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American Institute of Aeronautics and Astronautics
1
Space Transportation System and Mission Planning for
Regular Interplanetary Missions
Hao Chen
1
, Hang Woon Lee
2
, and Koki Ho
3
University of Illinois at Urbana-Champaign, Champaign, IL, 61820
This paper develops a computationally efficient and scalable mission planning
optimization method for regular space transportation missions, defined as a set of repeating
and periodic interplanetary transportation missions over a long time horizon after one or a
few setup missions. As more long-term manned missions to Mars are being conceptualized,
the need for a sustainable interplanetary transportation system has become increasingly
prominent. However, planning regular transportation missions with existing space mission
planning optimization formulations has a limitation in computational scalability in the time
dimension. The proposed partially periodic time-expanded network can address this
limitation of the past studies; it is shown to be computationally scalable and also capable of
generating solutions that are practically preferred. Properties of the proposed partially
periodic time-expanded network are analyzed, and a case study reveals that the total initial
mass in low-Earth orbit of regular missions approaches to the theoretical lower bound as the
number of transportation missions increases.
Nomenclature
= set of directed arcs
1
Ph.D. Candidate, Aerospace Engineering, Talbot Laboratory, MC-236, 104 South Wright Street, Urbana, AIAA
Student Member.
2
MS Student, Aerospace Engineering, Talbot Laboratory, MC-236, 104 South Wright Street, Urbana, AIAA Student
Member.
3
Assistant Professor, Aerospace Engineering, Talbot Laboratory, MC-236, 104 South Wright Street, Urbana, AIAA
Member.
American Institute of Aeronautics and Astronautics
2
= spacecraft payload capacity
c = cost coefficient vector
d = demand and supply vector
e = spacecraft design parameters
f = spacecraft fuel type (integer)
= network graph
= space mission cost
= gravitational acceleration on Earth (equal to 9.8 m/s2)
= concurrency matrix
= specific impulse
i = node index ()
= objective function
j = node index ()
= total number of regular space missions
= space mission index
= number of concurrency constraints
= spacecraft propellant capacity
= upper bound of propellant capacity
= set of nodes
= number of commodity types
= transformation matrix
s = spacecraft structure mass
= set of time steps
= length of setup phase
= length of each cycle in periodic phase
t = time index (integer)
= set of spacecraft
= spacecraft-type index
American Institute of Aeronautics and Astronautics
3
= set of time windows
x = commodity flow vector
y = spacecraft flow variable (integer)
= structural fraction
= propellant mass fraction
I. Introduction
s more government and commercial players show interest in Mars exploration, an affordable regular
interplanetary transportation system for the Earth-Moon-Mars system has become increasingly important for
sustainable human space exploration. One example of such a system is the regular cargo route to Mars proposed
by SpaceX using fully reusable rockets [1]. These space missions are designed to support the setup and periodic
resupply for future Mars colonization in a similar way as the logistics vehicles for the International Space Station
(ISS) resupply. Meanwhile, reusable vehicles and in-orbit infrastructure technologies such as propellant depots and
in-situ resource utilization (ISRU) plants in the cis-lunar and Martian system have been proposed to support space
exploration sustainably [2]. However, designing a long-term regular space transportation system involves greater
complexity than designing a short-term single mission. Particularly, when we take into consideration the deployment
and operation of in-orbit infrastructures that are used over many missions (e.g., ISRU plants), the analysis needs to
consider a long time horizon and thus requires extensive and sometimes prohibitive computational effort with
conventional methods.
Many past studies have been conducted on the topic of space logistics. The space logistics optimization models
such as the Interplanetary Logistics Model [3], the Exploration Architecture Model for In-Space and Earth-to-Orbit
(EXAMINE) tool [4], SpaceNet [5], and graph theory-based Space System Architectures Model [6] have been
proposed for space logistics mission planning. However, these studies either lack or have limited global optimization
capability for complex mission design problems that involve deployment and utilization of propellant depots and
ISRU plants. Moreover, these models were not formulated to consider the mission interdependency at the multi-
mission campaign level. In response to this background, Ishimatsu et al. proposed a Generalized Multi-Commodity
Network Flow (GMCNF) to optimize the space mission planning problem globally based on a static network [7].
A
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However, due to the linear nature of the static network, vehicles cannot be tracked through the network and the size
of vehicles may change during the mission. Cornes et al. improved the optimization fidelity through mixed-integer
linear programming (MILP) and a tracking algorithm [8]. To take into account space mission planning, spacecraft
design, and in-orbit infrastructure design concurrently, Ho and Chen further improved the GMCNF model considering
the time dimension (i.e., time-expanded GMCNF) and proposed an integrated space logistics mission planning
framework [9, 10]. This framework enabled nonlinear spacecraft and in-orbit infrastructure design through the
piecewise-linear function approximation. It also provided an insight into the significant reduction in the initial mass
in low-Earth orbit (IMLEO) through a campaign-level design rather than a mission-level design. Although these
existing mission planning formulations consider the deployment and operation of the in-orbit infrastructures
concurrently, they have significant limitations in time dimension scalability, and thus cannot be applied to long-term
space transportation design. In addition, these formulations are not suitable to find a sequence of repeating space
transportation missions.
On the other hand, time-expanded networks (TEN) have been widely used in different fields, including vehicle
and crew scheduling [11], airline fleet assignment [12-14], and ground transportation [15]. Among these works, fully
periodic TEN was proposed to solve long-term repeating mission planning problems. It has been utilized in truck fleet
sizing problems [15] and airline fleet assignment problems [12-14]. However, to take advantage of space resources
and reduce space exploration mission cost, a space infrastructure deployment setup phase is required before the
mission entering the repeating transportation phase [10]. Therefore, fully periodic TEN cannot be used in the
campaign-level space mission design directly. A similar problem to the setup phase in space missions can be found in
the literature on airline schedule recovery problems [16, 17], where TEN is used to consider a period of irregular
mission planning and then switching to a periodic repeating phase. However, the airline schedule recovery problems
mainly focus on the recovery period and do not consider the subsequent schedule optimization in the periodic phase
simultaneously.
In response to this background, this paper proposes an interplanetary transportation system mission planning
optimization framework by constructing a partially periodic TEN. The model is constructed based on Chen and Ho’s
previous space logistics optimization framework [9, 10] and has been reformulated to incorporate the scalability of
the regular transportation missions to make the mission planning formulation computationally preferred. This new
mission planning framework can also find repeating transportation solutions, which is preferred in practical missions.
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Thus, the proposed mission planning framework can provide a practical and computationally efficient method for
long-term interplanetary transportation mission planning.
The rest of the paper is organized as follows: Section II introduces the partially periodic GMCNF model and its
properties. Section III evaluates the performance of the partially periodic GMCNF utilizing a long-term Earth-Mars
transportation mission case study and compares it with the static and the fully time-expanded GMCNF models. Section
IV discusses the contribution and conclusion of this paper.
II. Methodology
A. Generalized Multi-Commodity Network Flow
In response to the emerging paradigm shift in space exploration from a set of isolated missions to an intricately
linked campaign, Ishimatsu proposed the GMCNF model to consider the interdependency among space missions in a
long-term space exploration campaign [7]. It models space missions as multi-commodity flows in a network, where
nodes correspond to orbits, planets, or celestial objects, and arcs correspond to the trajectories connecting the nodes.
The spacecraft, crew, scientific instruments and other kinds of payload (e.g., water, food, oxygen, propellant) are
considered as the commodities flowing along the arcs. An example network model for the Earth-Moon-Mars
transportation system is shown in Fig. 1.
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Fig. 1 An Earth-Moon-Mars logistics network graph.
The original GMCNF model proposed by Ishimatsu used a static network, which considered spacecraft design and
space infrastructure design linearly; it also considered all commodities as continuous variables. Thus, the original
static GMCNF model proposed by Ishimatsu is a linear programming formulation. To improve the optimization
fidelity in space logistics mission planning, Chen and Ho [10] proposed a piecewise-linear approximation approach
and big- method to convert the nonlinear spacecraft and space infrastructure design models into a mixed-integer
linear programming formulation. Therefore, the space logistics mission planning and spacecraft design can be solved
with higher fidelity as a MILP problem.
The following shows the mathematical formulation of static GMCNF including the spacecraft design model.
Consider a network that is made up of a set of nodes and a set of direct arcs . Each arc has an index to
show the commodity flow from node to node through spacecraft , where denotes the spacecraft type. There can
be multiple types of spacecraft design and multiple spacecraft for each type. The commodity flows are divided into
the outflow
and the inflow
. Corresponding cost vectors
are defined for each commodity outflow. If there
are types of commodities, both
and
are vectors. All commodity flows
are nonnegative. The
variable for the number of spacecraft,
, is considered separately from the general commodity flow variable
,
which makes it easier to consider mission planning and spacecraft design simultaneously. Corresponding cost vectors
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are also defined for spacecraft outflows. In each node , the commodity demand vector is defined based on the
specific space mission requirement, where a negative value means mission demand and a positive value means mission
supply. The demand vector is a vector. The spacecraft demand variable,
, is also defined separately.
Based on the aforementioned notations, the static GMCNF formulation, including the spacecraft design model, is
as follows:
Minimize:
(1)
Subject to:
(2a)
(2b)
(2c)
(2d)
(2e)
Table 1. Definition of indices, variables, and parameters
Name
Definition (dimension)
Network model
Node set
Arc set
Spacecraft set
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Index
Spacecraft index
Node
Variables
Commodity outflows/inflows. Commodities in
are considered as an integer
or continuous variables based on the commodity type. For example, the number
of crew members is an integer variable (i.e., ), while the other commodities,
such as propellant, payload, and human consumables, are considered as
continuous variables (i.e., ). ()
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)
Demand or supply of different commodities at each node. Demand is negative
while supply is positive. ()
Demand or supply of spacecraft at each node. (scalar)
Commodity transformation matrix. ()
Concurrency constraint matrix. ()
Table 1 lists the definition of network model sets, indices, variables, and parameters in the formulation. The
descriptions of objective function and constraints in the static GMCNF model are as follows.
American Institute of Aeronautics and Astronautics
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Eq. (1) is the objective function. Following the space mission cost metric used by Ishimatsu [7], the initial mass in
low-Earth orbit (IMLEO) is used as the mission cost metric for space missions in this formulation. As a widely
accepted mission cost metric, IMLEO has been used in many past space logistics studies [3, 7, 9, 10]. Although it is
useful to estimate the mission launch cost, it does not address the cost of spacecraft construction, spaceflight operation,
and infrastructure development. Furthermore, the increased number of vehicle flights and rendezvous in long-term
space missions can lead to reliability problems. Therefore, the resulting mission design of interplanetary space
transportation must be evaluated from different perspectives through multi-objective optimization. This paper,
however, focuses on the scalability issue in large-scale interplanetary transportation missions. IMLEO is used as the
single-objective to demonstrate the performance of the proposed method. Moreover, by changing the value and
definition of the cost matrix,
and
, other mission evaluation metrics can be considered as the objective if
necessary.
Eqs. (2a) and (2b) are the mass balance constraints, which means that the sum of commodity outflow and demand
is always lower or equal to the commodity inflow at node . Eq. (2c) is the commodity transformation constraint. The
propellant burning and propellant generation by ISRU are performed through this constraint. For example, the
impulsive propellant burning can be expressed as follows:
(3)
In Eq. (3), dry mass means the spacecraft structure mass. is the propellant mass fraction defined by the rocket
equation,
where is the change in velocity along the arc , is the specific impulse of
the propellant system, and is the standard gravity. Eq. (2d) is the concurrency constraint. It is a constraint coming
from the spacecraft capacity, such as the payload capacity and the propellant capacity. Assuming that the number of
concurrency constraint considered is , then , which is the concurrency constraint matrix, is a matrix. Eq.
(2e) is the spacecraft design model considered in network optimization. In the expression, is the spacecraft
structural mass; is the spacecraft design parameters, including payload capacity and propellant capacity; is the
type of fuel used by the spacecraft propulsion system. The spacecraft design model considered in this paper is
introduced in section III.B.
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The static GMCNF model does not consider the time dimension in the problem, which may lead to the “time
paradox”. For instance, the propellant from an ISRU plant may be used before it is generated [9]. To solve this issue,
Ho proposed a fully time-expanded network [9] to include the time dimension in the static GMCNF model. Using the
same notation as the static GMCNF model, the dynamic space logistics optimization formulation based on the fully
time-expanded GMCNF model is shown as follows:
Minimize:
(4)
Subject to:
(5a)
(5b)
(5c)
(5d)
(5e)
(5f)
This formulation is a fully time-expanded version of the static GMCNF model shown in Eq. (1) and Eqs. (2a)-
(2e). It is also a MILP formulation, where is the time index and is the set of time steps. Note that both t and Δt are
assumed to be integers in TENs, and holdover arcs are considered to have unit lengths. Compared with the static
GMCNF, there is an additional time window constraint in the fully time-expanded GMCNF, which is Eq. (5e). In Eq.
(5e), is the time window vector. It is a vector, where is the number of time windows for arc to . The
commodity flow is permitted only when the time window is open.
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The fully time-expanded GMCNF notably improves the space logistics optimization fidelity compared with the
static GMCNF model. However, it takes a significantly longer time to solve the problem compared to the static
GMCNF because of the additional time dimension. Numerical experiments in Ref. [10] showed that it takes about
2000 seconds to solve an optimization problem for campaign-level lunar mission design, containing three Apollo-like
short lunar missions, by Gurobi 6.5 solver on an Intel Core i7-4790 quadcore @3.6 GHz platform. When four lunar
missions are considered, it takes more than 200 hours to solve the problem. This is the time cost to optimize the
mission planning only considering the transportation missions within the cis-lunar system. If the subject of interest is
the Mars mission, the mission complexity and the time scale would increase much more significantly.
To solve a long-term space transportation mission planning problem efficiently while maintaining the model
fidelity, this paper takes advantage of the regular space transportation concept and develops a partially periodic TEN,
as introduced in detail in the next section.
B. Partially Periodic TEN
As shown in Fig. 2, the proposed logistics network consists of two general phases: an initial setup phase and a
periodic steady phase. The initial setup phase of the campaign is a one-time event that is dedicated to the construction
and deployment of the in-space infrastructure that would be later utilized and maintained during the periodic missions.
For instance, ISRU systems can provide resources to reduce mission cost and make transportation sustainable.
Propellant depots are in-orbit infrastructure systems that can refuel spacecraft passing by.
After a sufficiently long setup phase, the system enters a periodic steady-state phase, where the system repeats the
same transportation missions regularly. Here, we make the repeating period longer than the longest transportation arc
in the network. For example, for Earth-Mars missions, we can consider the repeating period as the launch window
cycle to Mars (i.e., ~780 days), which is longer than the longest transportation arc in the network (i.e., from Earth to
Mars ~210 days). Note that this assumption on the length of the repeating period can be relaxed easily if necessary.
With this repeating period, we can simplify the TEN into a partially periodic TEN, which only includes the setup
phase and one periodic cycle. The idea is to aggregate all the repeating periods into one periodic cycle and constrain
the inflow and the outflow of that cycle to be equal to each other. In this way, the mission planning solution for the
first regular cycle becomes identical to all the following cycles. As a result, after implementing the partially periodic
TEN, we only need to solve the mission planning problem for the setup phase and the first regular cycle. The result
obtained is then feasible for all future cycles.
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Fig. 2 Partially periodic TEN.
Assuming that the objective function of the setup phase is , while the objective function of the periodic steady
phase is . The total number of regular cycles considered in the optimization is . The mathematical formulation of
this partially periodic GMCNF model is shown as follows:
Minimize:
(6)
Subject to:
(7a)
(7b)
(7c)
(7d)
and
Eqs. (5a)-(5f)
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where is the time when the first regular cycle begins and is the length of a regular cycle. Eq. (6) is the objective
function. It is the sum of IMLEO of the setup phase and the regular transportation phase, which are calculated in Eqs.
(7a) and (7b) respectively.
Eqs. (7c) and (7d) are periodic flow constraints that constrain all the inflows to a regular cycle are equal to the
outflows, thus guaranteeing the repeating nature of the regular cycles. As shown in Fig. 2, outflows and inflows are
those commodity flows crossing the boundaries of regular cycles. A direct way to constrain cycling is adding terminal
constraints into the TEN,
(7c)
(7d)
where is the index of cycles. If we add Eqs. (7c) and (7d) into the time-expanded GMCNF problem, we solve the
problem shown in the upper half of Fig. 2. Eqs. (7c) and (7d) are terminal constraints that force the system to repeat
the same cycle. We can remove the redundant variables and constraints by solving the problem shown in the lower
half of Fig. 2 using Eqs. (7c) and (7d). With Eqs. (7c) and (7d), we are solving the partially periodic GMCNF problem
for the setup phase and the first regular cycle, as the commodity flow results in the first cycle are applicable for all
subsequent regular cycles.
Other commodity constraints (i.e., Eqs. (5a)-(5f)) are the same as the general fully time-expanded GMCNF model.
C. Properties of Partially Periodic TEN
1. Computational complexity
In this section, we discuss the computational complexity of the GMCNF models, which were formulated based on
different types of TENs.
Assuming that in a fully time-expanded network model there are different types of spacecraft, nodes,
commodity types, types of spacecraft concurrency constraints, and time steps, then the number of nodes and arcs
over the fully time-expanded GMCNF model are at most and , respectively. As a result, there are at most
variables and at most constraints. If we consider regular space missions, assuming a
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setup phase over time steps followed by transportation mission cycles each of which lasts over time steps,
, then, for a fully time-expanded network model, there are at most variables and at most
constraints. Following the same logic and notations, we can get the computational
complexity of the static network and the partially periodic network, as shown in Table 2.
Table 2. Computational complexity comparison
TEN Type
Variables
Constrains
Static
Fully time-expanded
Partially periodic
Compared with the static network, a caveat of the fully time-expanded network is that it requires a large number
of nodes and arcs, generated by the time dimension, which would lead to a large number of variables and constraints.
The complexity of the TEN is pseudo-polynomial. For the partially periodic network model proposed in this paper, as
shown in Eq. (6), Eqs. (7a)-(7d), and Eqs. (5a)-5(f), the computational complexity is much smaller than the fully time-
expanded network. With the increase of the number of cycles, the number of variables and constraints in the fully
time-expanded network increases linearly (with an offset due to the setup phase). However, the number of variables
and constraints in the partially periodic network stays the same as the increase in the number of cycles. Therefore, for
large-scale space transportation mission planning optimization problems (e.g., Earth-Mars transportation mission), the
partially periodic network has significantly less computational workload, thus greatly increasing the time dimension
scalability.
2. Bounds on Optimal Solutions
In this section, we discuss the bounds of the partially periodic network model solution. For the specific
minimization problem considered in this paper, we discuss the lower bound of the partially periodic network and the
relationship among static, fully time-expanded, and partially periodic network models.
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Consider the fully time-expanded GMCNF problem discussed earlier (shown in Eq. (4) and Eqs. (5a)-(5f)). The
following arguments show that it can be relaxed to a static GMCNF problem by aggregating nodes and arcs.
Since the cost coefficient
and
are constant over time, Eq. (4) can be written as
(8)
where
and
. They are the total commodity flows throughout the time horizon along
each arc.
Eqs. (5a) and (5b) can be aggregated over the time horizon and be relaxed to the following form,
(9a)
(9b)
where and
. They are the total commodity demand or supply throughout the time
horizon at each node.
Similarly, Eqs. (5c) and (5d) can be relaxed as follows:
(9c)
(9d)
The time window constraint Eq. (5e) is relaxed and eliminated through the aggregation of commodity flows
throughout the time horizon, thus eliminating the information about time windows.
The resulting formulation Eq. (8) with constraints in Eqs. (9a)-(9d) and the spacecraft design model Eq. (5f) leads
to a static GMCNF problem as a lower (relaxed) bound of the fully time-expanded GMCNF problem if both
formulations are feasible and bounded. This static formulation and the original fully time-expanded GMCNF
formulation are defined as the corresponding formulations of each other.
From the above derivation, the following important theorem has been proved.
Theorem 1. A lower bound of the optimal objective of a fully time-expanded GMCNF problem can be found by solving
its corresponding aggregated static GMCNF problem if both problems are feasible and bounded.
A more detailed proof and a generalization of Theorem 1 have been derived by Ho in Ref. [18].
From this theorem, it can be seen that a lower bound of a computationally expensive fully time-expanded GMCNF
problem can be found by a computationally cheaper aggregated GMCNF problem if both are feasible and bounded.
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In other words, , where is the optimal solution of the static GMCNF problem and is the optimal solution
of the fully time-expanded GMCNF problem, as shown in Eqs. (1) and (4), respectively.
Now, consider a partially periodic GMCNF problem (shown in Eq. (6), Eqs. (7a)-(7d), and Eqs. (5a)-5(f)). The
following arguments show that it can be relaxed to a fully time-expanded GMCNF problem.
First, consider the time-expanded GMCNF formulation with terminal constraints Eqs. (7c) and (7d). It is a cycling
time-expanded GMCNF problem shown in the upper half of Fig. 2. We can obtain its formulation by adding terminal
constraints (i.e., Eqs. (7c) and (7d)) into the fully time-expanded GMCNF problem. Additional terminal constraints
reduce the solution space of the optimization problem. Therefore, the fully time-expanded GMCNF problem provides
a lower (relaxed) bound of the cycling time-expanded GMCNF problem if both formulations are feasible and bounded.
Moreover, the cycling time-expanded GMCNF problem shown in the upper half of Fig. 2 is an equivalent problem
to the lower half by replacing the terminal constraints Eqs. (7c) and (7d) by Eqs. (7c) and (7d) and removing the
redundant variables and constraints. As a result, the fully time-expanded GMCNF problem is also a lower (relaxed)
bound of the partially periodic GMCNF problem [shown in Eqs. (6), (7a–7d), and (5a–5f)] if both formulations are
feasible and bounded. This partially periodic formulation and the original fully time-expanded GMCNF formulation
are defined as the corresponding formulations of each other. Note that the partially periodic GMCNF problem is not
necessarily feasible even when its corresponding fully time-expanded GMCNF problem is feasible.
From the above derivation, the following theorem has been proved.
Theorem 2. A lower bound of the optimal objective of a partially periodic GMCNF problem can be found by solving
its corresponding relaxed fully time-expanded GMCNF problem if both problems are feasible and bounded.
In summary, for the optimal solutions of the corresponding static, fully time-expanded, and partially periodic
GMCNF problems, expressed by , , and , there is a relationship: , if all of them are feasible and
bounded.
III. Case Study Results and Analysis
This section evaluates the performance of the partially periodic network by solving a long-term Mars space
transportation mission planning problem. The results of the partially periodic network are compared with the results
from the static and fully time-expanded networks.
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A. Mars Transportation Mission
Our case study considered in this paper is a Mars exploration with the cislunar transportation system. We model
the transportation network as a 7-node network. The and time of flight (TOF) are shown in Fig. 3. Since IMLEO
is the cost metric used in this paper, and TOF between Earth and LEO are not considered. Space mission time
windows, spacecraft flight and TOF are determined by interplanetary transportation trajectories. In this paper, we
use Hohmann transfer orbits as the transportation trajectories and the high-thrust liquid oxygen/ liquid hydrogen
(LO2/LH2) propulsion system in spacecraft for Mars transportation. The boiloff of liquid oxygen and liquid hydrogen
in space is also considered. As shown in Fig. 3, the spacecraft flight can take advantage of aerocapture to reduce the
propellant cost. The aeroshell and thermal protection system (TPS) can be reused in space. However, after the
spacecraft reentry to Earth or Mars, the aeroshell and TPS cannot be reused again; new aeroshell/TPS needs to be
launched. The mass of aeroshell/TPS is assumed as 40% of the total flight mass according to NASA Design Reference
Architecture 5.0 [19].
Fig. 3 Earth-Moon-Mars transportation network model.
The scenario considers a set of regular missions that transport cargo from Earth to Mars every 780 days when the
Earth-Mars system opens its repeating time windows. We consider the first Earth-Mars time window as the setup
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phase and each of the subsequent time windows as a regular repeating cycle. The mission demand and supply are
shown in Table 3.
Table 3. Demand and supply of Mars transportation mission
Payload Type
Node
Demand Time [day]
Supply [kg]
Payload
Mars
780 (repeating every 780 days)
-51,700 [18]
Payload, propellant, ISRU,
ISRU maintenance spares
Earth
All the time
+∞
The considered ISRU system is a lunar water ISRU whose productivity is assumed as 5 [kg propellant/ year/ kg
system]. The ISRU plant is supplied from Earth. To utilize the ISRU system, the transportation system must deploy
the ISRU plant first on the Moon or Mars. If some ISRU systems are in operation for the mission, the transportation
system also needs to supply its maintenance spares. We assume that ISRU maintenance requires the spares equal to
10% of system total mass every transportation mission (i.e., every 780 days). We also assume that two types of
spacecraft propulsive stages are designed and two spacecraft propulsive stages are available to be launched on Earth
for each mission. Spacecraft sizing is also part of the tradespace and is shown in detail in Section III.B. A summary
of mission parameters and assumptions is shown in Table 4.
Table 4. Summary of parameters and assumptions
Parameter
Assumed value
Propellant
LO2/LH2 [20]
420 [s] [20]
Type of spacecraft designed
2 types
Available spacecraft each mission
2 spacecraft each type
Aeroshell
40% of total mass [19]
Oxygen boil-off rate
0.016% [/day] [9, 10]
Hydrogen boil-off rate
0.127% [/day] [9]
Earth-Mars time window
780 [days] [19]
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ISRU productivity
5 [kg propellant/year/ kg system] [19]
ISRU maintenance
10% [/mission] [9, 10]
Moreover, to evaluate the effectiveness of a partially periodic GMCNF model, we need to set a baseline TEN
formulation for comparison. A naïve fully time-expanded network with uniform time steps (e.g., 1 day) would be
computationally expensive to analyze. In this paper, we use a cluster-based TEN as the baseline [9]. For the
interplanetary exploration mission, the system can be divided into several clusters (e.g., Earth/cis-lunar cluster and
Martian system), where only the transportation across the boundary of the clusters involves periodic time windows
and long flights. In a cluster-based TEN, we only consider the time window of the space flights between two clusters
and assume the short space flights inside each cluster as instantaneous transits. This method is known to give a good
approximation of the optimal solution of the naïve formulation with a reasonable computational effort [9]. In later
analysis cases, both the fully time-expanded GMCNF and the partially periodic GMCNF model are formulated based
on this cluster-based TEN formulation, although our theory developed in Section II applies to any other TEN
formulation as well.
B. Spacecraft design model
Our space logistics optimization includes spacecraft sizing as part of the tradespace. For the sizing model of
spacecraft, this paper utilizes a nonlinear regression model developed by Taylor [20] based on pre-existing spacecraft
elements. Although the original model includes the structure for payload, we do not consider that as a part of the
spacecraft design model in this paper. The payload is enclosed in aeroshell, which is considered separately. We are
only designing the propulsive stage using this spacecraft design model. Therefore, in the mission assumptions (shown
in Table 4), designing two types of spacecraft means designing two different sizes of propulsive stages. In addition,
due to the mixed-integer linear programming nature of the proposed optimization method, there is an underlying
assumption for the spacecraft flowing across the network: the spacecraft capacities can be additively combined and
the propulsive stages designed are modular and reconfigurable. In reality, the interoperability between the spacecraft
can be significantly more complex.
Denoting the spacecraft propellant capacity as , spacecraft fuel type as , and structure mass as , the spacecraft
propulsive stage structure mass is a function of propellant capacity and spacecraft fuel type, , shown as
follows [20]:
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10
where is the standard gravitational acceleration on Earth, 9.8 [m/s2]; is spacecraft impulsive burn time, set as
120 [s]; is the upper bound of spacecraft propellant capacity, assumed as 500,000 [kg] as in Ref. [20], which
defines the scope of application of this data-based spacecraft design model; is the specific impulse determined by
fuel type ; is the spacecraft structural fraction defined in Ref. [20], which is also determined by fuel type . In this
paper, LOX/LH2 is chosen as the spacecraft propellant as an example. As a result, according to Ref. 207], is 420
[s] and is equal to 0.079.
Although this spacecraft model is nonlinear, Chen and Ho developed a technique to convert the space mission
design problem with a nonlinear spacecraft model into a mixed-integer linear programming problem [10]. This
technique is based on a piecewise-linear approximation and big- method. We employ this approach in this paper as
well for efficient mission design optimization.
Note that, this paper uses the above nonlinear spacecraft stage design model as an example to illustrate the ability
of the proposed space logistics optimization method; our method can accommodate other spacecraft design models if
necessary. The focus of this paper is on the TEN method, and we do not claim the accuracy of the spacecraft design
model used.
C. Comparison of Optimization Formulations
This problem is solved in Python by the Gurobi 7.0 solver on an Intel Core i7-4790, Quad-Core 3.6 GHz platform.
The results of the average IMLEO for each regular space transportation mission (i.e., cycle) based on different network
models are shown in Fig. 4, and their computational times are shown in Fig. 5. The result from the partially periodic
GMCNF model is compared with the results from the static and fully time-expanded GMCNF models. The number
of missions in Fig. 4 and Fig. 5 corresponds to the number of regular cycles after the setup phase because each cycle
only contains one mission. Note that we only obtain the results for a fully time-expanded GMCNF model when the
number of space missions is less than four because the optimization becomes computationally infeasible with our
computational resource when the number of missions is large.
The results obtained from static, fully time-expanded, and partially periodic GMCNF models are compared in Fig.
4. As the number of regular space transportation missions (i.e., cycles) increases, the average mission IMLEO from
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both fully time-expanded and partially periodic GMCNF models are approaching to the result from static GMCNF
(i.e., lower bound). The result from fully time-expanded GMCNF model is slightly closer to the static GMCNF model
result compared with the partially periodic GMCNF model. The reason is that the fully time-expanded GMCNF model
does not have the periodic and terminal constraints, meaning that each period can have a different mission, while the
partially periodic GMCNF model assumes that the mission repeats infinitely. This also matches with our analysis in
Section II that proves that for the optimal objective, .
The comparison of computational time among three GMCNF models is shown in Fig. 5. We can find that as the
number of missions increases, the computational time of fully time-expanded GMCNF model increases dramatically
while the computational times of the partially periodic and the static GMCNF models remain at a low level. This
figure demonstrates that our partially periodic GMCNF model can achieve a better time dimension scalability than
the fully time-expanded GMCNF model.
In summary, we can see that the proposed partially periodic GMCNF model can provide a good approximation of
the fully time-expanded GMCNF model with a much lower computational cost.
Fig. 4 Mission IMLEO comparison among GMCNF formulations.
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Fig. 5 Problem-solving time cost comparison among GMCNF formulations.
D. Analysis of Partially Periodic GMCNF Results
With our developed partially periodic TEN formulation, we can analyze the long-term space campaign with regular
missions more computationally efficiently than traditional methods. This section analyzes the results from the partially
periodic GMCNF model.
The mission planning result of ten regular transportation missions to Mars is shown in Fig. 6. In the setup mission
phase, the ISRU system is deployed on the Moon. It starts to produce propellant on the Moon to support subsequent
space missions. When the second Earth-Mars flight time window is open, all spacecraft launched for Mars mission
fly to the LTO first. Then, some spacecraft fly to the Moon to transport the propellant produced on the Moon back to
the LTO. The spacecraft used to deploy the ISRU system in the setup mission phase assists this propellant
transportation. As a result, the propellant from Earth only needs to support the flights from the LTO to the LLO in the
Moon propellant transportation. Other spacecraft flights during this propellant transportation are covered by the
propellant generated on the Moon. After being refueled by the propellant from the Moon, the spacecraft that stay in
LTO temporarily during the propellant transportation fly to Mars. This regular space transportation mission is repeated
following the same mission planning and spacecraft design. This resulting solution can be preferred in practical
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missions to the global optimal solution with no repeating missions due to the simplicity of its repeating mission
architecture. The learning curve effect, which is not considered in this paper, can also be incorporated to further
identify the potential benefits of repeating space missions.
Fig. 6 Mars regular space transportation mission planning (10-mission case).
A sensitivity analysis of the campaign performance against ISRU productivity is shown in Fig. 7, which shows the
relationship between average mission IMLEO and ISRU productivity for different numbers of missions. It
demonstrates that a higher ISRU productivity can always lead to a lower mission cost. Moreover, the performance of
low-productivity ISRU is more sensitive to the total number of space missions. In Fig. 7, when ISRU productivity is
2 [kg/yr/kg system], the average mission IMLEO decreases more than 25% as the total number of transportation
missions considered increases from 1 to 15. However, the average mission IMLEO only decreases by about 14% when
the ISRU productivity is high (i.e., 8 [kg/yr/kg system]). This result shows that the impact of a large number of
missions is especially critical when the ISRU productivity is low. This type of analysis is not possible under previous
fully time-expanded network formulation for long-term space campaign due to the scalability issue but is possible
with our newly developed partially periodic TEN method.
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Fig. 7 Relationship between average mission IMLEO and ISRU productivity.
IV. Conclusion
This paper proposes a computationally efficient interplanetary space transportation mission planning optimization
framework by constructing a partially periodic time-expanded network. It takes advantage of the periodic nature of
regular space transportation missions, thus resulting in a practical and computationally efficient mission planning
method.
The analysis and result comparisons with the traditional static and fully time-expanded GMCNF models show
that the partially periodic GMCNF model could solve a long-term interplanetary mission planning problem
significantly faster than the previous mission planning frameworks while maintaining a reasonable level of fidelity.
This paper also derives the mathematical properties of the proposed partially static GMCNF formulation and its
relationship with the traditional static and fully time-expanded GMCNF formulation.
The proposed work can be useful for future large-scale space transportation campaign design, which contains
multiple interplanetary transportation missions. Moreover, the framework improves the mission planning
computational efficiency and enables a quick sensitivity analysis on space infrastructures and spacecraft in
interplanetary transportation missions. The proposed methodology can be further explored to consider the mixed
cargo/manned missions to Mars with consideration of life support systems.
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