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American Institute of Aeronautics and Astronautics
1
Analytical Optimization Method for Space Logistics
Zhengyu Chen
1
, Hao Chen
2
and Koki Ho
3
University of Illinois at Urbana-Champaign, Champaign, IL, 61820
Nomenclature
C = in-situ resource utilization (ISRU) infrastructure productivity, kg/kg plant/year
D = total mass to be delivered, including ISRU infrastructure and instruments, kg
G = gear ratio
K = total number of missions (i.e., stages of deployment)
M = ratio of infrastructure system mass over total delivered mass
= spacecraft payload mass, kg
= spacecraft structure mass, kg
= spacecraft propellant mass, kg
mPISRU,i = mass of propellant generated by ISRU infrastructure at the destination at stage i of deployment, kg
T = time horizon, years
= mission interval, years
= fraction of payload mass over initial spacecraft mass
= fraction of structure mass over initial spacecraft mass
= fraction of propellant mass over initial spacecraft mass
Subscripts
LEO = low-Earth orbit
IMLEO = initial mass in low-Earth orbit
ISRU = in situ resource utilization
1
Undergraduate Student, Department of Aerospace Engineering, 104 S. Wright Street. zchen131@illinois.edu.
2
PhD Candidate, Department of Aerospace Engineering, 104 S. Wright Street, AIAA Student Member.
hchen132@illinois.edu.
3
Assistant Professor, Department of Aerospace Engineering, 104 S. Wright Street, AIAA Member.
kokiho@illinois.edu.
American Institute of Aeronautics and Astronautics
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I. Introduction
USTAINABLE space exploration is becoming increasingly important as we travel beyond Earth orbit, and various
in-space resource logistics infrastructure technologies have been developed to support long-term space campaign
concepts; examples include in-situ resource utilization (ISRU) plants and propellant depots. These logistics
infrastructure elements can be massive and costly, and thus we need an effective and efficient strategy to deploy and
utilize them; such a mission design and planning problem tends to involve complex numerical optimization. This Note
aims to demonstrate the value of analytical optimization approaches as a quick approximation for solving complex
space logistics design and planning problems.
Literature has shown that multi-stage deployment (i.e., staged deployment) of in-space infrastructure systems can
achieve higher mass and cost efficiency than single-stage deployment. Oeftering proposed a concept of
“bootstrapping” deployment of space infrastructure [1]. In this concept, instead of deploying all infrastructure
elements at once, we deploy some infrastructure elements first and deploy more later utilizing the previously deployed
infrastructure. Ho et al. mathematically formulated this deployment strategy problem using time-expanded generalized
multi-commodity network flow (GMCNF) optimization [2-4]. This numerical optimization model can be used to
identify the best strategy for deployment and utilization of space infrastructure elements. For example, Ref. [2]
numerically showed the effectiveness of bootstrapping staged deployment of ISRU plants, where the resources from
the previously deployed ISRU plants are transported back to the Earth orbit and used to deploy more stages of ISRU
plants. Other numerical space logistics optimization and tradespace exploration methods have also been developed in
the literature to explore effective space mission design with in-space infrastructure [5-8].
However, one large limitation in those numerical optimization methods is the “black box” nature of the
optimization process. For a complex mission design problem, it is very difficult to have intuition and understanding
of the resulting optimal staged deployment concepts and the relationships among the parameters. For example, the
numerical optimization methods require the number of stages of deployment as an input, but we have little intuition
about what number of stages of deployment to choose (or whether there even exists such an optimal number of stages).
With the traditional numerical optimization methods, we can only find the optimal number of stages of deployment
by running numerous optimizations repeatedly with various numbers of stages, which can be computationally
expensive.
S
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This Note is one of the first attempts to analytically model and optimize the staged deployment strategy for space
infrastructure, particularly focusing on bootstrapping deployment of ISRU. We mathematically derive the conditions
where there exists an optimal number of stages as well as the analytical expression for that number. We validate our
analytical results against numerical optimization, and demonstrate its effectiveness. Utilization of the resulting
analytical conditions and expressions can provide a quick computationally efficient approximation of the optimal
deployment strategies of in-space infrastructure.
II. Analytical Model for Infrastructure Deployment
A. Problem setting
We consider a space mission design problem to deliver a certain mass [kg] of commodities including ISRU
plants and scientific instruments from the origin (e.g., low-Earth orbit (LEO)) to the destination (e.g., the Moon) in
missions (i.e., stages) over a time horizon [years] (i.e., the missions happen every
[years]). Only
these two generic types of commodities, ISRU plants and instruments, are considered for simplicity, and all the
supporting subsystems (e.g., power, thermal, structure, …) are included in either of them. Over the total mass , the
ratio of ISRU mass to be delivered is denoted as ; thus, [kg] of ISRU plants and [kg] of
instruments are delivered to the destination. We make an assumption
; this means that the mass of ISRU
plants is less than or equal to the mass of instruments, which is reasonable in many space missions.
The cost metric is defined as the total launch mass needed from the origin to deliver that mass leveraging ISRU
and reusable spacecraft structure. Note that although the total launch mass is often used as a measure of the space
mission cost, a higher-fidelity cost model is needed for realistic mission design. In the considered bootstrapping
concept, we first launch ISRU plants in stages, and then launch instruments using the generated propellant from ISRU.
One example of a bootstrapping strategy is shown in Fig. 1. After every mission of delivery to the destination (i.e.,
outbound trip), the spacecraft used for delivery stays at the destination; right before the next mission, part or all of that
spacecraft is used to deliver the propellant generated from ISRU back to the origin (i.e., inbound trip). These retrieved
ISRU-generated propellant and reusable spacecraft from the destination are combined with the additional propellant
and spacecraft from the origin to deliver the next stage of ISRU plants and/or instruments to the destination. This
concept is expected to reduce the total launch mass as we are leveraging the deployed ISRU infrastructure and reusable
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spacecraft for further deployment. The more propellant we can gain from the deployed ISRU, the less propellant we
need to launch from the Earth.
Figure 1. Concept of a bootstrapping strategy
A linear and continuous spacecraft model is utilized with the ratio between the payload mass, and the structure
mass, and the propellant capacity as , where. We also assume that any part of the structure mass
can be reused. This linear approximation, although commonly used in the literature [2,3,5], may not result in a realistic
spacecraft design; the impact of this approximation will be evaluated in Section IV compared with the numerical
optimization model with a realistic spacecraft design. With this inert mass fraction
and the gear ratio
, we can derive a well-known relationship in Eq (1).
(1)
Note that this relationship in Eq (1) is only true for the outbound trips. For the inbound trips, our payload bay is empty;
instead we are carrying back the propellant generated by ISRU in the tank (i.e., propellent cannot be carried in the
payload bay, although part of it is the “payload” to transport back).
The ISRU infrastructure plays an important role in the problem. After a stage of ISRU plant is deployed at the
destination, it can generate the propellant we need for transportation. In our study, we assume that ISRU has a
productivity [kg/year/kg plant], which means the infrastructure will generate kg of resources per kg infrastructure
every year. This is a common method to evaluate in-space resource generation plant in the previous research [2-5].
Also, to simplify the model, we do not consider the maintenance or degradation of the infrastructure. Additionally,
transportation time is assumed as negligible compared with the time horizon for staged deployment (i.e., days vs.
years).
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B. Cost for staged deployment strategy
This section shows calculation of the cost metric for our bootstrapping strategy. We deliver
[kg] at each
stage every
[years]. At stage , we deliver ISRU plants; At stage , we deliver
instruments. If is not an integer, we deliver both ISRU plants and instruments at stage , where is
a ceiling function.
If there is no ISRU, then the cost for each stage of deployment is simply
; however, after the first stage
of deployment, we want to take advantage of ISRU to reduce this cost. Namely, right before every trip at stage
of deployment, we want to first retrieve (i.e., transport back) the propellant generated by the ISRU plants already at
the destination. Denote the available propellant at the destination at stage of deployment as When
, the mass of the ISRU plants already at the destination is [kg]; when, it is
[kg] (i.e., all ISRU plants). If we assume that all resources generated before stage of deployment are consumed
or discarded due to the limit of tankage, the amount of available resource at the destination at stage of deployment
is that generated in one mission interval [years], which corresponds to [kg]
when or [kg] when.
In order to use this propellant generated at the destination, we need to transport it back to the origin using the
spacecraft whose propellant tank capacity is
. During this inbound trip, we need to consider two cases:
(i) meaning we have more (or equal) propellant available than our tank capacity; (ii) ,
meaning we have less propellant available than our tank capacity. For case (i), we can only carry full tank back to the
origin regardless of how much propellant is available there; thus the structure and propellant mass at the destination
for the inbound trip becomes the same as those for the outbound trip
[kg], and thus we retrieve
[kg] of structure and propellant at the origin. For case (ii), we can retrieve all the available propellant
[kg] back to the origin, which, combined with the structure mass needed to carry it, becomes
[kg] at the destination, and thus we retrieve
[kg] of structure and propellant at the origin. In summary,
the cost function for each stage of deployment is
American Institute of Aeronautics and Astronautics
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(2)
Thus, the total cost becomes
. With the above problem setting, we are able to find the analytical solution
for the optimal number of stages we need for deployment using the analytical model including its existence condition.
III. Analytical Results
This section discusses the result we get from the analytical model including the expression for the optimal number
of stages of deployment and the approximate cost.
Firstly, using Eq. (2), we define the optimal number of stages as:
(3)
which means is the number of stages with the lowest cost using a bootstrapping strategy. Note that, although not
explicitly shown, the cost function in Eq. (2) contains and , which depend on .
Noting that is a monotonically (weakly) increasing function of i, we need analyze three different possible
cases:
• Case I: : The ISRU-generated propellant is enough to fill the spacecraft tank for
all stages after stage 1 at the destination.
• Case II: : The ISRU-generated propellant is never enough to fill any spacecraft
tank at the destination.
• Case III: Neither of
the above happens; the ISRU-generated propellant is enough to fill the spacecraft only on or after some stage
of deployment. Note that because is constant when.
The following analyzes the cost expression for each case.
Case I: As is a monotonically increasing function of i, the condition for Case I implies for
. This condition can be written as
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Noting that
and
, and defining
, we can rewrite the inequality as:
Combing this inequality with , we can acquire the following conditions on for Case I. Note that the last
condition never leads to Case I because .
Summing the cost for each stage of deployment using Eqs (2) and (3), the cost function for Case I is:
(4)
This cost function is a monotonically decreasing function of.
Case II: As is a monotonically increasing function of i, the condition Case II implies for
. This condition can be written as
Noting that
,
, and , and using defined above, we can obtain the following conditions on
for Case II.
The first condition never leads to Case II because cannot be negative, and the last condition corresponds to .
Summing the cost for each stage of deployment using Eqs (2) and (3),
(5)
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Evaluation of this summation is not trivial as is not necessarily integer. In order to reach a simple analytical
solution, we derive the expression of the cost function assuming as an integer (e.g.,
), and then use that expression as an approximation for the general case; the later validation shows that
this approximation is effective. With this approximation, the cost function for Case II is:
(6)
As , this cost function is a monotonically increasing function of.
Case III: This case corresponds to when neither of Case I or Case II happens. This corresponds to the conditions:
In these cases, the stage cost is never limited by the spacecraft tank until stage of deployment, but at this stage or
later, the propellant we can retrieve is limited by the tank size. In order to find this , we need to solve the equation:
for. Noting that , and relaxing the integer constraint for simplicity, we get
Summing the cost for each stage of deployment using Eqs (2) and (3),
(7)
Applying the same approximation as in Case II for in the summation range, and noting that
, the cost
function for Case III is:
(8)
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Since in Case III, and
, we can derive , and thus we can conclude that this cost function
is a monotonically decreasing function of.
Summarizing all three cases, we can acquire the relations as shown in Fig. 2.
Figure 2. Summary of the relationship between each case of staged deployment strategy.
In summary, the analytical solution for the optimal K with minimum cost is as follows:
- If, the larger is, the lower the cost is.
- If
,
.
- If
, .
This analytical model can not only provide a quick simple formula for the optimal number of stages of deployment,
but also offer various intuition into the relationships of parameters. For example, by examining that a larger
implies more capable ISRU, we can find that when ISRU is too capable, we prefer more stages because we want to
take advantage of bootstrapping deployment of ISRU. We can also easily examine the sensitivity of the results against
different parameters. This model can be used for initial analysis prior to more complex space mission design.
IV. Validation against Numerical Optimization
In this section, we implement a case study for Cis-lunar space mission using the analytical optimization model and
validate it against the numerical optimization [4]. We use the initial mass in LEO (IMLEO) as the cost metric in this
example; thus the origin is LEO and the destination is the Moon.
For validation, we compare the cost estimate and the optimal number of stages between the analytical and
numerical optimization models with the parameters in Table 1. Two parameters are chosen to vary to demonstrate the
Condition
Corresponding case for each range of K
: Case I
[monotonically
decreasing]
: Case III
[monotonically decreasing]
: Case I
[monotonically
decreasing]
: Case III
[monotonically decreasing]
: Case II
[monotonically increasing]
: Case I
[monotonically decreasing]
: Case II
[monotonically increasing]
: Case II
[monotonically increasing]
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broad applicability of the proposed methods: (1) ISRU productivities; (2) inert mass fraction. These parameters are
chosen because they are key parameters for ISRU system design and spacecraft design.
Table 1. Validation parameters
Parameter
Assumed value
Spacecraft propellant capacity
70,000 [kg]
Inert mass fraction
0.05 – 0.2
Gear ratio G
4.61
Maximum number of available spacecraft
As many as needed for each mission
Propellant
LOX/LH2
Total ISRU plant demand on the Moon
50,000 [kg]
Total Instrument demand on the Moon
100,000 [kg]
Time horizon T
2 [years]
ISRU Productivity C
50, 5, 4, 1 [kg/kg plant/year]
ISRU Resources
Propellant (both LOX and LH2)
First, four cases of ISRU productivity are tested, each of which corresponds to each of the four conditions about
in Fig. 2. (Note that is a function of .) For this analysis, the spacecraft structure mass of 5,000[kg] (i.e.,
inert mass fraction 0.067) is used. To find the optimal number of stages in the numerical optimization, 12 optimization
cases are tested with different numbers of stages from 2 to 13. Table 2 summarizes the validation results. We can
observe that the analytical optimization model underestimates the IMLEO by at most 7% compared with the numerical
optimization model. This difference is partly due to the optimistic linear modeling of spacecraft; the numerical
optimization considers the integer numbers of spacecraft (i.e., we cannot reuse part of the spacecraft only), whereas
the analytical optimization model assumes that any part of the structure mass can be reused. Furthermore, the optimal
number of stages of deployment estimated by the analytical model matches with the numerical optimization well. This
demonstrates the effectiveness of the analytical model.
Table 2. Summary of the comparison for each condition of the bootstrapping strategy.
ISRU
Productivity
[kg/kg
plant/year]
Corresponding
Condition
Analytical Optimization
Model
Numerical Optimization
Model
Relative
error
IMLEO
[kg]
Optimal # of
Stages
IMLEO
[kg]
Optimal # of
Stages
50
776,560
or lower
No finite optimal
number
789,089
or lower
13
1.6%
5
820,138
3
852,697
3
3.8%
4
817,185
3
874,034
3
6.5%
1
906,313
2
941,297
2 or 3
3.7%
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In addition, the results of the analytical optimization model are also validated at different inert mass fractions. In
this comparison, 16 analytical optimization cases are tested with different inert mass fractions from 0.05 to 0.2 and
their results are compared with the numerical optimization. In the numerical optimization, the spacecraft propellant
capacity is fixed to be 70,000 [kg] but the structure mass is varied depending on the chosen inert mass fraction. The
results in the case of ISRU productivity = 1 [kg/kg plant/year] with 2 stages of deployment (which is the optimal
number of stages in this case) is shown in Fig. 3. The differences in the IMLEO results between the analytical and
numerical optimization models are within 10% for all cases. Thus, the analytical optimization model approximates
the numerical optimization results well at a wide range of inert mass fractions.
Figure 3. Comparison of analytical optimization results and numerical optimization results at different
inert mass fractions
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V. Conclusion
This paper proposes an analytical model to predict the cost and the optimal number of stages for an in-space
infrastructure staged deployment strategy. The developed analytical model for ISRU deployment is demonstrated to
predict the IMLEO reasonably well and estimate the optimal number of stages of deployment accurately without
needing to run the complex numerical optimization models many times. The analytical model is valuable as it can
provide a quick approximation for the key parameters and offer insights into the parameter relationships that the
“black-box” optimization process cannot. Future possible research directions include: (1) introducing a nonlinear
vehicle inert mass model; (2) considering a series of missions that deliver different mass and/or are unevenly-spaced;
and (3) employing alternative formulations such as maximizing the instrument mass given a fixed launch mass. We
hope that this Note will open up a new research direction in analytical optimization for space logistics campaign
planning and design.
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