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IAC C.1.1.11 Mission Design for Close-Range Lunar Mapping by Quasi-Frozen Orbits

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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 QFO. 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.
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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
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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
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2019 by International
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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+1µ
r1
+µ
r2
+µ(1 µ)
2,[2]
where r2
1= (x+µ)2+y2and r2
2= (x+µ1)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
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GEO. The final circular LLO is determined by
Φf=
krfk2r2(tf)2
kvfk2(vxfr2y(tf))2(vyf+r2x(tf))2
r2x(tf)(vxfr2y(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,
λxfyfλyf(xf+µ1) + λvxfvyfλvyfvxf= 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(r1Dcos 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 φ12ωvr1cos φ1vr1vθ1
r1
+vθ1vφ1sin φ1
r1cos φ1
,
[12]
˙vφ1=µmDsin φ1cos θ1
r3
ms
+µmsin φ1cos θ1
D2
+at1sin v12ωv1sin φ1ω2r1sin φ1cos φ1
vr1vφ1
r1
+v2
θ1sin φ1
r1cos φ1
,
where
rms =qr2
12Dr1cos φ1cos θ1+D2,
at1=Tmax
m
The set of equations [12] was normalized for numer-
ical propagation such that, D=ω= 1, µm=µ
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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 φ22ωvr2cos φ2vr2vθ2
r2
+vθ2vφ2sin φ2
r2cos φ2
,
˙vφ2=µeDsin φ2cos θ2
r3
ms
µesin φ2cos θ2
D2
+at2sin v22ωv2sin φ2ω2r2sin 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
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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
WZey
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 Lhycos 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
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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
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= 0to 90was 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.
IAC–19–C1.1.11 Page 7 of 11
70th International Astronautical Congress, Washington DC, USA, 21-25 October 2019. Copyright c
2019 by International
Astronautical Federation (IAF). All rights reserved.
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
IAC–19–C1.1.11 Page 8 of 11
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= 90LLO, 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= 45LLO 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)
027.15 210.12 38 56.3
4521.87* 217.88* 21* 0
9031.54 203.67 32 95.6
Table 3: Piece-wise optimal trajectories (GEO-
QFO); * denotes parameters until crash
The higher transfer time via a 90LLO compared
to a 0LLO can be attributed to the fact that the
final meridian angle of the 90LLO 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
IAC–19–C1.1.11 Page 9 of 11
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= 45LLO 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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... The set of nominal trajectories are computed using a combination of indirect optimal control formulation [25,26] and homotopy on the final boundary condition [27]. The problem is treated as a minimum-time transfer and is formulated using the set of MEEs. ...
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