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1
Integrated Space Logistics Mission Planning and Spacecraft
Design with Mixed-Integer Nonlinear Programming
Hao Chen
1
and Koki Ho
2
University of Illinois at Urbana-Champaign, Champaign, IL, 61820
This paper develops a campaign-level s pace logistics optimization framework that
simultaneously considers miss ion planning and spacecraft design utilizing mixed-intege r
nonlinear programming. In the mission planning part of the framework, deployment and
utilization of in-orbit infrastructures , such as in-orbit propellant depots or in s itu resource
utilization plants are also taken into account. Two methods are proposed: First, the mixed-
integer nonlinear programming problem is converted into a mixed-integer linear
programming problem after approximating the nonlinear model with a piecewise linear
function and linearizing quadratic terms. In addition, another optimization framework is
provided based on s imulated annealing, which separates the spacecraft model from mission
planning formulation. An example mission scenario based on multiple Apollo missions is
considered, and the results show a significant improvement in the initial mass in low Earth
orbit by campaign-level des ign as compared with the traditional mission-level design. It is also
shown that the mixed-integer linear programming- based method gives better-quality s olutions
than the simulated annealing-bas ed method, although the simulated annealing method is more
flexible for extension to a higher-fidelity spacecraft model.
Nomenclature
A = set of directed arcs
b = spacecraft flow vector (binary)
C = spacecraft payload capacity
f = spacecraft fuel type (integer)
G = network graph
= gravitational acceleration on Earth (equal to 9.8 m/s2)
H = concurrency matrix
= specific impulse
= node index ()
= objective function
= node index ()
M = spacecraft propellant capacity
1
PhD Student, Department of Aerosp ace Engineering, UIUC, AIAA Student M ember.
2
Assist ant Professor, Department of Aerospace Engineering, UIUC, AIAA M ember.
2
= oxygen in situ resource utilization plant mass
MUB = upper bound of propellant capacity
N = set of nodes
= oxygen in situ resource utilization plant production
= oxygen in situ resource utilization plant productivity
O = continuous variable denotes payload or propellant capacity
P = polyhedron formed by epigraph of piecewise function
= union of polyhedra
Q = transformation matrix
R = recession cone
s = spacecraft structure mass
t = time index (integer)
= spacecraft impulsive burn time
U = branching scheme
V = set of spacecraft
= set of breakpoints
v = spacecraft-type index
W = set of time windows
x = payload flow vector
y = spacecraft flow variable (integer)
= structural fraction
= propellant mass fraction
I. Introduction
S SPACE exploration projects become increasingly complex, the campaign -level logistics perspective becomes
more important for human s pace mission planning. In many pas t human space programs such as Apollo, every
mission was logistically independent us ing a so-called carry-along s trategy. This was efficient for short-term
missions to a relatively clos e destination. However, for a longer-duration multimiss ion space exploration campaign,
reusable in-orbit infrastructures can be more attractive. For example, it can be beneficial to have an in -orbit propellant
depot and/or a lunar in-situ resource utilization (ISRU) plant to refuel human missions to Mars or near-Earth objects
(NEOs) in return for their emplacement cost and the required maintenance. There are pros and cons for operati on of
those infrastructures . Therefore, it is important to perform an integrated campaign-level analys is considering both
mission planning and infrastructure design to evaluate whether it pays off to deploy those infrastructures .
In addition, we need a paradigm s hift in vehicle design as well. The conventional space programs such as Apollo
delivered relatively s imilar pay loads to the des tinat ion of each mis s ion throughout the campaign, and therefore a few
identical single-use vehicles were s ufficient for their logistics . For example, each Apollo vehicle carried a s imilar
amount of similar types of commodities (e.g., consumables, scientific payloads) in all missions. However, in a longer-
duration multimission space exploration campaign with infrastructure deployment and utilization, we need to deliver
different types and amounts of commodities to the destination throughout the campaign. For instance, in the beginning
of the campaign, the spacecraft may need to carry ISRU plants, whereas later on in the campaign, it may need to carry
A
3
the ISRU-generated propellants and its tank structure. Thus, for such campaign-level mission design, it is critical to
study and analyze the optimal fleet of multiuse vehicles, where each vehicle is flexible in its assigned function.
Many past studies have addressed space logistics mission planning, but few of them have considered in-orbit
infrastructure design and spacecraft model concurrently at a reasonable level of fidelity. One of the most recent studies
that analyzed space mission optimization is the generalized multicommodity network flow (GMCNF) formulation
developed by Ishimatsu et al. [1]. Their research modeled the s pace logistics planning problem as a linear
programming problem us ing a graph-theoretic approach, but this method did not include the emplacement cost for the
in-orbit infrastructures properly. Ho et al. improved the GMCNF by proposing a dynamic space logistics optimization
formulation that removed “time paradoxes” exis ting in the static network [2], referring to the propellant produced by
ISRU that could be used before the ISRU plants are deployed. In addition to this, Ho et al. cons idered the emp lacement
cos t of the in-orbit infrastructure. However, the infrastructu re and spacecraft model used in the analys is was a low-
fidelity linear model, which could result in an unrealistic s olution (e.g., an unrealis tically small vehicle size). Other
space logistic optimization models s uch as the interplanetary logistics model [3], the exploration architecture model
for in-space and Earth-to-orbit tool [4], and the s pace system architectures model based on graph theory [5] have been
proposed for space logistics mis s ion planning, but these studies did not consider the possibilities of using propellant
depots and ISRU plants , which can be critical for long-term s pace exploration.
On the other hand, various s tudies have focus ed on the detailed design of in -orbit infrastructures, but few of them
have also addres sed global optimization of space mission planning concurrently. For example, Oeftering [6] studied
detailed cislunar infrastructure des ign and Schreiner [7] performed a detailed sys tem-level ISRU plant sizing analys is
for the molten regolith electrolys is process. Also there were numerous architectures propos ed about on-orbit propellant
depots from both technological pers pectives [8-10] and economical perspectives [11-13]. However, those studies only
looked at the des ign of depot with its local transportation system rather than providing a design method cons idering
global optimization for the entire space logistics sys tem.
As shown previously, most of the past space logistics literature has not cons idered space mission planning, in-orbit
infrastructure and vehicle design concurrently at a reas onable fidelity. In res pons e to this background, this paper
proposes a system-level optimization formulation considering mis s ion planning, in -orbit infrastructure design , and
spacecraft design simultaneously, aiming to evaluate the cos t and benefit of reusable in-orbit infrastructures and
generalized designed spacecraft. The model is a mixed-integer nonlinear programming formulation based on a time -
expanded GMCNF model. This model s ignificantly advances Ho et al.’s previous model [2] and takes into account
higher-fidelity in-orbit infrastructure design models , including propellant depots and ISRU. We also add new
constraints for vehicles and crew members. The integrated optimization framework is shown in Fig. 1. This
optimization framework cons ists mainly of two parts : mission planning including infrastructure design, and spacecraft
des ign. Note that, propellant depots is not cons idered with a specific model in this paper; instead, the spacecraft model
is us ed for depot design, assuming a spacecraft staying in a certain orbit can s erve as a temporary propellant depot. A
similar concept to evolve the spacecraft to a propellant depot was proposed by the United Launch Alliance [14].
4
Fig. 1 Integrated optimization framework ([adapte d from [7, 15-17]]).
Our work will provide an important step to close the gap in the past literature and propos e an integrated
optimization formulation, considering both mission planning (including in -orbit infrastructure design) and spacecraft
des ign using a mixed -integer programming formulation. The developed formulation can be particularly useful for
evaluating the cost and benefit of cislunar infrastructure and for developing concrete space mission s chedules for
future long-term space exploration.
II. Methodology
The optimization formulation, aiming to minimize the total cos t of the whole multi-mission campaign, is
developed based on Ho et al.’s time-expanded generalized multicommodity network flow model [2]. To improve the
fidelity of the infrastructure modeling, we take into account a higher-fidelity ISRU model based on Schreiner’s system-
level ISRU plant sizing analysis [7], a high-fidelity spacecraft model, and new constraints for crew members.
This integrated optimization problem turns out to be a complex mixed-integer nonlinear programming (MINP)
problem and cannot be solved using the conventional formulations proposed by Ishimats u et al. [1] and Ho et al.[2]
due to their linear programming natures . The nonconvex nonlinear in-orbit infrastructure design model, which
specifically denotes ISRU in this paper, the complex nonlinear spacecraft design model, and the quadratic terms in
both objective function and constraints make it very challenging to solve the problem and find the global optimum
directly [3].
In order to solve the optimization problem efficiently, we developed two different methods .
The first method to solve this problem is converting the whole problem into a MILP problem. The nonlinear ISRU
and s pacecraft models are approximated by piecewise linear functions , to then be converted into a binary mixed -
integer programming formulation. The remaining quadratic terms in the miss ion planning formulation are linearize d .
This MILP formulation can always find the approximated global optimum. The detail of this MILP optimizatio n
5
method is introduced in Sec. II.B.2.
Another developed method s olves miss ion planning and s pacecraft model s eparately us ing s imulated annealing
(SA). Since the spacecraft model is separated from mission planning, no quadratic term exist s in the formulation. Only
the ISRU model needs to be approximated by a piecewise linear function. This SA -bas ed method cannot guarantee its
global optimality, but it can potentially provide a solution to the problem with a more realistic and complex spacecraft
model. The detail of this SA-based optimization framework is shown in Sec. II.B.3.
A. Modeling
This s ection introduces the s pace logistics, spacecraft, and ISRU modeling methods. The space logistics model
is developed bas ed on Ho et al.’s time-expanded GMCNF [2] while using higher-fidelity s pacecraft and ISRU models .
The spacecraft model us ed in this paper is developed based on Taylor’s integrated trans portation system design
framework [15]. This is a data-based model where all the data comes from preexisting spacecraft. The ISRU plant
considered in this paper is an oxygen ISRU developed by Schreiner [7] bas ed on their system-level ISRU plant sizing
analys is for the molten regolith electrolysis proces s .
1. Space Logistics Modeling
The space logistics problem can be solved as a multicommodity network flow problem. In the network, nodes
correspond to planets, celestial objects, or orbits; and arcs correspond to trajectories. Spacecraft, crew members,
scientific instruments, and other kinds of payloads are considered as commodities flowing along arcs. The left-hand
side of Fig. 1 shows a network graph of the Earth-Moon-Mars-NEO space logistics model [16-18]. A generalized
multicommodity network flow formulation has been developed with this network graph for space logistics
optimization [1, 2].
The space logistics model in this paper is developed based on a time-expanded GMCNF formulation. Solutions
of the time-expanded GMCNF provide schedules that determine when to transport which commodity from one node
to another. In past research, everything that flows over arcs is considered to be one kind of commodity, including crew
members, consumables, ISRU plants, spacecraft, etc. [1, 2]. This paper separates the spacecraft from other
commodities and its model is considered as a separate module so that its nonlinear mixed-integer feature can be
included efficiently.
The following shows the mathematic formulation of time-expanded GMCNF. Consider a time-expanded network
graph G that is made up of a set of nodes and a set of direct arcs , including both transportation arcs that connect
different nodes and holdover arcs that connect the node and itself. Each arc has an index (v, i, j), meaning that
spacecraft v flies from node i to node j. We assume that the spacecraft set is . Commodities flow over arcs are split
into outflow
and inflow
, where t is the time step. There are also cost coefficients
assigned to
outflow. Note that, if we assume there are types of commodities delivered in spacecraft,
and
are all
vectors, where each component shows the flows and costs of the corresponding commodity. Therefore, all
flows
are nonnegative. Most of the commodities in
are continuous variables , except crew members,
which form a discrete (i.e., integer) variable. Moreover,
represents those commodities delivered by spacecraft.
The spacecraft itself is also one kind of commodity denoted by
, which is a discrete (i.e., integer) variable.
Each node i has a demand vector , which is the demand or supply of each commodity at time step t. In the
demand vector, demand is shown by a negative value, whereas supply is shown by a positive value. Fig. 2 shows an
example of a full-time expanded network. For instance, the commodity in node k can take one of three paths:
1) Stay at node k over the holdover arc until t+4;
6
2) Be transported to node j at time t, and stay at node j until t+4;
3) Be transported to node j at time t, and then to node i at time t+2, and stay at node i until t+4.
The optimizer will choose the optimal path of these three.
Fig. 2 Full time-expande d network [16].
For each arc from node i to node j, a positive transit time is defined. Besides transit time, another important
parameter of time-expanded GMCNF is the possible departure time , which corresponds to the time
windows of arc i to j. T denotes the maximum time horizon of the campaign-level sequence of space missions. The
concept of the time window defines whether it allows commodity flows at a specified time step.
According to the aforementioned notations, the time-expanded GMCNF can be expressed by the followin g
formulation:
Minimize:
(1)
Subject to:
(2)
(3)
(4)
(5)
(6)
(7)
7
Table 1 Definition of indices, variables, and parameters
Name
Definition (dimension)
Index
Spacecraft index
Node
Time step
Commodity index
Concurrency constraint index
Variables
Commodity outflows/inflows. Commodities in
are cons idered as integer
or continuous variables based on the commodity type. For example, the number
of crew members is an integer variable, which is , whereas other commodities
(such as propellant, payload, and human consumables ) are considered as
continuous variables, which are . ()
Number of spacecraft flying along arc i to j. Integer variable. (scalar)
Structure mass of spacecraft . Continuous variable. (scalar)
Spacecraft design parameters, including payload capacity and propellant
capacity. Continuous variables. ()
Spacecraft fuel type. Integer variable. (scalar)
Parameters
Commodity cost coefficient. ()
Spacecraft cost coefficient. (scalar)
Demands or supplies of different commodities at each node. ()
Demand or supply of spacecraft at each node. (scalar)
Commodity transformation matrix. ()
Concurrency cons traint matrix. ()
Time window vector. (, where n is the number of time windows. It is
dependent on space missions)
Table 1 lists the definitions of indices, variables, and parameters. The detailed descriptions of the objective
function and constraints are as follows.
Equation (1) is the objective function. It gives the value of the initial mass in low Earth orbit (IMLEO), which is
the cost metric for space logistics in this paper.
Equations (2) and (3) are mass balance constraints that limit commodity flows to satisfy the demands of all nodes.
Equation (4) shows the transformation of commodity flows, such as propellant burning, propellant boiloff, ISRU
production, and crew consumables. For example, impulsive propellant consumption can be expressed as follows:
(8)
8
In Eq. (8), is the propellant mass fraction which is defined by the rocket equation,
, where
is the change in the vehicle’s velocity along the arc, is the specific impulse, and is the standard gravity.
Another example about propellant production from ISRU can be expressed as follows:
(9)
In Eq. (9), is the ISRU production rate. It is a function of ISRU plant mass , which is
introduced in detail in Sec. II.A.3.
Equation (5) is a spacecraft concurrency constraint, which corresponds to the upper bound of commodity flows
limited by s pacecraft propellant capacity or payload capacity. In this paper, we only consider spacecraft payload and
propellant mass capacities as concurrency constraints. Therefore, the number of concurrency constraints is equal to
two, which is indexed by . For example, the concurrency constraints about payload and propellant can be expressed
as following:
(10)
where and are the payload and propellant capacity of spacecraft . In past literature, Eq. (5) could also work
as a linear spacecraft model through an inert mass fraction [2]. Note that there is an underlying ass umption in this
formulation that the spacecraft capabilities of the components can be additively combined. In reality, the
interoperability between spacecraft can be significantly more complex.
Equation (6) is the time window of missions defined by time window matrix .
Equation (7) is spacecraft design model. It is introduced in detail next, in Sec. II.A.2.
The time index of this model is inspired by a biscale time-expanded network by Ho et al. [2]. In a biscale network
model, nodes are partitioned into clus ters such that transportation across the same pair of clusters has a common time
window [2]. The word “biscale” means a larger time-step s cale is us ed among clusters and a smaller time-step scale
is us ed at the intracluster level.
In this paper, based on the biscale time-expanded network, nonuniform time steps are used within clusters. If there
are time windows within clusters, the time s tep length of the holdover arc is determined by the intervals among the
time windows . For example, we have a network defined as shown in Fig. 3. Becaus e the time windows of node k only
open at time t, t+2 and t+8, there are only two holdover arcs for node k. The first one starts from time t and ends at
t+2; the second one starts from time t+2 and ends at t+8. The time-step lengths of these holdover arcs are two and six
in this example for node k. Nonuniform time steps can eliminate redundant time steps and increas e computational
efficiency.
9
Fig. 3 Nonuniform time-step time-expanded network.
2. Vehicle model
This paper us es a data-based spacecraft model developed by Taylor in an integrated transportation system [15].
This is a nonlinear regression function based on preexisting spacecraft. The relations hip among s tructure mass and the
three design variables is as follows [15]:
(11)
where is structure mas s (or dry mass); C is the spacecraft payload capacity; M is the propellant capacity; f is the
fuel type; is the gravitational acceleration on Earth, 9.8 m/s2; is the spacecraft impulsive burn time that is set
as 120 [s]; is the upper bound allowed for the propellant tank capacity, which is assumed as 500,000 [kg];
and are the structural fraction and specific impulse that are determined by fuel type f. Note that is the structure
fraction of the s pacecraft propellant tank defined in [15]. In this paper, only one kind of propellant liquid oxygen
(LOX)/kerosene is considered in the s pacecraft model. This is becaus e , compared with other propellants, liquid oxygen
(LOX)/liquid hydrogen (LH2), LOX/kerosene, and Monomethylhydrazine (MMH)/Nitrogen tetroxide (N2O4) have
relatively high s pecific impulse with a relatively low structural fraction (see Table A2). Considering the
abundant availability of oxygen on the moon from ISRU and the large boiloff of hydrogen over the long campaign,
LOX/keros ene is chos en.
Table 2 Spacecraft s izing comparis on
Spacecraft
Payload
Capacity
C, kg
Propellant
Capacity
M, kg
Propellant
Actual Structure
Mas s
s, kg
Nonlinear
Regres sion
, kg
Photographic
scaling [5]
s, kg
Centaur
0
20,830
LOX/LH2
2,462 [19]
3,131
3,215
S-IVB
0
107,725
LOX/LH2
12,014 [15]
13,513
16,625
HTV
6,000
2,000
MMH/N2O4
10,500 [20]
15,001
15,197
ATV
5,500
2,613
MMH/N2O4
10,300 [21]
13,986
14,257
Apollo LM DS
500
8,804
N2O4/UDMH
2,770 [15]
2,505
2,566
Apollo LM AS
250
2358
N2O4/UDMH
1719 [15]
1,005
687
Apollo CM
524
0
N2O4/UDMH
4841 [15]
1,254
1,150
Apollo SM
60
18413
N2O4/UDMH
6053 [15]
2,682
5,367
10
The spacecraft data used to develop this spacecraft model is listed in the Appendix. A comparison of this
spacecraft model with a conventional sizing method is shown in Table 2 [15, 19-21]. We can find that, as a nonlinear
regression model, this spacecraft sizing model has a higher fidelity compared with a conventional method such as
photographic s caling [5] when evaluating some historic spacecraft. However, limited by the accuracy of the spacecraft
model, this nonlinear spacecraft model output can also diverge from historic s pacecraft, s uch as the Apollo lunar
module (LM) absent stage (AS), the Apollo command module (CM), and Apollo service module (SM), as shown in
Table 2. Note that this paper does not intend to develop an accurate spacecraft model; instead, this paper proposes the
methods for space logistics optimization with a nonlinear spacecraft model as an input. The aforementioned spacecraft
model is chosen from the literature as an example to demonstrate our space logistics optimization method.
3. ISRU (oxygen) model
Only oxygen ISRU is considered in this paper because oxidizer is a major component of spacecraft mass . Studies
have shown that oxygen can be produced at a lower cost as compared with being delivered directly from Earth [14].
The integrated ISRU s ystem developed by Schreiner us ed a molten regolith electrolysis model and balanced the
tradeoff between optimal reactor performance and optimal excavato r des ign. The ISRU system plant includes the
reactor, Yttria-Stabilized Zirconia (YSZ) separator, excavator, hopper and feed s ys tem, oxygen liquefaction and
storage system, and power sys tem. For a detailed description of thes e systems, please see [7, 22]. Because a separate
cooler is designed to reliquefy oxygen that has boiled off in the s torage sys tem [7], the boiloff effect of oxygen
produced by ISRU is not cons idered. A ccording to Schreiner’s research [7], the ISRU production rate is increasing
with the ISRU plant mass. The relationship is shown in Fig. 4.
Fig. 4 The oxygen production level normalized by ISRU s ystem mass [7].
It can be expressed as the following function [7]:
(12)
where is the oxygen production rate in kilograms of oxygen (O2) per year per kilogram system mass and NO is
the oxygen production per year. If we assume ISRU plant mass as MO, then
11
(13)
Equation (12) is a regression equation that has a horizontal asymptote as . Based on Eq. (12) and the
data from Schreiner [7], we build another regress ion equation to get , which is shown as follows:
(14)
The comparison between the regression equation [Eq. (14)] and Schreiner’s model [7] is s hown in Fig. 5. It is
the relationship between the ISRU sys tem plant mas s and the oxygen production rate . Fig. 5 shows that Eq.
(14) fits the data of Schreiner’s model well. It is us ed to calculate oxygen production in our space logistics model.
Fig. 5 The oxygen production normalized by ISRU system mass.
Equation (14) is a piecewise nonlinear function. When , ISRU does not function. Becaus e the ISRU
system includes the reactor, excavator, storage sys tem, power s ystem, etc., if the total mas s of ISRU plant is too low,
the integrated ISRU s ystem does not contain enough components to start functioning as a unit. The domain of the
ISRU plant mass in this equation is from 0 to 10,000 kg. From Eqs. (13) and (14), we can get the relationship
between the oxygen production every year and the total mass of the ISRU plant deployed, which is
.
B. Optimization Methods
The campaign-level space logistics problem considering nonlinear spacecraft and ISRU model is a mixed-integ e r
nonlinear programming (MINP) problem. This s ection introduces the piecewise function approximation method that
linearizes the nonlinear ISRU and s pacecraft models , and then converts them into binary mixed-integer programming
formulations . Two approaches to solve this MINP problem are propos ed. Both methods us e the piecewise function
approximation method for approximating the ISRU model. The difference between the two methods lies in how to
12
deal with the nonlinear s pacecraft model. The first approach converts the whole problem into a MILP formulation ,
which is expected to solve this problem more efficiently. The second approach is developed bas ed on simulated
annealing. It separates the spacecraft design from the miss ion planning part, which is intended to create the ability to
consider higher-fidelity spacecraft models if neces s ary.
1. Piecewise function approximation method
The following s hows the method to approximate a s ingle variable nonlinear function by a piecewise linear
function, and then convert it into a binary mixed -integer programming formulation. Cons idering a nonlinear function,
a proper number of breakpoints inside the function domain is selected and then connected linearly, as shown in Fig.
6. This single variable nonlinear function is approximated by a piecewise linear function, which can be des cribe d as
follows:
(15)
for s ome :
,
, and
, where is a breakpoint.
Fig. 6 Piecewise function and its epigraph as the union of polyhedra.
To solve this piecewise function in our mixed -integer linear programming model, the next step is modeling the
piecewise function as a binary mixed-integer programming formulation, where is the domain of
. An appropriate approach to do it is to model its epigraph
as s hown in Fig. 6. The epigraph of a piecewise function can be cons idered as a union of polyhedra . Each
polyhedron P corresponds to an interval . Therefore, we can rewrite the piecewise function as follows:
(16)
Vielma et al. [23] reviewed and introduced several ways to convert a union of polyhedra into a binary mixe d -
integer programming formulation, including a convex combination model, logarithmic branching convex combination
13
(referred to as log) model, a multiple choice model, an incremental method, etc. As one of the most performant
formulations , the log model is chosen for piecewise linear approximation of spacecraft and ISRU models in this paper.
Following the theory des cribed by Vielma et al. [23], can be repres ented as the convex
combination of points for , where denotes the set of vertices (i.e., breakpoints ) of
polyhedron , plus a ray in . Then a continuous variable is ass umed for each
vertex and for each . As a result, a point can be repres ented as
for
and
such that
Moreover, we can further reduce the number of continuous variable and cons traints of the formulation by
identifying a binary branching scheme. As a result, we only need to define λ variables associated with the vertices (i.e.,
breakpoints) in union of polyhedra , rather than for each vertex in each polyhedron . This can reduce the number
of λ variables s ignificantly becaus e two adjacent polyhedra share the s ame vertex (i.e., breakpoint).
Finally, the resulting formulation is given by
,
(17a)
,
(17b)
,
, (17c)
where is a family of dichotomies, which is a branching s cheme of variables. This formulation is
called logarithmic branching convex combination model. The branching scheme is introduced by Vielma and
Nemhaus er [24], and the details of the log method can be found in [23]. Using the aforementioned method, an
example of building a log formulation is shown in the following.
For example, the nonlinear function in Fig. 6 can be approximated by a piecewise function us ing five
breakpoints, given by
(18)
Then, a point in this piecewise function can be express ed as a binary mixed-integer programming
formulation using the log formulation given by
(19a)
(19b)
(19c)
, , , , (19d)
2. Mixed-Integer Linear Programming Optimization Framework
From the MILP formulation [Eqs. (1)-(7)], we can see that, if the spacecraft model is cons idered together with
14
mission planning, even though it is approximated using a piecewise function and then converted into a binary mixe d -
integer programming formulation, there are s till quadratic terms . These quadratic terms exist in an objective function
[Eq. (1)], a commodity trans formation cons traint [Eq. (4)], and a concurrency constraint [Eq. (5)]. The reason is that
spacecraft design variables , , , and , are also variables in the MILP formulation. These quadratic terms are
the products of discrete and continuous variables. Becaus e the coefficient matrix of quadratic terms is not positive
semidefinite, to s olve this model, linearizing quadratic terms is a better choice.
Consider a nonlinear equation , where is a continuous variable representing the dry mass
, payload capacity , or propellant capacity of spacecraft . is a binary variable representing whether
the spacecraft is flying along the arc from i to j at time t. Note that represents the type of spacecraft, and there
are multiple spacecraft for the same type. Also, is the index of spacecraft of the same type. Assuming the maximu m
number of spacecraft can be used is n, we have
(20)
Therefore, using the big-method, product can be linearized as follows:
(21a)
(21b)
(21c)
(21d)
where M is a large constant.
As a res ult, after approximating the nonlinear model of ISRU and s pacecraft by piecewise function
approximation method and linearizing quadratic terms in concurrency constraints , all nonlinear terms are converted
into linear cons traints . The campaign-level space logistics optimization problem cons idering nonlinear s pacecraft and
ISRU production models finally becomes a mixed-integer linear programming problem.
3. Simulated Annealing Optimization Framework
SA is a conventional heuristic optimization method that cannot guarantee or certify its global optimality. Typically,
it takes a long time to achieve an acceptable s olution. This SA-based method is inspired by the optimization framework
developed by Taylor [15]. However, Taylor’s approach cannot take into account infrastructure such as ISRU because
the mission planning part of the framework uses a shortest -path-based approach. This paper improves this method by
using an arc-based formulation and s olves it as a multicommodity network flow.
The only difference between the MILP-based method and the SA-bas ed method is how to deal wit h the nonlinear
spacecraft model. The SA-based approach separates the spacecraft model from mis s ion planning. As a res ult, the
spacecraft design variables and are cons tant inside the MILP formulation. They are only variables in the
separated spacecraft model. In other words , the quadratic terms in the MILP formulation disappear. The nonlinear
ISRU model is s till designed concurrently with miss ion planning part. It is converted into a binary mixed-integ e r
programming formulation through piecewise linear approximation.
Because the spacecraft model is separated, the advantage of the SA approach is that it provides the possibility of
extending the spacecraft model in an easier manner when compared to the method of MILP formulation .
15
Fig. 7 Flow chart of S A-based approach.
A flow chart of SA is shown in Fig. 7. The mis sion planning section is actually the evaluation step of simulated
annealing. After selecting random neighbors in the spacecraft model, the performances of the spacecraft are evaluated
by a space logistics mission planning optimization utilizing these s pacecraft. The best space logistics s olution is
recorded. The optimization algorithm repeats until a stopping criteria (e.g., convergence, computation budget) and
then optimal spacecraft parameters and space logistics s olutions are output.
III. Results and Analysis
A. Model Validation
This section validates the network model using the Apollo 17 mission as an example, where no ISRU and
spacecraft design models are cons idered. Ins tead, the original spacecraft in Apollo mission is used in mission planning.
The Apollo 17 mission can be modeled as a four-node network as shown in Fig. 8, including the Pacific Ocean (PAC),
low Earth orbit (LEO), low lunar orbit (LLO) and lunar surface (LS). The time of flight (TOF) and for each
trans portation arc are also s hown in Fig. 8. We ass ume that one day is one time s tep.
Fig. 8 Apollo 17 network model.
Note that the propellant cost for transportation from Earth to LEO is not considered becaus e the IMLEO is
minimized in this paper. For a trans portation s ystem, the IMLEO is a widely accepted meas ure of the mission cos t [1,
3]. In some literature, the financial cos t is also us ed as a metric [5]. However, this paper focuses on limited resource
utilization and the interaction between the miss ion planning and s pacecraft des ign. To simplify the problem, the
IMLEO is used as a metric for mission cost in this paper.
Table 3 lists all s pacecraft used in Apollo 17, and Table 4 lists the demand and supply of the Apollo 17 mis s ion.
In this example, two astronauts , along with the s cientific experiment equipment, are sent to the lunar surface, and stay
there for one day; whereas a third as tronaut stays in lunar orbit for three days . Then, all three astronauts come back
with 110 kg of lunar samples. For this Apollo 17 mission, the time window is always open.
16
Table 3 Spacecraft us ed in Apollo 17
Spacecraft
Propellant capacity
M, kg
Isp, s
Payload capacity
C, kg
Structure mass
s, kg
Saturn V s econd stage
452,045
421
0
38,415
Saturn V third stage
107,725
421
0
12,014
Command module
0
0
524
4,841
Service module
18,413
314
60
6,053
LM des cent stage
8,804
311
500
2,770
LM ascent stage
2,358
311
250
1,719
Table 4 Demand and supply of Apollo 17 miss ion
Payload Type
Node
Demand Time, day
Supply
Crew,a no.
Lunar s urface
5
-2
Crew,a no.
Lunar orbit
4
-1
Crew Return,a no.
Lunar s urface
6
2
Crew Return,a no.
Lunar orbit
7
1
Crew Return,a no.
Earth
11
-3
Equipment,b kg
Lunar s urface
5
-420
Sample,c kg
Earth
11
-110
Crew, Equipment,b kg
Earth
0-11
+∞
Sample,c kg
Lunar s urface
0-11
+∞
aCrew and Crew Returning are separated in order to ensure that the crew shall reach the moon before returning to
Earth.
bEquipment denotes all scientific experiment equipment for lunar surface activities, including a Lunar Roving Vehicle
(LRV);
cSample denotes all the things we want to bring back from other planet, typically for scientific research.
The optimized solution using the GMCNF model for this example is depicted in Fig. 9. All commodities are
delivered together from LEO. Notice that crew members fly back to the Earth by LM descent s tage (LMDS) ins tead
of the A pollo command/ service module (CSM) in the original mission. This is a feasible solution since the payload
capacity of LMDS is large enough to contain three crew members and 110 kg of lunar samples. The s ize of the
propellant tank is also sufficient to support the return directly from lunar orbit. This is also an optimal solution
because one important function of the CSM is to bring crew members and lunar s amples back to the Earth. If the
LMDS can complete this work, the s ervice module would not be us ed because LMDS has a smaller structure mas s
which means less propellant cost. A similar result is observed in Taylor et al.’s s tudy [3]. Saturn V’s third stage
performs as a propellant depot after reaching the lunar orbit. The detailed mass flows are s hown in Fig. 10. The
consumables in Fig. 10 include water, oxygen, and food for the crew.
17
Fig. 9 Apollo 17 example.
Fig. 10 Commodi ty flows in optimal solution.
Table 5 compares the differences of payload and propellant at Earth orbit insertion between the optimized Apollo
17 mission and the original mis s ion [25]. Notice that no spacecraft design model is considered in this example. The
optimized Apollo mis sion us es the same vehicles as original mis s ion.
Table 5 Mas s differences comparison at Earth orbit insertion
Spacecraft
Class of mass
Original Apollo 17, kg
Optimized Apollo 17, kg
Saturn V third stage
Propellant
96,204
74,758
Structure mass
12,014
12,014
18
Command module
Crew member
3*100
— —
Consumables
124
— —
Equipment
— —
420
Structure mass
4,841
4,841
Service module
Propellant
18,413
— —
Structure mass
6,110
— —
LM descent stage
Crew member
— —
3*100
Consumables
— —
124
Equipment
420
— —
Propellant
8,804
8804
Structure mass
2,770
2,770
LM ascent stage
Propellant
2,358
— —
Structure mass
1,719
1,719
All spacecraft
Total mass
154,077
105,750
The GMCNF model actually selects a better solution for the Apollo 17 mission. For validation purpos es, we can
manually track what has changed as compared with the original mis sion. Starting from the original Apollo 17 mission ,
we remove the service module, the propellant in LM ascent stage (LMAS), and the propellant to trans port these
commodities to lunar orbit. Then, we add the propellant for the LMDS to support its return to Earth. Finally, the initial
mass in LEO becomes 105,614 kg. The optimal solution by the GMCNF model is 105,750 kg, which is only a 0.1%
difference from our manual validation results . Thus, we can conclude that the time-expanded GMCNF co uld find an
optimal s olution and has s ufficient accuracy to optimize the space logistic problem.
B. Campaign-level Space Logis tics Design
This s ection evaluates and analyzes the performance of the space logistics optimization framework for miss ion
planning an d spacecraft des ign. Both ISRU and spacecraft des ign models are taken into account. Moreover,
maintenance costs and the oxygen boiloff are also considered in this case. This is a MILP problem s olved in Python
by a Gurobi 6.5 solver on an Intel Core i7-4700MQ, quad-core 3.4GHz platform. All the following numerical
experiments in this paper are solved on this platform. The piecewise linear approximations for nonlinear ISRU and
spacecraft models are done through a Python -bas ed open-source optimization package named Pyomo.
In this section, we first perform an optimization for a single space mission as the baseline solution . Then, we
optimize a campaign, which contains multiple s pace mis sions, and compare it s result with the bas eline solution to
study the effect of campaign-level space logistics design. Thes e space miss ion optimizations are all completed through
the MILP-based method. Next, we compare the performance of the MILP-based method with the SA-based method .
Finally, we perform sensitivity analyses of ISRU productivity and spacecraft design.
1. Single mission Design with MILP
In this part, we optimize a single s pace mis sion as the baseline solution. It is us ed to demonstrate the effect of
campaign-level space logistics des ign in the next section. Based on the Apollo 17 mis s ion, a new large-scale lunar
mission is des igned as an example for optimization. The demand and supply are s hown in Table 6. In this case, 12
crews are delivered to the Moon with 4200 kg of equipment and come back with 500 kg samples. The as sumptions
19
are listed in Table 7 [26, 27]. For this single mission example, the time window is always open.
Table 6 Demand and supply of lunar mission
Payload Type
Node
Demand Time, day
Supply
Crew, no.
Lunar Surface
5
-12
Crew Return, no.
Lunar Surface
8
12
Crew Return, no.
Earth
13
-12
Equipment, kg
Lunar Surface
5
-4200
Sample, kg
Earth
13
-500
Crew, Equipment, kg
Earth
0-13
+∞
Sample, kg
Lunar Surface
0-13
+∞
Table 7 Summary of parameters and ass umptions
Parameter
As sumed value
Percentage of oxygen in propellant
71.91% LOX/kerosene
Oxygen boil-off rate
0.016%/day
Food cons umption rate [26, 27]
1.015 kg/day/crew
water cons umption rate [27]
6.37 kg/day/crew
oxygen consumption rate [27]
1.18 kg/day/crew
Crew mass (including space s uit)
100 kg/crew
ISRU maintenance
10%/year
Spacecraft maintenance
1%/flight
The percentage of oxygen in propellant in Table 7 is us ed to cons ider the production of oxygen ISRU plant and
the boiloff effect. Note that the oxygen consumed by crew members is very low as compared with the oxidizer in
propellant (LOX/kerosene). To simplify the problem, the oxygen produced by ISRU plant is only used for propellant
oxidizer.
The maintenance of ISRU is performed every mis sion, and the amount of ISRU maintenance mass is determined
by the ISRU plant mass. In addition, the maintenance of s pacecraft is performed in every flight and the amount of
needed maintenance mass is related to the spacecraft s tructure mass .
The solution of the single miss ion optimization is listed as follows. Two types of spacecraft are considered , and
two spacecraft can be used for each type.
Table 8 Spacecraft design of a single mission des ign
Spacecraft
Type
Payload Capacity,
C, kg
Propellant Capacity,
M, kg
Structure Mass,
S, kg
1
2,020
166,481
17,996
2
2,262
23,891
7,342
Table 8 lists the s pacecraft design in this case, and the mission planning s olution is depicted in Fig. 11. Because
20
the mission only lasts for 13 days , no ISRU plant is deployed on the moon. This solution is also an optimal solution
of the conventional carry-along strategy. All propellant and consumables used in the mission come from the Earth.
The spacecraft size is also shown in Fig. 11, corresponding to the structure mas s in Table 8. Even though the spacecraft
are designed to be multiuse without specific functions as signed for each type, they s till perform different roles in the
mission. Spacecraft 1, which is relatively larger, is mainly us ed to deliver equipment. Spacecraft 2, which is s maller,
is mainly used to deliver crews and samples back from the moon. The detailed mass flows are shown in Fig. 12, and
we can see that the IMLEO of this mission is 439,375 kg.
Fig. 11 Single lunar mission solution.
Fig. 12 Commodi ty flows of single lunar miss ion.
2. Campaign-Level Mission Design with MILP
This section s tudies the effect of campaign-level s pace logistics miss ion design. Three lunar missions are
combined together into a campaign. In each mission, 12 crews are delivered to the moon with 4200 kg of equipment
21
and come back to Earth with 500 kg of lunar s amples. All the ass umptions remain the same as before. For a multi-
mission campaign, the time windows are open at the start of each miss ion, which are determined by the mis sion
intervals. It takes about 2000 s to optimize this problem. The s olution of campaign-level mis s ion optimization with
one year mis sion intervals are shown in the following.
Table 9 Spacecraft design of the multi-mission campaign design s olution (interval: one year)
Spacecraft
Type
Payload Capacity,
C, kg
Propellant Capacity,
M, kg
Structure Mass,
S, kg
1
3,978
198,580
24,745
2
2,262
23,891
7,342
Table 9 lists the s pacecraft design of this case. The miss ion planning of this campaign is depicted in Fig. 13. We
can s ee that the size of spacecraft 2 is the same as it in the single miss ion case, as shown in Table 8. The reas on is that,
like the single miss ion case, spacecraft 2 is mainly used to deliver crews and samples. Becaus e the demands of crews
and samples do not change, the s ize of the s pacecraft 2 does not change. However, the size of spacecraft 1 is larger
because, in a multi-mission campaign, spacecraft 1 is used to deliver not only the equipment but also ISRU plants .
The detailed mass flows are listed in Fig. 14. From Fig. 14, we can see that the IMLEO for each mis s ion is
decreasing. The firs t mis s ion has the highest IMLEO because the ISRU plant is deployed in this mis sion. Starting
from the second miss ion, the oxidizer used in the spacecraft flying out of the moon is produced by the ISRU plant. It
relies less on the Earth res ource for propellant oxidizer, which is expens ive to deliver from Earth. As a result, the total
IMLEO of this multi-mission campaign is lower than three single miss ions, as can be seen by comparing Fig. 12 and
Fig. 14. If this mission is repeated in the fourth year, because there is no need to deploy ISRU plants again, the mission
cos t is still lower than a single lunar mission.
a) Mission-I
22
b) Mission-II
c) Miss ion-III
Fig. 13 Multi-miss ion lunar campaign s olution (interval: one year).
23
Fig. 14 Commodi ty flows of the multi-mission lunar campaign solution (interval: one year).
The mis s ion time intervals determine the total working time of the ISRU plant during the campaign. To s tudy
the effect of miss ion interval, the spacecraft structure mass and mis s ion planning results with res pect to the mission
interval are depicted in Fig. 15.
24
Fig. 15 Results of multi-mission lunar campaign des ign with respect to mission interval.
As shown in Fig. 15, the IMLEO first decreases quickly because the mission obtains lots of benefits from oxygen
ISRU plant with a longer mission interval. However, after the mission interval is longer than 10 years, the IMLEO
starts to decrease very s lowly with the increas e of mis s ion interval becaus e the oxygen ISRU can only produce
propellant oxidizer. No matter how much oxidizer can be produced by ISRU plant, we still need kerosene delivered
from Earth. When mission interval is long enough, the benefit from the oxygen ISRU plant diminishes.
Furthermore, two types of spacecraft are considered. There is a direct relations hip between the size of large
spacecraft (e.g. spacecraft 1) and the mass of the ISRU plant deployed. In Fig. 15, the variation pattern of the structure
mass of spacecraft 1 matches the variation pattern of the ISRU plant mass. Similar to the results in campaign-leve l
mission des ign, the large spacecraft is mainly used to deliver equipment and the ISRU plants. Becaus e the demand of
equipment is cons tant in each mission, the size of large spacecraft is mainly determined by the ISRU plants; however,
the s mall spacecraft is mainly used deliver samples and crew members. Even if all spacecraft are designed in a general
way (i.e., no specified function identified at the start of s pacecraft design), the solver identifies the functions for each
type of spacecraft.
3. Optimization Method Comparison
As discussed in the Methodology section (Sec. II), there are two optimization methods proposed in this paper:
the MILP method and the SA optimization framework. This section studies the influences of different optimization
methods . Thes e methods are us ed to solve the same campaign-level miss ion design problem defined previous ly.
In the previous section, the nonlinear ISRU model and nonlinear spacecraft model are finally converted into a
25
binary mix-integer programming formulation through piecewise functions . However, thes e models can initially be
assumed as linear to simplify the problem. For the linear ISRU model, we assume that the oxygen production rate is
7.5 kg of O2 per year per kilogram of ISRU s ystem mass. The linear spacecraft model is a linear simplification of Eq.
(11), which is assumed as
(22)
The comparison of different optimization methods is s hown in Table 10.
Table 10 Comparis on of different optimization methods (interval: one year)
Optimization method
IMLEO, kg
Optimization
Time, s
MILP
ISRU model
Spacecraft model
Nonlinear
Nonlinear
1,298,813
2,000
Nonlinear
Linear
1,424,484
1,705
Linear
Nonlinear
1,296,999
2,566
Linear
Linear
1,424,484
1,360
Simulated annealing
≈1,407,000
≈4,000
The IMLEO of the problem with the linear spacecraft model is much higher when compared with the norma l
case (i.e., all nonlinear). This is becaus e, in the nonlinear spacecraft model, the structure mass ratio of a large spacecraft
is lower than a small spacecraft. The structure mass increases linearly with the spacecraft size in Eq. (22), which is
developed bas ed on s mall s pacecraft; thus , the structure mass of a large spacecraft bas ed on Eq. (22) is larger than
reality. More propellant is consumed due to the large s tructure mass . On the contrary, the problem with linear ISRU
model has a lower IMLEO compared with a nominal scenario. The reason is the production rate of ISRU is increasing
with the ISRU plant mass. As s hown in Fig. 5, the ISRU would not work before the sys tem mass reaches 400 kg. Then,
the production rate is only 2.5 kg of O2 per year per kilogram of ISRU sys tem mas s , with a 400 kg ISRU plant and 5
kg of O2 per year per kilogram of ISRU system mass with a 1,000 kg plant. To achieve a production rate of 7.5 kg of
O2 per year per kilo gram of ISRU sys tem mass , more than 3,000 kg ISRU plant should be deployed. Thus, the ISRU
production rate of a linear ISRU model is typically higher than the actual case.
The optimization time cos t lis ted in Table 10 is the running time Gurobi s olver, ignoring the model input time ,
which is typically 1 or 2 min. Because a MILP problem is generally solved by a linear-programming-based branch-
and-bound algorithm, the calculation time is influenced by multiple factors, such as mission interval and ISRU or
spacecraft model. For this four-node network in the space logistics model, cons idering a campaign including three
lunar missions, there are approximately 100,000 cons traints, 1500 integer variables and 5400 continuous variables. In
Table 10, the optimization time of all linear MILP method is 1360 s when the mis sion intervals are one year. In fact,
it may take a longer time when the miss ion intervals change. When we compare this all-linear MILP method with our
method based on a piecewise linear function and MILP, we can find that our methods solve the space logistics problem
including nonlinear ISRU and s pacecraft models without much deterioration of computational efficiency. Our method
may bring in several binary variables and cons traints , but mos t of the cons traints and variables come from the miss ion
planning part, which is a GMCNF model. These cons traints and variables are not influenced by which ISRU or
spacecraft model cons idered. Therefore, no matter which ISRU or spacecraft model is considered, the optimization
time cost of this campaign-level mission s olved by the MILP method is at the same order of magnitude of
approximately 1500-2500 s.
26
The same problem is also s olved by the SA-based method. It cannot guarantee an optimal solution , but it is a
good method if the precise global opt imum is less important and an acceptable local optimum is wanted in a given
time horizon. The biggest advantage of SA in this problem is that s pacecraft model is separated from the mis s ion
planning part. As a result, high er-fidelity s pacecraft models can easily be considered, such as cons idering the fuel
types. Moreover, different s pacecraft can easily be ass igned different functions initially in a separate spacecraft model.
The solution of SA is influenced significantly by the cooling strategy and the initial value of the problem (e.g.,
initial spacecraft). The start temperature is set as 10,000 and the algorithm will s top after the temperature reaches 0.1.
The annealing schedule is an exponential cooling s cheme with a cooling rate assumed as 0.985. Thus, there are 762
iterations in total. The initial spacecraft in this paper is generate d by the spacecraft model randomly. Therefore, the
final s olution of the SA method may vary even for the same scenario.
The comparison of IMLEO between the MILP method and the SA method in different mis sion intervals is shown
in Fig. 16. Becaus e finding the optimal strategy to minimize IMLEO in s pace logistics is a minimization problem and
the SA method always achieves a feas ible (and suboptimal) solution, we can observe that the results from SA are
typically 8-15% higher than the results of the MILP method. The gap is dependent on the candidate spacecraft
generation method and cooling strategy.
We can improve the performance of the SA method by starting from a better initial solution with heuristics or
optimizing the cooling s trategy. Becaus e the spacecraft model is s eparated from the mission planning part, SA is a
better choice if a higher-fidelity s pacecraft model needs to be cons idered and the precise global optimum is less
important than finding an acceptable local optimum within given time.
Fig. 16 Comparison of IMLEO results by MILP and SA method.
4. Sensitivity Analysis
With the development of technology, ISRU could have a higher production rate and the s pacecraft could have a
27
lighter structure while maintaining sufficient s trength at the same time. On the other side, s ome uncertainty factors
may prevent the ISRU sys tem from working properly or the spacecraft may be assigned extra structure mass. This
section s tudies the sens itivity of this campaign -level mis sion design to the ISRU productivity and the spacecraft
structure mass using the MILP method.
The considered problem is the campaign-level mission design with three lunar missions under the s ame
assumptions as before. The mis s ion interval is one year. The IMLEO with res pect to ISRU productivity is s hown in
Fig. 17. The bas eline is the normal ISRU production rate in Eq. (14).
Fig. 17 Results of multi-mission lunar campaign des ign with respect to ISRU productivi ty (interval: one year).
As shown in Fig. 17, with the increase of ISRU productivity, the IMLEO of space mission is decreasing. When
the oxygen ISRU productivity is less than 80% of the normal rate, no ISRU plant is deployed. The IMLEO and
spacecraft design stay constant until the productivity reach 80%. Then, with the increase of ISRU productivity, the
IMLEO of the whole mission decreas es monotonically. When the ISRU productivity reaches 150% of the normal
production rate, the mass of ISRU plan deployed starts to decrease slowly. The s tructure mass of the large s pacecraft
(i.e., spacecraft 1) s till matches the variation pattern of the ISRU plant mass deployed in this case. Although the
structure mass of the s mall s pacecraft (i.e., spacecraft 2) varies in a certain range . One important note is that the
increase of ISRU production does not reduce IMLEO unlimitedly. When the ISRU productivity is high enough, the
mission cos t sensitivity to ISRU productivity is very low. With the increas e of ISRU productivity, the total IMLEO is
decreasing slowly. The reas on is that the oxygen ISRU plant only produces oxidizer for this mission. The
trans portation still needs kerosene supplied from Earth.
If we fix the ISRU productivity on the normal rate, the s ensitivity of the IMLEO with respect to the spacecraft
28
structure mass is shown in Fig. 18.
Fig. 18 Results of multi-mission lunar campaign des ign with respect to s pacecraft structure mass (interval:
one year).
The spacecraft mass ratio shown in Fig. 18 is the spacecraft structure mass divided by the structure mass in
bas eline [i.e., Eq. (11)]. Becaus e the spacecraft structure mas s influences the propellant cos t directly, with the increase
of spacecraft s tructure mass , the total IMLEO increas es significantly. When the s pacecraft structure mas s is low, there
is a tradeoff between enlarging the s pacecraft for ISRU deployment and using small s pacecraft without ISRU. When
the s pacecraft is more than 20% heavier than normal, it is not worth building a large spacecraft for ISRU deployment.
As a result, the structure mass of small spacecraft s tarts to increase and the division of roles between two types of
spacecraft becomes less clear. Both types of spacecraft start to deliver crews, equipment and samp les. Note that we
can s till obs erve that the s tructure mass of s pacecraft 1 matches the variation pattern of ISRU plant mass if ISRU is
deployed in the mis sion, even though the structure mas s is also increasing at the same time.
IV. Conclusion
This paper propo ses two optimization methods to solve the space logistics problem cons idering nonlinear
spacecraft and nonlinear ISRU models. First, the MILP optimization framework is developed based on the GMCNF
model. Piecewise functions are used to approximate the nonlinear spacecraft and ISRU models as well as to convert
the models into binary mixed-integer programming formulations. After linearizing the remaining quadratic terms , the
whole problem is converted into a MILP problem. This MILP-based optimization framework can always find the
29
global optimum of the problem with approximated nonlinear models. The second method is the SA-based optimization
framework, which is developed with the concern that a higher-fidelity spacecraft model may be considered. In this
method, the spacecraft model is separated from the mission planning part. This SA-based optimization framework can
never guarantee or certify its global optimality.
To illustrate the effect of both optimization methods, a campaign -level miss ion including three lunar missions is
optimized.
For the MILP method, the influences of miss ion interval are s tudied. The IMLEO of the space logistics system
firs t decreas es with the increase of mission interval until ISRU production cannot provide further benefits to the s pace
logistics system. Moreover, even though the s pacecraft are all designed as multiuse without specific functions
identified initially, they are all assigned s pecific functions in the res ult. The large spacecraft are typically used to
deliver equipment and ISRU plants, whereas the small ones are us ed to deliver crews and samples. Thus , the size of
the large s pacecraft is directly influenced by the mass of ISRU plant deployed in the miss ion (see Fig. 15, 17, and 18).
The influences of ISRU and spacecraft model fidelities are also studied, as shown in Table 10. A linear ISRU or
spacecraft model would not improve the computational efficiency becaus e the nonlinear models can be linearized into
binary mixed-integer programming formulations and most of the variables and constraints come from the mis sion
planning part, which is a GMCNF model. Therefore, the MILP method developed in this paper s olves the s pace
logistics problem, including nonlinear ISRU and s pacecraft models without much deterioration of computational
efficiency.
Besides the MILP method, a heuristic optimization algorithm based on SA is also propos ed. Compared with the
MILP method, the SA method cannot guarantee an optimal solution and the quality of the res ults is strongly dependent
on the initial s ettings . For the problem cons idering medium-fidelity spacecraft model, the MILP method is always a
better choice. However, the separated spacecraft model in the SA optimization framework makes it easier to cons ider
a higher-fidelity spacecraft model, such as considering fuel types and s pecifically predefined functions of each type
of spacecraft.
The ISRU productivity and spacecraft structure mass sensitivity are also studied. The IMLEO decreases
monotonously with the increase of ISRU productivity. The spacecraft s tructure mass is also an important factor in
determining the space logistics strategy. With the increase of spacecraft structure mass, IMLEO of the s ystem increases
significantly. When the spacecraft s tructure mass is too large, as compared with the normal condition, it is too
expens ive to build a large spacecraft for ISRU deployment and the specified functions of different types of spacecraft
also become less clear.
Future work includes cons idering uncertainties in the transport scheme, which leads to flexible time windows .
Moreover, the influences and interactions among different fuel types and ISRU plants deployed will make the problem
much more complex. The piecewise linear approximation mentioned in this paper for linearizing nonlinear ISRU and
spacecraft models is only s uitable for a single variable function. To solve the space logistics problem cons idering
higher-fidelity spacecraft and ISRU models by the MILP method, the formulation that converts the piecewise linear
function into a binary mix-integer programming formulation should be extended to a multivariable case.
Appendix: Parametric Spacecraft Model
This appendix lists preexis ting s pacecraft parameters upon which the spacecraft model of Taylor was based on
[15]. Table A1 lis ts the spacecraft parameters, and Table A2 lists the fuel type and corresp onding function values .
30
Table A1 List of spacecraft data us ed for spacecraft model [1 5]
Spacecraft name
Fuel identification
(ID)
Propellant
mass, kg
Payload
mass, kg
Structural
mass, kg
Saturn V third stage
2
107,725
0
12,014
Apollo command module (CM)
3
0
524
4,841
Apollo service module (SM)
3
18,413
60
6,053
Apollo LM DS
3
8,804
500
2,770
Apollo LM AS
3
2,358
250
1,719
Lunar crew exploration vehicle (CEV) CM
6
363
500
8,034
Lunar CEV SM
5
7,222
0
3,027
Altair DS
2
28,932
2,200
6,182
Altair AS
4
5,257
100
4,964
Earth departure stage (EDS)
2
226,693
0
22,500
Altair Cargo Carrier
0
0
15,000
1,000
Soyuz TM
3
900
255
7,250
Soyuz TMA
3
900
355
7,220
Progress M
3
900
2,350
7,450
Progress M1
3
900
1,800
7,150
Zvezda service module
0
0
10,000
20,000
STS-stage 2 (orbiter)
2
12,412
18,000
78,498
Soyuz-stage 2 (upper)
1
22,845
0
2,355
Proton-stage 2
3
46,562
0
4,115
ISS CEV CM (3 crew + cargo)
6
2,000
400
8,008
ISS CEV CM (6 crew)
6
2,000
0
8,079
ISS CEV (pressurized cargo)
0
0
3,500
7,683
ISS CEV SM
4
2,033
0
3,997
ATV: Automated transfer vehicle
5
2,613
5,500
10,470
HTV:H-II Transfer Vehicle
5
2,000
6,000
10,000
ISS CEV CM Prop
5
2,000
400
8,008
EDS (75 mt)
2
129,500
0
19,986
Table A2 List of fuel and corresponding parameters used for spacecraft model [15]
Fuel Type
ID
, s
α
LOX/keros ene
1
330
0.045
LOX/LH2
2
420
0.079
N2O4/UDMH
3
310
0.08
LCH4/LOX
4
318
0.958
MMH/N2O4
5
307
0.226
GOX/Ethanol
6
300
3.9353
31
Acknowledgements
We would like to t hank Patrick Sears and T ristan Sarton Du Jonchay for t heir reviews and the t houghtful suggestions for
improvement.
Reference
[1] Ishimatsu, T., de Weck, O. L., Hoffman, J. A., Ohkami, Y. , and Shishko, R., "Generaliz ed Multicommodity Network Flow
Model for the Earth–Moon–M ars Logistics System," Journal of Spacecraft and Rocket, Vol. 53, No. 1, Jan. 2016, pp . 25-38.
doi: 10.2514/1.A33235
[2] Ho, K., de Weck, O. L., Hoffman, J. A., and Shishko, R., “Dynamic M odeling and Optimization for Space Logistics Using
Time-Exp anded Networks,” Acta Astronautica, Vol. 105, No. 2, Dec. 2014, p p. 428-443.
doi: 10.1016/j.actaastro.2014.10.026
[3] Taylor, C., Song, M ., Klabjan, D., de Weck, O. L., and Simchi-Levi, D., “A M athematical Model for Interp lanetary Logistics,”
Logistics Spectrum, Vol. 41, No. 1, Jan-Mar 2007, pp . 23-33.
[4] Komar, D. R., Hoffman, J., Olds, A., and Seal, M ., “Framework for t he Parametric Sy st em Modeling of Space Exploration
Architectures,” AIAA Space 2008 Conference & Exposition, AIAA 2008-7845, San Diego, CA, Sep. 2008.
doi: 10.2514/6.2008-7845
[5] Arney, D. C. and Wilhite, A. W., “Modeling Sp ace System Architectures with Graph Theory,” Journal of Spacecraft and Rockets,
Vol. 51, No. 5, M ar. 2014, p p. 1413-1429.
doi: 10.2514/1.A32578
[6] Oeftering, R. C.,” A Cis-Lunar Propellant Infrastructure for Flexible Path Exploration and Space Commerce”, AIAA SPACE
2011 Conference & Exposition, AIAA 2011-7113, Long Bench, CA, Sep. 2011, ,.
doi: 10.2514/6.2011-7113
[7] Schreiner, S. S., "Molten Regolith Electrolysis Reactor M odeling and Optimization of In-Situ Resource Utilization Sy stems,"
M.S. T hesis, Aeronautics and Astronautics Dept., MIT, Cambridge, MA, 2015.
[8] Howell, J. T., M ankins, J. C., and Fikes, J. C., ”In-Sp ace Cry ogenic Prop ellant Depot Step ping Stone,” Acta Astronautica,
Vol. 59, No. 1-5, Jul.-Sep. 2006, p p . 230-235.
doi: 10.1016/j.actaastro.2006.02.019
[9] Notardonato, W., ” Active Control of Cry ogenic Prop ellants in Space”, Cryogenics, Vol. 52, No.4-6, Apr.-Jun. 2012, pp. 236-
242.
doi: 10.1016/j.cryogenics.2012.01.003
[10] Liggett, M. W., ” Space-Based LH2 Propellant Storage Sy st em: Subscale Ground Testing Results”, Cryogenics, Vol. 33,
No.4, Apr. 1993, p p . 438-442.
doi: 10.1016/0011-2275(93)90174-M
[11] Tay lor, T., Kistler, W. P., and Citron, B., ”On-Orbit Fuel Dep ot Deployment, Management and Evolution Focused on Cost
Reduction,” AIAA SPACE 2009 Conference & Exposition, AIAA 2009-6757, Pasadena, CA, Sep. 2009.
doi: 10.2514/6.2009-6757
[12] Woodcock, G. R., ”Logistics Supp ort of Lunar Base,” Acta Astronautica, Vol. 17, No. 7, Jul. 1988, pp. 717-738.
doi: 10.1016/0094-5765(88)90186-5
[13] Charania, A. C., and DePasquale, D., ”Economic Analysis of a Lunar In-Situ Resource Utilization (ISRU) Prop ellant
Services Market”, 58th International Astronautical Congress, IAC-07-A5.1.03, Hyderabad, India, Sep. 2007.
[14] Zegler, F. and Kutt er, B., ”Evolving to a Depot -Based Sp ace Transportation Architecture”, AIAA SPACE 2010 Conference &
32
Exposition, AIAA 2010-8638, Anaheim, CA, Sep. 2010.
doi: 10.2514/6.2010-8638
[15] Taylor, C., “Integrated Transportation System Design Optimization,” Ph.D. Dissertation, Aeronautics and Ast ronautics Dept.,
MIT, Cambridge, M A, 2007.
[16] Ho, K., “Dynamic Network Modeling for Spaceflight Logistics with Time-Expanded Networks,” Ph.D. Dissertation,
Aeronautics and Astronautics Dept., MIT, Cambridge, MA, 2015.
[17] Ishimatsu, T., "Generalized Multi-Commodity Network Flows: Case Studies in Space Logistics and Complex Infrastructure
Systems," Ph.D. Dissertation, Aeronautics and Astronautics Dept., MIT, Cambridge, M A, 2013.
[18] Ho, K., de Weck, O. L., Hoffman, J., and Shishko, R., “Campaign-Level Dynamic Network Modelling for Sp aceflight Logistics
for the Flexible Path Concept,” Acta Astronautica, Vol. 123, Jun.-Jul. 2016, p p . 51-61.
doi: 10.1016/j.actaastro.2016.03.006
[19] "Atlas V Launch Services User’s Guide," United Launch Alliance,
http://www.ulalaunch.com/uploads/docs/AtlasVUsersGuide2010.pdf [retrieved 31 M ay 2017].
[20] Miki, Y., Abe, N., Matsuyama, K., M asuda, K., Fukuda, N., Sasaki, H., "Development of the H-II Transfer Vehicle (HTV),"
Mitsubishi Heavy Industries Technical Review, Vol. 47, No. 1, Mar. 2010, pp. 58-64.
[21] de la Bourdonnaye, O. and Kinnersley, M., "The Automated Transfer Vehicle – A Valuable Asset for ISS Logistics," AIAA
SPACE 2010 Conference & Exposition, AIAA 2010-8620, Anaheim, CA, Sep. 2010.
doi: 10.2514/6.2010-8620
[22] Schreiner, S. S., Sibille , L., Dominguez, J. A., Hoffman, J. A., Sanders, G. B., and Sirk., A. H., “Development of a Molten
Regolith Electrolysis Reactor M odel for Lunar In-Situ Resource Utilization,” 8th Symposium on Space Resource Utilization,
AIAA SciTech Forum, AIAA 2015-1180, Kissimmee, FL, 2015.
doi: 10.2514/6.2015-1180
[23] Vielma, J. P., Ahmed, S., and Nemhauser, G., “Mixed-Integer Models for Nonseparable Piecewise Linear Optimization:
Unifying Framework and Ext ensions,” Operations Research, Vol. 58, No. 2, Oct. 2009, pp. 303-315.
doi: 10.1287/opre.1090.0721
[24] Vielma, J. P., Nemhauser, G. L., “M odeling disjunct ive constraints with a logarithmic number of binary variables and
constraints,” Mathematical Programming, Vol. 128, No. 1, Jun. 2011, pp. 49-72.
doi: 10.1007/s10107-009-0295-4
[25] “Apollo 17 Mission Report,” Apollo 17 M ission Evaluation Team, NASA Lyndon B. Johnson Space Center, JSC-07904,
Houston, TX, March 1973.
[26] Wickman, L., Nota, B., Keates, S., “Lunar Life Support System Study: Metabolic Energy and Water Considerations,” Space
2004 Conference and Exhibit, AIAA 2004-6025, San Diego, CA, Sep. 2004.
doi: 10.2514/6.2004-6025
[27] Jones, H. W., “Ultra Reliable Space Life Support,” AIAA SPACE 2012 Conference & Exposition, AIAA 2012-5121, Pasadena,
CA, Sep. 2012.
doi: 10.2514/6.2012-5121
