ion. Analysis of strategy dynamics and equilibrium points only evaluate broad decisions such as whether to pursue collaborative or independent architectures, rather than more detailed decisions governing system design and implementation. Thelogicalgapbetweenstrategicandnon-strategicdesigndecisionspresentsabarriertoadopting game-theoretic metrics to a tradespace exploration paradigm. The work presented here addresses this issue with a bi-level decision model. The lower-level design information determines each constituent’s utility given a particular implementation and strategic context. The constituent system utilities under each strategic context are thenused for game-theoretic analysis of the strategy dynamics; linking design decisions to their impacts at the strategy level. D. Research Objectives The broad objective of this work is to develop and evaluate a methodology for ﬁnding the “best” constituent systems foroperationinFSSandotherSoS.While“best”mayseemambiguous,itisanalogouswithvalueorutilitymaximization, the objective of all engineering and design. As previously noted, utility maximization under real-world conditions demandsrobustnesstocommonperturbations. Coordinationfailurebetweenconstituentsisauniqueperturbationfor SoS which means constituent actors must consider two new system characteristics. First, they must design their systems 6 to be robust to coordination failures to minimize the loss of utility if the SoS dissolves. Second, they must design their systems to increase the stability of the collaborative strategy for all constituent systems, reducing the likelihood of SoS dissolution. These tasks are related—a system actor who loses relatively little under coordination failure is more likely to choose the collaborative strategy than one who takes a great risk by collaborating. Twocomponentscomprisethiswork’smethodologyforﬁndingutilitymaximizingsystemsinthecontextofFSS. The ﬁrst component implements a game-theoretic equilibrium selection criterion within a simple FSS tradespace explorationprocesstodetermineifselecteddesignsarerobusttocoordinationfailureandcontributetofavorablestrategy dynamics. The next component broadens the initial scope to include economic factors such as contract terms and ﬁnancialresourceexchangesthataﬀectsystemactorpayoﬀstostudyhowtheproposedmethodologycanguideselection of economic variable values and how the broader scope impacts the utility and robustness of candidate designs. III. Baseline Model This section introduces a notional FSS application case, constituent system models, and results of a baseline tradespaceexplorationprocesstypicalofexistinganalysismethods. Alatersectioncomparestheresultsobtainedinthis section with those obtained when SoS unique strategy dynamics are considered. A. Application Case Conceptualdesignofcommercialcommunication(SATCOM)andearthobservation(EO)satellitesystemsisusedto illustrate the methodological concepts presented in this paper. Net present value (NPV), in dollars, measures attainment of individual objectives and is calculated from system cost and revenue time series, assuming a 10% discount rate. SATCOM system revenue depends on subscription users and EO system revenue depends on imagery resolution and its latency, measured as the time from request of an image of a speciﬁc location to the downlinking of that image. Both systems can increase revenue by forming a FSS. EO latency decreases, increasing system revenue, by utilizing the SATCOM system to relay image data to the ground. SATCOM revenue increases by selling the data transport service to the EO system. Figure 1 graphically displays the two systems operating independently and as a FSS. B. SATCOM System Model The SATCOM system is modeled as a Low Earth Orbit (LEO) constellation with a primary mission to provide internet connectivity services. Table 1 lists variable tradespace parameters. For design space simplicity, all SATCOM systems use a polar orbit with a streets of coverage constellation design. Constellation design evaluation uses the speciﬁed number of planes ( cB), satellites per plane ( #B), and elevation angle ( n) to determine the lowest operating altitude allowing constant coverage. First, Eq. (1)calculates the minimum altitude, 0<8=, to achieve a minimal street of coverage, where 'is the radius of the Earth. An optimizer varies the altitude to minimize the diﬀerence between the 7 Fig. 1 System operations diagrams. combinedlongitudinalcoverageofallthe streets(measuredinunitsofEarthcentral angle)and 2cradians. Altitudes below 400 km are not permitted; constellations that can maintain coverage below 400 km have greater overlap between antenna footprints. The altitude and minimum user elevation angle ndetermine the maximum slant range ( 'B) and coverage area ( \, deg2) of each satellite. 0<8==" 'cosn sinc 2nc #B# ' (1) Table 1 SATCOM system tradespace Variable Symbol Maximum Minimum Number of planes cB 60 4 Number of satellites per plane #B 80 8 Number of user channels  32 1 Per channel transmit power (W) ? 10 1 Minimum user elevation angle (deg) n 40 10 Modulation scheme 22 options Thenextstepdeterminesmaximumdatathroughputforeachsatellite. Steerableandshapeablebeamswith10% eﬀective coverage are assumed for all designs, i.e. 10% of the satellite ﬁeld-of-regard is covered by beams at maximum gain. Equation (2)calculates the area covered by each beam. Equation (3)calculates maximum transmit antenna gain ( , dBi) where [is transmit antenna eﬃciency [32]. \45 5=\10% (2) 8 =10log41253 \45 5 ¸10log¹[º (3) The speciﬁed transmit power ( ?) and modulation scheme determine the maximum per channel data rate as the minimum of the power limited and bandwidth limited rates. Per-satellite data rate is simply the channel rate multiplied by the number of channels. See the Appendix for an example link budget. The number of user subscriptions, as the minimumofthesupportablesubscriptions(basedondeployment)andthesubscriptionbaseforthemonthoftherevenue calculation,determinesmonthlyrevenue. Thenumberofsubscriptionssupportablebytheconstellationusesthetotal constellationthroughput,a10Mbpsuserdatarate,andaconservativeoversubscriptionratioof5:1[ 33]. Alogistic curve [34,35] using satellite internet market growth projections published in Ref. [ 36] caps the subscriber base, (10B4. Eq.(4)implements the logistic curve, where is the maximum number of subscriptionsin millions, 11is the scaling factor,Cis the time in months from the start of system development, and 12is the inﬂection point in months. (10B4= 1¸411¹C12º (4) The SATCOM system cost model uses parametric cost-estimating relationships (CERs) starting with the estimation of dry massfrom transmit power in Eq. (5). The costmodels assume the total communicationpayload transmit power, including gateway, user, and inter-satellite links (ISLs), is 200% of the user link transmit power with an additional 10% overheadforreceiversandotherpayloadelectronics. Theestimateofspacecraftdrymassdividesthepayloadpower bya0.46payloadpowerfraction[ 32]andmultipliesbyadensityfactor0.20W/kgandaredundancyfactorof0.8to account for the short, ﬁve year mission life [37]. <3AHB=?200%110% [)G046 020W kg08 (5) ThemodelusescommunicationsatelliteCERspublishedinRef.[ 32],Table11-5toestimatespacecraftnon-recurring engineering (NRE) and production costs. Production costs assume an 85% learning curve [ 38]. Estimates of total spacecraftmassassumeanadditional27%propellantmass[ 32]. Launchcostsassumedirectinsertion,mass-limited launcheswithatleastonelaunchperplane,AtlasV551performancedata[ 39],at$62Mperlaunch. A1:10gateway ground station tosatellite ratio determines the number ofground stations required [ 40]. The model implements CERs published in Ref. [ 32] to estimate costs for ground station infrastructure, operations, launch and orbital operations, and program level costs. The NRE period is 24 months, followed by 12 months of production and launch preparation, with the ﬁrst launch occurring 36 months from the start of the program. Subsequent launches occur monthly until the entire constellation is deployed. The model spreads costs uniformly by month over the phase in which they are incurred. The model incurs launch costs starting 24 months prior to each launch and groups production costs into lots by launch and 9 spreadsthemovera12monthperiod,workingbackwardsfromthelaunchdate. SeetheAppendixforadetailedcost model including the factors and CERs used. C. EO System Model ThemodelrepresentstheEOsystemasalow-Earthorbit(LEO)constellationwithcircular,polarorbits. Table2 listsvariabletradespace parameters. Themodeluses aﬁxedminimumelevationangleof40forimaging and15for communication with ground stations. The model uses Orekit 10.0∗and a Python wrapper†to measure latency for each architecture by propagating spacecraftorbitsandperformingspacecraft-to-targetgeometriccalculations. 50pointsdistributeduniformlyinlongitude, with a triangular distribution between 89North and South latitude, approximating a uniform global distribution, comprisethetargetset. Alltargetsentertheimagingqueueatthebeginningofthesimulation. Thesatellitewiththe minimum time to receive a target imaging command, collect the image, and downlink to a ground station is tasked with collection. The model selects the ﬁrst Wground stations from a list that prioritizes extreme latitudes for favorable access to all satellites in the system (see Appendix for all ground station locations). The time from simulation start to imagery downlink is logged as the latency. The simulation runs until the data for the last target is downlinked. Simulation replications consider ﬁve start times, each oﬀset by six hours, to mitigate eﬀects of initial conditions. The mean latency for all targets and all simulation trials represents the performance metric for a given architecture. Themodeldoesnotconsiderobservationrangeorangletoimproveexecutionspeedandtoenableagreedyalgorithm toscheduleimagingoperations. Theresultingmetriccapturesbestcaselatency,butnotdataquality. Inareal-world application one would likely sacriﬁce a small amount of latency for better quality data. However, a more complex targeting valuation model would complicate interpretation of results with additional assumptions. Therefore, resolution is simply determined by the Rayleigh diﬀraction limit and small angle approximation for a nadir target. Table 2 EO system tradespace Variable Symbol Maximum Minimum Number of planes c4 4 1 Number of satellites per plane #4 4 1 Aperture diameter (cm)  100 10 Altitude (km) 0 800 400 Number of ground stations W 5 0 A two objective multiplicative value function [ 41] computes monthly revenue with a maximum revenue of $200M in Eq.(6). An exponential single-attribute value function (SAVF) for latency represents the decay of information value ∗Available from CS GROUP at https://orekit.org †Available from CS Systèmes d’Information at https://gitlab.orekit.org/orekit-labs/python-wrapper 10 [42,43] in Eq.(7). An aggressive decay rate ( _) of130000s1requires system mean latency signiﬁcantly less than 24 hours for appreciable value generation. An exponential SAVF in Eq. (8)for resolution ( \) yields a value of 1 for resolutions better than 20 cm. %$=$200Maa\ (6) a=4_(7) a\=017\11(8) The EO system cost estimation process mirrors that of the SATCOM system with a few diﬀerences. EO system drymassisestimatedfromaperturediameter( )inEq.(9)usingaparametricrelationshipfromRef.[ 37]. NREand spacecraftproductioncostsareestimatedusingreconnaissancesatelliteCERsfromRef.[ 32]Table11-5. Lastly,the number of ground stations is an independent design variable, rather than a parameter determined by the number of satellites in the system. <3AH4=2287kg cm08 (9) D. Collaborative System Model A collaborative system utilizes excess SATCOM system capacity to transport EO system commands and data, increasingthepotentialrevenuestreamsfortheSATCOMsystemandthevalueoftheEOsystemdata. Thecollaborative systemmodelisimplementedwithupdatestotheEOlatencymodelandrevenuemodels,SATCOMrevenuemodel,and both cost models. TheEOsystemlatencymodelassumesimagingtasksareinstantaneouslytransmittedtotheEOsystemupondemand generation, and data is immediately available upon collection. Collaborative systems use the same targets and start times as the independent systems, but latency for each target is reduced to the time from simulation start to the ﬁrst possibleobservationofthetarget. EOmonthlyrevenueisreducedby$10Mtopayadatatransportservicefee. The collaborative EO system cost model increases system mass by a factor of 5% to account for inter-satellite link hardware, NRE cost increases by an additional 10% to account for software to enable open inter-satellite links and additional bus complexity from adding link hardware. Recurring costs increase as a result of the mass increase, plus an additional 5% complexity factor. TheSATCOMsystemstartscollectingthedatatransportservicefeeafteritdeploystheminimumnumberofsatellites 11 requiredtoaccessanyothersatelliteinLEOandoneormoreEOsatellitesaredeployed. ThebaselineSATCOMsystem networkisalatticewithphasedarraysforsteerablelinksthatareorthogonaltothevelocityvector. Inimplementation,a small area of each array would be used periodically as a low gain “listening” antenna to receive data transport requests from the EO system [ 44]. When a request is received and accepted, the array would focus a high gain beam at the EOsatellitetoreceivethe data[ 44]. Therefore,noadditional hardwarewasnecessarytoenableinteroperability. The onlychangetotheSATCOMcostmodelwasa10%increaseinNREtoaccountforthead-hocnetworkingprotocol implementation. E. Baseline Tradespace Exploration Thebaselineanalysisexecutestradespaceexplorationintwophases. Theﬁrstphaseusesaheuristicsearchtechnique, speciﬁcally a genetic algorithm, to ﬁnd the value-maximizing independent systems. The second phase searches for collaborativesolutionsthatmaximizeNPVundersuccessfulcollaborationforeachsystemactor(i.e.,asamulti-attribute objective). A collaborative design resulting in NPV greater than the independent designs for all systems is a viable candidate FSS. Figure 2 depicts the tradespace exploration process. Fig. 2 Tradespace exploration structural diagram. Tradespace exploration uses the Distributed Evolutionary Algorithms in Python (DEAP) eaMuPlusLambda genetic algorithm(GA)implementation[ 45]. Eachtrialstartswithapopulationof200designsandrunsfor100generations. The cross-over and mutation rates are set to 0.8 and 0.2 respectively. Initial trials use the tradespaces displayed in Table 1 and Table 2 to narrow the variable ranges, increasing the probability of ﬁnding the optimal designs in the subsequent trials. Two or three trials are conducted for each tradespace. Table3displaysthetopﬁveindependentEOsystemdesigns. All designsaresimilarwithtwoorthreeplanesand 12 three or four satellites per plane. No advantage is gained by employing more than three ground stations. Designs 1 and 2 are identical aside from the employment of two additional ground stations for design 2, which reduce NPV. Design 1 is the single Pareto eﬃcient design. Table 3 Top ﬁve independent EO systems to maximize NPV Designc4#4(cm)0(km)WNPV ($M) 1 3 3 100 400 3 346.75 2 3 3 100 400 5 311.42 3 3 3 95 400 3 303.72 4 3 4 100 400 3 275.70 5 2 4 100 400 3 273.74 Table 4 shows the top ﬁve independent SATCOM system designs. As was the case with the EO system, the highest value SATCOM system designs are clustered around a narrow set of design variables. Most high-value SATCOM systems select low transmitter power and large constellations, likely due to several factors. Large constellations can maintain coverage at the minimum altitude, maximizing launch vehicle capability, and reducing free-space losses. Additionally,theNREforasmallersatelliteislowerandlearningeﬀectsbeneﬁtlargeconstellationsmuchmore than smallones. Therefore,agivenpowerﬂux-densityismoreeﬃcientlyachievedwithlargenumbersofrelativelylow-ﬂying, low-power satellites than with small numbers of higher-ﬂying, high-power satellites. The relationship between required power ﬂux-density and satellite quantity also explains the preference for high minimum elevation angles, which require more satellites but reduce the maximum slant range and enable the use of higher gain antennas. Table 4 Top ﬁve independent SATCOM systems to maximize NPV Designc4#4n()?(W)Modulation Scheme NPV ($M) 1 41 80 40 1 32 8PSK, 2/3 Code Rate 9035.65 2 40 80 40 1 32 8PSK, 2/3 Code Rate 9034.57 3 42 80 40 1 32 8PSK, 2/3 Code Rate 9025.21 4 39 80 40 1 32 8PSK, 2/3 Code Rate 9021.90 5 38 80 40 1 32 8PSK, 2/3 Code Rate 9021.07 Amongst the EO systems designed for collaboration, the highest value designs in Table 5 favor large constellations withnogroundstations. CollaborationincreasesmaximumNPVby 103%overthehighestNPVindependentdesign. The complete elimination of ground stations is the most striking feature of the designs which maximize NPV under collaboration. Eliminatinggroundstationsmaximizeseﬃciency,butalsocreatesfunctionalfragility;afeaturethatis explored later. UnlikethehighestvaluecollaborativeEOsystems,theSATCOMarchitecturesinTable6arenearlyunchangedaside 13 Table 5 Top ﬁve collaborative EO systems to maximize NPV Designc4#4(cm)0(km)WNPV ($M) 1 3 4 100 400 0 704.38 2 3 3 100 400 0 685.78 3 2 4 100 400 0 662.45 4 3 4 95 400 0 622.02 5 3 3 95 400 0 606.41 fromtheadditionofinteroperableISLs. Thehighestvalueindependentandcollaborativesystemsareidenticalwith respect to other architectural decisions. Successful collaboration increases NPV by a modest 3.3%. Analysis presented in subsequent sections shows that the relatively invariant SATCOM system architecture beneﬁts collaboration because no modiﬁcation is made to the core system design which may inhibit system performance if coordination fails. Table 6 Top ﬁve collaborative SATCOM systems to maximize NPV DesigncB#Bn()?(W)Modulation Scheme NPV ($M) 1 41 80 40 1 32 8PSK, 2/3 Code Rate 9332.78 2 40 80 40 1 32 8PSK, 2/3 Code Rate 9331.70 3 42 80 40 1 32 8PSK, 2/3 Code Rate 9322.34 4 39 80 40 1 32 8PSK, 2/3 Code Rate 9319.03 5 38 80 40 1 32 8PSK, 2/3 Code Rate 9318.20 BaselineanalysisresultsshowcollaborativedesignscanachievesuperiorNPVthanindependentalternativesfor both EO and SATCOM systems; however, the results critically assume successful FSS formation. The analysis does not consider sources of risk from coordination failures that diminish the desirability of collaborative solutions, exempliﬁed in the potentially-fragile choice of eliminating dedicated EO ground stations. Furthermore, as a strategic decision, choosing to collaborate or remain independent is an interactive problem among the constituent system actors that cannotsimplybeaddressedwithprobabilisticanalysisalone. Analysisofstrategydynamicsinthefollowingsection demonstrates that other factors should be considered in design selection. IV. Collaborative System Design with Strategy Dynamics This section explains and illustrates fundamental game theoretic principles of risk dominance and applies it to SoSconceptualdesign. Riskdominanceisthencomputedforresultspresentedinthelastsection. Finally,anupdated tradespace exploration methodology is presented and applied to the FSS design case with the baseline design variables and an extended case with additional design space dimensions. 14 A. Risk Dominance in Bi-level Design Games Previous sections cited strategic robustness as a desirable property of constituent systems of FSS. In the context of this paper, strategic robustness is a SoS property that enables a system designer or owner to pursue a collaborative strategy thatincreases payoﬀ over an independentstrategy without decreasingexpected value whenstrategy dynamics are considered. The game-theoretic concept of risk dominance is used here to measure the strategic robustness of systemdesignsandstabilityoftheeﬃcientstrategyinaFSScontext. Thissectionintroducesriskdominanceandits application to engineering design. Grogan and Valencia-Romero [14]develop a methodology to apply game-theoretic principles to engineering design. Theirmethodologyusestheconceptofrisk-dominancefromequilibriumselectionliteraturetoquantifytherelationships between design decisions and strategic stability in a federated system. The weighted average log measure of risk dominance(')convertspayoﬀsfromamulti-actordesigngameintoasinglenumberthatindicateswhichstrategyisrisk dominant [13]. When comparing two or more SoS alternatives, the lowest 'value most strongly favors collaboration. Theriskdominanceconceptiseasilyillustratedbyapplicationtothecanonicalstaghuntgame[ 46]illustratedin Table 7 in normal form. It presents a coordination problem between two hunters who have the choice of pursuing hare (q) orstag (k). Ifboth hunters select thehare-hunting strategy, then each receives amoderate payoﬀ (2 utils). If both hunters pursue a stag-hunting strategy, then they each improve their gains (5 utils) relative to the hare-hunting strategy. However, if one hunter chooses to cooperate and the other defects, then the cooperating hunter receives nothing (0 utils) from their unsuccessful hunt. Whereas, in this case, the non-cooperative, defecting hunter receives a greater reward (4 utils) from hunting hare due to reduced competition. Table 7 Normal form representation of a stag hunt game Actor 1Actor 2 Hare (q) Stag ( k) +qq 2=2 +qk 2=0 Hare (q)+qq 1=2 +kq 1=4 +qk 2=4 +kk 2=5 Stag (k)+kq 1=0 +kk 1=5 Choosing a strategy in the stag hunt game must consider interactive eﬀects; speciﬁcally, the strategy chosen by the otherhunterandthepayoﬀforagivenstrategycombinationasafunctionoftheplayer’sselecteddesign. Bothhare/hare and stag/stag outcomes are Nash equilibria in a stag hunt game. There is no reason to pursue stag if one’s partner chooseshareand,alternatively,noreasontopursuehareifone’spartnerchoosesstag. Whilebothare“stable”solutions, the two equilibria produce very diﬀerent outcomes. Similar to how payoﬀ dominance identiﬁes the alternative with higher payoﬀs under successfulcoordination, risk dominance identiﬁes thealternative with higher (expected) payoﬀs 15 under a possible coordination failure. Tovisuallyexplainriskdominance,Fig.3(a)illustratesahunter’sexpectedpayoﬀasafunctionoftheprobability that the other hunter chooses to hunt stag (i.e. collaborates). The intersection point ( D8) of the two expected value lines displaysthethresholdprobabilitythatplayer 9collaborates( ?9)forwhichtheeﬃcient(expectedvalue-maximizing) strategy shifts from hunting hare ( q) to hunting stag ( k). Equation (10)computesD8for a payoﬀ function, +, where superscriptsindicateplayers 8and9hunthare(qq),8huntshareand 9huntsstag(qk),8huntsstagand 9huntshare (kq),andbothplayershuntstag( kk). Duetosymmetry, D1=D2=23. Withoutadditionalinformation,i.e.undera 5050chance the other hunter chooses to hunt stag, a rational decision maker will choose to hunt hare which yields an expectedpayoﬀof 205¸4¹105º=30whichexceedsthestagpayoﬀof 005¸5¹105º=25. Thisanalysis shows why choosing the hare-hunting strategy is viable and, under speciﬁc conditions, even rational or normative, despite stag/stag being the payoﬀ dominant equilibrium. D8=+qq 8+kq 8+qq 8+kq 8+qk 8+kk 8 (10) To connect with engineering applications, a design layer is added by assuming payoﬀs associated with the strategic decisions in Table 8 are also aﬀected by the hunting implement (design) selected by each of the actors. For example, a hunter who intends to hunt hare brings dogs, whereas a hunter who intends to hunt stag with a partner brings a spear. Table 8 Stag hunt design-strategy payoﬀs DesignStrategy Combination Hare, Hare Hare, Stag Stag, Hare Stag, Stag Dogs 2 4 0 0 Spear 0 0 0 5 Bow 1 1 1.75 4.5 Adding a third design, bow, changes the strategy dynamics. Though the bow has a lower payoﬀ than dogs if hunting hare and a lower payoﬀ than spear for a successful stag hunt, it signiﬁcantly reduces the downside risk of collaboration in the event of a failed stag hunt. A similar probabilistic analysis in Fig. 8, selecting the bow for a stag hunt, shows the collaborative strategy has higher expected payoﬀ for probabilities of cooperation above 13, graphically represented in Fig. 3 (b), increasing »+¼8at?9=05from 3.0 to 3.125 despite the bow option being Pareto dominated in each ﬁxed context. Cooperationistherationalstrategyifbothhuntersusethebowandnootherfactorsinﬂuencestrategyselection. The riskdominance metric 'formalizes theconcept of risk-informedstrategy selection whenthere are morethan twoplayersorasymmetricpayoﬀs. Severaldecadesofresearchhavedemonstratedthattheriskdominantstrategyhasthe greatest basin of attraction; that is, it is selected in repeated games between multiple players, updating their strategies to maximizepayoﬀineachroundofplay[ 47]. CarlssonandvanDamme [48]alsofoundthattheriskdominantequilibrium 16 (a) (b) With bow design option Fig. 3 Expected value of player strategies under uncertainty. should be selected in single-shot games if players allow the possibility of a strictly dominant equilibrium (this requires that the players are not certain the game is a stag hunt, an entirely reasonable assumption for a real-world ‘game’ in whichpayoﬀsareuncertain). 'iscomputedusingEq. (11)whereF8areinﬂuenceweightsbasedonaninﬂuencematrix andF8=05for two-player games (see Ref. [ 14] for a complete explanation of F8and). A positive 'indicates the independentstrategy( q)isriskdominantwhileanegative 'indicatesthecollaborativestrategy( k)isriskdominant. More importantly, 'provides a relative measure of collaborative stability that can be compared across scenarios. '=Õ 8=1F8¹ºlnD8 1D8(11) The'valueforthestaghuntscenarioinTable7is 069whichisreducedto 069afterselectingthebowdesign alternative in Table 8. The bow design makes the payoﬀ dominant strategy also risk dominant; in other words, the bowdesignunderthecollaborativestrategy( k)istherationalchoiceintheabsenceofadditionalinformationabout thelikelyactionsoftheotherplayer. InthecontextofSoSE, 'actsasacriterionforcollectivestabilityofindividual objectivesasafunctionoftheselectedsystemdesigns. Selectingasystemdesignthatreduces 'morestronglysupports collaboration either by increasing the upside potential of collaboration or by decreasing the downside risk of failed collaboration(as demonstratedwiththe bowdesign). However,as illustratedhere, risk dominancemay alsodecrease upside potential, demanding a multi-attribute evaluation process to balance the two desirable traits. B. Baseline Analysis of Strategic Robustness Returning to the the independent and collaborative EO and SATCOM systems presented at the end of Sec. III, the systemsdesignedforcollaborationappeartohavesuperiorvalue. Naïvecollaborativesystemdesignmethodologiesmay stop once the value maximizing designs have beendiscovered, assuming thatcollaboration will likely besuccessful 17 because mutual beneﬁt is maximized. However, simple mutual beneﬁt is insuﬃcient to determine the rational strategy. Table 9 shows the normal form game for the EO-SATCOM SoS with value-maximizing architectures selected for independent and collaborative strategies. Simulations of FSS coordination failure produce the oﬀ diagonal values. System cost, performance, and revenue determinations for all strategic outcomes follow the two-stage decision process in Fig. 4. Table 9 Normal form representation of EO-SATCOM SoS strategies EOSATCOM Independent Collaborative $9035.61M $9024.36M Independent $346.75M $346.75M $9035.61M $9332.78M Collaborative$68911M $704.38M Fig. 4 Two-stage decision ﬂow chart. The payoﬀs in Table 9 show the SATCOM system losses are relatively small if collaboration fails, whereas the EOsystemlossessigniﬁcantlyexceedupsidepotential. Thenormalizeddeviationlosses, D8,are0.743and0.036for theEOandSATCOMsystemsrespectively. Thediﬀerenceindownsidelosseshastwocauses. First,theEOsystem costs to enable collaboration are much more signiﬁcant than those of the SATCOM system. Second, maximization of the upside value of collaboration requires architectural changes that severely inhibit functionality in independent operations (the “best” collaborative EO design has no ground stations). The model implements a ground station leasing contingencyplanusingpubliclyavailablepricing‡forzerogroundstationcasesunderfailedcollaboration. Whilethe leased ground station recovers some capability and avoids initial investment costs, operation costs are signiﬁcantly higher relative to a dedicated installation and capability does not match that of the three ground stations implemented in the best independent design. These factors result in negative NPV for EO in the case of failed collaboration. Considering the information in Table 9, it appears that the rational strategy for each system owner is diﬀerent. However, mutual collaboration and mutual independence are both strict Nash equilibria. In other words, it would ‡Accessed 7 August 2020 at https://aws.amazon.com/ground-station/pricing/ 18 be irrational for the SATCOM system to pursue a collaborative strategy if the EO system is certainly notgoing to collaborate or for the EO system to pursue an independent strategy if the SATCOM system certainly isgoing to collaborate. Unfortunately,theactionsofotheractorscannotbeknownwithcertainty. Intheabsenceofanyinformation about the likelihood of the other actors actions, it is diﬃcult to readily identify the rational strategy. Thisiswhereriskdominanceishelpful. 'providesaquantiﬁablemeasure,indicatingtherelativepreferencefor strategy-design pairs, by taking into consideration the relative impacts of each actor on the other(s) as well as the expected values for each strategy. In this case, applying Eq. (11)shows'is111. Assuming no prior information about either actor’s preference for a particular strategy, the rational strategy for bothsystem actors is to collaborate. The relative losses of the SATCOM system are small, so they are likely to collaborate. Combined with the signiﬁcant upside gains for the EO system, the overall strategy dynamics favor collaboration. C. Tradespace Exploration with Risk Dominance WhiletheNPVmaximizingdesignsproducedbythebaselinemodelproducefavorabledynamics,guidingtradespace explorationwith aprioriknowledge ofriskdominancecanproduce designsthatmorestrongly favorcollaborationby sacriﬁcing some upside potential (eﬃciency) for reductions in downside loss (robustness). The results presented in this section are produced by an updated tradespace exploration methodology that adds simulation of systems under coordinationfailuretogeneratevaluestocompute '. Implementationof 'asameasureofthecollectivestabilityof individual objectives diﬀerentiates this work from previously proposed SoSE design methods. Figure 5 shows the modiﬁed structural diagram elements for the updated tradespace exploration methodology. The output is a Pareto eﬃcient front in three dimensions: 'and NPV for each of the two systems. Fig. 5 Tradespace exploration with risk dominance structural diagram. 19 Table 10 shows the Pareto eﬃcient designs. Figure 6 plots the expected value of each EO solution against the independentsolutionandFig.7showstheEONPVand 'dimensionsofthethree-dimensionaltradespaceforalldesigns with deﬁned '. Figure 6 clearly shows that design 3 from Table 10 sacriﬁces substantial upside potential relative to the other twoParetoeﬃcient designs, butalso has signiﬁcantlyreduced downside risk. Design 3is similar tothe optimal designforindependentoperations,givingitutilityrobustnesswhencollaborationfails. Figure7showsthattheviable EO system design space is fairly narrow and a subset of designs with positive NPV under the independent strategy. The narrow range of viabledesign variablesindicates astable regionin the spacewhich yieldshigh-value designsfor both strategies; in this case resulting in a number of designs which favor the collaborative strategy. Table 10 Pareto eﬃcient collaborative designs with risk dominance EO Design Planes Satellites Aperture Altitude Ground #%+4#%+B' D4 per plane diameter (cm) (km) stations ($M) ($M) 1 3 4 100 400 0 704.38 9332.78 1110.743 2 3 3 100 400 0 685.78 9332.78 1130.733 3* 3 3 100 400 3 486.35 9332.78 1170.717 *Same as the value-maximizing design for the independent strategy, with interoperable ISLs added. All SATCOM designs were Table 6, design 1 Fig. 6 Expected value for Pareto eﬃcient EO solutions with consideration of risk dominance. The resultsshowthat the primarydilemma is for theEO system. The D4values in Table10 indicate the minimum probability of SATCOM collaboration for which the corresponding design yields higher expected value than the independentstrategy. Notethatall D4exceed0.5,yetthe 'valuesindicatethatcollaborationisdominantbecauseit accounts for SATCOM’svery low DBvalue near 0.036, which strongly favors collaboration. However, D4is a valuable metric because it can diagnose the ability of the two systems to successfully engage without requiring global payoﬀ visibility, as 'does. EOdesign1yieldsthehighestupsideNPV.However,itisonlypreferredovertheindependentsolutionforprobabilities 20 Fig. 7 Projected EO system tradespace under the collaborative strategy. ofSATCOMcollaborationintherange 085?B0C2><1,whiledesign2ispreferredfor 0743?B0C2><085 anddesign3ispreferredfor ?B0C2><0743. ThisinformationenablestheEOsystemdecisionmakertoselectthe design that maximizes expected value tailored to an estimated probability of successful collaboration. Whilethediﬀerencebetween D4valuesandcorrespondingpreferencerangesarerelativelysmall,theywouldbe enhanced with increased decision maker risk aversion, strengthening the favorability of the design with the lowest downside risk. Such risk aversion could be incorporated in the tradespace exploration activity with a risk-adjusted utility metric. The most notable feature of this analysis is the clear trade between eﬃciency and robustness for the EO system. Whiledesign1yieldshighNPVundercollaboration,thedownsideissigniﬁcantwhenSATCOMpursuesanindependent strategy. Design3tradessomeupsideforasubstantialdecreaseindownsidebyreducingspacesegmentinvestment costs and adding ground stations, mimicking the highest value independent design. D. Extended Application Case: Eﬀect of Economic Design Variables on Strategy Dynamics Solutions to real-world engineering problems are intertwined with economic and mission factors that impact utility throughout the system’s life-cycle. Therefore, this section provides expanded analysis of the original problem with added economic design variables. Adding economic design variables aims to understand how the proposed tradespace exploration methodology can facilitate evaluation of technical and non-technical variables and determine the usefulness of including economic variables in design space exploration. Itisreasonabletoassumethatcontracttermsbetweenthesystemownersarepartofthedesignspaceofacollaborative system and that the utility of a given technical implementation is a function of economic factors as well as system 21 performance. Furthermore,economicdesignshouldbeassessedforimpactstostrategydynamicsbecauseeﬃciency alone doesnot create theconditions for collaboration. Conversely, simultaneousevaluation of technicaland economic variables can identify designs with greater strategic robustness, producing more stable dynamics, including designs that would not be considered when assessing technical variables only. Theextendedapplicationcase implementsthe monthlydata transportservicefeeanda developmentcost-share as economic design variables. Fixed fee results in the preceding section indicate the downside risk of failed collaboration for SATCOM is low as compared to EO. A new cost-sharing agreement assigns up to 100% of EO’s collaboration costs toSATCOMtobalancetheriskburdenforeachsystem. Themonthlyservicefeecanvarybetween$1Mand$20M. TheSATCOMownerdoesnotrecoupthecost-shareiftheybreakthecontracttopursueanindependentsolution. EO reimburses SATCOM for the collaboration investment if EO defects. The variable fee and cost-share enable SATCOM to collect higher data transport service fees to recoup their investment from EO without jeopardizing the federation. TheeconomicarrangementreducesEO’sdownsideriskanddisincentivizesSATCOM’sdefectionwithoutincentiviz- ing EO defection. Table 11 shows a normal form example representing a modiﬁcation to the highest NPV collaborative design examined earlier with the data transport fee raised to $16.2M and addition of 50% cost-share. Note that values are similar to Table 9 but with a transfer of $192M to EO in the event that EO collaborates and SATCOM defects. While the game isno longer strictly a stag hunt, it isstilla bipolar game where 'identiﬁes the risk dominantstrategy. The upside payoﬀs are almost unchanged but the D8values are substantially reduced and '=145. The cost-share agreementincreasesSATCOM’scommitmenttothecollaborativestrategy(afactthattheEOsystemownerknows,even ifSATCOM’sactualpayoﬀsareunknown),andthecostpenaltyimposedonSATCOMcanberecoveredbyanincreased data transport service fee, while still improving '. Table 11 Example normal form game for cost-share contract EO NPV ($M)SATCOM NPV ($M) Independent Collaborative 9035.61 9024.36 Independent 346.75 346.75 8843.92 9332.27 Collaborative49738 704.88 Additionoffeeandcost-sharevariablesproducemanyParetoeﬃcientdesignsforeachtechnicalimplementation withdesirablecharacteristics. Table12listsaselectionofhighperformingcollaborativesystemdesignswithadded service fee and cost-share design variables in the tradespace exploration. Note that the design column has been changed fromthenumericidentiﬁersintheprevioustables(withtheexceptionofdesignscarriedoverfromtheprecedingsection, includedforreference)todescriptivenamesforfurtherdiscussion. Inthecontextofdesignnames,min designsare 22 those which minimize the diﬀerence in upside of collaboration for both systems. Table 12 Select Pareto eﬃcient collaborative designs with fee and cost-share as design variables Design c4#4 0 W fee cost-share #%+4#%+B' (cm) (km) ($M) ($M) ($M) 1 3 4 100 400 0 10.00 0.00 704.38 9332.78 -1.11 2 3 3 100 400 0 10.00 0.00 685.78 9332.78 -1.13 3 3 3 100 400 3 10.00 0.00 486.35 9332.78 -1.17 min' 3 3 100 400 3 13.49 0.99 731.13 9087.99 -4.41 max NPV43 4 100 400 0 6.76 0.51 1000.04 9037.11 -1.30 max NPVB3 4 100 400 0 19.74 0.01 406.28 9630.87 -0.56 max up, min '3 4 100 400 0 17.56 0.99 852.44 9184.72 -1.80 max up, min 3 4 100 400 0 19.91 0.72 673.25 9363.91 -1.57 3x4, 3W3 4 100 400 3 17.56 0.99 660.86 9179.64 -2.65 2x4, min'2 4 100 400 3 13.49 0.99 707.54 9098.41 -2.52 2x4, min* 2 4 100 400 4 14.42 0.82 534.40 9187.49 -1.75 2x3, min'* 2 3 100 450 0 14.60 0.68 556.40 9260.86 -1.32 2x3, 1W** 2 3 100 450 1 14.60 0.68 437.11 9259.36 -0.86 Note: All SATCOM designs were Table 6, design 1 *Not on Pareto front **Not in the results produced by the GA, but a promising low cost solution. Figure8 displaysdesigns fromTable 12aswell aslineswhich representtherange ofupside valuesforwhich 'is deﬁned for each technical implementation on the Pareto frontier. Three of the designs in Table 12 are not on the Pareto frontier,sotheirtechnicalimplementationvaluerangelinesareexcluded. Twoofthesedesignsareapparentinthelower left quadrant of Fig. 8; the third is nearby and is located at (556.40, 9260.86). 'values are expressed in the color map, with the scale maximum at 0. Points with '¡0are colored the same as those with '=0. Note that'decreases as EO NPV increases across technical implementations because of the low downside risk for SATCOM relative to EO. Several designs which would not be otherwise feasible, such as the two plane, three satellite per plane EO designs, become feasible when economic variables are included in the tradespace exploration. Furthermore, the preference for lower-cost designs may be even greater for a risk averse decision maker because downside risk is limited to the investment cost, which is decreased. Both the addition of economic variables and the application of risk dominance to assess the resulting designs create insights over analyses that lack one or both of these features. The ﬁrst beneﬁt from implementing economic design variables is an expanded Pareto front with new technical implementations, such as the two plane, four satellite per plane EO systemdesign and the three plane, four satellite per plane, threeground station design that were not feasible under theﬁxedeconomicconditionspresentedabove. Additionally,economicvariablesenablebalancingoftheupsideand downsides between each actor such that those bearing the greatest risk (EO in this example) have the most to gain. This 23 Fig. 8 Pareto frontier, collaborative designs with fee and cost-share as design variables. Designs in Table 12 are represented by ‘+’ markers. Lines each represent the range of payoﬀs and 'for a single technical implementation that appears on the Pareto frontier. Note that the maximum 'value on any line is 2.6, but the color map maximum is set to 0 for clarity inﬂuences the dynamics towards strongly favoring collaboration; as demonstrated by the min 'design in Table 12, which is identical to design 3 apart from the economic variables. Lastly, the economic variables move the Pareto front such that design 3 is dominated by the max up, min Xdesign, yielding a superior option for both systems. As previously mentioned, the addition of economic variables greatly expands the number of Pareto eﬃcient designs. Discriminatingbetweendesignswithsimilartechnicalimplementationsrequiresanadditionalmeasure. 'quantiﬁes the basin of attraction for each strategy as a function of the game payoﬀs, which, in this context are a function of technicalandeconomicdesign. Therefore, 'indicateswhichsetofeconomicvariablesproducethegreatestbasinof attraction for the collaborative strategy. The four “max” designs in Table 12 illustrate this point well. All four technical implementationsareidentical. However,theﬁrsttwodividetheupsidebeneﬁtofcollaborationtomaximizegainsby one or the other systems, whereas the last divides the upside nearly equally between the two systems. The min design favors collaboration the most strongly of the three, but still does not represent the optimal arrangement in terms of maximizingthebasinofattractionfortheSoS.Themaxup,min 'designshiftsagreaterproportionofthepayoﬀtothe EOsystem,maximizingthefavorabilityofcollaborationforthemosteﬃcienttechnicalimplementation. Whileitis intuitive that the solutions that maximize the payoﬀ for one system with minimal beneﬁt for the other are not optimal, ' quantiﬁes this intuition and indicates which designs are “best” with respect to creating the conditions for the SoSto form and survive. 24 Theseresultsdemonstratetheadvantageofincludingeconomicdesignvariablesandapplyingtheriskdominance criterion to their assessment. Using only the upside payoﬀs as design selection criteria creates a dilemma for each system owner between pursuing a solution that maximizes their own payoﬀ and agreeing to a solution that is most attractivetotheothersystemowners,therebyfacilitatingmutuallybeneﬁcialcoordination. 'helpstosolvethisdilemma byprovidinganobjectivemeasureofthefavorabilityofcollaborationforagivensetofdesignsasaconsequenceof balancing the risk and upside potential for each system actor. V. Discussion This work demonstrates the application of the game-theoretic equilibria selection criterion, risk-dominance ( '), to satelliteSoStradespaceexploration. Resultsshowthatmulti-objectivetradespaceexplorationcombiningtraditional measures of utility with a measure of risk-dominance yields designs that are more robust to dissolution of the SoS. Furthermore, the results reinforce the idea that eﬃciency must be traded to gain robustness. In the case of the earth observationandsatellitecommunicationsSoSstudiedhere,thatmeansreducingthecostoftheEarthobservationsystem spacesegmentandincreasingthenumberofground-stationstoensurefunctionalityintheabsenceofdatatransport services from the satellite communications system. Addingeconomicvariablestothetradespace,intheformofadatatransportservicefeeandacost-sharecontract, expands the number of feasible alternatives and the feasible regions of the technical design space. 'is shown to beavaluablemetricfordiscriminatingbetweendesignswithcommontechnicalparametersanddiﬀeringeconomic parameters. While negotiation based methods may produce satisfactory results, 'quantiﬁes the stability of the collaborative strategy given the system design utilities. This enables the identiﬁcation of designs which most strongly support desirable strategy dynamics. The high-performing designs for both systems, whether independent or as an FSS, are architecturally similar to systemscurrentlyindevelopmentordeployment[ 49,50]. TheEOsystemsarecharacterizedbyrelativelylargenumbers ofsatellitesandlowaltitudeswhencomparedtolegacysystems. TheSATCOMarchitecturesarecomprisedofsatellites with many channels, low transmit power, narrow beams, and low operating altitude. This combination of features resultsina largenumberofsmall satellites. Thenumberof usersanddatarates heldconstant,largenumbers ofsmall satellites are superior to fewer large satellites because learning curve eﬀects result in much lower production costs. It is reasonable to conclude that this ﬁnding holds true for other satellite systems, assuming that the mission’s physics permit disaggregation. Several problem speciﬁc modeling assumptions are made to accelerate the simulation eﬀorts. For example, illumination conditions are not included in the models, a major consideration for imaging. Additionally, spacecraft and user demand models that enable simulation of data collection and transport would likely uncover orbital conﬁgurations, regions, and time periods, that are more or less conducive to interoperation between the EO and SATCOM systems. 25 Simulating these interactions between thousands of satellites would be computationally prohibitive. Therefore, this problem requires further consideration, and potential solutions need to be carefully weighed against the time it takes to generate results. While the speciﬁc results are contingent upon these and other modelling assumptions, the methodological contributions of the work are minimally impacted. Therearealsoassumptionsthatinﬂuencedresultsrelatedtothecorecontributionsofthiswork. Themodelassumes that each system is “locked-in” to the design choices that are made at the beginning of the program. This is not likely to bethecase. Relaxingthisassumptioncouldmakecomecollaborativedesignsmorefavorablewhentherearedesign pathstosolutionsthatarehighervalueundercoordinationfailure. Mostimportantly,theconclusionsreachedinthis workassumethatriskdominanceisanappropriatemeasureofstrategicstability,andinturn,thestrategicrobustness of system designs. While work in other ﬁelds indicates that this is the case [ 51], additional research on the bi-level decision dynamics of SoS design problems is required to fully understand its normative application to design. A priori consideration of signiﬁcant sources uncertainty is required to generate the reliable utility measures on which the methodology depends. Future work will address this gap in the current methodology by implementing games of incompleteinformation [ 52] andother uncertaintymodelling techniques; enabling computationally eﬃcient, up-front risk modeling and methodological realism for cases where partner utility functions are uncertain. VI. Conclusion ThisworkdemonstratedamethodologyforFederatedSpaceSystem(FSS)tradespaceexploration,implementing a risk dominant equilibrium selection method from game theory that yields system designs that are well-suited to collaboration. Robustnesstocommonperturbationsisessentialtogooddesign. FSS,likeothersystem-of-systems(SoS) withdistributeddecisionauthority,aresubjecttoasystemactorwithdrawingcooperation. Therefore,FSSengineers should design constituent systems to reduce strategic uncertainty and/or its impacts. The work assumes, ceteris paribus, that an equilibrium that is both risk and payoﬀ dominant is the rational choice for all system actors; a property this workreferstoasstrategicstability. Constituentsystemdesignsdeterminethepayoﬀsforeachsystemactorunderthe possiblestrategicoutcomes,andthereforestronglyaﬀectthestabilityoftheFSS.Themethodologycanbeimplemented to assessfactors aﬀectingconstituentsystem payoﬀs, yielding anobjective measureof design qualityas it relatesto strategic robustness. The methodology is implemented in the evaluation of a FSS comprised of an earth observation (EO) and a satellite communication(SATCOM)systemintwocases. ResultsshowthatoptimalEOandSATCOMsystemsarecomposedof many spacecraft in low orbits (relative to the nearest analogs from previous decades). While the details diﬀer between thesystems,thechiefreasonsforthesetwoarchitecturalcharacteristicsarethatmanysmallspacecraftarelessexpensive thanfewlargespacecraftduetolearningduringproduction,andshorterobservationandcommunicationrangesincrease performance for a set platform design. 26 Minimization of the risk dominance measure in the tradespace exploration activity yields system designs exhibiting utility robustness, with signiﬁcantly reduced utility losses when FSS formation fails. Constituent system designs achievedrobustnessbyindependentprovisioningofsomeofthefunctionalityprovidedbytheFSS.Theserobustness enhancements diminish FSS eﬃciency, but increase the overall favorability of collaboration, thereby strengthening theFSS.VaryingtheeconomicparametersdescribingtheinteractionbetweensystemsintheFSSincreasestherange ofpossibletechnicalimplementationsandimprovescollaborativedynamics. Theriskdominancemeasureindicates whichvaluesoftheeconomicvariablesmoststronglypromotecollaborativedynamics;aninsightunattainablewhen measuring only the utility of each system or the FSS. Rigorous testing and redundant systems have long been mainstays of spacecraft design. Collaborative systems introduce a new source of uncertainty that is not easily addressed through testing and other traditional engineering processes. As FSS become a reality, system designers and decision makers will need to understand the interaction betweensystemdesigndecisionsandthestrategicdecisionofwhethertocollaborateordesignastand-alonesystem. Strategic decisions interact with technical designs, economic factors, and uncertainty to create complex decision spaces. Systemdesigners,managers,andownersmustaccountfortheseinteractionswhenconsideringdecisionsatalllevels. This work has demonstrated how modelling system strategy decisions as games enables the use of the weighted average log measure of risk dominance, ', as a criterion in multi-system tradespace exploration activities, discovering designs that are robust to coordination failure and inﬂuence strategy dynamics to favor collaboration. Appendix Table 13 shows an example link budget. Note that transmitter power, coverage area, free space loss, required E b/No and spectral eﬃciency are inﬂuenced by design variables; additional parameters are in turn inﬂuenced by these. Table 14 shows the factors and constants used for the cost estimate. Equation (12)determines space segment non-recurring (NRE) costs for SATCOM by multiplying single spacecraft dry mass by an NRE factor and the inﬂation from the base year of the factor to 2020. j=A4B=<3AHB#'B¹1¸810º (12) Equation (13) applies an additional factor when the collaborative strategy is selected. j=A4Bk=j=A4B¹1¸ (!Bº (13) 27 Table 13 Example single channel link budget Parameter Value Units Downlink frequency ( 5) 11.7 GHz Bandwidth 250 MHz Tx power (?) 2.6 W Coverage area 6683 deg.2 Tx gain ( ) 31.41 dBi Tx line loss -4.50dB EIRP 31.06 dBW Free space loss -169.4 dB Atmospheric loss -3.51dB Rx antenna diam. 0.7 m User gain 36.07 dBi Rx line loss -2.00dB Rx carrier power -107.75 dBW System noise temp. 25.61dB-K Rx C/N o 95.25 dBW Req. E b/No 5.490dB Spectral eﬃciency 3.00- Link margin 1.0 dB Power limited rate 751.5 Mbps Bandwidth limited rate 632.4 Mbps 0parameter value from Ref. [32] 1parameter value from Ref. [40] Equation (14) calculates ﬁrst unit production cost (T1). j)1B=<3AHB%B¹1¸810º (14) Equations (15)–(16) calculate EO system space segment NRE and ﬁrst unit production costs respectively. j=A44>=<3AH4#'4¹1¸810º (15) j)14>=<3AH4%4¹1¸810º (16) Equations (17)–(18)areappliedtoincreaseEOsystemmassandcostrespectivelywhenthecollaborativestrategyis selected. <3AH4k=<3AH4¹1¸ (!<º (17) j=A44>k=<3AH4k#'4¹1¸810¸ (!2º (18) 28 Table 14 Cost estimating relationship factors and constants Factor Variable Value Units Tx eﬃciency [)G0.170- Power-to-mass ratio %"' 5.0 W/kg 2010 to 2020 inﬂation 8100.18581- Annual inﬂation 80.02051- Payload power fraction %% 0.460W/W Learning curve (0.853- First ground station (10.090- Subsequent ground station (=0.060- Systems engineering ( 0.200- Program management %" 0.150- Integration & test &)0.150- Launch & orbital operations support !$$( 0.0610- Other " ( 0.160- Operations $%( 0.050- Ground maintenance "#) 0.1350- Launch cost ! 62 $M Fuel fraction  0.270- Redundancy reduction ' 0.21- Open inter-satellite link complexity (!20.1 - Open inter-satellite link mass (!<0.05 - SATCOM non-recurring engineering #'B0.4290$M/kg dry mass SATCOM recurring engineering %B0.090$M/kg dry mass EO non-recurring engineering #'40.4020$M/kg dry mass EO recurring engineering %40.0820$M/kg dry mass 0from Ref. [32],1from Ref. [53],2from Ref. [37],3from Ref. [38] 4accessed 10 May 2020 from https://www.spacex.com/about/capabilities Equation (19) increases T1 costs from increased mass and complexity when the collaborative strategy is selected. j)14>k=<3AH4k%4¹1¸810¸ (!22º (19) Equation (20) calculates the cost of each satellite with learning eﬀects. jD=8C¹=º=j)1=¹1¸ln(ln 2ºj)1¹=1º¹1¸ln(ln 2º ==1¹#cº (20) Equation (21) calculates individual spacecraft wet mass for all systems. <F4C=<3AH¹1¸º (21) 29 Equation (22)calculatesmaximumlaunchmass, <;usingalinearapproximationofAtlasV551capabilitytoapolar orbit [39], where 0is altitude in km. <;=15888148560 (22) Equation (23)determines the number of launches, =;, as the ceiling of the total mass of all satellites in a plane divided by the launch vehicle capability to that orbit multiplied by the number of planes. =;=#<F4C8 <; c8 (23) Equation (24) calculates total launch cost as the number of launches multiplied by the single launch cost. j;=!=; (24) Equation (25) calculates the number of satellites per launch to determine the number of production lots. =B0CB=#8c8 =;(25) The number of ground stations is a design variable for the EO system. Equation (26)calculates the number of ground stations for the SATCOM system. WB=#BcB 10(26) Equation (27)calculates the ground NRE costs estimated based on the spacecraft T1 and NRE costs, where j=A4is space segment NRE cost for either system. Note that the independent spacecraft NRE costs are always used as the model assumes the ground system NRE would not increase substantially with the addition of open inter-satellite links to the space segment. j=A46=3=¹j=A4¸j)1º¹ (1¸ (=W8º (27) Equation(28)calculatesgroundoperationscostsasthesumofspacesegmentoperationsandgroundsystemmaintenance costs. The independent strategy space segment NRE costs are always used. j>?B=j=A46=3"#)¸¹j)1¸j=A4º$%( (28) Equations (29)–(30)apply wrap factors to the space segment NRE and each production lot respectively, where j;>C isthecostofeachproductionlotdeterminedbythenumberofsatellitesineachlotandthecostofeachsatellitewith 30 learning eﬀects applied as calculated in Eq. (20). j=A4=j=A4¹1¸(¸%"¸ &)¸" (º (29) j;>C=j;>C¹1¸(¸%"¸ &)¸" (¸!$$(º (30) Costs are ﬁnally spread uniformly over each system program phase as described in Sec. III.B and inﬂation, ( 8), applied proportionally to each element of the cost series. Table 15 shows the location of each ground station used in the EO models. Ground stations are always selected inascendingindexorderstartingwithnumber1. AllgroundstationslocationsarenearexistingKongsbergSatellite Service§or Swedish Space Corporation¶locations. Table 15 Ground station locations Index Latitude Longitude 1 78.20 15.30 2 -72.00 2.50 3 -52.93 -70.85 4 67.88 21.07 5 68.40 -133.5 Funding Sources ThismaterialisbaseduponworksupportedbytheNationalScienceFoundationunderGrantNo.1943433. The view expressed in this paper are those of the authors and do not represent the policy of the U.S. Air Force, the U.S. Department of Defense, or the U.S. Government.