Earth-moon Trajectory Optimization Using Solar Electric Propulsion

Transcription

1 Chinese Journal o Aeronautics (7) Chinese Journal o Aeronautics arth-moon rajectory Optimization Using Solar lectric Propulsion Gao Yang* Academy o Opto-lectronics Chinese Academy o Sciences Beijing 18 China Received 5 April 7; accepted 17 August 7 Abstract he optimization o the arth-moon trajectory using solar electric propulsion is presented. A easible method is proposed to optimize the transer trajectory starting rom a low arth circular orbit (5 km altitude) to a low lunar circular orbit ( km altitude). Due to the use o low-thrust solar electric propulsion the entire transer trajectory consists o hundreds or even thousands o orbital revolutions around the arth and the moon. he arth-orbit ascending (rom low arth orbit to high arth orbit) and lunar descending (rom high lunar orbit to low lunar orbit) trajectories in the presence o J perturbations and shadowing eect are computed by an analytic orbital averaging technique. A direct/indirect method is used to optimize the control steering or the trans-lunar trajectory segment a segment rom a high arth orbit to a high lunar orbit with a ixed thrust-coast-thrust engine sequence. For the trans-lunar trajectory segment the equations o motion are expressed in the inertial coordinates about the arth and the moon using a set o nonsingular equinoctial elements inclusive o the gravitational orces o the sun the arth and the moon. By way o the analytic orbital averaging technique and the direct/indirect method the arth-moon transer problem is converted to a parameter optimization problem and the entire transer trajectory is ormulated and optimized in the orm o a single nonlinear optimization problem with a small number o variables and constraints. Finally an example o an arth-moon transer trajectory using solar electric propulsion is demonstrated. Keywords: trajectory optimization; solar electric propulsion; analytic orbital averaging technique; direct/indirect method 1 Introduction * Since the Deep Space 1 spacecrat demonstrated the irst use o solar electric propulsion (SP) or an interplanetary mission [1] the application o low-thrust propulsion or uture space missions has been a popular research subject. More recently the SA Smart-1 spacecrat [] launched in 3 successully perormed the irst lunar mission using solar electric propulsion. It is well known that the spacecrat propelled by low-thrust SP engines is capable o delivering a greater payload raction *Corresponding author. el.: mail address: gaoy@aoe.ac.cn Foundation item: National Natural Science Foundation o China (1635) compared to the spacecrat using conventional chemical propulsion. However the low-thrust continuous control proiles can not be approximated by the impulsive velocity dierences. How to optimize low-thrust transer trajectories becomes a new challenge to the mission designers who want to use SP as the primary spacecrat propulsion. In the past two decades lots o researches on low-thrust arth-moon trajectories were conducted. For instance Golan and Breakwell [3] investigated minimum-uel lunar trajectories with the ixed transer time by patching arth- and moon-centered spirals at an intermediate point. Huelman [4] introduced a power-limited optimal guidance law or arth-moon transers in a planar restricted three-

2 Gao Yang / Chinese Journal o Aeronautics (7) body dynamics. Piersion and Kluever [5] solved optimal arth-moon trajectories using a three-stage approach. Later they obtained a series o arthmoon trajectories [6-8] in the classical restricted threebody dynamics. Herman and Conway [9] employed a parallel Runge-Kutta method to solve optimal arth-moon transers which start rom a circular orbit with 6.3 arth radii and end at a two-lunar-radii orbit. he initial thrust acceleration o 1 4 g (g is the arth sea-level gravitational acceleration) yields a 3-day light journey. Recently Betts [1] obtained an optimal trajectory rom a geostationary transer orbit (GO) to a high elliptic lunar orbit with the perilune radius o 378 km and the apolune radius o km using the direct transcription method or collocation method which creates a large-scale nonlinear optimization problem including variables and constraints. A less accurate approximate solution to this problem was obtained with the computational time o s (11.67 h) and the longer time is needed to obtain a more accurate solution. However complete transers rom low arth orbits (LO) to low lunar orbits (LLO) using solar electric propulsion with or without consideration o signiicant shadowing eect still have not been solved in above-mentioned papers. In Res.[7] and [8] Kluever and Piersion used the delbaum s method to approximate the spirals rom LO to a high arth orbit (HO) and rom a high lunar orbit (HLO) to LLO. Nevertheless the delbaum s method does not consider J perturbations and shadowing condition that signiicantly aects trajectory evolution in relatively low-altitude orbits around the arth and the moon. I Betts direct transcription and the parallel Runge-Kutta method [9] are utilized to solve a long-duration transer trajectory rom LO to LLO the dimension o the nonlinear optimization problem would be much larger. In addition it is diicult to consider the shadowing eect in the existing direct methods. his paper demonstrates a easible method to compute the low-thrust arth-moon trajectory rom a low earth circular orbit (5 km altitude) to a low lunar circular orbit ( km altitude). he transer rom HO to HLO is computed by a direct/indirect method utilized by Kluever and Piersion [5-8] but the equinoctial elements in a our-body (the arth the moon the sun and the spacecrat) dynamics including J perturbations are employed. Furthermore the low-thrust spirals rom LO to HO and rom HLO to LLO are computed by an analytic orbital averaging technique (AOA) in which the thirdbody perturbations are ignored but the signiicant J perturbations and shadowing eect are taken into account. he ephemeris o the arth the moon and the sun are obtained using JPL planetary and lunar ephemerides [11]. he moon s orbit around the arth is not a planar circular orbit; the moon s inclination varies between 18 and 8 every 15 years. he entire trajectory is assumed to be a burn-coast-burn sequence which is a good tradeo between uel consumption and transer time. he optimal control problem or the arth-moon transer trajectory is converted to a parameter optimization problem that is in turn solved by nonlinear programming sequential quadratic programming (SQP). he entire transer trajectory is optimized by a single nonlinear SQP problem including only a ew variables and constraints. quations o Motion.1 Deinitions o CI and MCI coordinates he CI coordinate (arth-centered inertial coordinate O e x e y e z e see Fig.1) reers to the J arth equatorial rame that is deined by the mean orientation o the arth s equator and ecliptic orbit at the beginning o the year. he O e x e y e plane is parallel to the mean arth s equator. he line ormed by intersection o the ecliptic orbit plane and the arth s equatorial plane deines the axis x e. he line on the irst day o autumn (starting rom the sun to the center o the arth) deines the positive direction o the axis x e which is called the vernal equinox. he axis z e is perpendicular to the O e x e y e plane and points to the north. he axis y e completes the Cartesian coordinate using the right-handed

3 454 Gao Yang / Chinese Journal o Aeronautics (7) principle. he MCI coordinate i.e. moon-centered inertial coordinate O m x m y m z m (Fig.1) is parallel to O e x e y e z e with dierent origins. In this paper the equations o motion in both CI and MCI coordinates are expressed by a set o equinoctial elements [1-14] which avoid singularities when the orbit s inclination and eccentricity equal to zero. Fig.1 Illustrations o CI MCI coordinates and thrust steering angles in the local rotating coordinate O R x R y R z R where the axis x R is rom the arth to the spacecrat the axis z R is perpendicular to the orbital plane and the axis y R completes the Cartesian coordinate O R x R y R z R ollowing the right-handed principle which is in the orbital plane and points to the velocity direction.. quations o motion in the CI coordinate he equations o motion in the CI coordinate or the arth-departure trajectory are x M( Jarth moon sun ) D (1) 1 m m /( gisp) where P/( gisp) () where x [ pghkl] is the vector o equinoctial elements in the CI coordinate and J arth moon and sun 1 are arth J oblateness acceleration moon s perturbation and sun s perturbation respectively in the CI coordinate. he elements in the matrices M and D are given in Appendix A. q.() is the mass low rate where is thrust amplitude and P I sp and are power speciic impulse and eiciency o the SP system respectively. he thrusting direction unit vector can be expressed in terms o local pitch and yaw steering angles sin cos cos cos sin (3) where the pitch angle () is measured rom the local horizon (the axis y R ) to the projection o the thrust vector onto the orbit plane and the yaw angle () is measured rom the orbit plane to the thrust vector (see Fig.1). he arth gravitational parameter and arth radius ( e and R e ) deine the non-dimensional units distance velocity time and acceleration in the CI coordinate as ollows e Re d e Re ve te ae Re / te R v (4) e he arth J perturbations can be expressed in the rotating coordinate O R x R y R z R J arth 3eJeR e 1( hsin L kcos L) 4 1 r (1 h k ) 1 ejere ( hsin Lkcos L)( hcos Lksin L) 4 r (1 h k ) 6 ejere (1 h k )( hsin Lkcos L) 4 r (1 h k ) (5) where r is the distance rom the arth center to the spacecrat. he gravitational accelerations o the sun and the moon expressed in the CI coordinate can be written as rsunsc r esun sun 1 sun (6) 3 3 rsunsc re sun rmsc r em moon m (7) 3 3 rmsc rem where r sunsc is the position vector rom the sun to the spacecrat r esun rom the arth to the sun r mscrom the moon to the spacecrat and r em rom the arth to the moon. he gravitational parameters o the sun and the moon are denoted by sun and m respectively. he vector sun 1 and moon in the CI coordinate should be transormed to and moon in the local rotating coordinate. sun 1.3 quations o motion in the MCI coordinate Likewise the equations o motion in the MCI coordinate or the moon-capture trajectory are e

4 Gao Yang / Chinese Journal o Aeronautics (7) where x M( Jmoon arth sun ) D (8) m x [ p g h k L] is the vector o equinoctial elements in the MCI coordinate and J moon arth and sun are moon J oblateness acceleration arth s perturbation and sun s perturbation respectively in the MCI coordinate. he elements in the matrices M and D are in the expressions as in the matrices M and D and the mass low rate is the same as in q.(). he local thrusting pitch and yaw steering angles have the same deinitions as in q.(3) but reerenced in the MCI coordinate. he moon s gravitational parameter and moon s radius ( m and R m ) also deine the non-dimensional units in the MCI coordinate as ollows m Rm d m Rm vm tm am Rm / tm R v (9) m he moon s J perturbations have the same ormulation as in q.(5) with the substitutions o J m m and R m or J e e and R e. he gravitational accelerations o the sun and the arth expressed in the MCI coordinate can be written as rsunsc r msun sun sun (1) 3 3 rsunsc rm sun m resc r me arth m (11) 3 3 resc rme where r msun is the position vector rom the moon to the sun r esc rom the arth to the spacecrat and r me rom the moon to the arth. Also sun and in the MCI coordinate should be transormed arth to and arth in the local rotating coordinate. sun 3 Five Segments o the arth-moon rajectory he entire arth-moon trajectory is divided into ive segments: (1) Burn arc rom LO to HO; () Burn arc rom HO to the start o the trans-lunar coast arc; (3) rans-lunar coast arc; (4) Burn arc rom the end o trans-lunar coast arc to HLO; (5) Burn arc rom HLO to LLO. Fig. illustrates trajectory segments and able 1 summarizes perturbations trajectory propagation methods and inertial coordinates in dierent trajectory segments. Segment (1) () (3) (4) (5) Fig. Illustration o ive trajectory segments. able 1 Dierent trajectory segments Perturbations Propagation method Coordinate arth shadow AOA CI arth J arth J moon Numerical CI gravity sun gravity integration Moon J arth gravity sun gravity Numerical integration MCI Moon shadow moon J AOA MCI It is well known that the low-thrust spirals rom LO to HO and those rom HLO to LLO consist o hundreds or even thousands o orbital revolutions which requires a ormidable amount o time to compute i trajectories are numerically integrated. Segments ()-(4) having relatively ewer orbital revolutions are easier to be numerically integrated or moderate duration. hus the AOA a ast trajectory propagation algorithm is used to compute the segments (1) and (5). For the segments (1)-(3) the trajectories in the CI coordinate are propagated orward in time while those in segments (4) and (5) in the MCI coordinate backward in time. Since the gravitational orces o the sun the arth and the moon in both CI and MCI coordinates are considered the optimization subroutine would automatically ind the proper location to switch rom the CI coordinate to the MCI coordinate. hus it is not necessary to speciy a ixed intermediate point requisite or the conventional patched conic method.

5 456 Gao Yang / Chinese Journal o Aeronautics (7) Computing arth-orbit Ascending and Lunar Descending rajectories Using AOA he tangential steering along the velocity direction is employed or the arth-orbit ascending trajectory rom LO to HO while the anti-tangential steering or the lunar descending trajectory rom HLO to LLO. ither the tangential or anti-tangential steering is the optimal control strategy to change the instantaneous rate o semi-major axis. he trajectory segments (1) and (5) computed by the AOA are accurate only or relatively low orbits about an attracting body where the J perturbations and signiicant shadowing eect are included but the insigniicant third-body perturbations are ignored. he AOA algorithm is described in Re.[15] and also briely presented in Appendix B. he primary advantage o the AOA lies in the multirevolution trajectories which can be quickly propagated while maintaining satisactory accuracy [15]. Note that the computational procedure o the mooncapture trajectory using anti-tangential thrust is opposite to that using tangential-thrust trajectory. Let the initial condition o classical orbit elements mass and time at LO be (1) (1) (1) (1) (1) (1) (1) a e i m t where the superscripts represent the segment index number and the subscript denotes the initial time o the corresponding trajectory segment. he AOA propagates the tangential-thrust trajectory and determines the orbital elements spacecrat mass and light time at a point where the semi-major axis is pre-deined by a (1) with the subscript denoting the terminal time o the corresponding trajectory segment (1) (1) (1) (1) (1) (1) (1) a e i m t Likewise or the segment (5) that is propagated backward in time i the terminal condition at LLO is (5) (5) (5) (5) (5) (5) (5) a e i m t he initial condition o the segment (5) can be obtained by backward integrating anti-tangentialthrust trajectory to a point where the semi-major axis is pre-deined by (5) a (5) (5) (5) (5) (5) (5) (5) a e i m t Note that the AOA does not consider true anomaly which is a variable to be optimized. 5 Optimization Method or rans-lunar rajectory Segments A direct/indirect method is used to obtain near optimal control steering or trans-lunar trajectory rom HLO to HLO. Note that this method is employed to solve an arth-orbit transer (Zondervan Wood and Caughey [16] ) low-thrust arth-moon transers in the classical restricted three-body dynamics (Kluever and Pierson [5-8] ) as well as interplanetary transers (Gao and Kluever [17] ). In this paper the direct/indirect method is used in terms o the equinoctial elements. According to the calculus o variation theory the Hamiltonian o optimal control problem takes the ollowing orm H M p D m (1) m g I sp where p includes J perturbations and third-body perturbations (in both CI and MCI coordinates). p g h k L [ ] is the costate vector associated with the corresponding equinoctial elements and m the costate variable associated with the spacecrat mass. he optimal control steering direction unit vector is obtained by setting H / with the constraint 1. * [ ] M M (13) aking the partial derivative o the Hamiltonian with respect to the states the costate equations are determined. H M * D x x m x M p p M (14) x x H * m M M (15) m m m It indicates that the optimal control is governed by costate variables whose dynamics are given by

6 Gao Yang / Chinese Journal o Aeronautics (7) q.(19). For simplicity the costate dynamics associated with the two-body dynamic model is used to govern the control steering to make deriving derivatives o p with respect to the states unnecessary. In act the third-body perturbations are time-varying unction without explicit derivatives with respect to equinoctial elements. However the equations o motion do include perturbations. he initial costate variables or terminal costate variables need to be guessed to satisy the necessary boundary constraints without regard to the transversality condition and time-varying condition which should be considered in a classical two-point boundary-value problem. he optimal objective unction and boundary conditions are all treated by nonlinear optimization programming. Since the costate variable associated with mass do not aect the optimal control the irst six costate variables ( m is not used) could be used. hus or the segment () the state equations are q.(1) and q.(16) and the costate equation is q.(17). P m where gisp gisp (16) H M * D x x x (17) Likewise or the segment (4) the state equations are q.(9) and q.(16) and the costate equation has the same orm as in q.(17) but reerenced in the MCI. he advantage o the direct/indirect method over the direct transcription method lies in ewer variables and constraints leading to saving on considerable time or computation. However the costate variables needed to be integrated have no intuitive physical meaning which makes optimization problem much harder to converge although some techniques such as the adjoint-control transormation [18] and the multiple-shooting technique [17] are helpul in improving convergence robustness. he initial condition o the segment () can be denoted by state variables x () and costate variables where x () expressed in equinoctial ele- () () () () () () ments [ p g h k ] is transormed rom (1) (1) (1) (1) (1) () (1) () (1) a e i and m m t t. he initial values o true longitude L () and costate variables need to be guessed. he segment (3) does not integrate the costate equations since there is no thrust during the coast arc. he initial condition o the segment (3) is the terminal condition o the segment (). In contrast the segment (4) is integrated backward and the terminal condition o the segment (4) is state variables x (4) and costate variables (4) (4) where x in equinoctial elements (4) (4) (4) (4) (4) [ p g h k ] are transormed rom (5) (5) (5) (5) (5) (4) (5) (4) (5) a e i and m m t t.he terminal values o true longitude L (4) and costate variables need to be guessed. Obviously a complete trajectory should satisy the ollowing constraint (3) (3) (3) (3) (3) (3) (3) (3) (4) (4) (4) (4) (4) (4) (4) (4) [ p g h k L m t ] [ p g h k L m t ] 6 Formulation o Nonlinear Optimization Problem In the Section 4 and Section 5 the AOA and the direct/indirect method are described and the initial/ terminal conditions are speciied or each burn segment. he next step is to ormulate a parameter optimization problem that is in turn solved by nonlinear programming sequential quadratic programming (SQP) in which the overall mission objective is to minimize the propellant consumption during the transer. he SQP variables and constraints are summarized as ollows SQP variables: 11 variables in the orward trajectory propagation or segments (1)-(3): Segment (1): LO departure date (1) (1) t LO s ; () Segment (): six initial costate variables or the () () arth-departure burn arc burn arc duration t -t () HO s longitude angle L ; (3) (3) Segment (3): coast arc duration t -t. 11 variables in the backward trajectory propagation or segments (4) and (5): Segment (4): six terminal costate variables (4) or the moon-capture burn arc burn arc duration t (4) - (4) t HLO s longitude angle (4) L ;

7 458 Gao Yang / Chinese Journal o Aeronautics (7) Segment (5): LLO departure date t (5) spacecrat (5) (5) mass at LLO m LLO s. 8 SQP equality constraints: 6 constraints: terminal states o segment (3) = initial states o segment (4); 1 constraint: date at the terminal time o segment (3) = date at the initial time o segment (4); 1 constraint: mass at the terminal time o segment (3) = mass at the initial time o segment (4). 7 Comparison with Direct ranscription Method he direct transcription method parameterizes the control history using discrete nodes which inevitably results in a large number o variables and constraints i the transer contains much more revolutions. In this paper with the help o costate equations the ormulated SQP problem involves only variables and 8 constraints which is generally regarded as a smaller dimensional nonlinear optimization. Another ocused point is the use o the AOA a semi-analytic method or trajectory propagation. Capable o propagating hundreds or even thousands o revolutions it is much aster than precise numerical integration. Compared with the direct transcription method the proposed method reduces computational time by orders especially or long-duration multi-revolution transers. his makes it possible to optimize the entire transer trajectory as a single nonlinear optimization problem. 8 Numerical Results An arth-moon trajectory rom a LO with 5 km altitude to a LLO with km altitude is presented. First the parameters o the spacecrat to be deined are: the input power o the solar electric propulsion P = 1 kw the eiciency =.65 the speciic impulse I sp = 3 3 s and the initial spacecrat mass set to be 1 kg. hus the calculated initial thrust-to-weight ratio (/m g ) equals to he parameters o the initial LO and the terminal LLO are speciied in able. he initial arth departure date is set to be Jan Note that the 9 inclination o the LLO is reerenced in the MCI coordinate deined in this paper. In act 9 might not be an ideal inclination o a LLO parking orbit or the spacecrat which should take into account the moon s obliquity angle angle between the moon s rotation axis and the arth equatorial plane or ecliptic plane and urther detailed analysis o lunar gravity ield. Falling out o the scope o this paper this problem will not be discussed. able Initial LO and terminal LLO Orbital elements Initial LO erminal LLO Semi-major axis R e R m ccentricity.1.1 Inclination/( ) Ascension o ascending node ree ree Argument o periapsis/( ) rue anomaly ree ree As described in previous sections the AOA computes trajectory segment rom the LO to a HO (a HO = 1R e ) in the CI coordinate. It goes the same way or the AOA to compute the trajectory segment rom the LLO to a HLO (a HLO = R m ) in the MCI. As shown in Re.[15] the lower the altitudes o HO and HLO the more accurate the solutions. However the low attitudes might result in more revolutions or the trans-lunar trajectory segment which will conceptually cause diiculties in converging i the indirect/direct method is used. But i the number o revolution is not excessively large or instance under 5 the indirect/direct method still works well without too many troubles guessing initial costate variables. he indirect/direct method was once used to solve a lot o transer problems with small numbers o revolutions [6-8]. On the base o previous researches and author s experience in trajectory optimization a HO = 1R e and a HLO = R m are selected in the ollowing example. he ormulated parameter optimization problem has a small number o variables and constraints. However it appears not easy to make initial guesses or SQP variables to obtain converged solutions at the irst run. As a result the ollowing steps are

8 Gao Yang / Chinese Journal o Aeronautics (7) taken to ind out the inal optimal solutions: Step 1 o generate a trajectory to rendezvous moon in the CI coordinate with a burn-coast-burn engine sequence exclusive o the lunar perturbation. In this step the lunar capture is not considered. Step o remove the second burn and adjust the duration o the irst burn and coast arcs to make the end o the coast arc in the vicinity o moon. Step 3 o generate a trajectory in the MCI coordinate to escape moon s gravity and try dierent values or SQP variables in the segments (4) and (5) to make 8 SQP equality constraints as close as possible. Step 4 o optimize the trajectories rom the LO to the LLO in a single SQP problem based on the solutions obtained rom Step 3. Ater perorming the Step 4 the obtained LO- LLO trajectories are summarized in able 3 which shows about 8 days let or the trans-lunar trajectory rom the HO to the HLO. he arth-orbit ascending trajectory takes about 161 days and the moon descending trajectory about 1 days. hrough AOA are obtained about 1 9 orbital revolutions rom the LO to the HO and about 94 revolutions rom the HLO to the LLO. It indicates that the trajectories computed by the AOA have majority o revolutions o the entire transer trajectory and take about almost hal the transer time. he time histories o semi-major axis eccentricity and inclination in both CI and MCI coordinates are presented in Figs.3-5 respectively rom which it is clear that semi-major axis increases in the CI coordinate and decreases in the MCI coordinate. From the LO to the HO the eccentricity is raised at irst to about.16 ollowed by slightly alling and then quickly increasing. In the mooncapture phase the eccentricity is about.14 at the HLO. he eccentricity does not change monotonically mainly because o the eect o shadow. Without the shadowing eect the trajectory rom the LO to the HO or rom HLO to the LLO should be a near-circular transer. he inclination change in the MCI coordinate is not signiicant since the spacecrat enters moon-capture trajectory with an optimally chosen orientation. Additionally the transer trajectories in the CI coordinate are presented in Fig.6 and the lunar capture trajectory in Fig.7. he control steering direction angles in both CI and MCI coordinates are presented in Fig.8 and Fig.9. able 3 Solutions or the LO-LLO trajectories Mission parameters Solutions LO departure date Dec Fig.3 ime history o semi-major axis rom LO to LLO. HO arrival date June 18 Duration rom LO to HO/day st burn arc duration (trans-lunar)/day 5.3 coast arc duration (trans-lunar)/day 9.49 nd burn arc duration (trans-lunar)/day 19.4 HLO arrival date Aug. 8 8 LLO arrival date Sept. 9 8 Duration rom HLO to LLO/day 1.37 otal transer time/day Final mass at LLO/kg Fig.4 ime history o eccentricity rom LO to LLO.

9 46 Gao Yang / Chinese Journal o Aeronautics (7) Fig.5 ime history o inclination rom HO to HLO. Fig.9 ime histories o control steering or trajectory rom capture to HO (MCI). 9 Conclusions Fig.6 rajectory rom HO to HLO in the CI coordinate. Fig.7 Moon-capture trajectory to HLO in the MCI coordinate. A easible method has been proposed to ind preliminary solutions or arth-moon transer trajectories rom LO to LLO using SP. he shadow eects oblateness o the arth and the moon the gravitational orces o the sun the arth and the moon are all taken into consideration and the trajectories are solved using a practical complex threedimensional dynamic model. he JPL planetary and lunar ephemerides are utilized to compute precise positions o involved celestial bodies. he signiicant shadowing eect at relatively low altitude orbits is considered to highlight the use o solar electric propulsion. By means o AOA and the direct/indirect method the optimal orbit transer problem can be converted to a parameter optimization problem that only involves a small number o SQP variables and constraints. he main advantage o the proposed method lies in its provision o a more eicient algorithm to optimize long-duration multi-revolution transer trajectories while keeping the solution satisactorily accurate. he obtained solutions can be used as a good initial guess or urther trajectory optimization problems in highidelity dynamic models. Reerences [1] Rayman M D Williams S N. Design o the irst interplanetary Fig.8 ime histories o control steering or trajectory rom HO to escape (CI). solar electric propulsion mission. Journal o Spacecrat and Rockets ; 39(4): [] Koppel C R Marchandise F stublier D et al. he SMAR-1

10 Gao Yang / Chinese Journal o Aeronautics (7) electric propulsion subsystem in light experience. AIAA Paper [3] Golan O M Breakwell J V. Minimum uel lunar trajectories or a low-thrust power-limited spacecrat. AIAA Paper [4] Guelman M. arth-to-moon transer with a limited power engine. Journal o Guidance Control and Dynamics 1995; 18(5): [5] Pierson B L Kluever C A. hree-stage approach to optimal low-thrust earth-moon trajectories. Journal o the Astronautical Sciences 1994; 4(3): [6] Kluever C A Pierson B L. Vehicle-and-trajectory optimization o nuclear electric spacecrat or lunar missions. Journal o Spacecrat and Rockets 1995; 3(1): [7] Kluever C A Pierson B L. Optimal low-thrust three-dimensional earth-moon trajectories. Journal o Guidance Control and Dynamics 1995; 18(4): [8] Kluever C A Pierson B L. Optimal earth-moon trajectories using nuclear electric propulsion. Journal o Guidance Control and Dynamics 1997; (): [9] Herman A L Bruce B A. Optimal low-thrust earth-moon orbit transer. Journal o Guidance Control and Dynamics 1998; 1(1): [1] Betts J rb S O. Optimal low thrust trajectory to the moon. SIAM Journal o Applied Dynamical System 3; (): [11] Standish M. he JPL planetary and lunar ephemerides D4/ L4. Bulletin o the American Astronomical Society 1995; 7: 13. [1] Walker M J H Ireland B Owens J. A set o modiied equinoctial orbit elements. Celestial Mechanics 1985; 36(4): [13] Betts J. Optimal interplanetary orbit transers by direct transcription. Journal o the Astronautical Sciences 1994; 4(3): [14] Kluever C A. Optimal low-thrust interplanetary trajectories by direct method techniques. Journal o the Astronautical Sciences 1997; 45(3): [15] Gao Y Kluever C A. Analytic orbital averaging technique or computing tangential-thrust trajectories. Journal o Guidance Control and Dynamics 5; 8(6): [16] Zondervan K P Wood L J Caughey K. Optimal low-thrust three-burn orbit transers with large plane changes. Journal o the Astronautical Sciences 1984; 3(3): using hybrid trajectory optimization method with multiple shooting. AIAA Paper [18] Dixon L C Bartholomew-Biggs M C. Adjoint control transormations or solving practical optimal control problems. Optimal Control Applications and Methods 1981; (4): [19] Battin R H. An introduction to the mathematics and methods o astrodynamics. AIAA ducation Serials. Washington DC: AIAA 1987: [] Neta B Vallado D. On satellite umbra/penumbra entry and exit positions. Journal o the Astronautical Sciences 1998; 46(1): Biography: Gao Yang Born in 1974 he received B.S. degree rom Beijing University o Aeronautics and Astronautics in 1997 and M.S. degree rom Chinese Academy o Sciences in. He received his doctoral degree in 3 rom the University o Missouri-Columbia USA. From 4 to 5 he was a postdoctoral research ellow in the University o Missouri-Columbia. Since June 5 he has been an associate research ellow at Academy o Opto-lectronic Chinese Academy o Sciences. His research interests include orbital mechanics spacecrat dynamics and control guidance and navigation trajectory optimization and space mission design. -mail: gaoy@aoe.ac.cn Appendix A: Matrices M and D he elements o matrices M D expressed by equinoctial elements are as ollows M M M M M M M M M M M M M M M M M M M M11 M p p w M 13 p M1 sin L p 1 M [( w1)cos L ] w [17] Gao Y Kluever C A. Low-thrust interplanetary orbit transers

11 46 Gao Yang / Chinese Journal o Aeronautics (7) p g M3 ( hsin Lkcos L) w p p 1 M31 cos L M3 [( w1)sin Lg] w p M33 ( hsin Lkcos L) w M M M 41 M4 51 M5 61 M6 p s M43 cos L w p s M53 sin L w 1 p M63 ( hsin Lkcos L) w w D [ D6] D6 p p where w1 cos L gsin L is the arth gravitational parameter and s 1h k. he equinoctial elements can be obtained in terms o classical orbital elements (a e i ): p a(1 e ) ecos( ) g esin( ) h tan( i/ )cos k tan( i/ )sin L where is true anomaly. Appendix B: Analytic Orbital Averaging echnique In this section equations are deduced in the CI coordinate which in the MCI coordinate can be deduced likewise. In order to obtain the analytic incremental changes using tangential-thrust in classical orbital elements it is preerable to start with the Gauss planetary equations [17] : da a esin a p r (B1) d t h* h* r de 1 1 psin r [( pr)cos re] (B) d t h* h* d pcos ( p r)sin r (B3) d t h* e h* e d na 1 r [ r (cos e ) (1 )sin ] (B4) dt r nae a where p a(1 e ) h* p n / a and r p/(1 ecos ). When the thrust acceleration ( r and ) is much smaller than the gravitational acceleration the derivative o eccentric anomaly with respect to time is approximated by removing the 3 thrust term d na (B5) dt r Ater transorming eccentric anomaly to true anomaly by sin 1 e sin 1 ecos cos e cos (B6) 1 ecos the derivatives o the irst ive classical orbital elements (a e i ) with respect to eccentric anomaly can be computed by dividing the Gauss planetary equations (B1)-(B4) by q.(b5): 3 da a e r sin 1 e (B7) d de a r (1 e ) sin ( cos d cos ) 1 (B8) e e e d d a cos i r (cos e) 1e d d e ( e ecos )sin (B9) he thrust is so low that it can be assumed that over an orbital arc the classical orbital elements and acceleration are kept constant. he steering that maximizes the rate o semi-major axis (i.e. tangential steering) can be derived by setting (d a/ d ) / and (d a/ d ) / sin esin 1 e cos 1e cos 1e cos (B1) he analytic expressions or integrals with respect to eccentric anomaly are: da a d 3 d in 1 cos d e (B11) de a d (1 e ) in 1 e cos d d e 1 1e cos d [ln(sin 1e cos e)] d a d 1e d e 1 1e cos sin ( ecos ) in (B1) (B13)

12 Gao Yang / Chinese Journal o Aeronautics (7) where in is acceleration amplitude. Note that it is unable to ind analytic expressions or the terms 1 1 e cos d and d. he 1 e cos ollowing approximations can be utilized or these unctions to be integrated. 1e cos 1 e (1 1 e )sin (B14) 1 1e cos 1 e cos e cos e cos 1e cos 1e c (B15) o make the unctions and their approximations as close as possible c =.8 is chosen by way o trial-and-error tests. he corresponding integrals are then obtained in the ollowing analytic orms: 1e cos d 1 e (1 1 e )(.5.5sin ) 1 e 1 e cos 1 cos d d e 1 ec.5.5sin (B16) (B17) Furthermore the averaging changes o orbital elements due to J perturbation are as ollows: da de di (B18) dt dt dt Re d 3 J ncos i dt a (1 e ) Re d 3 J n(5cos i 1) dt 4 a (1 e ) (B19) (B) he increments in classical orbital elements and time including J perturbations and cylindrical shadow [] per revolution are approximated as: dz dz d dt n t n (B) en z d (B1) ex where z = [ a e i ]. he mass loss per revolution is computed by arth shadow entrance and exit angles. P 1 m [ ( en e sin en ex e sin ex )] ( gi ) n sp (B3) With y [ z t m] and y [ z t m] the elements in the (i+1)th revolution is computed only in terms o the elements in the (i)th revolution y y y (B4) i1 i i A terminal semi-major axis can be speciied or the stop condition o the orbital averaging. he states at the terminal semi-major are obtained by interpolating the orbital elements during the last revolution a a y y ( y y ) (B5) i i i1 i ai1 ai