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AAS 21-547 Eclipse-Conscious Transfers to Lunar Gateway Using Ephemeris-Driven Terminal Coast Arcs, 2021 AAS/AIAA Astrodynamics Specialist Conference, Big Sky, Montana


Abstract and Figures

A novel indirect-based trajectory optimization framework is proposed that leverages ephemeris-driven, "invariant manifold analogues" as long-duration asymptotic terminal coast arcs while incorporating eclipses and perturbations during the optimization process in an ephemeris model; a feature lacking in state of the art software like MYSTIC and Copernicus. The end-to-end trajectories are generated by patching Earth-escape spirals to a judiciously chosen set of states on pre-computed manifolds. The results elucidate the efficacy of the proposed trajectory optimization framework using advanced indirect methods and by leveraging a Composite Smooth Control (CSC) construct. Multiple representative cargo re-supply trajectories are generated for the Lunar Orbital Platform-Gateway (LOP-G). The results quantify accurate ∆V costs required for achieving efficient eclipse-conscious transfers for several launch opportunities in 2025 and are anticipated to be used for analogous un-crewed lunar missions.
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AAS 21-547
Sandeep K. Singh*, Brian D. Anderson, Ehsan Taheriand John L. Junkins§
A novel indirect-based trajectory optimization framework is proposed that leverages ephemeris-
driven, “invariant manifold analogues” as long-duration asymptotic terminal coast arcs while
incorporating eclipses and perturbations during the optimization process in an ephemeris
model; a feature lacking in state of the art softwares like MYSTIC and Copernicus. The end-
to-end trajectories are generated by patching Earth-escape spirals to a judiciously chosen set
of states on pre-computed manifolds. The results elucidate the efficacy of the proposed tra-
jectory optimization framework using advanced indirect methods and by leveraging a Com-
posite Smooth Control (CSC) construct. Multiple representative cargo re-supply trajectories
are generated for the Lunar Orbital Platform-Gateway (LOP-G). The results quantify accu-
rate Vcosts required for achieving efficient eclipse-conscious transfers for several launch
opportunities in 2025 and are anticipated to be used for analogous uncrewed lunar missions.
Low-thrust electric propulsion systems are suitable for long-duration deep space missions. A variety of
engines have been developed to cater to low-thrust trajectories like solar electric propulsion (SEP) [1], and
beamed power [2]. Although the working concept may be derived from different underlying principles of
physics, low-thrust engines are typically characterized by high specific impulse (Isp) values, which directly
corresponds to high propulsion efficiency, and therefore less propellant is required for delivering the same
amount of V. Trajectory design using low-thrust propulsion systems is a complex process mainly due to a
departure from natural dynamics caused by the continuous forcing function incorporated in the mathematical
model. Additionally, planet-centric phases of trajectories can be characterized by many orbital revolutions
(revs) and numerical convergence becomes a challenge for numerical solution methods. Local extremals
often occur, which can complicate the trajectory optimization process [3].
Optimization methods can be broadly classified into indirect and direct methods [4, 5], where indirect
methods require numerical solution of analytically derived necessary and sufficient optimality conditions [6,
7], while direct methods involve an iterative procedure leading to sequentially improving some parameterized
approximation of the optimal solution. Indirect methods have undergone significant recent enhancements
to alleviate associated challenges, with concepts like homotopy [8, 9, 10, 11] and arc length continuation
methods employed to make the problem more amenable to numerical treatment. Often times, a judicious
choice of the element space used to define the dynamical model also improves convergence [11, 12].
Most state of the art trajectory design software tools provide robust high-fidelity solutions for optimal
transfers (e.g., Copernicus [13], with its built-in direct and indirect solvers and MYSTIC [14] that uses a
non-linear static/dynamic optimal control algorithm). However, Copernicus and MYSTIC do not incorporate
eclipse-induced coast arcs constraints within the optimization. Copernicus and MYSTIC do not currently
incorporate eclipses during the optimization process.
*PhD Candidate, Department of Aerospace Engineering, Texas A&M University, College Station, Texas 77843-3141, AAS Member.
Current employment is Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109.
Assistant Professor, Department of Aerospace Engineering, Auburn University, Auburn, AIAA Senior Member.
§Distinguished Professor, Department of Aerospace Engineering, Texas A&M University, College Station, Texas 77843-3141, Honorary
Fellow AIAA.
Another vibrant research area in the realm of low-thrust transfers leverages insights from invariant man-
ifolds of the many families of periodic orbits existing in simplified dynamical models like the Circular Re-
stricted Three-Body Problem (CR3BP). For instance, Anderson and Lo [15] studied the role of invariant
manifolds in the dynamics of low-thrust trajectories passing through unstable regions of the three-body prob-
lem. Dellnitz et al. [16] used the reachable sets concept coupled with invariant manifolds of libration orbits
to solve for a low-thrust Earth-Venus transfer. Vaquero and Howell [17] studied the leverage provided by
resonant-orbit manifolds for designing transfers between Earth-Moon libration-point orbits. More recently,
Singh et al. [18] demonstrated a methodology to leverage stable/unstable manifold pairs of a halo orbit
around the Earth-Moon L1for end-to end transfers in the cis-lunar space. They also extended their work
to study the behavior of the invariant manifolds of Southern L2Near Rectilinear Halo Orbits (NRHOs) in a
multi-body system for a more accurate representation of the manifolds, and using them as terminal coast arcs
for trajectory design [19]. Similar approaches to low-thrust trajectory design also appear in [20, 21].
Although trajectory design in cislunar space seems well studied, real mission design requires eclipse-
conscious trajectories especially for spacecraft equipped with SEP engine for analyzing impact of eclipses
on mission cadence due to battery depth of discharge. Typically, engines can only be ON in ‘full-Sun’ and
power is consumed during eclipses to operate the spacecraft. Moreover, the use of invariant manifolds as
long-terminal coast arcs (LTCAs) reduces the problem complexity significantly, where the resulting two-
point boundary-value problems (TPBVP) are easier to solve rather than an end-to-end transfer. The pre-
computation of ‘invariant manifold analogues’ for an ephemeris-corrected quasi-periodic libration-point orbit
provides more accurate patch-point states than the ones provided in [19].
In this paper, we first present a methodology for pre-computation of stable/unstable manifold analogues for
an ephemeris-corrected quasi-periodic orbit, and demonstrate this procedure using the ephemeris-corrected
9:2 NRHO of the Southern Earth-Moon L2family. Secondly, we present an in-depth analysis towards selec-
tion of patch-points, which determines the judicial initial state for the LTCA and the terminal state anchor
vector for the first phase of the transfer. Fuel-optimal transfers are formulated from two ‘super Synchronous
Geostationary Transfer Orbits’ (sGTOs) to a selected patch-point using an indirect optimization method. The
Composite Smooth Control (CSC) [22] method is exploited to cover multi-functional discontinuous switches
to both the thrust profile due to eclipse and fuel-optimality conditions. The resulting trajectories patched with
the LTCA, provide accurate, efficient and eclipse-conscious transfer trajectories with an ephemeris-driven
LTCA. Transfer opportunities have been listed for transferring a spacecraft from the considered sGTOs to the
quasi-periodic 9:2 NRHO of the southern L2family in the year 2025 with their respective fuel cost and eclipse
behavior. Finally, the impact of an early departure for one such transfer has been quantified elucidating the
differences between a typical fuel-optimal transfer and eclipse-conscious fuel-optimal transfers.
In this section, an approach is presented for computing high-fidelity perturbed analogues for CR3BP invari-
ant manifolds in an ephemeris-driven point-mass gravitational model (“Ephemeris Model”). These “invariant
manifold analogues” are not invariant in the strict sense, because the dynamical model is non-autonomous.
Using nearly-periodic segments of orbits in a high-fidelity model and approximating them as periodic in
order to compute their invariant manifold analogues has been employed with success in missions such as
Genesis [23]. The advantage of this approach is that the structure of the manifolds is driven by the stability
properties of the orbit in the Ephemeris Model, and the trajectories “on” the manifold satisfy the Ephemeris
Model equations of motion. This method assumes the user has a “near-periodic” high-fidelity orbit solution
available. An exact periodicity is unlikely due to the already discussed complexities of the Ephemeris Model.
Equations of Motion and Variational Equations
Without loss of generality, i= 1,2is chosen to represent the primary and secondary bodies from the
CR3BP. Naturally, the chosen bodies should approximate the behavior of bodies orbiting a common barycen-
ter. When defining the simplified rotating frame, we will choose a fixed angular rate and axis of rotation and
allow this axis to define the z-axis of the frame. Thus, = [0,0, ω]>and ˙ω= 0, where the angular rate (ω)
is chosen to be the assumed constant mean motion of the primaries with the axis of rotation along the angular
momentum vector at a chosen reference epoch (t). Considering, three basis vectors listed in Eq. (1), Q0in
Eq. (2) is defined as the rotation matrix from the inertial frame to the rotating frame at t. The triad of unit
vectors, with components in the Earth-Moon barycentric inertial frame are
|R2(t)R1(t)|,ˆe2=ˆe3׈e1,ˆe3=(R2(t)R1(t)) ×(˙
|(R2(t)R1(t)) ×(˙
3>,ˆei=unit vectors with Cartesian non-rotating components (2)
Using an additional simple time-varying rotation matrix Qz(t), defined in Eq. (3),
Qz(t) =
cos(ω(tt)) sin(ω(tt)) 0
sin(ω(tt)) cos(ω(tt)) 0
0 0 1
the complete rotation matrix Q(t)as Q(t) = Q0Qz(t)can be formed. The coordinates in this rotating
frame to inertial coordinates can be related using Eq. (4). The displacement vectors relative to the chosen
center care ρi=RiRcand ρ=RRc. The relative displacement and angular velocity vectors with
rotating components are
The inertial point mass gravity forcing function is well known and leads to the inertial equations of motion
(Eq. (5)),
R=F(t, R) = X
where µiis the gravitational parameter of body i. The gravitational forcing function can alternatively be
expressed as a function of the displacements relative to an accelerating center (which can serve as an idealized
inertial origin), as is shown in Eq. (6) as
φ(t, ρ) = X
Finally, the forcing function, when the time derivative of the displacement vectors are considered (relative
to the rotating frame) requires additional Centrifugal and Coriolis acceleration terms due to the simplified
definition of the frame and is shown in Eq. (7) as
r=f(t, r,˙
r) = X
| {z }
| {z }
which can be split into the gravitational forcing terms, fg(t, r,˙
r)and the kinematic terms due to the frame
rotation, fr(t, r,˙
r). The state vector and its time derivative are defined in Eq. (8) as
f(t, r,˙
In order to compute the variational equations, we require the Jacobian of Eq. (8), expressed with simplified
3x3 blocks in Eq. (9).
J(t, r,˙
Finally, the variational equations in the form of the well-known State Transition Matrix (STM) differential
equation, Eq. (10), is used for propagation, with initial value problem initialized with Φ(t0, t0)as a 6×6
identity matrix. ˙
Φ(t, t0) = J(t, r,˙
r)Φ(t, t0).(10)
Orbit Manifold Parameters
Let the nearly-periodic orbit have a period of Ti, where iis the rev number in consideration. The flow
around a “fixed point” can be characterized by studying the Monodromy matrix M(t0), which serves as a
local linearization of the flow and can be computed as
M(t0) = Φ(t0+Ti, t0),(11)
using Eq. (10). In the current context, the apsidal period was chosen to define the period for each quasi-
periodic revolution. Then, computing the eigenvalues of M(t0)would ideally yield the properties: λ1=
12,λ3=λ4= 1,λ5=λ
6, where |λ1|>|λ2|, and λ
6indicates the complex conjugate of λ6. Since
the orbit is “near-periodic”, the eigenvalue set only approximates these characteristics. The eigenvectors
associated with the real eigenvalues λUλ1and λSλ2represent the unstable and stable eigenvectors,
respectively. These are labeled as, ξU,0ξU(t0)and ξS,0ξS(t0). Initial states for the unstable or stable
“invariant manifold analogue” at t0, assuming the state of the orbit is x0at t0, can be generated using the
following relation
xU(t0) = x0+sξU,0,xS(t0) = x0+sξS,0,(12)
where s=±1is the “sense” of the manifold and is the desired small offset magnitude along the manifold.
Due to the option of the sense, the unstable manifold can evolve in one of two directions, as can the stable
manifold. To produce an approximation of a manifold surface, it is standard practice to compute initial
manifold states on several sampled times (various choices for t0and corresponding states x(t0)) on an orbit
rev. The above process can be repeated for every desired point on the orbit, or alternatively, the state transition
matrix of the orbit itself can be used to approximately propagate the stable and unstable eigenvectors from a
reference state on the orbit to another state on the orbit. This is applied as
ξU(t)Φ(t, t0)ξU,0
|Φ(t, t0)ξU,0|,ξS(t)Φ(t, t0)ξS,0
|Φ(t, t0)ξS,0|.(13)
Once several initial states have been generated on the chosen manifold these states are then all propagated
using the dynamics in Eq. (8), with unstable manifolds propagated forward in time, and stable manifolds
backward in time.
Discretization of the Near-Periodic Orbit
In order to generate the manifold analogues, several points on the near-periodic rev should be analyzed to
reveal the “optimal” initial boundary condition. This process was carried out by first considering a discretiza-
tion of states on the selected near-periodic rev followed by generation of the corresponding Monodromy
matrix. A spectral decomposition of M, reveals the stable/unstable eigenvector directions, thereby enabling
computation of the manifold analogues. Consider a near-periodic rev on an ephemeris-corrected, 9:2 reso-
nant NRHO of the southern L2family. A simple methodology for extraction of the discrete states include
discretization at uniformly spaced time intervals or uniformly spaced intervals in the eccentric anomaly, E.
Discretization in time leads to a sparse scattering of points near the periapse region, which often leads to a
restricted mapping of the manifold analogues. A more uniform spatial scattering achieved by discretization
in Eleads to a more comprehensive and robust mapping in our case.
Eclipse-conscious trajectories are defined as trajectories where the engine throttle is turned ‘OFF’ (Thrust
= 0) whenever the spacecraft is in the shadow of an occulting body. An event-trigger handling capability is
required during the numerical optimization, which implicitly constrains the admissible control when applying
Pontryagin’s Minimum Principle (PMP). Several shadow models have been studied extensively [24, 25, 26].
In this paper, a conical shadow [25] model has been adopted for analyzing the manifold analogues as well as
designing eclipse-conscious optimal trajectories.
Figure 1: Schematic for the Conical Shadow Model.
The conical shadow model assumes spherical shapes of the occulting body and the Sun as is depicted in
Figure 1. The two bodies are viewed as overlapping discs by the spacecraft. For the spacecraft to be in at
least a partial eclipse shadow, the occulting body must block some of the angular view of the sun from the
spacecraft. Defining the parameters Apparent Solar radius (ASr), Apparent Occulting Body radius (ABr) and
Apparent Distance (AD) as the following,
ASr = sin1RS
||rS/sc|| ,ABr = sin1RB
||rB/sc||,AD = cos1rT
B/sc rS/sc
||rB/sc|| ||rS/sc|| ,(14)
where rS/sc =rSrsc and rB/sc =rBrsc. An implicit time-varying function, fec (rS/sc(t),rB/sc(t))
can be defined, which determines the eclipse condition as: Eclipse occurs when the sum of the apparent
angular radii exceeds the apparent angular distance. The function fec is expressed as,
fec(rS/sc (t),rB/sc (t)) = ASr +ABr AD. (15)
This time-varying function is evaluated at every time-step during state propagation and the sign (+ / -)
triggers the onset or exit from the eclipse event (Thrust: OFF / ON). For simplicity, we do not distinguish
between partial or total eclipse; we adopt the rule that the thrust should be OFF in even a partial eclipse, and
ON only in “full sun”. Note that additional relatively simple conditions on ASr, ABr and AD can be used
to distinguish umbra, penumbra, and antumbra. We use these conditions for post processing, but they do not
apply to the dynamics or optimization in our model.
Using the procedure described in the preceeding sections, stable invariant manifold-analogues were gener-
ated for all “apo-apo” revs of the ephemeris-corrected 9:2 Southern L2NRHO. Several Earth-periapse states
on the generated trajectories were extracted for further analysis to gauge viability. Figure 2 shows stable
manifold analogues for 100 discrete points on the “apo-apo” rev starting ‘2025 JAN 02 20:02:28.677’.
These points were analyzed using phase portraits to pick the ‘best’ patch-point condition for trajectory
design. The primary consideration for selecting the patch-point, and thereby the manifold, was distance from
(a) Earth Mean Equator J2000 (EMEJ2000) frame. (b) Moon-centered rotating (MCR) frame.
Figure 2:Patch-point candidates on stable manifold analogues.
Earth (Rd). Other considerations were maximum eclipse duration (tec) and differences in the argument of
periapse (ω), eccentricity (e), and inclination (i) between the geocentric departure orbit and the patch
condition. While minimizing any combination of [Rd,ω, e, i], enable efficient transfers in terms of
fuel consumption, tec is also an important parameter that can determine viability of the LTCAs and render
otherwise valid choices impractical. Once an attractive patch-point is selected, it can either be held fixed or
can be subject to further refinement in an optimization process. Also of significance, the nature of temporal
and spatial behavior of manifold trajectories in a high-fidelity model implies that each subsequent state on the
manifold propagates as a part of the original ballistic trajectory, enabling tracking via guidance algorithms.
This provides robustness to missed thrust-arcs in the preceding phases of the converged trajectory. Figure
Jan 25 Feb 25 Mar 25 Apr 25 May 25
Jun 25
Jul 25 Aug 25 Sep 25 Oct25 Nov 25 Dec 25
1 2 3 4 5 6 7 8 9 10 11 12
Max. Eclipse Duration (tec) (mins.)
Figure 3:tec for stable manifold analogues in year 2025.
3 depicts the maximum duration of eclipse (tec) experienced by the spacecraft while coasting on the stable
manifold analogues. Five ‘best’ cases in ascending order of Rdwere considered for each rev in YR 2025.
Note that several of the candidate LTCAs are devoid of eclipses and an a priori knowledge enables a judicious
choice of the manifold analogue and associated patch-condition for various launch period scenarios. It is
apparent that the worst case scenario in terms of eclipses occur in June 2025 with a maximum tec = 250
mins. Although this includes both ‘UMBRAL’ (full shadow) and ‘PENUMBRAL’ (partial shadow) eclipse
components, we are interested in the total eclipse duration to provide this essential analysis and demonstration
of indirect trajectory optimization subject to eclipse constraints.
Another important consideration, as discussed above is ω=|ω0ωpp|, where ω0and ωpp are the
arguments of periapse of the initial geocentric orbit and the target patch-point condition in the EMEJ2000
frame. Transfers dealing with a minimal ωhave been found to be efficient and cheaper due to minimal effort
required to rotate the major axis of the osculating orbits, leading to the the final optimal osculating orbit before
insertion onto the manifold coast. Typical initial geocentric departure orbits have ω=ωnom
pp = 0° or 180°.
Therefore, in order to minimize ω,ωpp ⊂ {L1, L2}where {L1, L2} ∈ Ris a small neighborhood around
pp (±30°), depending upon the value of ω0. Figure 4 shows an ωpp vs. Rdphase portrait of all the candidate
patch-point conditions, segregated by the ‘MONTH’ of the year 2025 when the spacecraft gets inserted into
the NRHO. Note that the colorbar depicts the coast-time in days on the LTCAs.
Figure 4:ωpp vs. Rdfor candidate patch-points in the YR 2025.
In addition to the primary advantage of minimizing ω, analysis of the ωpp vs. Rdphase portrait also
introduces a flexibility in terms of mission launch window analysis and eclipse duration. In essence, it equips
mission designers with multiple patch-points to choose from, depending on the departure epoch and ωnom
pp .
An informed choice enables designers to circumvent many iterations of trajectory design in order to get a
favorable “Spacecraft - Sun - Occulting Body” geometry.
Establishing the target NRHOs and propagation of their associated manifolds are performed using synodic
Cartesian frames. It is known, however, that the set of Cartesian coordinates are not suitable for efficient con-
vergence of indirect optimal planet-centric, many-spiral trajectories during the Earth escape/capture phases
[12, 27, 28, 29]. Therefore, the set of modified equinoctial elements (MEEs) [30] are used to formulate time-
and fuel-optimal low-thrust trajectory optimization problems.
Let x= [p, f, g, h, k, l]>denote the state vector associate with MEEs. Their dynamics can be written as
x(t) = A(x, t) + B(x, t)a,(16)
where AR6×1denotes the unforced vector part of the dynamics and BR6×3denotes the control
influence matrix with their explicit forms given in [12]. The total acceleration vector, a, expressed in the
Local-Vertical Local-Horizontal frame attached to the spacecraft can be written as
a=ap+ρp[aSun +aMoon],(17)
where ap,aSun, and aMoon denote accelerations due to the propulsion system, and perturbing accelerations
due to the Sun and Moon, respectively. In Eq. (17), ρp[0,1] denotes a continuation parameter that is used
to gradually include the non-linear third-body perturbations due to the Sun, and Moon. Two-body dynamics
corresponds to When ρp= 0, whereas ρp= 1 corresponds to the high-fidelity model. Point-mass gravity
models are used for planetary perturbations and perturbation due to solar radiation pressure is ignored. The
acceleration due to the propulsion system can be written as
where δo[0,1] is the engine throttling input, ˆ
αdenotes the thrust unit direction vector, and δs[0,1]
reflects eclipse events. In Eq. (18), Tdenotes the maximum thrust value and mdenotes the spacecraft total
instantaneous mass. The time rate of change of mass of the spacecraft can be written as
where c=Ispg0is the constant effective exhaust velocity. In this work, it is assumed that specific impulse
and the maximum thrust value of the thruster of the spacecraft remain constant during the entire maneuver.
Minimum-Time Formulation
For minimum-time formulation the cost functional is written as
dt, (20)
subject to :
Equations (16),(17) &(19),x(tf)˜
x(t0) = x0, m(t0) = m0,
where ˜
xddenotes the set of desired (target) MEEs. Let λ= [λp, λf, λg, λh, λk, λl]>denote the costate vector
associated with the MEEs and let λmdenote the costate associated with mass. The Hamiltonian associated
with the minimum-time problem can be written as
HMT = 1 + λ>[A(x, t) + B(x, t)a]λm
Extremal (denoted by ‘*’ in superscript) control inputs, ˆ
α, and throttle factor, δ, become
||B>λ||, δ
o= 1.(21)
For minimum-time problems, the optimal control strategy is to operate the thruster at its maximum capac-
ity; however, the eclipses have to be considered. Assuming ρsas a smoothing parameter, the eclipse factor
can be written as δs=1
2[1 tanh(fecs)], where fec (see Eq. (15)) is treated as a “distance measure”
associated with eclipse events. Since final time is free, the final value of the Hamiltonian has to be zero,
HMT(tf) = 0. The Euler-Lagrange equation is used to derive the costate dynamics using the Hamiltonian
associated with the minimum-time problem as ˙
λ>=(∂HMT /∂x)and ˙
λm=∂HMT /∂m.
Minimum-Fuel Formulation
For minimum-fuel formulation the cost functional is written as
cδsδodt, (22)
subject to :
Equations (16),(17) &(19),x(tf)˜
x(t0) = x0, m(t0) = m0.
Formulation of the optimal control is straightforward. The Hamiltonian associated with the minimum-fuel
problem can be written as
cδo+λ>[A(x, t) + B(x, t)a]λm
PMP has to be used characterize extremal (denoted by ‘*’ in superscript) control inputs, ˆ
α, and throttle
factor, δ, as
||B>λ||, δ
o= arg min
δo[0,1] HMF(x, δo,λ).(23)
The optimal δodepends on the switching function, SF , defined as
o=(1,for SF > 0,
0,for SF < 0,SF =c||B>λ||
The Euler-Lagrange equation is used to derive the costate dynamics ˙
λ>=(∂HMF /∂x)and ˙
∂HMF /∂m. An additional admissible control constraint, resulting in eclipse-conscious solutions is δs= 0
during the course of the eclipse event. This can be implemented either by including a logical check on the
sign of fec from Eq. (15), or via PMP by restricting the admissible set δo. Derivation of the costate dynamics
(for both time- and fuel-optimal problems and in the presence of perturbations) is achieved using a symbolic
code as outlined in [22].
Optimization Algorithm
Equation (18) is the total acceleration delivered by the propulsion system and δoand δsdenote the com-
bined engine throttling input due to application of the PMP and encountering of eclipse events, respectively.
Since there are two sources that can lead to discontinuity in the thrust profile, the CSC methodology [22, 31]
is used to alleviate non-smoothness issues. Assuming ρoand ρsas the respective smoothing parameters, the
optimal δcan be written as δ=δsδowith δs=1
2[1 tanh(fecs)]. Here, fec (see Eq. (15)) is treated as a
“distance measure” associated with eclipse events and SF is the standard thrust switching function. The pro-
posed high-fidelity, Hybrid Eclipse-Conscious Trajectory Optimization Routine (HECTOR) is summarized
using the two Algorithms given below.
Algorithm 1: Hybrid Eclipse-Conscious Trajectory Optimization Routine (HECTOR) - TIME OPTI-
Result: Minimum time (t
f), Converged co-state vector (λ2b
Departure States on the sGTO : (xo),Arrival States on the LTCA : ( ˜
while MT do
Initial Guess: tf(Time of Flight) & λ(t0)
Propagate: Eqs. (16),(19) & Adjoint equations with ρp= 0, δ
o= 1, δs,ˆ
MT =||MT|| =||x(tf)˜
xd, HMT (tf)||;
In Algorithm 1 of the HECTOR, the minimum-time problem is solved. The unknown values are the
initial costates and total time of flight. The value of is set to 1.0×1012 . The resulting TPBVP is
solved using a standard single-shooting method, however, the eclipses are incorporated into the formulation
through the δsfactor. Any coast arc during the minimum-time maneuver is due to (penumbral) eclipse events
and for the rest of the maneuver the thruster is always ON. The next step is to find fuel-optimal solutions
as outlined in Algorithm 2 below. However, the time of flight has to be greater than the time of flight
of the minimum-time solution. Thus, a scalar parameter γis considered. The value of γdepends on the
problem. Here, it is considered to lie in a range γ[1.2,1.4]. In order to simplify the problem, the initial
problem corresponds to restricted two-body Earth-spacecraft dynamics (denoted by superscript ‘2b’). As a
consequence, the associated Hamiltonian, H2b
MF takes a simpler form (since ρp= 0).
In addition, the smoothing parameter ρsis set to 1 to handle the eclipse transitions. The value of ρsis
lowered to below a certain ρ
svalue to get sharp transitions at the entry and exit of eclipse-induced coast arcs.
Then, a step is initiated to decrease the value of ρobelow a certain threshold, ρ
oin order to obtain bang-off-
bang thrust profiles. The value of ρ
oin our simulations is set to 1.0×106. The final step is to introduce
perturbations through ρp. When, ρpis equal to one, the solution associated with the high-fidelity model is
obtained. While the value of ρpis increased, it is ensured that the final solution corresponds to ρp= 1. We
emphasize that the Hamiltonian is updated according to the considered dynamics to take into account the
contribution of the perturbing accelerations with the costate differential equations updated accordingly [22].
Algorithm 2: Hybrid Eclipse-Conscious Trajectory Optimization Routine (HECTOR) - FUEL OP-
Result: Eclipse-Conscious Fuel-Optimal Transfer Trajectory in the HFM
Fuel-optimal: tMF
f=γ t
fγ > 1;Initial Guess:λ2b
0(ρs= 1)
while ρsρ
while 2b
MF do
Propagate: Eqs. (16), (19) & Adjoint equations with δ,ˆ
MF =||2b
MF|| =||x(tMF
xd, λm(tMF
0=λ2b(Converged Value), ρs= 0.9ρs;
Initial Guess :λ2bS
while ρoρ
while 2bS
MF do
Propagate: Eqs. (16), (19) & Adjoint equations with δ,ˆ
MF =||2bS
MF || =||x(tMF
xd, λm(tMF
0=λ2bS(Converged Value), ρo= 0.9ρo;
Initial Guess :λhfS
while ρp1do
while hfS
MF do
Propagate: Eqs. (16), (19) & Adjoint equations with
s, δ
α) + ρp(aSun +aMoon);
MF =||hfS
MF|| =||x(tMF
xd, λm(tMF
0=λhfS(Converged Value), ρp= 1.1ρp;
A final fuel-optimal TPBVP is solved with ρp= 1.
A representative problem was solved for transferring a spacecraft from a ‘Super Synchronous GTO’
(sGTO) to the 9:2 NRHO of the Southern L2family, using the stable invariant manifold analogues as LT-
CAs. This problem was aimed at providing accurate and efficient transfer trajectories for resupply cargo
missions to the Lunar Gateway, proposed to be stationed in the said NRHO by 2024 [32]. With consideration
to the timeline, the ephemeris-corrected NRHO revs in the year 2025 were considered to generate manifold
analogues, having the full transfer occur in the year 2025.
Target Orbits and Spacecraft Parameters
Geocentric sGTOs are of great commercial value. The near-circular regime of the sGTOs has perigee above
the synchronous altitude, a region termed as the GEO graveyard belt [33]. These orbits find use as storage
and disposal location for derelict geosynchronous satellite debris. For the current mission, we assume that the
resupply spacecraft has been initially placed in a geocentric elliptical orbit of the super synchronous domain.
As the name suggests, a spacecraft in this orbit would orbit the Earth at a faster angular rate than the Earth’s
rotation rate near perigee, and slower at apogee. The orbits in this domain typically have a somewhat larger
apogee than a GTO, as shown in Figure 5. Table 1 lists the orbital elements for the two orbits of the sGTO
category selected as the initial orbits for the representative problem.
Figure 5: Schematic depicting the domain of Geocentric orbits.
The two starting orbits (sGTO1and sGTO2) differ only in their argument of periapse as is given in the
Table 1. The choice of sGTO with ω=ωnom
pp = 0° or 180° was made depending on the departure epoch to
enable a favorable eclipse condition and therefore an efficient transfer. Perigee is the ideal position to thrust in
order to raise apogee and vice versa, while on the other, eclipses are frequently longer around apogee than the
perigee, but can be more likely to occur at perigee than apogee depending on the 3D geometry and time of the
year. These considerations are pertinent to designing transfers analogous to the representative problem and
give rise to two competing effects, namely, efficacy of thrusting and eclipse duration. Additionally, fuel-cost
for in-plane rotation of the transfer spirals is another criterion for selection of the departure orbit as discussed
in the previous section.
Table 1: Classical Orbital Elements of the sGTO.
Orbit a(km) e i (°) (°) ω(°) ν(°) Period (hrs.) Center
sGTO144364 0.65 27 11.3044 180 0 25.832 Earth
sGTO244364 0.65 27 11.3044 0 0 25.832 Earth
The terminal orbit for the mission is the ephemeris-corrected 9:2 resonant NRHO of the Southern L2
family. Cartesian states for the Deep Space Gateway (DSG) platform in this orbit are available in the SPICE
kernels (BODY ID ‘-60000’) with respect to the Earth’s center (BODY ID ‘399’) [34]. The available states
span 15 years from ‘JAN 2, 2020’ to ‘FEB 11, 2035’, with the trajectory being continuous in position but has
repeated, small corrective velocity adjustments (1.86 mm/s). The dynamical model used for differential
correction was an n-body gravity model, with eclipse avoidance properties achieved by a judicious choice of
initial condition. Since the orbit is quasi-periodic, every rev has a slightly different orbital period with the
average being 6.562 days.
Additionally, the resupply spacecraft was assumed to have an initial mass (m0) of 1000 kg, with the engine
generating a maximum thrust (Tmax) of 0.5 N (approximately equivalent to 2 NASA NEXT engines [35])
resulting in a maximum thrust acceleration of 5.0×104m/s with a constant specific impulse of Isp = 1500 s.
Efficient transfers from sGTO1/sGTO2to the NRHO via pre-computed stable invariant manifold analogues,
leveraging them as LTCAs were designed for the assumed spacecraft properties. The transfer from the sGTO
to the patch point was solved using the indirect formulation of the OCP, described in the previous section, in
an eclipse-conscious, fuel-optimal sense. While sGTO1was the choice for departure orbit for transfers with
insertion dates in January, February, March, October, November and December, for insertion dates in May,
June, July, August and September, sGTO2was the departure orbit.
Transfer : Eclipses on Fuel-optimal Coast Arcs
Eclipse-conscious transfers are mindful of all shadow encounters of the spacecraft. The formulation using
CSC enables the algorithm to recognise onset and termination of eclipses on all intermediate spirals and
enforces an ‘OFF’ condition for the engine. These eclipse regions on the transfer spirals can occur in between
two consecutive thrust arcs ‘THRUST - ECLIPSE - THRUST’, between a thrust and a coast arc ‘THRUST
- ECLIPSE - COAST’ or between consecutive coast arcs ‘ COAST - ECLIPSE - COAST’. In this section, a
transfer opportunity is presented, where eclipses occurr in between consecutive coast arcs.
The spacecraft on this transfer trajectory gets inserted in the NRHO on ‘25 JAN 2025 18:39:50.103’. The
total transfer time was 106.59 days with the spacecraft coasting on the LTCA for 50.59 days. The total V
for the transfer was 1641.867 m/s. The transfer trajectory is shown in Figure 6, plotted in the EMEJ2000
frame. Other important transfer events are : departure on ‘11 OCT 2024 04:17:37.145’ and patching with the
LTCA on ‘06 DEC 2024 04:17:37.145’.
Figure 6: Full Transfer : sGTO1- 9:2 NRHO (EMEJ2000).
Note that highlighted ‘green’ regions of the spirals depict the part of the trajectory when the spacecraft
is in Earth’s umbral shadow and the ‘cyan’ regions depict the penumbral shadow regions. Here, all the
shadow regions coincide with the fuel-optimality-driven coast arcs (due to PMP). For the given departure and
insertion epochs, the time-varying Sun-Earth-Spacecraft geometry results in such a favourable situation from
a fuel-optimality point of view. Since, the eclipse-induced coast arcs which would have been forced coasts in
the midst of neighboring thrust arcs, are on fuel-optimal coast arcs, the optimal thrusting sequence remains
intact. This efficacy would be elucidated upon comparing the Vcost with the former.
The bang-off-bang throttle sequence for the transfer trajectory is depicted in Figure 7. The ‘black’ colored
discontinuous function represents the eclipse condition (0: No Eclipse, 1: Umbral, 2: Penumbral), whereas
the ‘red’ plot depicts the corresponding thrust sequence. The ‘blue’ curve represents the switching function
used for tangent hyperbolic smoothing. The blown-up image in Figure 7 clearly shows the eclipse arcs being
0 10 20 30 40 50 60 70 80 90 100 110
TOF (Days)
SF,Thrust, Eclipse condition
Thrust (N)
Eclipse Condition
Figure 7: Thrusting sequence, eclipse condition, and switching function vs. time.
out of phase with the thrust arcs, and lying on the intermediate fuel-optimal-driven coast arcs. Thus, there are
no eclipse-induced coast arcs in this transfer trajectory. An eclipse on the selected LTCA is also evident from
the figure, which occurs 65.46 days after departure. Finally, as mentioned before, the frequency of eclipses is
Figure 8: Thrusting Sequence and Eclipse Condition.
also an important parameter along with their duration. Figure 8 depicts this data displayed in a histogram plot
with separate bins depending on eclipse duration. The spacecraft encounters a total of 19 eclipses enroute to
the NRHO from the sGTO1for the mentioned departure and arrival epochs. It encounters 7 eclipses in the
130 - 140 mins range, 6 in the 120 - 130 mins range, 5 in the 110 - 120 mins range and 1 in the 80 - 90 mins
range. The duration of the longest eclipse encountered was 136 mins. We mention that even when eclipses
do not prevent optimal thrust-on arcs, sufficiently long eclipses for solar electric propulsion systems lead to
deep battery discharge (mission dependent) and may require a sunshine coast to re-charge the battery upon
exit from the shadow. Note the spacecraft computer, sensors, communications system and attitude control
system must continue operations in shadow. Long eclipses are generally to be avoided. While “long” eclipse
duration is mission dependent, more than 150 minutes hours in shadow is frequently considered unacceptable,
and greater than 120 minutes is frequently undesirable. These issues must be studied in a mission-specific
context to finalize the trajectory design during the preliminary phases of mission/vehicle design.
Figure 9: Full Transfer: sGTO1- 9:2 NRHO (EMEJ2000).
Transfer : Eclipses on Fuel-optimal Thrust Arcs
A representative transfer is presented in this section, where the eclipse arcs occur in between consecutive
thrust arcs. Theoretically, eclipse-driven coast arcs result in loss of thrusting efficiency, more so, if such coast
arcs appear near the periapse region of the intermediate transfer spirals, where the spacecraft velocity is high.
The resulting throttling sequence represents a significant departure from the ideal fuel-optimal behavior,
where the optimality conditions manifest into a regular distribution of the thrust arcs around the periapse
region for maximum efficiency. This concept is elucidated in the numerical solution presented.
The spacecraft on this transfer trajectory departs from sGTO1and is inserted in the NRHO on ‘19 MAR
2025 01:45:00.464’. The total transfer time was 135.39 days with the spacecraft coasting on the LTCA for
79.39 days. The total V for the transfer was 1991.916 m/s. The transfer trajectory is shown in Figure 9, plot-
ted in the EMEJ2000 frame. Other important transfer events are : Departure on ‘03 NOV 2024 16:23:36.467’
and Patching with the LTCA on ‘29 DEC 2024 16:23:36.467’.
The bang-off-bang throttle sequence is depicted in Figure 10. Since the eclipse-driven coast arcs happen
during thrust arcs, it leads to forced coasts for the eclipse duration. Comparing it with Figure 7, it is apparent
that there are no eclipses on the selected LTCA. The blown up image in Figure 10 clearly depicts the switch in
the throttle sequence due to the eclipse-driven coast arcs. The frequency of ‘Earth - occulted’ eclipses, being
an important consideration for accurate mission design, has been portrayed using a histogram plot in Figure
11 with separate bins depending on the eclipse duration. The spacecraft encounters 15 eclipses enroute to the
NRHO from the sGTO1for the mentioned departure and arrival epochs. It encounters 1 eclipse each in the
80 - 90 mins range and 100 - 110 mins range, 2 eclipses in the 110 - 120 mins range and 11 eclipses in the
120 - 130 mins range. The duration of the longest eclipse encountered was 126 mins.
Summary : Transfer Opportunities in 2025
In this section, opportunities for feasible and efficient transfers of a spacecraft with the assumed parameters
from a sGTO to the 9:2 NRHO have been listed. The transfers were solved for one insertion opportunity
every month of the year 2025. Figure 12 depicts all the spatial trajectories with respect to the Earth Mean
Equator (EMEJ2000) frame. Note the counter-clockwise movement of the eclipse arcs as the insertion date
is swept throughout the year. All candidate trajectories shown in Figure 13, were solved for using the same
methodology with the LTCAs and associated patch-points pre-computed and identified. The ‘best’ LTCA
for each transfer was identified by picking the one with a patch-point (periapse point) having an argument of
periapse as close to the sGTO ωas possible, among the top five cases having the least radius with respect to
0 20 40 60 80 100 120
TOF (Days)
SF, Thrust (N), Eclipse Condition
Thrust (N)
Eclipse Condition
Figure 10: Thrusting sequence, eclipse condition, and switching function.
Figure 11: Thrusting sequence and eclipse condition.
the Earth. The nature of eclipses encountered by the spacecraft enroute has been plotted as binned histograms
in Figure 13.
(a) Trajectories for January - July 2025.
(b) Trajectories for August - December 2025.
Figure 12:Eclipse-conscious trajectories: YR 2025.
Table 2 summarizes important transfer parameters for all the mission scenarios. The duration of the longest
eclipse as well as the number of eclipses encountered depend on the ‘Sun-Earth-Spacecraft’ geometry for the
duration of the transfer. These results present mission designers with efficient eclipse-conscious trajectories
for future resupply missions to the Lunar Gateway, which is beyond the scope of state-of the art trajectory
design tools. The fuel cost presented is comparable to analogous results presented in [19].
Figure 13: Eclipse binning for Mission Scenarios in YR 2025.
Table 2: Summary of departure and arrival times, V, maximum eclipse duration and number of eclipses
for different mission scenarios in YR 2025.
Departure Epoch Arrival Epoch V (m/s) Max. Eclipse (mins) #Eclipses
11 OCT 2024 04:17:37.15 25 JAN 2025 18:39:50.11 1641.867 135 19
05 OCT 2024 04:00:27.53 21 FEB 2025 00:19:30.41 1729.059 120 17
03 NOV 2024 16:23:36.47 19 MAR 2025 01:45:00.47 1991.916 126 15
16 JAN 2025 14:12:43.83 04 MAY 2025 00:41:05.43 1956.159 64 8
01 FEB 2025 23:50:15.69 19 JUN 2025 02:58:55.25 2189.803 176 27
02 APR 2025 01:36:17.10 21 JUL 2025 21:10:58.86 1979.373 164 19
01 MAY 2025 06:28:26.86 17 AUG 2025 01:44:34.07 1817.332 124 17
26 MAY 2025 23:20:11.50 12 SEP 2025 01:44:34.07 2054.525 86 17
15 JUN 2025 15:44:17.11 28 OCT 2025 17:18.58.24 2103.743 124 29
11 JUL 2025 04:23:01:44 23 NOV 2025 10:55:04.80 1900.061 130 27
19 AUG 2025 13:18:07.01 19 DEC 2025 14:58:07.03 1772.081 242 28
Early Departure
Purely fuel-optimal trajectories typically result in a lower V cost for a larger time of flight for the same
rendezvous type transfer. The limiting case of a barely reachable terminal state is, however, simultaneously
a minimum-time and minimal-fuel trajectory, if the time of flight established from a minimum-time transfer
is the specified final time. As an aside, this duality between minimum-fuel and minimum-time extremals is
exploited in the recent paper by Taheri and Junkins [36] to construct optimal switching surfaces considering
an infinite family of maximum thrust specifications. In this work, we are dealing with eclipse-conscious
trajectories, a part of which is solved using the fuel-optimal formulation. Due to the optimizer being aware
of shadow regions the spacecraft encounters, it is important to perform a “launch-period” study to analyze
the trade off between fuel-cost and time of flight for such eclipse-conscious trajectories. Note that in the
results presented below, the family of trajectories has the same number of revolutions with a larger time of
flight due to sliding the departure date backwards in time. For this analysis, the ‘best’ transfer opportunity
among the candidate trajectories presented in the section above in terms of fuel cost, frequency of eclipse
and maximum eclipse duration was picked as the nominal trajectory. The spacecraft departs sGTO2on ‘16
JAN 2025 14:12:43.83’ and inserts into the NRHO on ‘04 MAY 2025 00:41:05.433’, incurring a Vcost of
(a) Nominal trajectory. (b) Trajectory for departure on et9P.
Figure 14:Effect of early departure on transfer trajectories.
1956.159 m/s. Enroute to the NRHO, the spacecraft encounters 8 distinct eclipses with the duration of the
longest encountered eclipse being 64 mins. A family of transfers were solved by sliding the departure date
backwards in time with a step size of the orbital period of sGTO2(TsGTO2). Left plot in Figure 14 depicts
the nominal transfer trajectory and the right plot depicts the trajectory for the analogous transfer but an early
departure (et9P =etdep 9TsGTO2). Here, etdep is the ephemeris time associated with the nominal departure
epoch (‘16 JAN 2025 14:12:43.83’). Notice the longer final spiral for the early departure trajectory on the
right plot in Figure 14. The trajectory is nearly identical in terms of the thrust and coast arcs up to the last but
one spiral. This is because of the eclipse-induced coast arcs occurring in the same region of the trajectory,
thereby causing a near-identical loss of thrusting efficiency. Since the number of revs are fixed, the optimizer
forces the spacecraft to accommodate the residual time of flight on the final spiral. Table 3 lists the impact of
early departure on fuel cost and eclipses.
Table 3: Early Departure Results: sGTO - 9:2 NRHO.
Departure etiP Departure Epoch V (m/s) Max. Eclipse (mins) #Eclipses
et1P JAN 15 12:22:49.39 2001.921 64 8
et2P JAN 14 10:32:54.95 2037.897 62 9
et3P JAN 13 08:43:00.51 2063.117 64 9
et4P JAN 12 06:53:06.06 2084.662 64 9
et5P JAN 11 05:03:11.62 2114.921 64 9
et6P JAN 10 03:13:17.18 2167.921 64 10
et7P JAN 09 01:23:22.74 2219.465 62 10
et8P JAN 07 23:33:28.30 2255.739 64 10
et9P JAN 06 21:43:33.85 2278.431 62 10
It was observed that, eclipse-conscious, fuel-optimal transfers for this case, countered the intuition of the
inverse trade-off between time and fuel cost for purely fuel-optimal class of transfers. It is evident from Table
3 that, an early departure or a longer time of flight, for eclipse-conscious transfers with the same number
of revs, lead to a larger fuel cost. For an approximately 9-day sliding of the departure date from nominal,
the Vcost increased by 322.272 m/s. The nature of eclipses encountered changed marginally, with the
maximum eclipse duration reducing to 62 mins, whereas the number of eclipses increased to 10.
A methodology to generate efficient, eclipse-conscious transfers leveraging ephemeris-driven asymptotic
long terminal coast arcs has been presented. Using favorable periapse states as patch-points on a set of pre-
computed manifolds, an algorithm is devised to solve end-to end transfers using indirect formalism of optimal
control. The resulting solutions present accurate transfer trajectories, with a knowledge of eclipse transitions
of the spacecraft, solved in a high-fidelity model.
The algorithm was applied to generate transfers from a Super Synchronous GTO to the 9:2 Near Rectilinear
Halo Orbit of the southern L2family, with a focus on providing accurate trajectories for “cargo re-supply”
type missions to the Lunar Gateway. Specifically, efficient transfer trajectories have been presented for 11
distinct launch dates in the year 2025. Interestingly, the interaction of “fuel-optimal” thrust-coast-thrust
sequence with “shadow-induced” coast arcs were found to be illuminating and captured using two separate
examples. It was observed that in the cases where eclipses interfere with the optimal thrust-coast-thrust
sequence, the thrust and coast arcs are forced to redistribute, due to the loss of thrusting efficiency vis-a-vis
the “shadow-induced” coast arcs. The fuel cost was found to depend significantly on the position of shadow-
regions on the intermediate spirals, which typical fuel-optimal trajectories are oblivious to. The frequency
of such regions along with the duration of the maximum eclipse, important considerations towards an actual
mission design for charging of on-board batteries, have been presented. It is anticipated that these results will
find advantageous applications, especially for future un-crewed missions.
We are pleased to acknowledge the Jet Propulsion Laboratory, Air Force Research Laboratory, Dzyne,
Inc., and Texas A & M University for sponsorship of various aspects of this research. This work was com-
pleted at Texas A&M University. A part of this research was carried out at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the National Aeronautics and Space Administration
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Full-text available
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Application of indirect optimization methods to solve constrained optimal control problems is not a straightforward task. This study presents a new construct --- the Uniform Trigonometrization Method (UTM) --- that enables handling problems with control bounds and state path constraints within the indirect formalism. First, three different classes of problems are discussed with: 1) bang-bang and singular control arcs, where the control appears in a linear form, 2) control constraints in which the control appears in a non-linear form, and 3) state path constraints. Accurate solutions are achieved in all classes of problems with a simpler formulation compared to the conventional indirect approach. To demonstrate the potential of the UTM, a complex aerospace problem with bounded control (angle of attack) and state path (maximum heat rate) constraints is also solved. The results demonstrate the utility of the UTM for making challenging constrained optimal control problems amenable to numerical treatment with minimal modifications to the structure of the problem.
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Finding a solution to a low-thrust minimum-fuel transfer trajectory optimization problem in the circular restricted three-body problem (CRTBP) is already known to be extremely difficult due to the strong nonlinearity and the discontinuous bang-bang control structure. What is less commonly known is that such a problem can have many local solutions that satisfy all the first-order necessary conditions of optimality. In this paper, the existence of multiple solutions in the CRTBP is fully demonstrated, and a practical technique that involves multiple homotopy procedures is proposed, which can robustly search as many as desired local solutions to determine the best solution with the optimal performance index. Due to the common failure of the general homotopy method in tracking a homotopy path, the parameter bounding fixed-point (PBFB) homotopy is coupled to construct the double-homotopy, which can overcome the difficulties by tracking the discontinuous homotopy path. Furthermore, the PBFB homotopy is capable to achieve multiple solutions that lie on separate homotopy path branches. The minimum-time problem is first solved by the double-homotopy on the thrust magnitude to infer a reasonable transfer time. Then the auxiliary problem is solved by the double-homotopy and the PBFP homotopy, and multiple local solutions with smooth control profiles are obtained. By tracking homotopy paths starting from these solution points, multiple solutions to the minimum-fuel problem are obtained, and the best solution is eventually identified. Numerical demonstration of the minimum-fuel transfers from a GEO to a halo orbit is presented to substantiate the effectiveness of the technique.
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A central problem in orbit transfer optimization is to determine the number, time, direction, and magnitude of velocity impulses that minimize the total impulse. This problem was posed in 1967 by T. N. Edelbaum, and while notable advances have been made, a rigorous means to answer Edelbaum’s question for multiple-revolution maneuvers has remained elusive for over five decades. We revisit Edelbaum’s question by taking a bottom-up approach to generate a minimum-fuel switching surface. Sweeping through time profiles of the minimum-fuel switching function for increasing admissible thrust magnitude, and in the high-thrust limit, we find that the continuous thrust switching surface reveals the N-impulse solution. It is also shown that a fundamental minimum-thrust solution plays a pivotal role in our process to determine the optimal minimum-fuel maneuver for all thrust levels. Remarkably, we find that the answer to Edelbaum’s question is not generally unique, but is frequently a set of equal-Δv extremals. We further find, when Edelbaum’s question is refined to seek the number of finite-duration thrust arcs for a specific rocket engine, that a unique extremal is usually found. Numerical results demonstrate the ideas and their utility for several interplanetary and Earth-bound optimal transfers that consist of up to eleven impulses or, for finite thrust, short thrust arcs. Another significant contribution of the paper can be viewed as a unification in astrodynamics where the connection between impulsive and continuous-thrust trajectories are demonstrated through the notion of optimal switching surfaces.
Conference Paper
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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.
A key challenge in low-thrust trajectory design is generating preliminary solutions that simultaneously specify the spacecraft position and velocity vectors, as well as the thrust history. To mitigate this difficulty, dynamical structures within a combined low-thrust circular restricted 3-body problem (CR3BP) are investigated as candidate solutions to seed initial low-thrust trajectory designs. The addition of a low-thrust force to the CR3BP modifies the locations and stability of the equilibria, offering novel geometries for mission applications. Transfers between these novel equilibria are constructed by leveraging the associated stable and unstable manifolds and insights from the low-thrust CR3BP.
Equipping a spacecraft with multiple solar-powered electric engines (of the same or different types) compounds the task of optimal trajectory design due to presence of both real-valued inputs (power input to each engine in addition to the direction of thrust vector) and discrete variables (number of active engines). Each engine can be switched on/off independently and “optimal” operating power of each engine depends on the available solar power, which depends on the distance from the Sun. Application of the Composite Smooth Control (CSC) framework to a heliocentric fuel-optimal trajectory optimization from the Earth to the comet 67P/Churyumov-Gerasimenko is demonstrated, which presents a new approach to deal with multiple-engine problems. Operation of engine clusters with 4, 6, 10 and even 20 engines of the same type can be optimized. Moreover, engine clusters with different/mixed electric engines are considered with either 2, 3 or 4 different types of engines. Remarkably, the CSC framework allows us 1) to reduce the original multi-point boundary-value problem to a two-point boundary-value problem (TPBVP), and 2) to solve the resulting TPBVPs using a single-shooting solution scheme and with a random initialization of the missing costates. While the approach we present is a continuous neighbor of the discontinuous extremals, we show that the discontinuous necessary conditions are satisfied in the asymptotic limit. We believe this is the first indirect method to accommodate a multi-mode control of this level of complexity with realistic engine performance curves. The results are interesting and promising for dealing with a large family of such challenging multi-mode optimal control problems.
As is well known in celestial mechanics, coordinate choices have significant consequences in the analytical and computational approaches to solve the most fundamental initial value problem. The present study focuses on the impact of various coordinate representations of the dynamics on the solution of the ensuing state/costate two-point boundary-value problems that arise when solving the indirect optimal control necessary conditions. Minimum-fuel trajectory designs are considered for 1) a geocentric spiral from geostationary transfer orbit to geostationary Earth orbit, and 2) a heliocentric transfer from Earth to a highly eccentric and inclined orbit of asteroid Dionysus. This study unifies and extends the available literature by considering the relative merits of eight different coordinate choices to establish the state/costate differential equations. Two different sets of orbit elements are considered: equinoctial elements and a six-element set consisting of the angular momentum vector and the eccentricity vector. In addition to the Cartesian and spherical coordinates, four hybrid coordinate sets associated with an osculating triad defined by the instantaneous position and velocity vectors that consist of a mixture of slow and fast variables are introduced and studied. Reliability and efficiency of convergence to the known optimal solution are studied statistically for all eight sets; the results are interesting and of significant practical utility.
Application of optimal control principles on a number of engineering systems reveal bang-bang and/or bang-off-bang structures in some or all of the control inputs. These abrupt changes introduce undesired non-smoothness into the equations of motion, and their ensuing numerical propagation, which requires special treatments. In order to alleviate these induced difficulties, a generic smoothing technique is proposed that is straightforward to implement while providing additional flexibility in the rate of change of the control. The proposed technique does not affect the standard derivation of the so-called indirect methods (i.e., optimality conditions, form of the cost functional, and the structure of the switching function), which makes it ideal for programming purposes. In essence, the smoothing is applied directly at the control level. In many cases, modest smoothing can be introduced with full visibility of the frequently near-negligible loss of performance relative to smoothing the discontinuous controls. The utility of the proposed method is illustrated via two different problems: 1) a minimum-fuel multiple-revolution low-thrust interplanetary trajectory design problem, and 2) a rest-to-rest minimum-time attitude reorientation problem. Numerical results indicate that the domain of convergence of the TPBVP enlarges significantly where random initialization of the unknown co-states is sufficient to get a feasible solution. This solution can be used in a numerical continuation method (and over the smoothing parameter) to guide the solution towards the solution of the original problem.