðr 1 þ r a2 Þ V f ¼ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=r a2 1=a 2 cos i rel ¼ cos i 1 cos i 2 þ sin i 1 sin i 2 cos 1 cos 2 þ sin i 1 sin i 2 sin 1 sin 2 where, r 1, r a1 are defined as the erigee and aogee of the initial orbit, resectively. r a2 is the aogee of the orbit of the destination object. i 1 and 1 reresent the inclination and RAAN (right ascension of ascending node) of the initial orbit. a 2 is the semi-major axis of the orbit of the destination object, i 2 and 2 reresent the inclination and RAAN of the ð6þ orbit of the destination object. Based on the Hohmann orbit transfer, we chose 30 candidate exloration sequences with minimum velocity increment (Table 2) where A1 indicates the number of the first asteroid in the exloration sequence (excluding Earth) and A2 the second, and A3 the third. In interlanetary missions, rolonging the flight time or using the gravity-assist technique, which can increase or decrease the orbit energy of the sacecraft, reduces the fuel cost. The results in Table 2 where DV indicates the sum of DV1 (from Earth to A1), DV2 (from A1 to A2), DV3 (from A2 to A3), and DV4 (from A3 to Earth) can only be regarded as a reference. Here DV1, DV2, DV3, and DV4 reresent the velocity increment of each leg and can be comuted according to the Eq. (6).

5 May 2009 K. ZHU et al.: Trajectory Otimization of Multi-Asteroids Exloration with Low Thrust Flight Time and Synodic-Period þ! þ M ð14þ For a sacecraft to rendezvous with a celestial body, it must arrive at the osition with the same velocity, so launch date has some constraints. If a sacecraft is launched randomly, it consumes more fuel to chase the destination object. We should take account of the synodic-eriod and the hase difference between the dearture and arrival bodies in the reliminary trajectory design. For a low-thrust mission, the range of the orbit transfer is finite. If the sacecraft transfers from one celestial body to another celestial body, the hase difference must be as small as ossible to rendezvous in a short time. Otherwise, the sacecraft will send fuel in imlementing orbit transfer, and in chasing the rendezvousing body, which will influence the mission quality and increase the cost. If a sacecraft with mass m i is launched from the i-th asteroid at t 0, and will arrive at the j-th asteroid where the sacecraft mass is m j, the flight time is t ¼ t 1 t 0. Using Eq. (5), we can obtain the energy change of the sacecraft for the orbit transfer as: E ij ¼ m i 2a i m j 2a j The work done by the low thrust can be calculated by: Z tj W i!j ¼ ~T ~v dt ð8þ t i Then, the least energy required for the transfer can be written as: m j 2a j ð7þ m i 2a i T vt ð9þ During the whole rocess, the mean velocity is aroximated as: ffiffiffiffiffiffiffiffiffi v =a i þ ffiffiffiffiffiffiffiffiffi. =a j 2 ð10þ For reliminary analysis and estimation of the erformance, the magnitude of the thrust is assumed to be invariable and the direction accords with the local velocity. Therefore, the mass consumed can be obtained as: _m ¼ T ð11þ I s g 0 m j ¼ m i þ _mt ¼ m i T t ð12þ I s g 0 m i t ¼ rffiffiffiffiffi rffiffiffiffiffi T þ! 1 1 ð13þ a i a j a i a j a j I s g 0 When the sacecraft transfers from one asteroid to another, to avoid large energy consumtion, the hase difference should be small. Assuming that the eccentricity and the inclination of the orbits are small, the hase of sacecraft in eclitic lane with resect to equinox on a Kelerian orbit can be given aroximately as: where and! are defined above. indicates the hase of sacecraft and M is the mean anomaly given by: rffiffiffiffiffiffi M ¼ M 0 þ nt ¼ M 0 þ t ð15þ where M 0 is the initial mean anomaly. The angular velocity of the sacecraft can also be aroximated as: n ¼ 1 rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi þ ð16þ 2 a 3 i a 3 j Then, we obtain: i þ! i þ M i þ 1 rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi þ t 2 a 3 i a 3 j rffiffiffiffiffiffiffi ¼ j þ! j þ M j þ t þ 2k k ¼ 0; 1;... ð17þ a j 3 t ¼ 2ð2k þ j þ! j þ M j i! i M i Þ rffiffiffiffiffiffiffi r ffiffiffiffiffiffiffi! ð18þ a i 3 where the subscrit j denotes the variables related to the arrival asteroid and the subscrit i denotes the variables related to the leaving asteroid. T indicates the magnitude of thrust. g 0 is standard acceleration due to gravity at Earth s surface and I s reresents the secific imulse of the engine. v is defined as the mean velocity of the transfer orbit. The sacecraft finally rendezvous with Earth. The hase difference of the asteroid and Earth shall be zero during the 10 years from launch. So a sequence with large hase difference and large synodic-eriod is excluded from consideration because it is difficult for the sacecraft to return to Earth. Consequently, the candidate asteroids must take account of the synodic-eriod and initial hase different within 10 years. The sequences are shown in Table 3. The data in the lower-left area of Table 3 indicate the hase difference at the eoch of MJD57388 ( ) that shows the degree of the asteroids in the row in the front of asteroids in the column. The data in the uer-right area of Table 3 indicate the difference of the angular velocity with unit of degree er Julian year. The constraint of the launch window is 10 years and then the first asteroid with large synodic-eriod with Earth will be excluded. Here, the first synodic-eriod is less than 15 years. Considering the constraint of flight time, we must estimate the synodic-eriod between the first asteroid and the last asteroid. In addition, the synodic-eriod of the three asteroids with Earth should increase along the order of rendezvous. Otherwise the sacecraft will reduce the orbit energy, requiring more fuel to rendezvous with the next asteroid that has rendezvoused with Earth before this rendezvous. The ossible sequences are shown in Table 4 where the numbers of the sequences are the same as in Table 2. Here D1, D2, D3, and D4 indicate the ursuing time comuted through the hase difference and synodic-eriod. In a sequence, D1 is the time of Earth ursuing the first asteroid and D2 a j 3 a 3

6 52 Trans. Jaan Soc. Aero. Sace Sci. Vol. 52, No. 175 Table 3. Initial hase difference and angular velocity difference. (unit of lower-left: degree, unit of uer-right: degree er Julian year) No Earth Earth Table 4. Exloration sequences according to synodic-eriod with Earth. No. D1 D2 D3 D4 No. D1 D2 D3 D is the time of the first asteroid ursuing the second asteroid. D3 and D4 are the analogies. The stay-time of the sacecraft must be longer than 60 days at each rendezvous. The sacecraft cannot send too long time during a single leg because the total flight time is less than 10 years which is also a maximum constraint on the difference of D1 and D4. Adding the launch window of 10 years, the maximum ursuing time in the sequence must be less than 20 years. Because there is no sequence satisfying all the constraints, we selected candidate sequences 1, 21, 24, and 30 sequences from Table 4. Sequence 21 has the best time sequence that increases gradually, but the first time is beyond 10 years. Sequences 1, 24, and 30 have no incremental synodic-eriod with Earth, but the first time is less than 10 years. 6. Numerical Simulation Here, we select the asteroid whose synodic-eriod with Earth is within 10 years as the first object to be exlored and the asteroid whose synodic-eriod with Earth is from 10 to 20 years, as the last object. Based on the above energy relation in Table 2 and the constraints of launch window and flight time, we aly exloration sequence 1 listed in Table 2 and 4 in this aer shown below: Earth-A88(1991 VG)-A76(2006 JY26)- A49(2000 SG344)-Earth Trajectory otimization belongs to a global otimization, consisting of finding the global otimum of a given erformance index in a large domain, tyically characterized by the resence of a large number of local otima. In this aer, we use a hybrid algorithm of PSO (article swarm otimization) and DE (differential evolution) that were both develoed in the 1990s and are now alied widely in various fields. PSO 13) was develoed for otimizing continuous non-linear functions by Kennedy and Eberhart based on the action of foraging birds. It has been studied widely in the recent years and many modified PSO algorithms have been used. In general, these modified algorithms are all about how to adjust the arameters and enhance the diversity so as to search the whole solution domain. DE 14) was introduced by Storn and Price and resembles the structure of an EA (Evolutionary Algorithm), but differs from traditional EAs in generation of new candidate solutions and use of a

7 May 2009 K. ZHU et al.: Trajectory Otimization of Multi-Asteroids Exloration with Low Thrust 53 Table 5. Orbit elements in heliocentric inertial coordinate of MJD Object a (AU) e i (deg)! (deg) RAAN (deg) M (deg) (A88) (A76) (A49) Earth Note: (a, e, i,!, RAAN, M) are defined as classical orbit elements. Table 6. Parameters of hybrid algorithm. Table 7. From Earth to A88. F CR c1 c2 NP Iteration PSO + DE greedy selection scheme. It has great otential in many numerical benchmark roblems and real world alications. In many test suites, the DE algorithm outerforms other methods and has the advantage of fast convergence rate and low comutational consumtion of function evaluations. Because the above global otimization algorithms are sensitive to selection of tuning arameters, this aer combines these two algorithms to achieve stable erformance. In the former 30 iterations of every 50 iterations, PSO is first alied to extend the search ability. In the latter 20 iterations of every 50 iterations, DE is alied to converge raidly. In the rocess of arameter otimization, the aroximate dearture date and arrival date are fixed based on the above hase analysis. But two arameters date errors are set to be otimized to obtain the accurate dearture date and arrival date. In the aer, a continuous lowthrust is assumed as the fixed imulse in a small interval and every leg is divided into 13 intervals considering the tradeoff of recision and comutation time. Here, the magnitude and direction of the low thrust are to be otimized in the constraint range. We transfer the control otimization roblem into a arameters otimization roblem and continuously integrate the dynamics equation along the trajectory to avoid matching adjoint variables. Nevertheless, the final rendezvous recision cannot be controlled. Finally, the attern search algorithm is used to hel the above reliminary solution satisfy the recision requirements. The algorithm is a local otimization algorithm for unconstrained otimization using ositive sanning directions. It directs the search for a minimum through a attern containing at least n þ 1 oints er iteration, where the vectors reresenting the direction and distance of each oint relative to the current iterate form a ositive basis in R n. We finally aly the attern search tool in the Matlab Ò software to get the otimal solution. The erformance of the algorithm is sensitive to the selection of the routine s tuning arameters. This aer rovides the otimal solution and also rovides the tuning arameters in Table 6 where NP is the oulation size, CR is the crossover robability, F is the mutation scaling factor and c1, c2 are two weighting factors. The otimal arameters during every leg are shown in Tables 7 to 10. Launch date (MJD and calendar): ( ) Hyerbolic velocity excess (km/s): [ , , ] Arrival date (MJD and calendar): ( ) Initial mass (kg): Arrival mass (kg): Based on the otimal solution, we show the interlanetary trajectory in Figs. 5 to 8. Finally, the object function can be comuted as: J ¼ m f þ K min j¼1;3ð j Þ ¼ 1564:60 m i max 2000 þ 0:2224: :5 ¼ 0: Conclusion Table 8. From A88 to A76. Launch date (MJD and calendar): ( ) Stay-time at A88 (JD): Arrival date (MJD and calendar): ( ) Dearture mass (kg): Arrival mass (kg): Table 9. From A76 to A49. Launch date (MJD and calendar): ( ) Stay-time at A76 (JD): Arrival date (MJD and calendar): ( ) Dearture mass (kg): Arrival mass (kg): Table 10. From A49 to Earth. Launch date (MJD and calendar): ( ) Stay-time at A49 (JD): Arrival date (MJD and calendar): ( ) Dearture mass (kg): Arrival mass (kg): Based on the GTOC-3, this aer analyzes the energy relation of orbit transfer and determines the exloration sequence according to the hase difference and the synodic-eriod with Earth. The method can be used in the reliminary design of the interlanetary trajectory. Trajectory design is a global otimization roblem that

8 54 Trans. Jaan Soc. Aero. Sace Sci. Vol. 52, No. 175 Fig. 5. Trajectory from Earth to A88. Fig. 7. Trajectory from A76 to A49. Fig. 6. Trajectory from A88 to A76. Fig. 8. Trajectory from A49 to Earth. generally can obtain the otimal solution by meta-heuristic search algorithms, such as GA (genetic algorithm), PSO (article swarm otimization), DE (differential evolution), SA (simulated annealing), EA (evolutional algorithm), etc. This aer uses a hybrid algorithm of PSO and DE to obtain the otimal solution with low thrust. Last, the local otimization algorithm (attern search) is alied to obtain a more accurate solution to satisfy the required recision of 1000 km in osition and 1 m/s in velocity. A combination of the global otimization algorithm and classical local otimization rovides ideal method for trajectory design. If Earth flybys are considered during the flight, the solution sace is extended exonentially and a more efficient algorithm is required in future. Acknowledgments This work is suorted by the National Natural Science Foundation of China (No and No ). References 1) Ross, I. M., Gong, Q. and Sekhavat, P.: Low-Thrust High-Accuracy Trajectory, Otimization, J. Guid. Control Dynam., 30, 4 (2007), ) Cui, P. Y., Ren, Y. and Luan, E. J.: Low-Thrust, Multi-Revolution Orbit Transfer under the Constraint of a Switch Function without Prior Information, T. Jn. Soc. Aeronaut. Sace Sci., 50 (2008), ) Ulybyshev, Y.: Continuous Thrust Orbit Transfer Otimization Using Large-Scale Linear Programming, J. Guid. Control Dynam., 30, 2 (2007), ) Betts, J. T.: Very Low-thrust Trajectory Otimization Using a Direct SQP Method, J. Comut. Al. Math., 120 (2000), ) Coverstone-Carroll, V., Hartmann, J. W. and Mason, W. J.: Otimal Multi-objective Low-thrust Sacecraft Trajectories, Comut. Meth. Al. Mech. Eng., 186 (2000), ) Olds, A. D., Kluever, C. A. and Cules, M. L.: Interlanetary Mission Design Using Differential Evolution, J. Sacecraft Rockets, 44, 5 (2007), ) Kluever, C. A.: Otimal Low-Thrust Three-Dimensional Earth-Moon Trajectories, J. Guid. Control Dynam., 18, 4 (1995), ) Hull, D. G.: Conversion of Otimal Control Problems into Parameter Otimization Problems, J. Guid., Control Dynam., 20, 1 (1997), ) Petrooulos, A. E., et al.: 1st ACT Global Trajectory Otimisation Cometition: Results Found at the Jet Proulsion Laboratory, Acta Astronautica, 61 (2007), ) htt://www2.olito.it/eventi/gtoc3/ 11) Kechichian, J. A.: Otimal Low-Thrust Orbit Geostationary Earth Orbit Intermediate Acceleration Orbit Transfer, J. Guid. Control Dynam., 20, 4 (1997), ) Gao, Y.: Advances in Low-Thrust Trajectory Otimization and Flight Mechanics, Doctor Dissertation, University of Missouri-Columbia, ) Kennedy, J. and Eberhart, R. C.: Particle Swarm Otimization, Proceeding IEEE International Conference on Neural Networks, Piscataway, 1995, ) Storn, R. and Price, K.: Differential Evolution-A Simle and Efficient Heuristic for Global Otimization over Continuous Saces, J. Global Otimization, 11 (1997),