A preliminary architecture optimization for in-space assembled telescopes
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Citation Sanchez, William David, et al., "A preliminary architecture
optimization for in-space assembled telescopes." 70th International Astronautical Congress (IAC), October 21-25, 2019, Washington, D.C.
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A Preliminary Architecture Optimization for In-Space Assembled Telescopes William D. Sanchez𝑎*, Keenan Albee𝑎, Rosemary Davidson𝑏, Ryan de Freitas Bart𝑏,
Alejandro Cabrales Hernandez𝑏, Jeffrey Hoffman𝑐
𝑎 Ph.D. Candidate, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, USA 02141
𝑏 Master’s Student, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, USA 02141
𝑐 Professor of the Practice, Department of Aeronautics and Astronautics, Massachusetts Institute of Tech- nology, 77 Massachusetts Avenue, Cambridge, MA, USA 02141
* Corresponding Author, [email protected]
Abstract
The current trend towards larger diameter space-based and ground-based telescopes reflects both im- provements in manufacturing technology and the need for more light-gathering capability. Although ground telescopes can continue to grow in diameter using previous manufacturing and assembly techniques, space- based telescope mirror diameters are limited by the fairing size of a single launch vehicle. Looking towards the future, the demand for larger diameter primary mirrors is expected to quickly outgrow the size of a single launch vehicle fairing. In this case, the only viable option for a larger diameter space telescope will be on-orbit assembly. This paper provides a preliminary framework to optimize the architectural trade-space of in-space assembled telescopes as well as a metric to quantify the relative cost of the designs. Key parameters driving the architecture of such a system were identified and enumerated. These include primary mirror segment size, raft (i.e., unit of segments ready for assembly) geometry and configuration, in-space assembly location, and launch vehicle selection. The results of the paper are presented through a Pareto Analysis which ultimately describes the optimal architecture against the trade-space considered. This includes de- sign of fuel-efficient trajectories generated from the Circular Restricted Three-Body problem for transfer of components to the assembly and mission locations (e.g., Earth-Moon L1, Sun-Earth L2). Furthermore, an optimization scheme is demonstrated for launch vehicle packing/manifesting with constraints on component selection, payload limitations for reaching the desired assembly point, and scheduling of launch vehicle and components.
Keywords: in-space, telescopes, assembly, packing, optimization Acronymns/Abbreviations
EML1 Earth-Moon Lagrange Point 1 iSAT in-Space Assembled Telescope OTA Optical Telescope Assembly PM Primary Mirror
SEL2 Sun-Earth Lagrange Point 2 SLS Space Launch System UoS Utility of Science VoA Value of Assembly
NRHO Near Rectilinear Halo Orbit
1 Introduction
The objective of this study can be stated succinctly as follows: to provide a preliminary framework to opti- mize the architectural trade-space of in-space assem- bled telescopes (iSAT), as well as a metric to quan- tify the relative complexity of the designs — in par- ticular, by developing a technique to optimize the number of segments into which a mirror of a given
diameter can be divided, using all available launch options. Ultimately, this study aims to contribute to a knowledge base and set of analysis tools that will one day inform decision makers on the feasibility and cost-benefit trade-offs of an iSAT mission in light of heritage monolithic and deployable observatories.
1.1 Motivation
In general, larger telescopes equate to better science, providing better spatial resolution, spectral coverage, and signal-to-noise ratio than their smaller counter- parts [1]. Furthermore, the signal-to-noise ratio of an astronomical observation over a specific time in- creases exponentially with aperture diameter, which translates into a reduction in integration time, in- creasing spectral resolution, and the ability to map a greater amount of the celestial sphere [2].
The scientific demand for these capabilities are expected to out quickly outpace the ability of engi- neers to pack the observatory into a single launch vehicle fairing. In this scenario, multi-vehicle launch
campaigns are necessary. The largest diameter tele- scope that can fit into the upcoming planned launch vehicle configurations is about 14 to 15m [3], mean- ing a 20m telescope will necessarily require multiple launches and in-space assembly. Furthermore, stud- ies of the James Webb Space Telescope (JWST) cost and schedule can be used anecdotally to inform con- siderations for an iSAT mission [4]. Although iSAT and JWST are quite different, their linking factor is the relative complexity of both systems compared to heritage missions. For JWST, designing the telescope to be deployable in order to fit inside the fairing, in addition to all other new technology and complex- ity drivers, had a major impact on mission cost and schedule. The lessons learned from JWST can be ap- plied to the iSAT study to ensure that all potential new and complex technologies are modeled in early mission planning stages.
1.2 Background
Traditionally, cost estimates for new observatory concepts are extrapolated from earlier missions us- ing parametric models such as those provided by Stahl [5], Bely [6], and Meinel [7, 8]. These models are formed using exhaustive amounts of information from previous space telescope missions, but to date, these missions are still relatively small in number.
While this is the current state-of-the-art for tele- scope cost estimation (without access to proprietary information), all of the programs used in this model are vastly different from any proposed iSAT tele- scope — they are less than half of proposed iSAT primary mirror diameters (∼ 20𝑚), and are not as- sembled in space. This costing method is further disqualified because there will be many more cost drivers than simply primary mirror diameter for an iSAT mission. Additionally, increased segmenta- tion and modularization, while helpful in reducing ground manufacturing costs, will have an unknown impact on assembly costs. Using this method to model the cost of an in-space assembled telescope would extrapolate these parametric relationships be- yond a reasonable scope. Furthermore, these rela- tionships are determined using primarily monolithic telescopes, at least for the space-based telescopes in the database. Even if ground-based segmented tele- scopes were used to add a segmentation relationship, this modeling method does not capture the amount of complexity - expected to be a major cost-driver for iSAT - associated with a large telescope that is assembled in orbit by robots or astronauts in space.
2 Methodology
A Pareto analysis was performed to evaluate and find optimal solutions to the architectural trade-space of iSA, the details of which are described below.
Limitless combinations of parameters can be used to define an architecture of an in-space assembled telescope. However, a Pareto analysis that is both well-defined and tractable necessitates careful selec- tion of a minimum set of parameters which uniquely and independently define an architecture, that also reflects the key cost drivers of the program. Ulti- mately, four of these “driving parameters” were cho- sen to satisfy the requirements at the depth of this study. These are, (1) the primary mirror diameter, (2) degree of segmentation of the primary mirror, (3) method of segmentation of the primary mirror (raft geometry), and (4) assembly location.
To evaluate and search for optimal architectures, a Pareto analysis assigns each configuration a benefit
— defined as utility of science (UoS) — and a cost.
Each architecture once evaluated is represented by a point on the Pareto graph, Figure 1. The point of maximum benefit and zero cost is named the utopia point, and though infeasible, serves as a useful ref- erence for the analysis. Iconically, the architectures making up the points the farthest left on the graph together make a curve — the Pareto front — repre- senting the optimal set of architectures for a given benefit. All architectures positioned to the right of the Pareto front are suboptimal by the metrics con- sidered.
Figure 1: Pareto graph example
For this study, benefit (UoS) is defined as the pri- mary mirror diameter of the architecture — reflect- ing the in-space assembly objective of maximizing light-gathering capability. The cost metric defined
for this analysis — normalized total complexity ¯𝐶 — was constructed to quantify the relative complexity and risk between successful architecture deployments.
Complexity 𝐶, reflects the degree of difficulty to as- semble the system once in space and is quantified by the number of connections between rafts needed to assemble the telescope primary mirror (see section 2.1). Of additional importance in evaluating the com- plexity and risk of an architecture is the total num- ber of launch vehicles 𝐿𝑉 required to transport all telescope components to their assembly destination.
Normalized Total Complexity for a given architec- ture is the weighted average of these two contributing costs normalized by their maximum value over all ar- chitectures evaluated. This relationship is expressed by Equation 1,
𝐶 = 𝛼¯ 𝐶
max 𝐶 + 𝛽 𝐿𝑉
max 𝐿𝑉 (1)
where 𝛼, 𝛽 ∈ [0, 1] such that 𝛼 + 𝛽 = 1. The ad- vantage of this metric over standard financial cost lies in its ability to capture the relative difficulty in executing an architecture while being free from both bottom-up cost estimation from proprietary data and monolithic telescope cost models that dubiously ap- ply to segmented in-space telescope models. Note, however, that the 𝐿𝑉 selection itself is embedded in a minimum cost optimization. The launch mani- festing subroutine–and the cost function of the larger optimization–can be adjusted to the taste of the mis- sion designer; the approach shown here is just one of many possible cost function selections.
2.1 Driving Parameters
For this study, four independent driving parameters were selected to define a given iSAT architecture. Pri- mary mirror diameter — also the Pareto analysis ben- efit/UoS — is the first. Measured as the diameter of the circular mirror approximated by the segmented arrangement (Figure 2a), this parameter defines the size of the system, affecting both the mirror and its supporting structures. The second driving parame- ter is the number of segments along the chord; that is, the number of segments along the diameter of the primary mirror (e.g., Figure 2b shows five). This pa- rameter independently defines the degree of segmen- tation of the primary mirror in a way that is easy to visualize (as opposed to total number of segments composing the mirror) and is always represented by an odd integer number, which simplifies the analysis.
From the selected primary mirror diameter and the number of segments along the chord, the dimensions of each segment can easily be computed (see Section 2.2)
The third driving parameter is raft geometry and directly relates to how the mirror is segmented. A raft is defined as a single unit (i.e. component) of the pri- mary mirror assembly. Design of rafts, segmentation, and their deployment/assembly is an active area of re- search and hosts of options could potentially exist of varying complexity. For simplicity, this analysis was restricted to three basic shapes, each built from the familiar hexagonal segments seen on the James Webb Space Telescope (Figure 3). Thus, though the num- ber of segments along the chord defines the degree of segmentation, it is the raft geometry that ultimately defines the size and mass of the assembly components.
Note that it is assumed that the segments composing a raft are already connected. Recall that complexity is defined as the number of sides that need to be con- nected between rafts. Thus, while both a 1 segment raft and a 7 segment raft (top and bottom Figure 3 respectively) have the potential to contribute a com- plexity of 6 to the assembly, a 3 segment raft only has the potential to contribute a complexity of 3.
Figure 2: Driving parameters 1 & 2: (a) primary mirror diameter (b) number of segments along chord
Figure 3: Driving parameter 3: raft geometry (three variants based on number of hexagonal segments con- sidered)
The fourth and final driving parameter is assem- bly location. Currently only Sun-Earth L2 (SEL2) and Earth-Moon L1 (EML1) are considered. It is
assumed that SEL2 is the operating location of the observatory. EML1 is an intermediary point that provides low-energy transport to SEL2 from the in- variant manifold solutions of the Circular Restricted Three Body Problem. Assembly in Earth orbit is not considered due to the large amount of energy it would take to transport the final assembled system to SEL2.
For more information refer to Section 3.2.
Figure 4 illustrates a high-level overview of the analysis procedure. The first step is to define all ar- chitectures to be considered in the analysis. This is accomplished by specifying upper bounds, lower bounds, and step size for the first three driving pa- rameters. A single assembly location is specified for the fourth driving parameter. An algorithm then takes the bounds of all values and enumerates all possible architectures and stores them into a data structure. The analysis then begins by evaluating each architecture one-by-one by first, deconstructing the OTA — including primary mirror and supporting structures — into launch vehicle packing components (detailed in Section 2.2). The Transport Trajectory Design phase (Section 3.2) then provides a menu of optimal path, time-of-flight, and ∆𝑉 of transport to the assembly location on a set of launch windows.
This information is provided to the Launch Mani- festing subroutine (Section 3.1); ∆𝑉 is mapped back to provide mass constraints for any given launch ve- hicle within any given time window. Using the set of components to be launched, a Launch Manifest- ing Optimization analysis (Section 3.1) is performed to cost-optimally pack the components in the fleet of available launch vehicles and assign launch windows.
If any of the broken down OTA components are too large or too massive to be transported on any of the available launch vehicle options, the architecture is considered a failure and is discarded. Similarly, the architecture is discarded if the OTA breakdown pro- duces segments smaller than 0.5𝑚.
Figure 4: Pareto analysis overview
Figure 5: OTA breakdown according to raft geometry
2.2 OTA Breakdown
For each architecture in the analysis, the OTA break- down stage deconstructs the observatory primary mirror into rafts composed of hexagonal segments.
The size, total number, and geometry of these rafts are determined by the architecture’s driving param- eters.
The composition of the primary mirror is best vi- sualized as being composed of a single segment at the center surrounding by rings of rafts. In the case of the 7 segment raft geometry, this single center seg- ment is replaced by a single center raft. Figure 5 illustrates a primary mirror constructed by a center segment (raft) surrounded by two rings of rafts for all raft types considered.
Simple geometric relationships exist to compute the segment length, width, total number of segments, and total number of raft rings about the center ele- ment. For instance, given the number of rings, the number of rafts (for the 1 and 7 segment variant) can be computed from the solution of the finite sum of the natural numbers to obtain 3(𝑟 + 1)𝑟 + 1, where 𝑟 is the number of rings. From these values, raft length and width are also computed. Complexity 𝐶 from Equation 1 is also computed in this step by tallying all raft sides that share a connection. Again, a simple geometric relationship emerges for the 1 and 7 seg- ment raft case, where it is found that 𝐶 = 9𝑟2+ 3𝑟.
Ultimately, the dimensions of the broken down components determine if an architecture is to be dis- carded. The packing algorithm optimizes stowage of the rafts and supporting structures in a series of launch vehicle fairings. If it discovered during this process that a component does not fit within any fairing the analysis is terminated and the architecture flagged as a failure. Equally important in the packing
and launch analysis is mass. The surface area of each raft can be easily calculated following the OTA break- down through knowledge of the segment’s length and width. The surface area is multiplied by the segment area density provided by Stahl [9] of 27𝑘𝑔/𝑚2 to ob- tain an approximation of the mass of each raft.
A reference iSAT observatory design provided by the Jet Propulsion Laboratory served as a reference for the non primary mirror elements required for launch — including but not limited to the space- craft bus, backplane structures, instruments, and sunshade. The size (and mass) of these elements were scaled by the ratio of the primary mirror (mass) of the analyzed architecture to the JPL reference ar- chitecture. The raft support structures were scaled directly by the reference raft length and width.
3 Theory and Calculation
The algorithmic design of the optimal selection and packing of launch vehicles with iSAT components and their low-energy transfer to their assembly location is documented in this section.
3.1 Launch Manifesting Optimization
A summary of the problem statement, brief litera- ture review, and the solution approach are provided below.
3.1.1 Background and Problem Statement An on-orbit assembled telescope, like other multi- component missions, requires many segmented com- ponents to be launched at varying times, potentially on different rockets. The framework here addresses this optimal launch manifesting problem, in the sense of minimizing the cost of using a number of launch vehicles to pack a selection of components with con- straints mass, and volume. These constraints are in- duced by launch vehicle volume and payload mass capability, the latter of which is time-varying due to the variation in required ∆𝑉 over time. The deci- sion variables of this problem are: which and how many launch vehicles are used; their launch dates;
and which components are placed in a vehicle. That is, the cost-optimal launch manifest must be deter- mined: which components are placed on which launch vehicle, and when, called the launch manifest ℳ.
Decision variable x𝑖𝑗𝑘 at launch time 𝑡𝑖 captures the number of component packings of type 𝑗 loaded on rocket type 𝑘 to launch.
The optimization problem to solve can be stated:
minimize
x𝑖𝑗𝑘
= 𝜃⊤x𝑖𝑗𝑘 ∀𝑖 subject to
l𝑘 ∈ ℒ
𝒞𝑗𝑘∈ l𝑘≤ maxMass 𝒞𝑗𝑘∈ l𝑘≤ maxVolume
𝑐𝑗 ∈ 𝒞𝑗𝑘 meetDimensions 𝑐𝑗 ∈ 𝒞𝑗𝑘 meetPriority
(2)
With the following definitions:
ℒ set of launch vehicle types 𝜃 cost vector for rocket variety as-
sociated with 𝑥𝑗𝑘
x𝑗𝑘
number of component packings 𝑗 on launch vehicle 𝑘 to launch l𝑘 launch vehicle 𝑘
cj component of packing 𝑗 𝒞𝑗𝑘
component packing 𝑗 on launch vehicle 𝑘
Note that superscript 𝑖 indicates the 𝑖th launch window available for this launch campaign. Addi- tional constraints can be added as needed, for ex- ample, restricting the center of mass of the pay- load to be below a certain height. The functions maxMass and maxVolume ensure that each compo- nent packing fits volume- and mass-wise on the 𝑘- th rocket type. meetDimensions checks that indi- vidual components meet rough volume constrains.
Finally, meetPriority ensures that components are launched, roughly, in an assigned priority order (i.e.
more important components are launched first).
3.1.2 Prior Work
A few attempts at similar problems have been made.
Morgan et al. is by far the most similar, and was an inspiration for the baseline version of this ap- proach [10]. Morgan et al. propose using an exhaus- tive enumeration of all possible component packings, and then running a linear optimization. A heuris- tic version uses greedy packing with possible refine- ment steps to follow. However, the scenario of inter- est in this paper involves many more components, as well as trajectory, scheduling, and volume con- straints. Gralla et al. have also looked at the prob- lem in terms of an interplanetary mission, consider- ing a direct enumeration of packing options and an in-progress integer programming solution [11]. How- ever, direct enumeration for a significant number of
Figure 6: The entire process from generating component packings (a), to order component packings (b), repeated selection of the best admissible component packing (c), to refinement (d) to eliminate wasted ca- pacity. Colored shapes represent individual components; the blue shading in (d) represents use of mass and volume capacity.
components quickly becomes infeasible. Other pro- posals include [12] and a variety of proposals that im- plicitly deal with launch manifesting but do not use an optimizing approach or instead use hand-designed missions.
As [10] points out, this problem is NP-hard since the bin-packing problem is a special subcase. The bin packing problem states, given a number of objects 𝑚 with volumes 𝑉𝑚, find the minimum number of bins 𝑛 each of volume 𝑉𝑏𝑖𝑛such that 𝑛 is minimized when packing all 𝑚. Using a solution gadget (a special simplification to show a result covers a subcase) the launch optimization problem reduces to bin packing:
consider when there is only one type of launch vehi- cle and there are only volume constraints (1D only).
This problem is therefore NP-hard, with substantially more complexity than classic bin packing.
3.1.3 Approach
Thus, heuristic approaches are one way of approx- imating a sub-optimal solution in feasible runtime.
Morgan et al. proposes a heuristic approach for rock- ets with payload mass constraints only, but relies on an enumeration of all possible combinations of com- ponent packings. Here, a “greedy” approach is used with the additional constraints mentioned above, de- tailed in Algorithm 1. GreedyPack pregenerates a set of fairing packing options and ranks them by the de- sired metric. GeneratePackings avoids enumerating all packing options by only offering a few component packings for each rocket, e.g. by including a small number of component packings of differing composi- tions to choose from. As long as all components are included in at least one component packing option, a post-processing step will deal with underutilization later on.
One approach to sort component packings uses
cost per 𝑚 + 𝛾𝑉 , where 𝛾 is a relative weighting between mass and volume of all the components in the packing. The algorithm then cycles to choose the “best” packing available that does not exceed the number of components needed, and continues until all components are assigned. This process tends to choose the most cost effective rockets, but can lead to underutilization since packings are only considered one-by-one and the full enumeration of component packings is not available. A post-processing step aims to reduce wasted mass and volume capacity. To ac- count for launch priority, the approach is run in multi- step fashion over a set of desired launch windows and with the aid of a trajectory optimization subroutine to provide mass constraints. The optimization over component packings created by GeneratePackings can also be written as an integer program as in [13], but this approach is not presented here for brevity.
Algorithm 1 Greedy Pack
1: procedure GreedyPack(𝒞𝑡𝑦𝑝𝑒𝑠, 𝒞, ℒ, ΔV)
2: 𝒞𝑗𝑘← GeneratePackings
3: for 𝑥𝑖∈ 𝒳𝑗𝑘 do
4: 𝑠𝑐𝑜𝑟𝑒𝑖← 𝑚𝑐𝑜𝑠𝑡𝑖
𝑖+𝛾𝑉𝑖 5: end for
6: SortScoreBestFirst(𝒞𝑗𝑘)
7: while ConstraintsNotSatisfied do
8: if WithinConstraints(𝒞𝑗𝑘) then
9: 𝑥𝑗𝑘← 𝑥𝑗𝑘+ 1
10: else
11: Next(𝒞𝑗𝑘)
12: end if
13: end while
14: 𝑐𝑜𝑠𝑡 ← GetCost(𝑥𝑗𝑘)
15: ℳ ← 𝑐𝑜𝑠𝑡, 𝑥𝑗𝑘
16: end procedure
Figure 7: Fairing packings are first considered for pairwise combination, eliminating one of the under- utilized pair.
Figure 8: Fairing packings are spread between all other packings if constraints allow, removing very un- derutilized vehicles.
Suboptimal choices will be made, particularly be- cause not all packing options can be generated to op- timize over. A convenient and rather effective heuris- tic for the post-processing is to Repack components between component packings, Figure 6. This ap- proach is mainly inspired by Morgan et al. Com- ponent repacks are attempted in a two-step process:
(1) attempt to merge pairs of launch vehicle pack- ings, then (2) spread components between remaining rockets to eliminate other rockets. This effectively re- duces underutilization. Figure 7 shows the first step of this process. This allows moderately underutilized rockets to be eliminated.
Figure 8 shows the second step of this process.
This allows less underutilized rockets to be filled en- tirely, if another very underutilized rocket can be re- moved. In practice, this leads to consistently high uti- lization of vehicle capacity with little computational overhead for calculation.
The refining process mixes rockets of different pri- ority levels. For certain situations this is acceptable, such as storing components for a time near the as- sembly site of the on-orbit structure. One potential workaround is only to repack within a certain priority level then, if desired, mix priority levels. Additional heuristics may be beneficial: this one in particular has proved successful for the results shown in Section 4.
Algorithm 2 Repack
1: procedure Repack(ℳ𝑖)
2: for ℳ𝑚∀𝑚 do
3: for 𝑙𝑛∈ ℳ𝑚 ∀𝑛 do
4: for 𝑙𝑝 ∈ ℳ𝑚∀𝑝 ̸= 𝑛 do
5: for 𝑐𝑞 ∈ 𝑙𝑛 ∀𝑞 do
6: 𝑙𝑝← CheckIfMovable(𝑐𝑞, 𝑙𝑝)
7: end for
8: if Movable ∀𝑝 then
9: Remove(𝑙𝑛)
10: Reassign(𝑐𝑞)
11: end if
12: end for
13: end for
14: end for
15: for ℳ𝑚∀𝑚 do
16: for 𝑙𝑛∈ ℳ𝑚 ∀𝑛 do
17: for 𝑐𝑝 ∈ 𝑙𝑛 ∀𝑝 do
18: 𝑙𝑡𝑎𝑟𝑔𝑒𝑡𝑝 ← CheckIfMov-
able(𝑐𝑝, ℒ ∈ ℳ)
19: end for
20: if Movable ∀𝑝 then
21: Remove(𝑙𝑛)
22: Reassign(𝑐𝑝)
23: end if
24: end for
25: end for
26: end procedure
3.2 Transport Trajectory Design
One of the parameters for this study was the selec- tion of assembly location. Two locations were stud- ied: Sun-Earth Lagrange point 2 (SEL2) and Earth- Moon Lagrange point 1 (EML1) namely because the former will be the location of future space telescopes [14] [15], and the latter is the proposed destination for the NASA Lunar Gateway [16]. Assembling in LEO was not considered in this study as transport- ing a fully assemble large telescope would require a specialized tug to provide a large enough ∆𝑣 to enter a transfer trajectory to either SEL2 or EML1. In- stead, it has been shown that transfer ∆𝑣 between the EML1 environment and SEL2 orbits can be on the order of cm/s [17].
The nominal target trajectory for SEL2 and EML1 were both of the northern halo families, as they have both been the subject of extensive study.
The transfer trajectory which generates a dynamical solution from Earth to the nominal halo orbit was accomplished using an invariant manifold trajectory design approach. This approach — closely following [15, 18, 19] — uses Dynamical Systems Theory to generate low-energy trajectories that reach close to
earth and lie on the stable manifold of the halo orbit.
The following assumptions were made in the tra- jectory generation formulation. First, the problem was reduced to the classic Circular Restricted 3-Body Problem phase space in which the primary and sec- ondary bodies were either the Sun and the Earth, or the Earth and the Moon. This assumption is also made in [14, 15] in order to generate a first cut solution. The effects of Solar radiation pressure, Earth’s oblateness, and other n-body effects were ig- nored. These effects are known to change the loca- tion of the Lagrange point and the period of the orbit [15, 20, 21], as well as necessitate the performance of mid-course correction maneuvers. However, for a first cut solution, these effects on the total ∆𝑣 needed to reach the destination are not significant. Addition- ally, the objective of the trajectory design process was only to provide transport of the assembly components to a defined Halo orbit. The problem of phasing — i.e., ensuring all satellite components arrive in close proximity to each other — was not addressed. Fi- nally, a drag penalty of 1.7𝑘𝑚/𝑠 was added to the results to account for the drag needed to combat the atmosphere when launching from Earth.
It should be noted that the approach used in this paper was a discrete optimization of several candi- date transfer trajectories. This approach is locally- optimal as the algorithm searches over a discrete set of low energy solutions.
The main steps for generating a set of manifold trajectories are as follows,
1. Obtain the initial conditions for a halo orbit 2. Propagate the halo orbit for at least one period,
while also integrating the variational equations to obtain the state transition matrix Φ(𝑡, 𝑡0), which relates the state at the terminal inte- gration time t 𝑡 to the initial conditions 𝑡0: 𝑥(𝑡) = Φ(𝑡, 𝑡0)𝑥(𝑡0)
3. Obtain the eigenvector of Φ(𝑡, 𝑡0), vs, associ- ated with the stable manifold at several points along the orbit and perturb the current state by a small distance along that eigenvector 𝑥′(𝑡) = 𝑥(𝑡) + 𝑑vs
4. Propagate the system backwards in time and check if the trajectory gets sufficiently close to earth
Having a set of candidate manifold trajectories allows an optimizer to choose the best corresponding transfer trajectory based on the launch date. The transport function accepts the following set of inputs
Input Description
Target loc. Location of Halo orbit (SEL2 or EML1)
Launch loc. Location of departure from earth (Cape Canaveral, French Guiana, etc.)
Time of launch Initial time of when launch can happen
Launch window range of days over which launch can occur
The function is designed so that the user can spec- ify the latitude and longitude of the launch location.
Additionally, the function is structured so that addi- tional target locations or orbit types can be imple- mented in the future, such as the Near Rectilinear Halo Orbit (NRHO) — the planned operating orbit for the NASA Lunar Gateway [16]. Finally, the user selects the initial possible time of launch as well as the associated launch window. This can also be fed as a constraint from the Launch Manifesting algorithm which may prioritize the arrival of certain parts of the satellite as compared to others. Additionally, the variability of this feature can also be used to aid in limiting the search space to make the trajectory func- tion run faster.
The transfer trajectory function performs a dis- crete optimization to determine when is the best time to launch the vehicle with respect to ∆𝑣 magnitude.
To do this, the algorithm takes a discrete time step starting from the initial allowable time of launch till the end of the launch window and performs the fol- lowing computations,
1. Convert the current time to Julian date 2. Compute the position of the Sun, Earth and
Moon utilizing JPL’s planetary ephemerides DE430 [22]
3. Compute the position and velocity of the launch location with respect to the barycenter rotating frame in which the halo orbit was computed 4. For each trajectory inside the computed mani-
fold trajectories set,
(a) Generate a trajectory from launch location to an intersection point with the manifold trajectories using a boundary value prob- lem solver
(b) Compute the total ∆𝑣 and transfer time for each trajectory
5. Select minimum ∆𝑣 maneuver for that specified time
The choice of the discrete time step can cause a computational bottleneck. A time step of 30 min- utes was selected to balance this constraint and the fineness of the search. After all time steps are exe- cuted, the algorithm searches for the minimum ∆𝑣 and returns the following information,
Output Description
Time of Launch Time in which to launch the rocket
Total ∆𝑣 Total ∆𝑣 which includes the magnitude of ∆𝑣 to launch the rocket and magnitude of
∆𝑣 to insert onto the mani- fold trajectory
Travel time Total travel time from launch to arrival in halo orbit
Additionally, the algorithm returns a data packet with information on the index of the manifold trajec- tory used, the time at which to perform the delta-v maneuver to enter the manifold, and the ∆𝑣 vectors for launch and transfer. Results for this trajectory showed that the cost to reach SEL2 or EML1 were on the same order of magnitude and vary by about 5% depending on time of year.
4 Results
The results of this analysis are presented in two parts.
First, the launch manifesting algorithm is demon- strated for two test cases, highlighting its capabilities for broader use in the study. Finally, the results of the Pareto analysis are presented.
4.1 Launch Manifesting Test Cases
A test case with a large in-space assembled telescope was conducted (122 components and 19 different com- ponents types, 20m primary diameter with 2.86m seg- ment size). Realistic component parameter values were used, along with 7 launch vehicle options. In particular, the Space Launch System (SLS) was used, constituting a significantly different class of launch vehicle from the others available. The dimensional- ity of this problem exceeds the capability of a brute force solution, and becomes challenging for a human expert to assign optimally. Rockets and components considered are listed in Appendix B.
Figure 9 visualizes the launch vehicles selected and their utilization. A utilization of 94.02% for mass and 66.46% for volume was obtained. The compo- nents used have three different priority levels, corre- sponding to the three possible rocket launch dates.
The total cost of this scenario is $3,068 million, using launch vehicle cost estimates from the FAA [23].
A contrasting scenario is shown in Figure 10. A utilization of 84.97% for mass and 74.87% for vol- ume was obtained. In this example, small compo- nents (mass- and volume-wise) are able to fit entirely within more economical launch vehicle choices: the Falcon 9 and Falcon 9 Heavy. This scenario has 82 components, 19 component types, and 7 launch vehi- cle options. The primary diameter is 12m with 2.4m segment size, for a total cost of $686 million.
Drastic changes occur when components violate a constraint that requires jumping to a vastly different rocket class (in terms of capability and cost). If a con- straint leaves the mid-size rocket class, the heavy-lift SLS is the only remaining option. Though expen- sive on a dollar per kg basis, additional volume can siphon off components from other rockets. Strictly in terms of complexity, this may be seen as beneficial.
The utopia scenario is to launch the largest compo- nent possible on the biggest possible rocket for little cost—realistically, these factors must be balanced by careful component design and manifesting.
4.2 Pareto Analysis Results
The driving parameters selected for the analysis are summarized in Table 4.2. The primary mirror diam- eter is swept from its minimum to maximum value by a step size of 1𝑚 and the number of segments along the chord parameter is incremented by 2 segments to maintain symmetry across the mirror.
Driving Parameter Values
PM Diameter 1𝑚 − 25𝑚
Segmentation 1-51 segs. along the chord Raft Geometry {1seg, 3seg, 7seg}
Assembly Location {EML1, SEL2}
Table 1: Driving parameters selected for the study The results of the trade-space analysis are de- picted in a sequence of Pareto graphs. As intro- duced in Figure 1, each successful architecture is rep- resented by a single point on the graph spanned by the benefit and cost metrics considered — primary mirror diameter and normalized total complexity re- spectively. More architecture defining parameters ex- ist than can be depicted on a single plot. Thus multi- ple figures of the same trade-space are presented, each with a color map highlighting a different cost influ- encer. In order, these are total number of rafts per architecture (Figure 11), the optimized total number of launch vehicles required per architecture (Figure
Figure 9: Fairing packs generated for a 20m space telescope component list. Note that the optimizer is limited by both mass volume constraints, and attempts to spread the allocation of components within a time window so that the fewest, most economical rockets possible are used. Plot sizes are scaled to rocket dimensions. The heading above each rocket indicates {name, priority number}.
Figure 10: Fairing packs generated for a 12m space telescope component list. Smaller, more cost effective rockets are selected since they are able to satisfy all constraints.
12), and of those, the total number of SLS launch vehicles selected (Figure 13).
Figure 11: Pareto results highlighting total number of rafts per architecture
The Pareto front, shown explicitly in Figure 11, consists of the optimal set of architectures out of those considered. A table enumerating the architec- ture parameters along a segment of the Pareto front is located in the Appendix A.
Figure 12: Pareto results highlighting total number of launch vehicles required per architecture
Figure 13: Pareto results highlighting total number of SLS launch vehicles selected per architecture
5 Discussion
Pareto front architectures reflect the cost metric con- sidered. Given the definition of normalized total com- plexity as presented in Section 2, it was anticipated that for a given primary mirror diameter, an opti- mal architecture is that which minimizes the num- ber of raft connections (complexity) and the num- ber of launch vehicles required to transport the com- ponents to their assembly location. That is to say, architectures constructed from the largest available rafts packed into the largest available launch vehi- cles. This is what was observed. It can be seen in Figure 11 that the architectures occupying the Pareto front possess the lowest number of — and thus largest size — rafts1. In general, for a given primary mirror diameter, as one moves towards increasingly subopti- mal architectures by increasing normalized total com- plexity, the number of rafts increase (size decreases).
Similarly, it can be observed from Figure 12 that the Pareto front architectures require the lowest number of launch vehicles for a given PM diameter, and those requiring more are increasingly suboptimal.
The largest stratification in the data can be ob- served by highlighting the total number of SLS launch vehicles selected by the manifesting algorithm — this can be seen both graphically in Figure 13 and in the tabulated data located in the Appendix. Even more, it is revealed that the large gap in normalized total complexity between the Pareto front points and sub-
1Additional plots highlighting total number of segments and number of segments along the chord can be generated but are not included in this paper since they are not independent from the total number of rafts, which more directly influences the cost metric.
sequent suboptimal architectures — especially in the 12𝑚 − 21𝑚 range — is due to a jump in the number of SLS requested for launch. It should be noted that the launch manifesting algorithm seeks to minimize the financial cost incurred at launch by packing the assembly components into the least expensive fleet in which component mass and volume fit. This is a fac- tor that is not reflected in the Pareto analysis cost.
Thus, the final Pareto front point at a given PM di- ameter can be interpreted as the lowest risk archi- tecture — fewest required launches — out of the set of those packed as inexpensively as possible. Thus, given the aforementioned observations and the fact that SLS is the largest capacity launch vehicle con- sidered in the analysis, it is unsurprising that the set of optimal architectures are those that pack the largest rafts possible into the maximum number of SLS launch vehicles.
Though these observations are relevant, the tools and framework developed are considered to be the most significant outcome of this study. As was pre- viously discussed, the results largely reflect the cost metric chosen. However, the utility of the architec- ture analysis tools lie in their independence from the defining metrics. Future applicable financial cost esti- mating models and more accurate methods of quan- tifying risk can easily be plugged-in as the field of in-space assembly matures. Furthermore, more ex- otic raft geometries selections, manufacturing con- straints, scientific constraints, spacecraft bus design, and methods of robotic assembly are all critical fac- tors currently not considered. Adding these complex- ifying elements corresponds to adding steps in the Pareto analysis shown in Figure 4. Thus, the ulti- mate vision is that this “plug and play” framework allows for the continued evolution of this tool set par- allel to the development of the field and increasingly supports informed architecture decision making.
6 Conclusions
The tool created for this analysis can be used to enu- merate the optimal set of architectures for a given comparison metric — the goal was to find design solutions that minimized complexity and maximized utility of science. For iSAT, these metrics were the normalized total complexity and primary mirror di- ameter, respectively. The results were manifestations of variations in several driving parameters, includ- ing primary mirror diameter, raft geometry, num- ber of segments, and assembly location. The opti- mization of these architectures was weighted equally with the launch packing optimization to define the Pareto front with the lowest normalized complex-
ity. A tool for performing approximate cost-optimal launch manifesting under a variety of constraints was presented. A method of performing orbital trajectory optimization to SEL2 and NRHO was also presented.
Future work for this study can include the addi- tion of further cost and analysis tools, whether for iSAT or other unique missions. To truly define the Pareto front for an in-space assembly mission, opti- mizations for in-space assembly, manufacturing, and ground testing methods are necessary. This will be especially useful for missions that have the ability to perform in-depth cost analyses, as the Pareto front can be used to help make important design decisions.
Acknowledgments
This work was supported by the Jet Propulsion Laboratory as part of the broader NASA iSAT (in-Space Assembled Telescope) Study, subcontract number 1613399. This work was also supported by the NASA Space Technology Research Fellow- ship program, grant numbers 80NSSC17K0077 and NNX16AM72H. The authors would specifically like to thank Dr. Nick Siegler, Dr. Rudranarayan Mukher- jee, and Prof. David Miller for their guidance on the project.
Appendix A (Selected Pareto Front Architec- tures)
The architecture parameters of a subset of the Pareto front (PM = 12𝑚 − 21𝑚) are tabulated below, where 𝑃 𝑀 is the primary mirror diameter in meters, ¯𝐶 is normalized total complexity, and 𝑆𝐿𝑆/𝐿𝑉 is the number of selected SLS launch vehicles over the to- tal number of launch vehicles required. Futhermore, rafts, segs., and chord correspond to the total num- ber of rafts, segments, and number of segments along the chord dictated by the architecture. The tables are divided by raft geometry as illustrated in Figure 14.
PM 𝐶¯ SLS/LV rafts segs. chord
12 0.268 0/8 7 7 3
13 0.334 0/10 7 7 3
14 0.068 2/2 7 7 3
15 0.068 2/2 7 7 3
16 0.068 2/2 7 7 3
17 0.068 2/2 7 7 3
18 0.134 3/4 7 7 3
19 0.134 3/4 7 7 3
20 0.304 2/9 19 19 5
21 0.337 2/10 19 19 5
Table 2: 1 segment raft: selected architectures along the pareto front
Figure 14: Pareto analysis results highlighting raft type selected for each architecture depicted in the graph
PM 𝐶¯ SLS/LV rafts segs. chord
12 0.134 1/4 6 19 5
13 0.168 1/5 6 19 5
14 0.068 2/2 6 19 5
15 0.068 2/2 6 19 5
16 0.068 2/2 6 19 5
17 0.068 2/2 6 19 5
18 0.134 3/4 6 19 5
19 0.136 3/4 12 37 7
20 0.169 3/5 12 37 7
21 0.236 3/7 12 37 7
Table 3: 3 segment raft: selected architectures along the pareto front
PM 𝐶¯ SLS/LV rafts segs. chord
12 0.134 1/4 7 49 7
13 0.168 1/5 7 49 7
14 0.068 2/2 7 49 7
15 0.068 2/2 7 49 7
16 0.101 3/3 7 49 7
17 0.068 2/2 7 49 7
18 0.137 3/4 19 133 11
19 0.137 3/4 19 133 11
20 0.237 3/7 19 133 11
21 0.237 3/7 19 133 11
Table 4: 7 segment raft: selected architectures along the pareto front
7 Appendix B (Launch Vehicles Considered)
Launch Vehicle ∼ Cost [$M] PD[m] FH[m]
F9 Block FT 62 4.6 6.7
FHeavy 100 4.6 6.6
SLS 1200 7.5 19.1
Delta IV Heavy 350 4.57 12
Atlas V 551 92 4.57 7.63
Antares 80 3.9 7.52
where ∼ Cost is the approximate cost of the launch vehicle in millions of U.S. Dollars, PD is the payload diameter in meters, and FH is the faring height in meters.
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