AIAA/AAS Astrodynamics Specialist Conference and Exhibit AIAA 2002-4729 5-8 August 2002, Monterey, California DESIGN AND OPTIMIZATION OF LOW-THRUST GRAVITY-ASSIST TRAJECTORIES TO SELECTED PLANETS Theresa J. Debban,* T. Troy McConaghy,† and James M. Longuski‡ School of Aeronautics and Astronautics, Purdue University West Lafayette, Indiana 47907-1282 Highly efficient low-thrust engines are providing new opportunities in mission design. Ap- plying gravity assists to low-thrust trajectories can shorten mission durations and reduce propellant costs from conventional methods. In this paper, an efficient approach is applied to the design and optimization of low-thrust gravity-assist trajectories to such challenging targets as Mercury, Jupiter, and Pluto. Our results for the missions to Mercury and Pluto compare favorably with similar trajectories in the literature, while the mission to Jupiter yields a new option for solar system exploration. Introduction 5 HE design of low-thrust gravity-assist trajec- 4 T tories has proven to be a formidable task. 3 Many researchers have responded to the challenge, 2 Earth Flyby but in spite of these efforts a perusal of the litera- Venus Flyby Nov. 9, 2016 1-16 1 Dec. 2, 2015 ) ture reveals a paucity of known results. U . 0 Recently, new software tools have become (A available for the design and optimization of low- y −1 thrust gravity-assist (LTGA) trajectories. Petro- −2 Earth Launch 9 10,12 May 17, 2015 poulos et al., Petropoulos and Longuski, and −3 Petropoulos11 have developed a design tool which −4 Jupiter Arrival July 28, 2019 patches together low-thrust arcs, based on a pre- −5 scribed trajectory shape. This method allows −6 −4 −2 0 2 4 6 highly efficient, broad searches over wide ranges x (A.U.) of launch windows and launch energies. A new, Fig. 1 Earth-Venus-Earth-Jupiter trajectory. complementary tool, developed by Sims and Flanagan14 and McConaghy et al.,7 approaches the trajectory optimization problem by approximating Methodology low-thrust arcs as a series of impulsive maneuvers. The first step in designing an LTGA trajectory In this paper, we apply these tools to mission to a given target is to choose a sequence of grav- design studies of LTGA trajectories to Mercury, ity-assist bodies. While this problem has been Pluto, and Jupiter (see Fig. 1). We demonstrate examined for conic trajectories,17 no prior work with these examples that our method provides an proposes a method for path finding in the case of efficient way of designing and optimizing such LTGA missions. In our examples, we draw from trajectories with accuracy comparable to Refs. 13 two different resources for selecting paths. For the and 15. Mercury and Pluto missions, we take LTGA tra- jectories from the literature and attempt to repro- _____________________________ duce those results. The path to Jupiter, however, is devised from studies of conic trajectories to the *Graduate Student. Currently, Member of the Engi- gas giant.18 neering Staff, Navigation and Mission Design Sec- Once a path has been chosen, we perform the tion, Jet Propulsion Laboratory, California Institute next step: path solving. Employing a shape-based of Technology, Pasadena, CA 91109-8099. Student approach described below, we perform a broad Member AIAA. search over the design space to discover sub- †Graduate Student, Student Member AIAA, Member optimal LTGA trajectories. Evaluating the merits AAS. of these trajectories allows us to select a candidate ‡Professor, Associate Fellow AIAA, Member AAS. for the third step: optimization. We optimize our 1 American Institute of Aeronautics and Astronautics Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. trajectories to maximize the final spacecraft mass, To reduce mission costs, the minimum total which in turn may increase the scientific value of a propellant mass fraction is desired. Since time of mission. flight is not included in the cost function, it must be evaluated in conjunction with the total propel- Broad Search lant mass fraction. This, of course, requires engi- We desire a computationally quick and effi- neering judgment and a balancing of factors for a cient method of searching a broad range of launch particular mission. It is up to the mission designer dates and energies. For this process, we use a two- to judiciously select the most promising trajecto- body patched-arc model. These arcs can be either ries to use as initial guesses for optimization. coast only, thrust only, or a combination of the two. The coast arcs are standard Keplerian conic Optimization sections. The thrust arcs, however, are represented Once good candidate trajectories are found, by exponential sinusoids9 which are geometric we optimize them with the direct method devel- curves parameterized in polar coordinates (r, θ ) as oped by Sims and Flanagan.14 Our software, GALLOP (Gravity-Assist Low-thrust Local Op- = θ + φ timization Program),7 maximizes the final mass r k0 exp[k1 sin(k2 )] (1) of the spacecraft. The essential features and as- sumptions of GALLOP are as follows (see Fig. where k , k , k , and φ are constants defining the 0 1 2 2). shape and, consequently, the acceleration levels of the arc. These exponential sinusoids can be • propagated analytically, which eliminates the need The trajectory is divided into legs between for time-consuming numerical propagation. The the bodies of the mission (e.g. an Earth- mathematical and algorithmic details are worked Mars leg). • out in Refs. 9-12, and the algorithms are imple- Each leg is subdivided into many short mented as an extension of the Satellite Tour De- equal-duration segments (e.g. eight-day sign Program (STOUR).19 segments). • To evaluate the many candidate trajectories The thrusting on each segment is modeled found by STOUR, we employ a cost function that by an impulsive ∆V at the midpoint of the computes the total propellant mass fraction due to segment. The spacecraft coasts on a conic the launch energy, thrust-arc propellant, and arri- arc between the ∆V impulses. 7 • val V∞ (if a rendezvous is desired). Tsiolk- The first part of each leg is propagated for- ovksy’s rocket equation is used to account for the ward to a matchpoint, and the last part is departure and arrival energies.20,21 For the launch propagated backward to the matchpoint. • V∞ (∆V1 in Eq. 2), we use a specific impulse of Gravity-assist maneuvers are modeled as 350 seconds (Isp1) to represent a chemical launch instantaneous rotations of the V∞. vehicle. A low-thrust specific impulse (Isp2) of 3000 seconds is applied to the arrival V∞ (∆V2) The optimization variables include the because we assume that the low-thrust arcs will launch V∞, the ∆V on the segments, the launch, remove any excess velocity in rendezvous mis- sions. Finally, the thrust-arc propellant mass frac- Matchpoint Planet or target body tion given by STOUR (pmf) yields the total propel- Segment midpoint lant mass fraction (tmf): Impulsive ∆ V Segment boundary − ∆ − ∆ = − − V1 V2 tmf 1 (1 pmf ) exp exp (2) gI sp1 gI sp2 For flyby missions, ∆V2=0. While Eq. 2 is only an approximation of the true propellant costs, it serves quite adequately to reduce the candidate trajectories to a manageable number among the myriad (up to tens of thousands) of possibilities produced by STOUR. Fig. 2 LTGA trajectory model (after Sims 14 and Flanagan ). 2 American Institute of Aeronautics and Astronautics flyby and encounter dates, the flyby altitudes, the mized trajectory (using SEPTOP, an indirect flyby B-plane angles, the spacecraft mass at each method) has approximately 6.5 total revolutions body, and the incoming velocity at each body. around the sun, 5.75 of which occur on the Ve- The initial spacecraft mass can also be deter- nus-Mercury leg of the mission. The “spiraling” mined with a launch vehicle model so that the in this trajectory is necessitated by the limited 7,13,14 injected mass is dependent upon the launch V∞. thrusting capability of a single SEP engine. There are also two sets of constraint func- We began our quest for an initial guess for tions. One set ensures that the ∆V impulses can GALLOP by performing a search in STOUR. be implemented with the available power. The We searched over a five-year period that in- other set enforces continuity of position, veloc- cluded the launch date in Sauer’s optimized tra- 13 ity, and spacecraft mass across the matchpoints jectory. Our Earth-Venus trajectory leg was a (see Fig. 2). The optimizer uses a sequential pure thrust leg, while the Venus-Mercury leg quadratic programming algorithm to maximize was a coast-thrust leg. We allowed the space- the final mass of the spacecraft subject to these craft to coast for more than one full revolution constraints.7 before starting to thrust at 0.68 AU. The coast Reference 7 provides more details on the time allows the exponential sinusoid geometry to broad search and optimization procedures. Our be better aligned with the geometry of Mercury’s 11 method is also demonstrated on simple examples orbit. The additional coast revolution also pro- in Ref. 7. In this paper, we tackle more challeng- vides GALLOP with more time to thrust. ing problems using more mature versions of our Initially, we based our trajectory selection software. on the lowest total propellant mass fraction. Our attempts to optimize this case in GALLOP, how- Optimized Trajectories ever, were hindered by the thrust limitations of The success of our approach is illustrated in one SEP engine. Our trajectory had just under the following examples. The first two, missions five total revolutions around the sun with only to Mercury and to Pluto, are attempts to match or four complete revolutions on the Venus-Mercury improve upon optimized results in the literature.