IAC-11-C1.1.1

TRAJECTORY TOUR OF THE TROJAN ASTEROIDS GENERATEDVIA AN OPTIMAL LOW-THRUST ALGORITHM

Jeffrey R. Stuart∗

Purdue University, United States of Americajrstuart@purdue.edu

Kathleen C. Howell†

Purdue University, United States of Americahowell@purdue.edu

Asteroids in the vicinity of the Sun-Jupiter equilateral equilibrium points, commonly termed“Greek” or “Trojan” asteroids, offer insight into the primordial composition of the solar system.An increasing number of studies examine transfers to Near Earth Objects and to the equilateralLagrange points, but comparatively little investigation of transfers within the neighborhood ofthe L4 and L5 points has been completed. Low-thrust trajectories that are optimized in termsof propellant consumption potentially offer years or even decades of mission lifetime while si-multaneously ensuring a large number of asteroid encounters. This investigation considers thenearly continuous force from a variable specific impulse engine and its incorporation into a primervector-based, optimal transfer and rendezvous strategy. A shooting procedure yields rapid so-lutions and generates a framework for an investigation into the trade space between flight timeand consumption of propellant mass. Rendezvous between approximated asteroid paths in theplanar circular restricted three-body problem is initially investigated, but the methodology is notlimited by the dynamical regime and is readily extendable to the quasi-periodic paths representingvarious asteroids as well as three-dimensional examples in an ephemeris model.

NOMENCLATURE

x, χ State Variable Vectorst Timer, v Position and Velocity Vectors,

Sun-Jupiter Rotating Framem Spacecraft Massfn Natural System AccelerationT ThrustP Engine Poweru Thrust Direction Unit Vectord1, d2Sun and Jupiter DistancesMS Solar MassMJ Jupiter Massµ Mass ParameterIsp Specific Impulseγ Asteroid Approximation AngleC Approximation Rotation Matrixτ0 Asteroid Time-like ParameterTD Thrust DurationJ Cost FunctionH Problem Hamiltonian

∗Ph.D. Student, School of Aeronautics and Astronautics†Hsu Lo Professor of Aeronautical and Astronautical Engi-

neering

λ Co-statesψ Boundary ConditionsS Switching FunctionX Full State VectorF Constraint Vectorn Number of Rendezvous ArcsτS Total Sequence DurationCT Intermediate Coast DurationτO Observation Time

I. INTRODUCTION

Interplanetary objects such as asteroids and cometsoffer insight into the primordial composition of thesolar system and are, therefore, the subject of muchscientific interest. Near Earth Objects (NEOs) haverecently gained attention as opportunities for mannedsample return missions.1 Likewise, missions to themain belt asteroids and the vicinity of the Sun-Jupiterequilateral points offer opportunities for encounterswith multiple objects while retaining the option ofrobotic sample returns. Such a design concept allowsfor a broad survey of asteroid types and, therefore,a more complete view of solar system history. This

IAC-11-C1.1.1 Page 1 of 11

investigation focuses on potential long-term roboticmissions to the vicinity of the Sun-Jupiter L4 libra-tion point, however the design methodology is largelyindependent of dynamical regime as well as initial andtarget states; therefore, the low-thrust concept andprocedure are readily incorporated into diverse mis-sion design strategies.

High-efficiency, low-thrust propulsion systems areparticularly attractive for missions to the Sun-Jupiterequilateral equilibrium points because of the relativelystable natural gravitational dynamics in these regions.Fuel-optimal, low-thrust trajectories, realized by con-stant specific impluse systems in non-linear dynamicalregimes, typically require coasting arcs and the carefulbalancing of engine capability with transfer time. Theinclusion of additional coasting arcs requires engineshut-downs and restarts that may be operationally in-efficient and generally infeasible. Therefore, a variablespecific impulse (VSI) engine that varies the optimalthrust magnitude is selected to simplify the genera-tion of rendezvous solutions.2 Accordingly, no coastingarcs are required for rendezvous and the initial gener-ation of optimal trajectories is less restricted in termsof thrust duration. Examples of VSI engines includethe Variable Specific Impulse Magnetoplasma Rocket(VASIMR) currently under development by the AdAstra Rocket Company3 and the Electron and Ion Cy-clotron Resonance (EICR) Plasma Propulsion Systemsat Kyushu University in Japan.4

In general, the computation of locally fuel-optimaltrajectories is approached by posing an optimal con-trol problem. The possible formulations to solve theproblem include a low-dimension but less flexible in-direct approach using optimal control theory5–7 or ahigher-dimension but more robust direct approach.8,9

(A detailed discussion of indirect and direct trajec-tory optimization methods is addressed in a survey byBetts.10) Since the object of this investigation is theanalysis of a rather large number of preliminary trajec-tory designs, the indirect method is employed while thedirect method is reserved for later, more detailed tra-jectory construction. Relatively short times-of-flight(compared to long-duration spiral trajectories) as wellas continuation methods aid in achieving robust con-vergence.

In this paper, independently generated fuel-optimalrendezvous arcs between asteroids are sequenced tocreate a large number of candidate tours. Similar in-vestigations have been completed by Izzo11 and Cana-lias,12 though these authors implement global searchalgorithms that produce single tours comprised of briefflyby encounters, optimized end-to-end. In contrast, inthis preliminary investigation, asteroid tours are con-structed via a three step process:

1. Generate a number of locally optimal thrust seg-ments that link asteroid pairs of scientific interest.

2. Combine these independent solution arcs to yieldan ample space of possible tours with an easilyapproximated propellant cost and time of flight.

3. Select candidate tours with desirable characteris-tics for a higher-fidelity study.

The resulting trajectories offer tours on the order ofmonths and years in the vicinity of the asteroids, al-lowing for extensive scientific observation and datacollection. For this analysis, the spacecraft is assumedto originate within the L4 region, so transfer costs fromEarth to the Trojan asteroid group are not included inthe propellant estimates or mission times. The algo-rithm is illustrated by designing two tours within theasteroid swarm near the Sun-Jupiter L4 point, denotedthe “Greek camp”, but the concepts and implementa-tion details are readily extendable to other systemsand mission objectives.

II. SYSTEM MODEL

Two key steps are initially necessary to successfullyformulate the rendezvous problem, namely the defini-tion of the physical environment to model the dynam-ics of the system and the construction of the initial andtarget states. The model for the unpowered spacecraftdynamics is independent of the optimization strategyand can, therefore, be adjusted to introduce variouslevels of fidelity.

II.I Equations of Motion

The Sun-Jupiter system is modeled as a Circular Re-stricted Three Body Problem (CR3BP) with the Sunas one primary and Jupiter as the second. For thispreliminary analysis, the perturbing effects of otherSolar system bodies are neglected, including the targetasteroids. The equations of motion are then formu-lated within the context of a rotating reference framewhere x is directed from the Sun to Jupiter, z is nor-mal to the orbital plane of the primaries and parallelto orbital angular momentum, and y completes theright-handed set. The origin of the coordinate sys-tem is the Sun-Jupiter barycenter. Incorporated intothe forces that influence the motion in this system areterms that arise from the thrusting of the Variable Spe-cific Impulse (VSI) engine. The system of equationsare non-dimensionalized to aid numerical integrationefficiency: computed results are converted to dimen-sional quantities by the proper use of the characteristicquantities and spacecraft parameter values. The char-acteristic quantities are the Sun-Jupiter distance, themass of the primaries, the characteristic time, and theinitial spacecraft mass. The spacecraft state vector is

IAC-11-C1.1.1 Page 2 of 11

then defined as:

χ =

rvm

(1)

where r is the position vector relative to the barycen-ter, v is the velocity vector with respect to thebarycenter as viewed by a rotating observer, and mis the instantaneous mass of the spacecraft. The equa-tions of motion are then:

χ =

rvm

=

v

fn(r,v) + Tmu

− T 2

2P

(2)

where T is thrust magnitude, P is engine power, uis a unit vector defining the thrust direction, and fnrepresents the natural acceleration of the spacecraft.Furthermore, denote the six-dimensional vector thatincludes position r and velocity v by the vector x,where x =

[x y z x y z

]T . Thescalar elements of fn are then expressed in terms ofthe rotating frame as:

fn =

2y + x− (1−µ)(x+µ)

d31− µ(x+µ−1)

d32

−2x+ y − (1−µ)yd31

− µyd32

− (1−µ)zd31

− µzd32

(3)

where d1 and d2 are the distances from the Sun andJupiter to the vehicle, respectively, that is

d1 =√

(x+ µ)2 + y2 + z2 (4)

d2 =√

(x+ µ− 1)2 + y2 + z2. (5)

The mass parameter µ is

µ =MJ

MS +MJ(6)

where MS and MJ are the masses of the Sun andJupiter, respectively. The power P is defined as ascalar value between zero and a maximum availablepower level specified by the engine model, such that

0 ≤ P ≤ Pmax. (7)

Then, the engine thrust T is evaluated via

T =2PIspg0

(8)

where Isp is the engine specific impulse and g0 =9.80665 m/s2, the gravitational acceleration at the sur-face of the Earth. Further information on the systemand spacecraft parameters is available in Table 1.

Quantity Value

Solar mass (MS), kg 1.9891×1030

Jupiter mass (MJ), kg 1.8986×1027

Mass parameter (µ) 9.53816×10−4

Sun-Jupiter distance (l∗), km 7.78412×108

Characteristic Time (t∗), sec 5.95911×107

Characteristic Time (t∗d), days 6.89712×102

Initial spacecraft mass (m0), kg 500Maximum engine power (Pmax), kW 1.0

Table 1: System and spacecraft parameter values.

II.II Periodic Orbits

Families of stable planar periodic orbits exist in thevicinity of the equilateral libration points, that is L4

and L5. Near L4 and L5, analytical approximationsare numerically targeted and yield two families ofplanar periodic orbtis associated with each librationpoint, a short-period and a long-period family.13,14

Members of the L4 short period family approximatethe trajectories of the target Trojan asteroids for theten-year span from October 3, 2021 to October 3,2031. The ephemeris trajectories of the asteroids ap-pear in the Sun-Jupiter rotating frame in Fig. 1;15 notethat these tracks are approximately elliptical, as arethe short period L4 periodic orbits. Furthermore, theasteroid motion is assumed to be confined to the xy-plane for the selected low-inclination asteroids.

Fig. 1: Projections of ephemeris asteroid trajecto-ries onto xy-plane, October 3, 2021 to October 3,2031. Motion is clockwise.

To represent the asteroid tracks in terms of peri-odic orbits constructed in the CR3BP, select shortperiod orbits of approximately the same major axis,then rotate and translate the position and velocity ofa particle moving along such a path to more closelyalign with the orbits of the actual asteroids. The cor-responding rotation matrix is

C =

cos γ − sin γ 0sin γ cos γ 0

0 0 1

(9)

where the angle γ is selected such that the major axes

IAC-11-C1.1.1 Page 3 of 11

of the periodic orbit and the asteroid track align. Theorbits are transformed as follows:

• Rotate the velocity on the CR3BP orbit by

vapp = CvNP (10)

where vNP is the velocity of the new numericallyproduced (NP) periodic orbit with respect to therotating frame.

• Subtract the coordinates of the L4 point from theposition vector, resulting in the vector rLC locat-ing the particle with respect to L4.

• Rotate the vector rLC such that

rANP = CrLC (11)

where rANP is the position on the newly alignednumerically produced (ANP) periodic orbit.

• Add the L4 coordinates plus an additional set ofconstants such that the altered orbit and the as-teroid track nearly overlap, resulting in

rapp = rANP +[L4,x + ∆xL4,y + ∆y

](12)

with the quantities ∆x and ∆y determined viavisual inspection. A more rigorous method ofselecting the values of ∆x and ∆y may be imple-mented, but the current process is sufficient forpreliminary investigation.

The result of this procedure is a numerically gener-ated position rapp and velocity vapp that approximatesthe ephemeris state of the asteroid. In Figure 2, theephemeris trajectories are plotted along with theirassociated approximations, while the parameters fortranslating and rotating the approximate orbits aresummarized in Table 2.

Fig. 2: Ephemeris asteroid trajectories (dashed)and approximate periodic orbits (solid) in xy-plane.

Finally, all approximate asteroid states must reflectthe actual asteroid state with respect to time. There-fore, a non-dimensional, time-like parameter, τ0, is

introduced, one that is defined to be zero at the epochOctober 3, 2021. All future times, and the corre-sponding asteroid states, are then uniquely determinedby τ0. An approximate asteroid state is determinedby (i) forward propagation of the numerically gener-ated L4 short period orbit; then, (ii) application ofthe transformation. The initial states correspondingto the periodic orbits in the Sun-Jupiter CR3BP thatare used to approximate the asteroid paths are listedin Table 3.

III. TRAJECTORY OPTIMIZATION

The optimal asteroid rendezvous problem is solved in-directly using the calculus of variations to reformulateit as a two-point boundary value problem (2PBVP).The 2PBVP is then solved numerically using a shoot-ing method; the initial conditions corresponding tothe transfer are determined along with the time evolu-tion of the subsequent spacecraft position and velocitystates as well as the engine operational states by theEuler-Lagrange equations. However, the thrust du-ration TD must be specified when a VSI engine isemployed. If no limit is placed on either the thrustduration or the minimum mass consumed, the opti-mization process drives TD and Isp to infinity whileconsuming zero propellant mass.

To fully define the optimization problem, the perfor-mance index and the boundary conditions must alsobe specified. To arrive at the target asteroid with themaximum final spacecraft mass for a specified thrustduration, the performance index J is defined

max J = mf . (13)

The boundary conditions and the Hamiltonian are ad-joined to the performance index, so that Eq. (13) isexpanded to become the Bolza function

max J ′ = mf +νT0 ψ0 +νTf ψf +∫ tf

t0

[H−λT χ]dt (14)

where H is the problem Hamiltonian, λ is a co-statevector, the terms ψ are vectors comprised of bound-ary conditions, and the vector terms involving ν areLagrange multipliers corresponding to the boundaryconditions. The co-state vector is then

λ =

λr

λv

λm

(15)

where λr and λv are three-dimensional vectors com-prised of the position and velocity co-states, respec-tively, and the scalar λm is the mass co-state. Theinitial and final boundary conditions are

ψ0 = xI − xI(τ0) = 0 (16)

IAC-11-C1.1.1 Page 4 of 11

Asteroid Major axis, km Period, years γ, deg ∆x, km ∆y, km

659 Nestor 3.631×108 11.90238 12 0.5×108 -0.3×108

1143 Odysseus 1.829×108 11.90307 135 0.8×108 -0.55×108

1869 Philoctetes 1.373×108 11.90318 17 1.75×108 -1.4×108

5012 Eurymedon 2.722×108 11.90279 12 0.85×108 -0.5×108

5652 Amphimachus 2.024×108 11.90302 30 -0.5×108 0.3×108

Table 2: Orbital characteristics for selected L4 asteroids and approximate orbit translations and rotations.

659 Nestor 1143 Odysseus 1869 Philoctetes 5012 Eurymedon 5652 Amphimachus

x, ×108 km 2.01486 3.14342 3.24797 2.50477 2.90869y, ×108 km 7.58718 6.74125 6.89185 7.24476 6.98964z, ×108 km 0 0 0 0 0vx, km/sec 0.69678 -0.81724 -0.26322 0.05311 -0.32769vy, km/sec 1.37988 0.96886 0.69570 1.19029 0.99555vz, km/sec 0 0 0 0 0

Table 3: Periodic orbit states at epoch October 3, 2021, τ0 = 0.

andψf = xT − xT (τ0 + TD) = 0 (17)

where the subscripts I and T indicate the states as-sociated with the current asteroid and target asteroid,respectively. Equation (16) is implicitly satisfied bydefining xI as the state along the current asteroid tra-jectory as defined by the parameter τ0. The final, ortarget, boundary conditions in Eq. (17) are satisfiedby solving the boundary value problem.

The calculus of variations is employed to define sev-eral properties of the 2PBVP and acquire the deriva-tives of the co-states. The problem Hamiltonian is

H = λT χ = λTr v+λTv

[fn(r,v)+

T

mu

]−λm

T 2

2P(18)

where the value of H is constant over the trajec-tory. The optimal controls emerge by maximizing theHamiltonian with respect to the controls T , P , and usuch that

P = Pmax (19)

T =λvPmax

λmm(20)

u =λv

λv(21)

where λv=||λv||. From these controls, the Hamilto-nian is reformulated and Eq. (18) is rewritten as

H = λTr v + λTv fn + S · T (22)

where S is the switching function

S =λv

m− λmT

2Pmax. (23)

The Euler-Lagrange conditions for optimality modifythe performance index in Eq. (14). With the refor-mulated Hamiltonian, that is, Eq. (22), the following

equations of motion for the co-states emerge

λ = −(∂H

∂χ

)T=

−λTv

(∂fn

∂r

)−λTr − λTv

(∂fn

∂v

)λv

Tm2

(24)

where the initial state for λm is set to unity to reducethe number of variables to be determined.

All that remains to define the 2PBVP is construc-tion of the transversality conditions that ensure localoptimality. The first differential of the Bolza function,Eq. (14), with respect to the time-like parameter τ0supplies the condition

λTrv0

∂xI(τ0)∂τ0

− λTrvf

∂xT (τ0 + TD)∂τ0

= 0. (25)

To determine the partial derivatives in Eq. (25), applythe transformation from Section II.II to the velocityand acceleration on the short period orbit that ap-proximates the asteroid trajectory. Thus,

∂xI(τ0)∂τ0

=[CI 00 CI

]{vI(τ0)fn,I(τ0)

}(26)

∂xT (τ0 + TD)∂τ0

=[CT 00 CT

]{vT (τ0 + TD)fn,T (τ0 + TD)

}(27)

where the subscripts I and T are again associated withthe initial and target orbits, respectively, and the ve-locity v and natural dynamics fn are evaluated onthe short period orbit prior to transformation. The2PBVP is then completely formulated to numericallyproduce the set of design variables

X ={

τ0λrv0

}(28)

that satisfy the constraints

F (X) ={

ψfλTrv0

∂xI(τ0)∂τ0

− λTrvf

∂xT (τ0+TD)∂τ0

}= 0

(29)

IAC-11-C1.1.1 Page 5 of 11

with λrv0 and λrvfdefined as the co-state vectors at

the beginning and end of a thrust arc, respectively.The optimization problem can be solved using anyboundary value problem or equation solver, for exam-ple the fsolve function in MATLAB.

IV. MISSION APPLICATIONS

A mission to the vicinity of the Sun-Jupiter “Greek” or“Trojan” asteroid families will almost certainly entailrendezvous with and observation of multiple objects.A strategy is proposed for rapidly generating a largenumber of candidate asteroid tours and analyzing theresulting design space for trade-offs in terms of fuelconsumption, mission time, and scientific opportunity.This trajectory evaluation scheme yields only approxi-mate propellant costs and is, therefore, intended solelyfor preliminary design analysis. However, overall per-formance comparisons may still be assessed and spe-cific trajectory concepts are readily transitioned tohigher fidelity models that offer more accurate esti-mates of propellant consumption.

For baseline mission design, the transfer of thespacecraft from Earth to the Sun-Jupiter L4 regionmust also be addressed. This leg of the mission alsoprovides a wealth of opportunity for optimization andtrajectory design. For instance, a Hohmann transferfrom Earth to the Sun-Jupiter L4 point is completedin 2.73 years and requires a total 14.44 km/s of ∆vfor Earth departure and matching the L4 point veloc-ity upon arrival. On the other hand, gravity assistmaneuvers using solar system bodies such as Earth,Venus, Mars, and Jupiter extend the flight time inreturn for lowered ∆v costs. As an example, Cana-lias constructs an Earth-Venus-Earth-Earth flyby se-quence that transfers the spacecraft to the Sun-JupiterL4 region in 10 years for 9.17 km/s, where this cost in-cludes Earth departure and asteroid swarm arrival.12

On the other hand, the low-thrust propulsion systemmay be employed to assist in the transfer from Earthto the L4 region, as proposed by Desjean.16 For thisinvestigation, however, the spacecraft is assumed havealready completed this transfer and arrived at the Tro-jan asteroid swarm with a remaining mass of 500 kg.

IV.I Rendezvous Sequence Generation

The optimization procedure in Section III yields a sin-gle rendezvous segment connecting two asteroids andresulting in minimum propellant consumption for aspecified thrust duration. The initial and terminalstates along these arcs correspond to approximated as-teroid positions and velocities such that the spacecrafttransfers from the vicinity of one asteroid to that ofanother. However, once a point solution is generatedfor a single specified thrust duration TD, a simple con-tinuation scheme is applied that produces trajectories

over a large range of thrusting times. The continu-ation process updates the value of TD and uses thepreviously computed solution as the initial guess forthe subsequent 2PBVP. The complete set of thrustarcs that is determined via the continuation scheme,termed a “family”, represents a set of options for asingle pre-determined asteroid-to-asteroid link withina design space relating engine operation time and pro-pellant consumed for a spacecraft transfer.

Families of rendezvous segments are independentlygenerated for a variety of initial and target aster-oid pairs (e.g., 659 Nestor to 5012 Eurymedon, 1869Philoctetes to 1143 Odysseus, etc.). Note that for ev-ery thrust arc segment within these families, the initialspacecraft mass is assumed to be m0 = 500kg, orm0 = 1 non-dimensional unit. The propellant con-sumed during each interval of engine operation mustbe incorporated into an equivalent cost for any po-tential tour scenario comprised of several rendezvousarc segments. Accordingly, for a rendezvous sequencecomprised of n thrust intervals, the approximate pro-pellant mass consumed mcons is obtained via

mconsumed = m0

(1−

n∏i=1

mi

m0

)(30)

where mi is the arrival mass in kilograms at the endof the ith independently generated thrust arc.∗ Asidefrom propellant consumption, other key design consid-erations of a proposed tour concept include the timerequired to complete the overall rendezvous sequenceand the time interval within close proximity to the as-teroids of interest. The duration of the total sequence,τS , is assessed via

τS =n∑i=1

TDi +n−1∑i=1

CTi (31)

where CTi is the length of time near the target asteroidalong the ith thrust arc, that is

CTi = (τ0)i+1 − (τ0 + TD)i (32)

with (τ0)i+1 representing the time at thrust initiationon the (i+ 1)th arc and (τ0 +TD)i corresponds to theend of the thrust interval along the ith arc. The timeinterval in the vicinity of the intermediate asteroids,termed the “observation time” τO, is evaluated as

τO =n−1∑i=1

CTi (33)

and reflects the time the spacecraft spends loiteringnear the asteroids and, therefore, available for scien-tific observation, data collection, and any other perti-nent mission activity.∗For the case of impulsive maneuvers, an equivalent total

mission cost is ∆vtot =∑p

i=1 ∆vi where ∆vtot is the total im-pulsive ∆v and ∆vi is the equivalent value for one maneuver.

IAC-11-C1.1.1 Page 6 of 11

Equations (30), (31), and (33) are incorporated intoan iterative procedure that evaluates all possible com-binations of individual segments from the rendezvousfamilies without the necessity to re-optimize each spe-cific sequence. So, for example, combining a familywith 10 rendezvous paths from 5012 Eurymedon to1869 Philoctetes with a 10-member family from 1869Philoctetes to 1143 Odysseus yields 100 candidate so-lutions for an asteroid tour originating at Eurymedon,terminating at Odysseus, and including an interme-diate coast in the vicinity of Philoctetes. These 100possible sequences are then filtered for canditate toursexhibiting the desired balance of propellant consumed,observation time, and sequence time. Additionally, theset of sequences can be examined to gain additional in-tuition concerning the overall design space.

IV.II Sample Trajectory Sequences

The first step in the tour design process is the gen-eration of rendezvous families between the target as-teroids. At the initiation of this process, the asteroidorder along the tour is not yet available, so locallyoptimal thrust arc families between asteroid pairs aregenerated and stored on an individual basis. An orga-nizational framework for constructing an asteroid tourappears in Fig. 3, where the red arrows denote that afamily of low-thrust rendezvous segments between theasteroids has been computed.

5652 Amphimachus

5012 Eurymedon

1869 Philoctetes

1143 Odysseus

659 Nestor

Fig. 3: Target asteroids with arrows indicatingcomputed rendezvous trajectory families, i.e., theasteroid-to-asteroid links.

For a reasonable likelihood of a successful design, itis necessary to ensure that the optimal departure andarrival dates for the thrust intervals are realizable todeliver an end-to-end trajectory. Therefore, the avail-able asteroid rendezvous pairs appear in Table 4 alongwith the range of departure dates over which fami-lies of asteroid-to-asteroid low-thrust segments havebeen computed. The maximum engine operation timeacross the family representing a specified link is alsonoted in the table. To confirm the validity of a selectedtour sequence, simply check that the latest departure

date plus the interval of engine operation for the ith

rendezvous segment does not conflict with the earliestdeparture date on the (i + 1)th thrust arc. This re-striction does omit some valid trajectory options, butstreamlines the design process for this preliminary in-vestigation.

Asteroid Earliest Latest Max. TD,Pair Departure Departure yrs

Amph. → Nest. 06/2019 08/2020 4.15Amph. → Odys. 07/2025 05/2028 2.27Amph. → Odys. 06/2037 04/2040 2.27Eury. → Amph. 12/2031 04/2032 3.40Eury. → Nest. 09/2025 11/2026 2.08Eury. → Phil. 08/2026 06/2027 3.21Nest. → Amph. 01/2033 08/2033 3.21Nest. → Eury. 12/2024 11/2026 2.08Nest. → Phil. 06/2026 01/2027 3.21Phil. → Eury. 11/2026 03/2027 3.40Phil. → Nest. 04/2026 11/2026 3.40Phil. → Odys. 10/2031 04/2033 2.83

Table 4: Families of low-thrust arcs between as-teroid pairs; dates of earliest and latest departureon low-thrust arcs across the family. (Dates are inMM/YYYY format.)

As is apparent in Table 4, some of the departuredates lie outside the ten-year time frame originallyspecified to create the approximations to the asteroidtracks. For this analysis, the transformation valuesin Table 2 are assumed to remain valid beyond theoriginal ten-year mission window. Increased fidelity isaccomplished by updating the periodic orbit transfor-mation based on τ0 or by incorporating the ephemeristrajectories directly. Additionally, there are two 5652Amphimachus to 1143 Odysseus rendezvous families,with a twelve-year interval between them. These twofamilies are nearly the same set of solutions, shiftedin phase by one full revolution of the short periodorbits. Similarly, all the rendezvous families can beshifted by whole periods corresponding to the approx-imating orbits subject, of course, to the phase shift inthe positions of the asteroids and the similarity of thetransformed orbit to the asteroid track. Thus, toursdesigned within the ten-year window between October3, 2021 and October 3, 2031 are expected to retainsome of their properties if shifted in discrete twelve-year intervals around the nominal dates.

The next step in the trajectory design process is thecombination of individual segments into a sequence ofthrust arcs between asteroids and coast arcs in thenear-vicinity of the asteroids for observations. Twosample tours are examined:

1. Tour ANPO: 5652 Amphimachus to 659 Nestor to1869 Philoctetes to 1143 Odysseus;

2. Tour NEAO: 659 Nestor to 5012 Eurymedon to5652 Amphimachus to 1143 Odysseus.

In each tour, the spacecraft is initially assumed tobe in the local vicinity of the first asteroid. These

IAC-11-C1.1.1 Page 7 of 11

prospective tours are assessed in terms of the trade-off between propellant consumption, engine operationduration, overall end-to-end trajectory length, and theloiter time near the intermediate asteroids.

IV.II.I Sample Sequence 1: Tour ANPO

Sample trajectory sequence ANPO originates with thespacecraft in the vicinity of 5652 Amphimachus andterminates at 1143 Odysseus, with intermediate coastsnear asteroids 659 Nestor and 1869 Philoctetes. Theapproximated asteroid tracks as well as the pathsalong the thrust segments in the rendezvous familiesthroughout the tour are displayed in Fig. 4. The col-ors of the individual segments within the families thatlink pairs of asteroids as they appear in the figure indi-cates the engine operation time along that thrust arc;all three families use the color scale indicated in theplot. The dots mark the departure and arrival pointson the approximated asteroid tracks. For all poten-tial tours, the spacecraft spends approximately equalintervals near the asteroids 5652 Amphimachus and1869 Philocetes.

Fig. 4: Asteroid tracks and families of thrust seg-ments for ANPO tours.

The propellant mass performance along each familyof thrust segements is plotted in Fig. 5. The num-bering in the legend corresponds to the families ofthrust segments in Fig. 4. Note that in each ren-dezvous family the final spacecraft mass generally in-creases with increasing thrust duration TD, indicatingthat less propellant is consumed along longer thrustarcs. The results from the independently generatedrendezvous families are combined as described in Sec-tion IV.I, with the resulting trade space as illustratedin Fig. 6. The plots in Fig. 6 represent explicit valuesof the propellant consumption, trajectory duration,and scientific observation time at the intermediate as-teroids 5652 Amphimachus and 1869 Philocetes whileindirectly revealing information about the individualthrust arcs comprising the tours. The colors of thepoints in Fig. 6 reflect the thrust durations within thesecond rendezvous family, TD2, where black identifies

600 800 1000 1200 1400 1600480

482

484

486

488

490

492

494

496

498

500

TD, days

mf,k

g

1st

2nd

3rd

Fig. 5: Spacecraft mass at the end of the thrust arcin each of the three ANPO thrust arc families.

the smallest values of TD2 and white the largest. Like-wise, the point size reveals the relative values of TD3,that is, the length of time along the segments in thethird thrust arc family, with larger points indicatinglonger engine operation time, TD3. For a selected tra-jectory, the duration of the corresponding rendezvousarc from the first family of thrust segments, TD1, isdetermined from TD2, TD3, τS , and τO via Eqs. (31)-(33). Note also that the mass consumed appears interms of decakilograms (1 dakg = 10 kg).

Fig. 6: Trade space for ANPO tours: consumedpropellant, overall tour duration, and observationtime.

Upon examination, the trade space exhibits a clearand relatively linear trade-off between low propellantconsumption but higher sequence duration τS com-puted via Eq. (31) and lower asteroid scientific ob-servation time τO, from Eq. (33). For example, thetour with the lowest approximated fuel cost (16.323kg) also corresponds to the longest sequence intervalat 15.113 years and the lowest loiter time, i.e., 4.916years. This candidate tour is comprised of the longestduration thrust arcs from the three independent fam-ilies. This trajectory is re-generated from end-to-endto offer a more accurate estimate of the design metricsfor comparison to the approximated design parame-

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ters. So, when solving the 2PBVP, the spacecraft massat the initiation of thrust on the (i + 1)th segment isequal to mi, the spacecraft mass at the terminationof the ith rendezvous thrust arc. Accordingly, the re-generated trajectory is optimized from end-to-end andthe thrust segments are no longer independent. Thisre-converged tour consumes 16.192 kg of propellant,while the overall interval is again 15.113 years and theasteroid science observation duration is 4.916 years.Thus, in comparison to the actual trajectory, the ap-proximated tour estimates a relatively small additional0.811% in the propellant consumed and differences inthe sequence time and observation time, i.e., τS andτO, on the order of seconds. On the other end of thetrade space, the candidate tour with an approximatepropellant consumption of 42.077 kg and τS = 14.389years and τO = 8.724 years actually requires 40.983 kgof propellant when re-computed to optimize the fullend-to-end sequence. The mass over-estimate of thecombination procedure in Section IV.I is then 2.668%,while the overall tour duration and loiter time decreaseby approximately 5 minutes for the end-to-end trajec-tory.

IV.II.I Sample Sequence 2: TOUR NEAO

In the second example tour, the spacecraft originatesalongside 659 Nestor in its orbit, then transfers tothe vicinity of 5012 Eurymedon, 5652 Amphimachus,and ultimately 1143 Odysseus. The rendezvous seg-ments and the approximated asteroid tracks appear inFig. 7. Consistent with the previous sample tour, thecolor scale indicates the engine operation time for eachthrust arc and the dots mark the departure and arrivalpoints along the approximated asteroid tracks. Again,the scientific observation time in the rendezvous se-quence is divided nearly evenly between the two inter-mediate asteroids.

Fig. 7: Asteroid tracks and thrust trajectories forNEAO tours.

The spacecraft arrival mass corresponding to each ofthe individual thrust arc families is plotted in Fig. 8,where the numbers match that in Fig. 8. Two of the

families exhibit increasing arrival mass with increas-ing thrust duration TD, but the third transfer familydisplays a distinct local peak at approximately 550days of engine operation. Thus, on the final leg ofthe tour, there is a point when increasing the thrustduration is detrimental in terms of both propellantconsumption and mission operation time. The results

200 400 600 800 1000 1200 1400460

465

470

475

480

485

490

495

500

TD, days

mf,k

g

1st

2nd

3rd

Fig. 8: Spacecraft mass at end of the thrust arc ineach of the three NEAO thrust arc families.

from the unconstrained rendezvous segments are com-bined and the trade space is illustrated in Fig. 9. As inFig. 6, the values of propellant consumption, sequencelength, and asteroid observation time are explicitlyavailable in Fig. 9. Indirect information on the in-dividual thrust segments comprising the tour is alsoavailable, i.e., the second thrust duration is indicatedby the color of the points, with white transitioningto blue then red and eventually yellow as TD2 in-creases, while the point size reflects TD3, the lengthof the third rendezvous arc, with larger points indi-cating longer TD3. In contrast to the trade spacerepresenting the ANPO tour, the NEAO tours displaya non-linear relationship between mcons and τO with aclear minimum at intermediate values of asteroid loitertime. The relationship between propellant consump-tion and overall tour length is again approximatelylinear, however the tour with the minimum approxi-mate fuel cost, 5.722 kg, possesses an overall durationτS of 16.386 years and an observation time of 9.550years, neither of which are extreme values within thetrade space. This candidate tour is composed of thelongest duration thrust arcs from the first two ren-dezvous families and a moderate value for TD3. Thecandidate trajectory is re-optimized across the entiretour to provide a more accurate estimate of the designmetrics. The reconverged sequence consumes 5.704 kgof propellant, while τS and τO are altered only on theorder of seconds. So, for this sample tour, the approx-imated solution over-estimates the required propellantby 0.314%.

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Fig. 9: Trade space for NEAO tours: consumedpropellant, overall tour duration, and observationtime.

V. CONCLUSIONS

A strategy to estimate propellant cost and other per-formance metrics for a low-thrust trajectory effectedby a variable specific impulse system has been devel-oped and applied to the generation of asteroid ren-dezvous tours within the Sun-Jupiter L4 Trojan as-teroid swarm. Indirect optimization methods are em-ployed to rapidly yield a large number of rendezvousarcs between selected asteroids. These independentlygenerated asteroid-to-asteroid trajectory arcs are thencombined to quickly develop a trade space that offerssome insight into the tour design and allows for theselection of candidate trajectories for further analy-sis. The procedure is not limited by dynamical regimeand is readily extended to other mission architectures.In general, the approximation process is expected tobe conservative and, thus, slightly over-estimates therequired propellant. For the sample tours examined,the combination algorithm yields fuel mass estimateswithin 3% of identical trajectories optimized from end-to-end while timing results are accurate on the orderof minutes.

There are several avenues for further investigationand refinement. In particular, higher fidelity model-ing of asteroid motion will increase the accuracy ofthe resulting designs for both the asteroid-to-asteroidarcs as well as the end-to-end tour trajectory. Addi-tionally, the approximation procedure may be appliedto other scenarios requiring multiple low-thrust arcs,whether as part of a baseline or an extended trajectorydesign. Furthermore, the combination process may beimproved by implementing concepts from hybrid sys-tems theory, such as reachability analysis, as well asmulti-dimensional and constrained optimization.

ACKNOWLEDGEMENTS

This work was conducted at Purdue University and issupported by the Purdue Research Foundation. Manythanks to Wayne Schlei, who helped immensely withthe trajectory and trade space images.

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