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70th International Astronautical Congress, Washington DC, USA, 21-25 October 2019. Copyright c

2019 by International

Astronautical Federation (IAF). All rights reserved.

IAC–19–C1.1.11

Mission Design for Close-Range Lunar Mapping by Quasi-Frozen Orbits

Sandeep Kumar Singh

Graduate Research Assistant, Department of Aerospace Engineering, Texas AM University, TX 77843,

USA, sandymeche@tamu.edu, Corresponding Author

Ehsan Taheri

Assistant Professor, Department of Aerospace Engineering, Auburn University, Auburn, Alabama, 36849,

USA, etaheri@auburn.edu

Robyn Woollands

Mission Design Engineer, Mission Design and Navigation, Jet Propulsion Laboratory, California Institute

of Technology, Pasadena CA 91109, USA, Robyn.M.Woollands@jpl.nasa.gov

John Junkins

Distinguished Professor, Department of Aerospace Engineering, Texas AM University, TX 77843, USA,

junkins@tamu.edu

The presence of extremely low-altitude, lunar quasi-frozen orbits (QFOs) has given rise to interesting mission

opportunities. These QFOs are ideal for close-range, high-resolution mapping of the lunar south pole, and

their inherent stability translates into minimal station-keeping eﬀorts. Despite the aforementioned desirable

characteristics, designing transfer trajectories to these QFOs poses signiﬁcant diﬃculties, speciﬁcally, for

spacecraft equipped with low-thrust electric engines. A solution strategy is proposed, within the indirect

formalism of optimal control, for designing minimum-time trajectories from a geosynchronous orbit to a

candidate low-altitude, lunar QFO. The classical restricted three-body dynamical model of the Earth-Moon

system is used to achieve more realistic results. The diﬃculties in using indirect optimization methods are

overcome through a systematic methodology, which consists of patching three-dimensional minimum-time

trajectory segments together such that the spacecraft terminates in a prescribed highly stable quasi-frozen

orbit. Application of a pseudo-arc-length continuation method is demonstrated for a number of lunar capture

phases consisting of up to 38 revolutions around the Moon.

keywords: Indirect Optimization, Continuous-thrust, Optimal Trajectories, Minimum-time, Quasi-Frozen

Orbits,

1. Introduction

Humans have been fascinated by the Moon ever

since the dawn of their existence on Earth. What be-

gan as merely staring in awe, turned into observing

with telescopes and ﬁnally in the 20th century, the

ﬁrst humans were able to visit Earth’s Moon. Our

knowledge about our nearest celestial neighbour has

grown over time. The earliest foray into moon explo-

ration dates back to 1959, when a Soviet spacecraft,

Luna 2, made the ﬁrst landing on the Moon. This was

followed by a series of robotic, uncrewed exploration

eﬀorts, and culminated with the Apollo Missions from

1961 to 1971. More recently, missions like the Lunar

Prospector Mission 1in 1998, Lunar Reconnaissance

Orbiter2in 2009, and the GRAIL mission 3in 2011

have signiﬁcantly improved our understanding of the

Moon’s topography and its gravity ﬁeld.

One of the most exciting scientiﬁc discoveries

made by the Moon Mineralogy Mapper (M3) aboard

ISRO’s Chandrayaan 1 in 2008, which was later con-

ﬁrmed in 20184by NASA, was the presence of water-

ice deposits in the deep craters near of the lunar

South Pole. The presence of water has reignited in-

terest in lunar missions especially for extremely high-

resolution scientiﬁc imaging and mapping. Such mis-

sions would aid in the search for suitable base sites,

where the water-ice deposits could be used as an in-

situ resource. NASA plans to send the next man

and ﬁrst woman to the Moon by 2024, via the lunar

“Gateway”5. Thus, a hi-resolution close-range map-

ping mission to the lunar South Pole is of particular

interest in the study performed in this paper.

Frozen orbits around the Moon have been stud-

ied6–9 because they are excellent candidates for low-

IAC–19–C1.1.11 Page 1 of 11

70th International Astronautical Congress, Washington DC, USA, 21-25 October 2019. Copyright c

2019 by International

Astronautical Federation (IAF). All rights reserved.

cost station-keeping missions due to their stability.

Unfortunately, they are extremely rare due to the

“lumpy” gravity ﬁeld of the Moon and only exist

at particular inclinations. QFOs exist in the neigh-

bourhood of the frozen orbit conditions and display

a high degree of inherent stability. In a recent re-

search, Singh et al.10 found a number of extremely

low-altitude QFOs that can be maintained with min-

imal station-keeping eﬀorts for envisioned lunar map-

ping missions.

Low-thrust propulsion systems have long been

identiﬁed as eﬃcient means for performing orbit

transfers due to their capability in delivering a

greater payload fraction compared to other conven-

tional propulsion systems11. Low-thrust transfer tra-

jectories are typically characterized by longer ﬂight

times12;13 compared to Hohmann or any other high-

energy transfer, therefore, making them ideal for

cargo, instrument transport or scientiﬁc missions.

The dynamics of a spacecraft in the Earth-Moon

system can be modelled in several ways. London14

and Stuhlinger15 used patched conic approximations

at the sphere of inﬂuence. Szebehely 16 , Zhang et

al.17 and Taheri and Abdelkhalik18 took into ac-

count gravitational inﬂuences of the Earth and Moon,

solving trajectory optimization problems in a Cir-

cular Restricted Three-Body Problem (CR3BP) for-

mulation. The work in Zhang et al.17 used Pon-

tryagin’s Maximum principle (PMP)19 while Taheri

et al.18 used a Finite Fourier series shape-based

method. Minimum-fuel planar transfers have been

studied by Enright and Conway20 and Pierson and

Kluever21 . While the former used collocation meth-

ods the latter used a structured thrust sequence,

i.e., thrust-coast-thrust sequence. Three-dimensional

(3D) minimum-fuel transfers have been studied by

Golan and Breakwell22 for power limited spacecraft,

which resulted in a variable thrust trajectory with-

out coasting phase. Kluever and Pierson23 presented

a “hybrid” direct/indirect method to solve for opti-

mal 3D trajectories. Topputo et al.24 studied transit

orbits connecting two circular parking orbits around

the Earth and Moon by solving two three-body Lam-

bert arcs. More recently, the eﬀect of Sun’s pertur-

bation has been coupled with existing dynamics and

fuel-optimal, impulsive Earth-Moon transfers stud-

ied in the Bi-circular Restricted Four-Body Prob-

lem (BR4BP) by Topputo25, Qi and Xu26, Filho

and da Silva Fernandes27 . Low-thrust minimum-fuel

transfers considering BR4BP were studied in 2018 by

Palau and Epenoy28 .

In this paper, a continuous low-thrust, minimum-

time optimal control problem (OCP) is formulated in

the context of a CR3BP using an indirect optimiza-

tion method, which leads to a two-point boundary-

value problem (TPBVP). PMP is used to com-

pute the optimal thrust-direction and second order

strengthened Legendre-Clebsch condition is used to

resolve the sign ambiguity. The entire trajectory is

divided into three phases, Earth-escape phase, Moon-

capture phase and QFO insertion phase. A separate

OCP is formulated and solved for each phase where

a proper frame is chosen to describe the dynamics.

The ﬁrst phase has been solved in the Earth-centred

rotating frame while the last two phases are solved in

a Moon-centred rotating frame.

The rest of the paper is organised as follows. Sec-

tion 2 presents formulation of the overall problem,

introduces the coordinate systems, equations of mo-

tions (EOMs), and summarizes the ensuing TPBVP

associated with each phase of the problem. Dynamics

of the problem are discussed followed by a discussion

on the proposed methodology. A discussion on con-

tinuation methods is presented. Numerical results are

presented along with a number of conclusions.

2. Problem Formulation

Our goal is to design a minimum-time trans-

fer trajectory from a geosynchronous orbit (GEO)

to a QFO for a spacecraft equipped with an elec-

tric thruster. The complexity of the optimization

problem is attenuated by splitting the entire tra-

jectory into three phases: 1) an Earth-Escape Cis-

lunar Transfer (EECT) phase, a Moon-Capture (MC)

phase and a Lunar-Spiraling (LS) phase as shown in

Figure 1.

Fig. 1: (top) EECT and MC phases patched at the

boundary of the Moon’s Sphere of Inﬂuence (SOI);

(bottom) LS phase from the Low Lunar Orbit

(LLO) to QFO.

IAC–19–C1.1.11 Page 2 of 11

70th International Astronautical Congress, Washington DC, USA, 21-25 October 2019. Copyright c

2019 by International

Astronautical Federation (IAF). All rights reserved.

A common practice in solving such complex prob-

lems (with multiple bodies and highly non-linear dy-

namics) is to solve a simpler problem ﬁrst. In the con-

text of minimum-time trajectory optimization prob-

lem, the “easy” problem corresponds to a planer dy-

namics, where a minimum-time trajectory optimiza-

tion problem from the GEO to an equatorial LLO is

formulated and solved using a single-shooting scheme.

The EOMs are expressed in the Earth-centered syn-

odic Cartesian rotating frame. The altitude of the

LLO can be reduced to within an acceptable range.

In the next step, the resulting trajectory is split

into two phases and the EOMs are redeﬁned for each

phase with respect to a rotating frame centred at the

respective primary body. The point of intersection of

the trajectory with the planar projection of the lunar

SOI is deﬁned as the patch-point. In order to capture

3D transfer trajectories around the moon, spherical

coordinates are used to represent dynamics. The set

of state and co-states are mapped into their respec-

tive values in the moon-centered spherical coordinate

system.

In this work, the EECT phase is assumed to be

planar and the 3D formulation is only used for trans-

fers between the patch-point and a LLO. For these

two phases, the 3D EOMs are expressed in terms of

the coordinates of a spherical rotating frame centred

at the Moon. A continuation on the latitude leads to

solutions for various inclined LLOs. The details are

provided in the Results section.

Finally, for the LS phase, EOMs are deﬁned us-

ing Modiﬁed Equinoctial Elements (MEEs) 29 . The

large number of revolutions (≈20+) expected in this

phase was the prime factor in choosing a proper set

of elements12;30 . MEEs are distinctly advantageous

especially for multi-rev transfers as shown by Pan et

al.31. The next sections deﬁne the EOMs for the dif-

ferent phases described above and elucidate the OCP

formulation using indirect approach.

2.1 Planar, Cartesian CR3BP formulation

The Earth-Moon trajectory is governed by CR3BP

dynamics. The formulation is available in many

texts , and mentioned here for the sake of comple-

tion. Using appropriate non-dimensionalization, the

dynamics depends only on the mass ratio µ, deﬁned

as

µ=mM

mE+mM

,[1]

where, mMand mEdenote the mass of Moon and

Earth, respectively.

A synodic rotating frame around the barycenter

of the Earth-Moon system is considered, where the

non-dimensional mass of the Earth is (1 −µ) and

that of the Moon is µ. The Earth and Moon lie at

ﬁxed coordinates (−µ, 0) and (1 −µ, 0), respectively.

With these assumptions, the potential function of the

CR3BP can be written as

Ω = x2+y2

2+1−µ

r1

+µ

r2

+µ(1 −µ)

2,[2]

where r2

1= (x+µ)2+y2and r2

2= (x+µ−1)2+y2.

For a continuous-thrust control, the acceleration due

to the low-thrust engine can be written as,

al=uTmax

m,[3]

where Tmax is the magnitude of maximum thrust pro-

vided by the engine, u= (ux, uy) is thrust unit steer-

ing vector, and mis the spacecraft mass. The state

dynamics are given by

˙x=vx,

˙y=vy,

˙vx= 2vy+ Ωx+alx,

˙vy=−2vx+ Ωy+aly,

˙m=−Tmax

Isp g0

,

[4]

where Ωx, Ωyare the derivatives of Ω with respect

to xand y, respectively and Isp denotes the speciﬁc

impulse of the engine, g0is the acceleration due to

gravity at sea level, al={alx, aly}are the x and y

components of control acceleration respectively. The

minimum-time OCP is deﬁned as

minimize J=tf,

subject to:

Equation [4],

Φ0=0,

Φf=0,

tffree,

[5]

where Φ0and Φfare the initial and ﬁnal constraints,

respectively. Let r0= [x0, y0]Tand v0= [vx0, vy0]T

denote the initial position and velocity vectors, re-

spectively. The initial conditions in Equation [5] are

written as follows

Φ0="r0−[rGEO −µ, 0]T

v0−[0,qµ

rGEO −rGEO]T#,[6]

where rGEO is the magnitude of the radius of the

IAC–19–C1.1.11 Page 3 of 11

70th International Astronautical Congress, Washington DC, USA, 21-25 October 2019. Copyright c

2019 by International

Astronautical Federation (IAF). All rights reserved.

GEO. The ﬁnal circular LLO is determined by

Φf=

krfk2−r2(tf)2

kvfk2−(vxf−r2y(tf))2−(vyf+r2x(tf))2

r2x(tf)(vxf−r2y(tf)) + r2y(tf)(vyf+r2x(tf))

[7]

where vxfand vyfare the x and y velocity com-

ponents at tf,krfkis the magnitude of LLO ra-

dius and kvfk=qµ

krfk. Let X= [x, y, vx, vy]T

and Λ= [λx, λy, λvx, λvy]Tdenote the state and co-

state vectors, respectively. The Hamiltonian associ-

ated with the cost functional deﬁned in Eq. [5] can

be written as

H=ΛT˙

X.[8]

The Euler-Lagrange equation gives the co-state dy-

namics,

˙

Λ=−∂H

∂XT

,[9]

whereas PMP gives the optimal control as

ux=−λvx

kλvk, uy=−λvy

kλvk,[10]

where kλvk=qλ2

vx+λ2

vy. There are ﬁve unknowns

(Λ(t0) and tf), while Φfprovides three terminal con-

ditions. The transversality conditions give the other

two necessary conditions,

λxfyf−λyf(xf+µ−1) + λvxfvyf−λvyfvxf= 0,

1 + H(tf)=0.

The resulting TPBVP was solved to attain a direct

transfer to LLO starting from GEO. Deriving EOMs

in polar coordinates is identical to that in spherical

coordinates in two-dimension (2D)32 . In the next two

sections, the EOMs and formulation of OCPs for the

EECT and MC phases are presented.

2.2 Earth-Escape Cis-lunar Transfer (EECT) phase

The 3D EOMs for the spacecraft in Earth-

centered, rotating, spherical coordinate system gov-

ern the EECT phase. The Earth-Moon plane deﬁnes

the x-y (primary) plane.

Figure 2 depicts the spherical coordinate system

where the set {r1, θ1, φ1}form the Earth-centered

spherical coordinates. The states are radial position

r1, longitude angle θ, latitude angle φ, radial veloc-

ity vr1, longitudinal plane velocity vθ1and latitude

plane velocity vφ1. The state vector is denoted by

Fig. 2: Spherical co-ordinate system.

X1whereas the corresponding co-state vector by Λ1

such that,

X1= [r1, θ1, φ1, vr1, vθ1, vφ1]T,

Λ1= [λr1, λθ1, λφ1, λvr1, λvθ1, λvφ1]T.[11]

The EOMs for the EECT phase are

˙r1=vr1,

˙

θ1=vθ1

r1cos φ1

,

˙

φ1=vφ1

r1

,

˙vr1=−µe

r2

1

−µm(r1−Dcos φ1cos θ1)

r3

ms

−µmcos φ1cos θ1

D2+at1sin u1cos v1

+2ωvθ1cos φ1+ω2r1cos φ1+v2

θ1

r1

+v2

θ1

r1

,

˙vθ1=−µmDsin θ1

r3

ms

+µmsin θ1

D2+at1cos u1cos v1

+2ωvφ1sin φ1−2ωvr1cos φ1−vr1vθ1

r1

+vθ1vφ1sin φ1

r1cos φ1

,

[12]

˙vφ1=−µmDsin φ1cos θ1

r3

ms

+µmsin φ1cos θ1

D2

+at1sin v1−2ωv1sin φ1−ω2r1sin φ1cos φ1

−vr1vφ1

r1

+v2

θ1sin φ1

r1cos φ1

,

where

rms =qr2

1−2Dr1cos φ1cos θ1+D2,

at1=Tmax

m

The set of equations [12] was normalized for numer-

ical propagation such that, D=ω= 1, µm=µ

IAC–19–C1.1.11 Page 4 of 11

70th International Astronautical Congress, Washington DC, USA, 21-25 October 2019. Copyright c

2019 by International

Astronautical Federation (IAF). All rights reserved.

while µe= (1 −µ). For a minimum-time OCP, the

Hamiltonian is deﬁned as

H1=ΛT

1X1.[13]

Applying the necessary optimality conditions for an

indirect approach, the co-state dynamics is derived

as,

˙

Λ1=−∂H1

∂X1T

.[14]

The PMP stationarity condition yields the direction

cosines for the thrust acceleration vector components

as

sin u1cos v1=−λvr1

kλv1k,

cos u1cos v1=−λvθ1

kλv1k,

sin v1=−λvφ1

kλv1k,

[15]

where kλv1k=qλ2

vr1+λ2

vθ1+λ2

vφ1is the norm of

the velocity co-state vector λv1= [λvr1λvθ1λvφ1]T.

The sign ambiguity is resolved by applying the

strengthened Legendre Clebsch condition. The ini-

tial conditions for the EECT phase are deﬁned by

GEO whereas, the ﬁnal condition is the Cartesian

state vector at the boundary of the lunar SOI,

{rSOI vSOI}obtained from the solution of Section

2.1 and transformed to an Earth-centered, rotating,

spherical coordinate system.

Note that the thrust steering angle, u, is measured

positive above the local horizontal plane to the pro-

jection of the thrust vector onto the local longitude,

radial direction vertical plane. vis measured positive

above the local vertical plane to the thrust vector and

v∈[−π

2

π

2].

2.3 Moon Capture (MC) Phase

The development of the EOMs for this phase is

similar to EECT phase except that equations are

expressed with respect to a Moon-centered rotating

frame instead. With reference to Figure 2 the set

{r2, θ2, φ2}is for the Moon-centered rotating frame.

The states are radial position r2, longitude angle

θ, latitude angle φ, radial velocity vr2, longitudinal

plane velocity vθ2, latitude plane velocity vφ2all ex-

pressed with respect to the Moon-centered coordinate

system. The state and co-state vectors are denoted

by X2and Λ2, respectively, as

X2= [r2, θ2, φ2, vr2, vθ2, vφ2]T,

Λ2= [λr2, λθ2, λφ2, λvr2, λvθ2, λvφ2]T.[16]

The EOMs for the MC phase are

˙r2=vr2,

˙

θ2=vθ2

r2cos φ2

,

˙

φ2=vφ2

r2

,

˙vr2=−µm

r2

2

−µe(r2+Dcos φ2cos θ2)

r3

ms

−µecos φ2cos θ2

D2+at2sin u2cos v2+ 2ωvθ2cos φ2

+ω2r2cos φ2+v2

θ2

r2

+v2

θ2

r2

,

˙vθ2=µeDsin θ2

r3

ms

−µesin θ2

D2+at2cos u2cos v2

+2ωvφ2sin φ2−2ωvr2cos φ2−vr2vθ2

r2

+vθ2vφ2sin φ2

r2cos φ2

,

˙vφ2=µeDsin φ2cos θ2

r3

ms

−µesin φ2cos θ2

D2

+at2sin v2−2ωv2sin φ2−ω2r2sin φ2cos φ2

−vr2vφ2

r2

+v2

θ2sin φ2

r2cos φ2

,

[17]

where

rms =qr2

1+ 2Dr2cos φ2cos θ2+D2,

at2=Tmax

m.

A similar formulation of the Hamiltonian gives,

H2=ΛT

2X2.[18]

The necessary optimality conditions give the co-state

dynamics similar to the previous development,

˙

Λ2=−∂H2

∂X2T

.[19]

Application of the stationarity condition yields,

sin u2cos v2=−λvr2

kλv2k,

cos u2cos v2=−λvθ2

kλv2k,

sin v2=−λvφ2

kλv2k,

[20]

where kλv2k=qλ2

vr2+λ2

vθ2+λ2

vφ2is the norm of

the velocity co-state vector λv= [λvr2λvθ2λvφ2]Twith

IAC–19–C1.1.11 Page 5 of 11

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2019 by International

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the second order strengthened Legendre Clebsch con-

dition resolving sign ambiguities in the thrust vector

direction cosines.

The initial conditions for the MC phase are deﬁned

by the Cartesian state vector at the boundary of the

lunar SOI, {rSOI vSOI}while the ﬁnal condition is

that of a circular LLO of radius Rlf.

2.4 Lunar Spiraling (LS) Phase

This sub-problem is formulated using the MEEs.

MEEs, as an element set, are singularity free

for all trajectories with inclinations less than

π. The element set is written as XMEE =

[P, ex, ey, hx, hy, L]Twhere Pis the semilatus

rectum, [ex, ey] is the eccentricity vector, [hx, hy] is

the inclination vector and Lis the true longitude.

In order to scale the problem, P is normalized by

Rm, mass is normalized by mLS0, while tis normal-

ized by qRm

g0, where Rmis the equatorial radius of

the Moon, mLS0is the mass at the start of the LS

phase and g0is 9.805 m/s2. With this normalization,

the 3D point mass EOMs can be represented as

˙

XMEE =b+Tmax

mLS0mg0

MδT,

˙m=−Tmax

mLS0c,

[21]

where δTis the unit vector in the thrust direction and

c=Isp

rRm

g3

0

. The vector band matrix Mare deﬁned

as

b= [0,0,0,0,0,pµP W

P2

]T,

M=sP

µ

02P

W0

sin Lcos L+ex+cos L

W−Zey

W

−cos Lsin L+ey+sin L

W

Zex

W

0 0 Ccos L

2W

0 0 Csin L

2W

0 0 Z

W

,

[22]

where the scalars W,Zand Care deﬁned as,

W= 1 + excos L+eysin L,

Z=hxsin L−hycos L,

C= 1 + h2

x+h2

y.

[23]

The performance index is deﬁned for the minimum

time problem as,

JLS =tfLS ,[24]

and the Hamiltonian is deﬁned as,

HLS =ΛLSTXMEE .[25]

The terminal time tfLS is free and the initial condition

correspond to the circular LLO condition achieved

at the end of the MC phase. The ﬁnal conditions

are the predeﬁned QFO conditions with a free ﬁnal

longitude, Lf. The optimal thrust direction should

be opposite to the direction of MTΛLS to minimize

the Hamiltonian according to the minimum principle

i.e.,

δT=−MTΛLS

kMTΛLSk.[26]

The seven unknowns for an OCP in this phase

are ΛLS(t0) and tfLS and must satisfy the ﬁve state

terminal conditions and the two transversality con-

ditions associated with free ﬁnal longitude and free

ﬁnal time,

λL(tfLS )=0,

1 + HLS(tfLS )=0.[27]

The solution to this OCP gives the Lunar Spiral

transfer trajectory from the intermediate circular

LLO to the desired QFO. Numerical results are pro-

vided in the next section.

3. Numerical Results

The following table deﬁnes the parameters of the

problem and boundary conditions considered for mis-

sion design.

m0Isp Tmax rGEO rLLO

(kg) (s) (N) (km) (km)

250 3000 0.5 N 42378 4100

Table 1: Mission Design Parameters.

The QFO around the Moon is deﬁned by the fol-

lowing classical orbital element set,

a(km) e i (rad) Ω (rad) ω(rad)

2509.9 0.2996 1.5707 6.2789 4.7106

.

Table 2: QFO element set10

3.1 Planar transfer (Cartesian)

A Minimum-time planar transfer from GEO to a

4100 km circular LLO was solved using the develop-

ment in Section 2.1. The convergence was achieved

IAC–19–C1.1.11 Page 6 of 11

70th International Astronautical Congress, Washington DC, USA, 21-25 October 2019. Copyright c

2019 by International

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by carrying out a natural parameter continuation on

the terminal radius value of the Lunar orbit. The

number of revolutions (Nrevs) appear naturally albeit

randomly owing to the Cartesian formulation.

-0.2 0 0.2 0.4 0.6 0.8 1

X (DU)

-0.4

-0.2

0

0.2

0.4

0.6

Y (DU)

Departure

Earth

Moon

LT Traj

target

0.95 1 1.05

X (DU)

-0.04

-0.02

0

0.02

0.04

Y (DU)

Fig. 3: GEO to LLO - Single Shooting, Cartesian.

Figure 3 shows the minimum-time transfer trajec-

tory. The total transfer time was 16.92 days and the

ﬁnal mass of the spacecraft in the LLO was 225.15 kg

with 24.85 kg of propellant being used up.

0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03

X (DU)

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

Y (DU)

Moon

10000 km

6400 km

4147.2 km

2361.03 km

Fig. 4: Continuation on LLO altitude.

Figure 4 shows the progression of orbits and in-

crease in Nrevs with decrease in the ﬁnal altitude of

circular LLOs. The state vectors at the crossing of

the minimum-time, low-thrust trajectory and the pla-

nar projection of the lunar SOI are

rSOI = [0.8255,0.05624] DU,

vSOI = [0.5809,0.2378] VU.

3.2 3D transfer (Spherical)

Minimum-time 3D transfer from GEO to a 4100

km polar circular LLO was solved using the develop-

ment in Section 2.2 and 2.3.

For the same cost functional, i.e., minimum trans-

fer time, the state/co-state vectors can be trans-

formed to polar coordinates and used as initial con-

ditions for numerical optimization of the trajectory

in the spherical coordinate system. The polar tra-

jectory is essentially a 2D projection of the spherical

trajectory. Note that only the Moon-centered “SOI

boundary to LLO” segment, i.e., the MC phase was

modelled and numerically optimized in 3D.

-0.2 0 0.2 0.4 0.6 0.8 1

X (DU)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Y (DU)

Departure

Patch Point

Earth

Moon

LT - EECT

LT - MC

-0.05

0

0.05

Z (DU)

0.2

-0.2 00

Y (DU)

X (DU)

0.2 0.4 -0.2

0.6 0.8 -0.4

1

Departure

Patch Point

Earth

Moon

LT - EECT

LT - MC

Fig. 5: GEO to LLO - 3D, Spherical (top) X-Y view;

(bottom) 3D view.

Figure 5 shows the optimal minimum-time 3D

transfer from GEO to a polar LLO (both X-Y pla-

nar and 3D views). A continuation on the latitude

angle starting from φ2= 0◦to 90◦was performed.

0 10 20 30 40 50 60 70 80 90

Orbit Inclination (Deg)

17

17.5

18

18.5

19

19.5

20

20.5

21

tfmin

(Days)

Fig. 6: Continuation on orbit inclination.

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2019 by International

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The total transfer time for the polar case was 20.59

days and the ﬁnal mass of the spacecraft in LLO

was 219.76 kg (i.e., 30.24 kg of propellant). Figure 6

shows the total optimal transfer time with continua-

tion on LLO orbit inclination.

0 10 20

time (DAYS)

0

0.5

1

r (DU)

0 10 20

time (DAYS)

-40

-20

0

20

(rad)

EECT phase

MC phase

0 10 20

time (DAYS)

-2

0

2

(rad)

0 10 20

time (DAYS)

-0.4

-0.2

0

0.2

0.4

0.6

Vr

0 10 20

time (DAYS)

-1

0

1

2

V

0 10 20

time (DAYS)

-1

0

1

V

Fig. 7: State evolution - Spherical coordinates.

Figures 7 and 8 shows the state and co-state time

histories for the optimal 3D polar trajectory from

GEO to LLO, respectively. The discontinuous plots

are due to the separate OCPs being solved in dif-

ferent frames. Achieving convergence in the highly

non-linear MC phase is very challenging, even af-

ter applying frame transformation and continuation

techniques need to be employed, as was done in this

work.

0 10 20

time (DAYS)

0

5

10

r

104

EECT phase

MC phase

0 10 20

time (DAYS)

-2

-1

0

1

0 10 20

time (DAYS)

-10

0

10

0 10 20

time (DAYS)

-60

-40

-20

0

Vr

0 10 20

time (DAYS)

-1000

-500

0

V

0 10 20

time (DAYS)

-1000

0

1000

V

Fig. 8: Co-state evolution - Spherical coordinates.

3.3 3D transfer (MEE) - LS phase

The minimum-time trajectory associated with the

LS phase was obtained following the developments in

Section 2.4. Since a low-thrust minimum-time orbit

transfer between a 4100 km circular LLO and the

elliptical QFO is expected to have many spirals (≥

20), convergence is hard to achieve.

In this work a novel arc-length continuation based

on the ﬁnal boundary conditions was implemented.

Initially, a solution is obtained for an easier problem

to a set of intermediate conditions, which is slightly

oﬀset from the initial conditions. The solution for the

transfer orbit is gradually morphed into a minimum-

time orbit connecting the LLO initial condition and

the targeted ﬁnal condition by performing a sweep

on the homotopy parameter. The homotopy method

can be described as follows

eη=ηef+ (1 −η)e0,

e0=eLLO +eLLO,[28]

where e0is a perturbed terminal boundary condition

obtained via a small multiplier for which a con-

verged minimum-time solution is easy to achieve and

known, eηis the terminal boundary condition at the

current homotopy parameter ηand efis the targeted

QFO boundary condition. The arc-length continua-

tion scheme is used on the homotopy parameter, η,

where its value is swept from 0 to 1.

The irregularities in the homotopy path were cir-

cumvented by re-solving the problem for a slightly

larger homotopy parameter along with a larger guess

for ﬁnal time than that used for the previously con-

verged solution. This is akin to the double homotopy

method described in Pan et al. 33 where the authors

performed a continuation on thrust. The methodol-

ogy was found to be extremely eﬀective for a continu-

ation on the ﬁnal conditions such that the spacecraft

transfers to the QFO orbit.

Fig. 9: Transfer via 0 ◦LLO (η= 1).

Figure 9 shows the Y-Z view of the LS phase start-

ing from an equatorial LLO and terminating in the

desired QFO (see Table. 2). Figure 10 shows the 3D

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70th International Astronautical Congress, Washington DC, USA, 21-25 October 2019. Copyright c

2019 by International

Astronautical Federation (IAF). All rights reserved.

trajectory. Figure 11 shows converged minimum-time

trajectories in the LS phase for intermediate values

of η. A large number of revolutions around the Moon

were observed in the LS phase.

Fig. 10: Transfer via 0 ◦LLO - 3D, (η= 1).

Fig. 11: Trajectories for intermediate η- 0 ◦LLO.

For i= 90◦LLO, an intermediate ellipse was ob-

served to perform a large meridian change towards

eventually satisfying the QFO meridian boundary

condition with respect to the Moon, see Figure 12.

Fig. 12: Transfer via 90 ◦LLO - 3D, (η= 1).

Another case, of transfer via i= 45◦LLO was

studied. It was observed that one of the intermediate

ellipses dive into the Moon and hence the piece-wise,

minimum-time transfer was found to be infeasible in

this case, see Figures 13 and 14.

Fig. 13: Transfer via 45 ◦LLO - until crash.

This is because the solution is obtained by for-

mulating and solving an un-constrained optimization

problem. A re-formulated problem with constraints

on closest approach distance might lead to a feasible

trajectory for this case but is yet to be investigated.

0 2 4 6 8 10 12

time (days)

0

5

10

15

20

Radius (*Rm)

iLLO = 0

iLLO = 45

iLLO = 90

Lunar Surface

Intermediate Ellipse

dives into the lunar

surface for iLLO = 45

Fig. 14: Spacecraft Altitude vs. time.

Table 3 summarizes the piece-wise, time-optimal

trajectories across three separate segments. Three

diﬀerent cases of intermediate LLOs were considered

for transfer with inclination i∈ {0◦,45◦,90◦}.

LLO ‘i’ Ptf(days) mf(kg) Nrevs Rc(km)

0◦27.15 210.12 38 56.3

45◦21.87* 217.88* 21* 0

90◦31.54 203.67 32 95.6

Table 3: Piece-wise optimal trajectories (GEO-

QFO); * denotes parameters until crash

The higher transfer time via a 90◦LLO compared

to a 0◦LLO can be attributed to the fact that the

ﬁnal meridian angle of the 90◦LLO was kept free.

Thus, the spacecraft entered the LLO with a phase

shift with respect to the Quasi-frozen conditions. In

order to rectify this, a long amplitude intermediate

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70th International Astronautical Congress, Washington DC, USA, 21-25 October 2019. Copyright c

2019 by International

Astronautical Federation (IAF). All rights reserved.

spiral was required for the minimum-time trajectory

(see Figure 12).

The distance of closest approach, Rcis 56.3 km for

the equatorial case and 95.6 km for the polar case,

making these feasible piece-wise time-optimal trajec-

tories.

4. Conclusion

Low-thrust, piece-wise, minimum-time orbit trans-

fers, from geosynchronous orbit to a desired lunar

quasi-frozen orbit were achieved using a systematic

methodology. The entire trajectory was split into

three phases, each patched with the ﬁnal bound-

ary states of the previous phase. A novel homotopy

method was implemented on the terminal conditions

for the longest and ﬁnal phase of the trajectory, from

a low lunar orbit (LLO) to the desired QFO. This

improved convergence for optimal trajectories with a

high number of revolutions. The main conclusions,

speciﬁc to the considered spacecraft parameters in

Table 1, are

•A feasible, piece-wise optimal minimum time

transfer from GEO to the desired QFO involv-

ing a planar EECT and MC phase and a 3D

LS phase via a 4100 km equatorial circular LLO

took 27.15 days and requires 39.88 kg of propel-

lant.

•The same transfer via polar 4100 km circular

LLO took 31.54 days and required 46.33 kg of

propellant.

•Transfer via i= 45◦LLO led to intermediate

ellipses diving into the Moon. Minimum-time

control strategy for transfers to the extremely

low-altitude QFOs were found to be infeasible

for this case.

Inclusion of path constraints in the formulation of the

LS phase OCP would make the methodology more

robust, and is a subject of future investigation. Im-

plementation of a multiple-shooting technique is also

an avenue of future work.

5. Acknowledgement

This work was completed at Texas A&M Univer-

sity and funded by the Jet Propulsion Laboratory,

California Institute of Technology, under contract

with the National Aeronautics and Space Adminis-

tration.

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