This Article Presents the Rudiments of Constellation Design to Familiarize

Home , Inclined orbit

This articlepresents the rudiments ofconstellation design to familiarize the reader with some of this field’smethods, issues, and concerns. Two exotic constellations are under consideration for various missions.

The traveling-salesman problem The need for satellite constellationsis a natural outgrowth of the need for global communications and intelligence gathering. A single satellite can only observea spherical cap regionof the Earth (as illustrated in Figure l), called the instantaneous nadir-pointing coverage circle (also tational harmonics,produce an array of constel- referred to as a footprint) of the satellite. The lations with specific propertiesto support Various coverage circle’s area,A, is a function of altitude, mission constraints. Akey point is that constella- h, and the Earth’s radius,R (6,378.14 km), tion design isnot the mere repetitionof a satellite orbit as a template. A(h)= 2a2(1 - COS 0)= 2d2h/(R + h), (1) The aggregate of satellites andorbits presents a totally new systemsproblem with a combina- which is always smaller than the hemisphere’s torial growth in complexity for the most ele- area, 2d2.Hence, we can get a quick estimate mentary analysis questions. The fundamental of the lower boundfor the number, N, of satel- performance metric is constellation geometric- lites with altitude, h, required to provide instan- coverage statistics.Ergodic theory, the multires- taneous global coverageby dividing the Earth’s olution visual calculus, and dynamical systems surface byA(h): theory are some new approachesto constellation design problems. New infrastructures and new N > 4d2/ A(h) = 2 + 2Wh. (2) paradigms willfurther develop this technology. The importance of communications satellite Although Equation 2 is not a tight estimate, it constellations cannot be overstated. In one fell tells us right away that we must always have swoop, sucha constellation can provide an under-more than two satellites to provide simultane- developed region withouta modern communica- ous global coverage.Thus satellite constellations tion infrastructure with aninstant modem com- are essential to provide any kindof global satel- munications network.The social and economic lite service. implications of this technology are enormous. At first glance, it might seemthat constellation design is merelythe act of replicating multiple copies of a single satellitein slightly different or-

1521-9615/99/$10.000 1999 IEEE bits. Althoughthe end result might seemto support that idea, the design process is actually very

58 COMPUTINGIN SCIENCE & ENGINEERING Satellite I

~ ~~ ~ Figure 2. Elliptical orbit in the orbitplane.

Basic orbital mechanics To intelligently discuss constellation design, let’s cover some basic orbital mechanics to describe the geomeuy and provide metricsto measure performance. I will touch upon only circular and el- Figure 1. The nadir-pointing coverage circle of a liptical orbits,as depictedin Figure 2. The semi- satellite on Earth. major axis, a, is half the distance of the axis through the ellipse’s longest part.The semiminor axis, b, is halfthe distance of the axis through the difficult. It is a much more organic process some-ellipse’s shortest part. The eccentricity, e = cia, what akinto the buildmg of a multicellular organ- measures the ellipse’s shape.For a circular orbit, ism. In its most basic form, each theof cells is an the eccenmcityis 0 because inthis case, the axes a identical satellite. For more complex missions, and b are equal, andc degenerates to 0. In the lim- satellites with different capabilities, such as differ-iting case whena approaches infinity witha finite entiated cellulargroupings, might be neceswy. Butb, c also approaches infinity, and we obtaina par- even inthe simplest case of identical satellites in abolic orbit with eccenmcity equalto 1. Thus, for similar orbits,the constellation performanceanaly- elliptical orbits, the eccentricity always falls be- sis can be extraordinarily complex dueto the com- tween 0 and l. The satellite’s locationon theel- binatorial effectsof multiple satellitesMung with liptical orbit is given by angleuin Figure 2 (called multiple locationson the ground. the true anomaly). The closest approachto Earth Even before the current interest in constella- is called perigee,the farthest is called apogee. tions, ground-station plannersfaced the difficult From Keplerian orbit theory, without pertur- problem of scheduling communications passesfor bations, elliptical orbits arealways confined to a multiple satellites.One way to see whythis is such plane centeredon Earth known as the orbit plane. a hard problem isto compare it to the well-known To provide space with coordinates,a fundamental traveling-salesman problem: Givena list of cities, plane must be chosento represent the q plane. what is the optimal path for a salesman to visit For Earth satellites, this plane is usually the each city once? Now suppose the salesman is a equator’s plane withthe x-axis pointed in the di- satellite and the path traversed is a fixed orbit. rection of the constellation Aries, also knownas What orbit, if it is even possible, providesthe op- the vernal equinox direction. Our orbit plane timum pathto visit the list of cities? Now increase might be inclined with respect to the equator, the number of satellites, each ain different orbit. and the angle between these two planesis simply What combination of orbits provides the opti- called the inclination, usually denoted by i. The mum coverageof allground stations?The once- orbit plane intersects the equatorial plane in a simple graph-theory problem has not only in- line known as the line of nodes. The line of nodes creased in size and combinatorial complexity, butintersects the orbit in theequatorial planein two the paths must now satisfythe constraint of a set points. The point where the satellite moves into of differential equations for artificial satellites. the +Z hemisphere is called the ascending node; Fortunately for most applications, designerstyp- the othernode is calledthe descending node. Q ically use certain well-understood satelliteorbits generally denotes the angle between the x-axis for constellation design. and the ascending nodes. Finally,the last para-

JANUARY-FEBRUARY 1999 59 sooner than expected. This tends to make the satellite reachits ascending node sooner than it t‘ would withoutthe perturbation. Hence the node appears to move backward (seeEquation 3) and the effect is called nodal regression. In other words, the orbital plane is precessing due to J2 perturbation. For a space shuttle to orbit at 200 km altitude and 28.5 degrees (deg) inclination, the nodal regression is roughly -7 deg per day, which isquite significant.

dWdt = -2.06474 x lOI4 cos(t) (1 - e2)-2 deglday. (3 1 A second effect dueto32 perturbation is called the precession of the argument of perigee. 32 causes the perigee to rotate around the orbit’s

~~ ~ normal vector, which is thealso orbit plane’s nor- Figure 3. The orbital inclination, ascending node, mal vector (see Equation4). The orbital period is and the argument of perigee. shortened because as the orbit moves from node to node, it reaches the nodal crossings fasterdue to the equatorial bulge’s gravitational forces: meter required to specify the orbit is the angle o, called the argtcment ofpengee, between the as- dddt = 1.03237 X 1014 (4- 5 sin2(t]) cending node and the perigee. These angles are (1 - e2)-2 deg/day. (4) indicated on theinclined orbit in Figure 3. The e, six parameters, a, i, Q, m, and Malled the . ..,. . Qrblial &&o;gy:: ;:: .. .: classical orbital elements-completely charac- .. terize a conic orbit. Now that we have the basic tools for orbital mechanics, let’s examinesome of the space in- Perturbations dustry’s most useful orbits. For Earth-orbit design, we must consider three major perturbations: atmospheric drag,the Cirnrlm orbit. Circular orbits arethe most use- gravitation ofthe sun and moon, andthe Earth’s ful because of their symmetry andease of analysis. equatorial bulge. For typical constellations,av- They move with uniform speed, andwe can fre- erage orbit altitude is generally assumed to be quently estimate their coverage geometry with high enough to avoid the atmospheric drag that good, closed-form approximations. However, would causethe orbit to decay ifunmaintained. note that therate at which circularorbits cover Similarly, the altitude is assumedto be no higher ground is not uniform. In other words, a circular than thatof geosynchronous orbits (35,863 km) orbit’s projected velocityon the Fundis not uni- so that luni-solar perturbations are small enough form due to the Earth’s rotation. All points on and might be ignored during the design stage. Earth rotate at the same angular velocity around In actual satellite operation, these forces mustbe the North Pole. However, the actual velocityat modeled and included in the orbit calculations the equator is faster than that at a higher latitude. to the degree of accuracy that the mission re- This is because latitude circles have different radii, quires. But during typical constellation design, but the same angular rate. Because velocityis an- the main effect that must be included isthe so- gular rate times radius,the points at different lat- called32 term, named after the coefficient of a itude circles have different velocities. planet’s geopotential harmonic expansion. Three distinct cirmlar-orbit classes have emerged The32 term corresponds to the planet’s equa- in orbital nomenclature: the geosynchronousEarth torial bulge and hasthree important effects on orbit (GEO), the lowEarth orbit (LEO) for orbital the orbit. The equatorial bulge might thought be altitudes in the 100 to 1,000 km range, and the of as anadditional tire of massaround a spherical medium Earth orbit (MEO) for altitudes between planet’s equator.This extra mass tendsto pull the GEO and LEO. The cut-off point betweenLEO satellite down towards the equatorial plane and ME0 is not, of come, well-defined. A related

60 COMPUTING IN SCIENCE h ENGINEERING acronym for highly elliptical orbit (HEO) torefers orbits with high eccentricity.A final acronymof some use is the geosynchronous transfer orbit (GTO), which has a perigee altitude(usually 200 km) typical of a LEO, and the apogee altitudeof a GEO. It is an intermediate orbit usedto transfer a satellite fromLEO into GEO through the well- known Hohmann transfer depicted in Figure4. Geosyncbronous orbit. Geosynchronous orbits areperhaps the best known of allorbits due to their extreme usefulnessfor the communications satellite industry. The idea is as simple as it is elegant: find a circular orbit around the equator that moves at the same angular rate as the Earth’s rotation. A satellite inthis orbit has a period of 24 hours and would hover over the same point on theequator. Equation 5 provides the relations for orbital period: Figure 4. The CTO-Hohmann transfer of a satellite from a circularLEO to the high-altitude CEO. Period = 27rSqrt(a3/p) (5)

From this, we can compute a geosynchronous tude, with an argument of perigee at -90 deg orbit’s radius to be 42,241 km. p is the Earth‘s (over the south pole). Critical inclination is se- gravitational parameter, 398,601.2 km3/sec2. lected to fix the perigee over the south pole and Elliptical orbit. Elliptical orbits are much moremaintain the observation geometry over the difficult to analyze becausethe orbital velocity is northern hemisphere, where this orbit spends not uniform. The satellite movesthe fastest near most of its time. Figure 5 plots the Molniya or- perigee andthe slowest near apogee. This is eas- bit’s groundtrack. Note how the apogee is over ily observed from Equation 6 (an orbit’s energy both the continental US and Russia. This per- and the fact that energy is conserved): mits the satellite to gather data overthe US and transmit it while over Russia. Energy = d/2 - p /r, (6) Periodic groundtrack dit.The geosynchronous orbit and the Molniya orbit bothhave pe- where r and v are the orbit’s instantaneous ra- riodic groundtracks.For the geosynchronous or- dius and velocity. When the orbital velocity’s bit, the groundtrack is justa point. The Molniya nonuniformity is combined with nodal and orbit has a U-shaped one. For planetary obser- perigee precessions andthe Earth’s rotation, the vations, scientistsoften must take data overthe coverage analysisbecomes extremely difficult. same ground locations repetitivelyunder vary- However, the motion’s nonuniformity implies ing conditions. It is possibleto balance nodal re- that the satellite will linger over some regions gression with the Earth’s rotation rate to ensure and rush through others. If the72 effects canbe the groundtrack repeats after several orbits. controlled, the coverage’s nonuniformity can be Sun-synchronous orbit. To maintain the same exploited to great advantage. The next class of lighting conditions for observing a planet, it orbits provides the control. might be necessaryto maintain a constant geom- What makes the elliptical orbit particularly etry between the sun and the satellite. We can difficult to handle undery2 perturbation is the achieve this by adjusting the nodal regression perigee’s rotation. From Equation 4, we note rate that precesses the orbital planeto match the that when the inclination is S3.4 deg, the sun’s one deg/day motion. Such an orbit will al- perigee is fixed. This special inclination is called ways visit the same latitude at roughly the same critical inclination. Critically inclined elliptical time of day, thereby providing a near-constant orbits are very useful.The most famous of these lighting condition to observe the same location. is the Molniya orbit, which isa highly elliptical Figure 6a shows this geometry. In Figure 6b, we 12-hour-period orbit the Soviets originally de- see the orbitfrom the sun’s point of view, which signed to observe the northern hemisphere. A shows that the Earth never occultsthis orbit. typical design sets the perigee at 200-km alti- This might be very useful if,for example, the

JANUARY-FEBRUARY 1999 61 the satellite to linger overNew York on the day side aslong as possible and spend minimum time on Earth‘s night side. The critical inclination guarantees that the perigee andthe apogee’s location are both fixed.’

Basic constellation types Now that I have gonethrough the orbital zoo,let’s put these cellstogether and designa multicellular organism: the constellation. Recallthat the chief reason why constellations are neededthat is one Figure 5. Molniya orbit’s groundtrack. satellite can cover onlya limited portion of the Earth at any particularinstant. For global coverage, whether for observationor communications, a constellation is theonly solution. Fortunately,this mission requiresa lot of power and the solar pan- solution fitsin well with the current technological els must alwayspoint at the sun. This geomeuy trend to decentralize control and distribute the is maintainedthroughout the year. Perturbations process acrossa network The key differences are due to the Earth’s orbit ellipticity will causethe that the constellation isa wireless network whose geometry to shift slightlyif it is not compensated nodes arenot fixed and that operates in a less ac- by maneuvers. This is generally acceptable for cessible andmore challenging environment. most missions. Besides providing constant power,One of the key driversfor current constellation the solar geometry‘sconstancy also impliesa more design is the need for global wirelesstelephony stable thermal environmentthat helps with hard- services. Geosynchronous networks have beenin ware design and mission operations. service for some time,but these behemoth satel- Sun-synchronous orbits tendto be near-polar or- lites are extremely costly to build, launch, and bits with inclination greater than 90 deg. From maintain. The current drive for smaller, cheaper Equation3,weseethatforthenodaldrifttobepos- satellites and lower cost has driventhe network itive, the inclination mustbe greater than 90 deg. from a geosynchronous altitude of over 3 5,000 For typical Earth satellites usingthis orbit, thealti- km down to around 1,000 km for mobile satellite tude is generally around1,000 km, which yieldsa constellations. This drop in altitude represents near-polar orbit with inclinations around100 deg. many cost differences: satellites are cheaper to There are sun-synchronousorbits whose peri- launch into orbit, the telecom and power re- gees are fixed due to judicious selectionsof the quirements might be reduced, and the overall orbital eccentricity and balancing the effects of satellite hardware is cheaper to build. But the the 32 and j’3 harmonics. These are known as downside isthat now insteadof three to 10 geo- frozen orbits. Frozenorbits for Earth further re- synchronous satellites, close to 1,000 satellites quire o = 90 deg and for Mars,o = -90 deg. might be requiredto provide global coverage. Euipso-type orbit. Until recently, constellation In principle, constellation zoology is much design has beenrestricted mostly to military or more complex becausearbitrary combinations scientific applications. Civilian telecommunica- of orbits might be consideredin building a con- tions have very different requirements and de- stellation. But in practice, due to the enormous sign drivers. In particular, for business applica- difficulty in analyzing constellation performance, tions, peak usage tends to occur between 9 am a few have evolved andemerged as useful design and 5 pm (the day side). Designers decidedthat concepts. I reiterate here the enormous differ- an ideal situation would have the communica- ence and technical challengea network presents tions satellites bunch up over the day side and to the analyst as opposedto the task of analyz- move quickly through the night side. This is ing a single network node. The combinatorial achieved witha critically inclined, highly ellipti-complexity exhibitedin the factorial growth of cal, sun-synchronous orbit with apogee placed the number of branches in the analysis tree is a near the desired locationon Earth (such as New major factor for the increase in difficulty. York City). The sun synchronicity ensures the Another high-level driverin constellation de- satellite will always visitNew York at about the sign is the system’s communications design.Al- same time of day. The high eccentricity enables though this is an obvious fact, its impact on the

62 COMPUTING IN SCIENCE & ENGINEERING Midnight (a)

Figure 6. (a) 6 am view ofsun-synchronous orbit fromabove theNorth Pole, and (b) 6 am view ofsun-synchronous orbit from the sun’s view.

constellation design is crucial and cannot be design a communications networkon the ground. overemphasized. For example, whether the con- A first attempt might beto uniformly dividethe stellation’s satellites have intersatellite commu- region witha Cartesian grid and placea communications links can make a big difference in the nications node at each of the grid points.On the number of satellites the constellation requires. sphere, asideiiom the regular solid polygons well- This, inturn, has a large impact on total system known to the Greeks, there is no simple wayto di- cost. Unfortunately, system trades of this type vide the sphere into regular Cartesiangrids. Even cannot be fully addressedin this limited article. if this were possible,there is no means at present to maintain a network of satellites at fixed nodes. Constellations withcirccu1a.r orbits Mathematically, this is the challenging geometric Circular orbits are well-understood and providecombinatorial problemof tiling on the sphere. the most uniform and stable design component The next bestthing is to preserve as muchsym- for a constellation. In the case where 24-hour uni- metry as possible by usingthe same template cir- form global coverageis desired, a constellation cular orbit repeatedly in a regular and sensible with circular orbits providesan optimal solution. fashion. Three parameters describethe Walker constellation:* T/P/F. Tis the total number of Geosyncbronous constellations. From Equa- satellites, P is the number of orbit planes, andF is tion l, we can estimate a geosynchronous satel- the phasing parameter.The Tsatellites are equally lite’s coverage circleto have an 81-deg half-cone divided amongP planes with the same inclination. angle. This implies that with three geosynchro- The planes are evenly spacedby 360/P deg. The nous satellites, we can cover between 81- and phase-angle offiet is given by 360 F/T deg to en- -81-deg latitude of the entire world geometri- sure a more optimal packingof Earth’,,coverage cally. The NASA TDRSS system usedthis de- circles. For example, Motorola’s Iridium constel- sign to provide a global communications net- lation design is basedon the Walker constellation. work for NASA missions. From the spacecraft Stn?m-of-mverage nmStell/ttion. Streets of cov- and ground communications systems design, erage approaches the problem slightly differently more spacecraft might be required to provide from the Walker alg~rithm.~,~ basic The idea isto service withhigher quality or greater through- design the coverage afor single plane oforbits first, put capacity. In general, such a system would re- then extend it to the entire globe by duplicating quire at most tens of satellitesas opposed to the the satellite plane repeatedly.Start with an orbit thousand-satellite system usingLEOS or MEOs. plane and populateit with Ksatellites evenly spaced Walker constellation. Suppose you wished to with overlapping coverage circlesas shown in Fig-

JANUARY~EBRUARY1999 63 ure 7.This ensures theK satellites aroundthe or- vers should be carefully considered. Raisingthe bital plane cover an entire band.This band is called perigee while keepingthe semimajor axis (hence a street of coverage. The K- period) constant solves the decay problem but satellite plane is duplicated mul-greatly lessensthe elliptical orbit coverage strat- tiple timesuntil various streets egy's effectiveness. Consequently, most design- of coverage coverthe equatorial ers tend to shy away from elliptical orbits, espe- region. This guarantees an an- cially for LEO constellations where the drag nular region between two lati-problem can be very serious indeed. tudes (determinedby the inclination altitude)and is completely covered. Constellation performance metric The two most critical satellite functions are

rt"cI observation and communication. In both in- Street of stances, the service-frequency availabilityis the coverage When nonuniform coverage crucial metric for performance. Of course, the Figure 7. Street of coverage. is required,elliptical orbits are word available is loadedin that it depends highly much more efficient. The on the nature of the service required and the Molniya orbit is a good exam- hardware involved.An infrared detector is avail- ple becauseit is designedto spend mostof its 12- able to observe a patch of dark sky only when it hour period over the northern hemisphere. In is pointed away from the sun, Earth, or moon, fact, it was designed to gather information over but a camera photographing the Earth must the US for 12 hours and then relay the data over view it in sunlight. The most primitive memc is the Soviet Union during the next 12-hour orbit. geometric coverage: how often is the region in Thus if one requires 24-hour data gathering, at the spacecraft's line of sight and availablefor ob- least two satellites are needed. For redundancy servation bysome instrument? Even this simpli- and smoother transitions between satellites tak- fied metric is not trivial for a single satellite. ing data, a three-or-more satellite constellation in Molniya orbit is desirable. They can be designed to have the same groundtrackbut observe the same locationsat different times ofday. does a satellite see a ground station? A view pe- With a colleague, I designed the Nuonce con- riod is the amount of time a satellite is visibleto stellation (NonuniformOptimal Network Com- a ground station during a flyover. Ofcourse, in- munications Engine, using elliptical LEOS) tegrating the orbit and accumulating the times specificallyto target the western hemisphere dur- when the satellite sees the ground station easily ing business hours.'The constellation usedcrit- simulates this. When there are multiple satellites ically inclined, sun-synchronous, elliptical orbitsand ground stations, once again the combina- with the apogee overthe northern hemisphere. tions overwhelm the situation even with simple Recall that sun-synchronous orbits always cross propagators. the same latitudes during the same time of day. Instead of attaching the view circleto a satel- The Nuonce constellation concentratedthe orbit lite groundtrack's nadirpoint, one can attachthe planes that cross New York City between 9 am view circle to the ground station. Visibility be- and 11 am, and 1 pm to 5 pm. This concentrates tween satellite and ground station is achieved the coverage during the busiest times over busi- only if the satellite nadir crossesinto the station- ness hours where peak support is needed. This view circle asin Figure 7.This picture brings to independently developed constellation concept mind the Poincare Recurrent Theorem, which resembles Draim's copyrighted Ellipsoorbit. states that given a point x wandering ergodically Elliptical orbits are generally more difficultto in a box ofvolume 1, the probability of finding x handle. In addition to the problem with the ar- in regionA in the box is equal to A's volume. Put gument of perigee precession, the perigee pas- another way, the percent of time x spends in A sage itself is problematical todue drag effectsthat equals Volume@) x 100. If this holds for the cause the orbit to decay or lose its desirable char- satellite view-period problem, then the station- acteristics. Of course, thesecan be corrected with view-circle's area divided by the area the satel- maneuvers. But when you havea 1,000-satellite lite groundtrack band (Figure 8) covers should constellation, operational simplicity becomesa provide coverage probability, hence the long- critical cost issue-any means to avoid maneu- term average of the coverage viewperiod. Un-

64 COMPUTING IN SCIENCE h ENGINEERING fortunately, the satellite groundtrack‘smotion on Earth is not ergodic-the Recurrence Theorem does not apply. However, the reason it fails is due to the fact that satellite groundtracksdo not mme uniformly across Earth. They tend to bunch up near the poles. What if you could spreadthem apartas you go up in latitude closerto the pole? Indeed, using a weight functionto compute the area can this. do Mathematically, this is finding a measure (weight function) that is invariant underthe groundtrack‘s flow. In other words, an area weighted withthe measure remainsconstant under the flow of the groundtrack I found sucha measure for circular orbits and derived an integral that provides the Figure 8. Two stations with the same amountof probability fora satellite to be in a given ground view periods. station’s view.’ This integral replaces the differential-equation dynamics requiredto generate the station-view periods neededto compute conven- Suppose we use an equal-area project (such as tional algorithm probability. the Peter’s projection of the sphere) onto a rec- When using this approach, it becomes practi- tangular grid. Now every point in the rectangle cal to ask global questions suchas, given all satel- has the same infinitesimal area. On the com- lites in a circular orbit with a 1,000-km altitude, puter, we might consider a pixel as an infinitesi- what arethe best locationsto place a ground sta- mal. Thus, if we generate a plot showing where tion for maximum contact time betweenthe satel- the satellite groundtrack passed through the sta- lite andthe station?From this theory, we see that tion-view circle, just bycomputing the number stations at latitudes nearthe poles tendto provide of pixels in the view circle under an equal area the most contact time (because the satellite projection allows us to compute the coverage groundtracks tend to bunch up near the poles). quickly. Furthermore, the map projections han- Figure 8 shows the view circles of two stations, dle the data-organization problem. Once we one near the equator and one near the satellite compute maps with coverageinformation, they footprint band’s upper edge.The surprisingfdct is can be operated upon algebraically to produce that both stations have the same probability of further statistical products. seeing the satellite even though the view-circle One limitation to this approach isthat an esti- intersections withthe band appear different un- mate’s accuracy dependson the map’s resolution. der the standard sphere measure. To overcome this drawback, a multiresolution map function replacesthe pixel-based map.This also separatesthe computation completely from the visualization. Nevertheless, due to its con- a geometric approach to coverage computation ception, we still call this method multiresolution might be useful. Typically, coverage analysis is vimal calculus, which is currently under patent produced by first discretizing the sphere into a review at JPL. set of polygons. The propagated groundtrack is checked against the polygons to determine Global coverage enatysls whether it is insideor outside, and statistics are The reason for ergodic method develoDment accumulated to generate coverage history. Sphereand visual calculus to facilitate global-cover- discretization isnot a trivial mattera uniform age-information computation. In thecase of er- finite partition is hard to come by becausethere godic theory application, it enables a quick, are only five known classical solids. Asidefrom global calculation ofthe station-view-period per- the problem of how to uniformly partition the formance so that a planner can easily decidethe sphere, the task of keeping track ofthe statistics best location to place a ground station based on can quickly becomea computational nightmare. knowledge of the satellites it is supposedto ser- An answer to this problem is visual calculus vice. Prior to theergodic method, such a com- (based on the fact that an infinitesimal polygon putation would have been nearly prohibitive due provides a uniform discretization ofthe sphere). to the large amounts of computation required.

JANUARY-FEBRUARY 1999 65 Figure 9. Revisit-time map of a double constellation withradar instruments.

Similarly, the visual calculus is designedto pro- bits of the same radius to always intersect twice vide global statistics of visibility, revisit,other and per orbit. For elliptical orbits, the intersection coverage data. In Figure 9, I provide a revisit- patterns might be more complex, but it is virtu- time map computed using visual calculusfor a ally impossible to enforce a rigid polygon for- complex mission.In this instance, two constella- mation without propulsion and control. The tions each carry differentinstruments. The two cost of maintaining a rigid formation for long constellations are at different altitudes to opti- periods is prohibitive using the brute-force ap- mize their respective instrument performance. proach. The only viable ones are the string-of- The coverage analysisrequires modeling instru- pearls formation, where a series of satellites fol- ment performance, which is highly sensitiveto low one another in the same orbit. Another geometry. Visual calculus provides a language formation conceptis to use heliocentric orbits hdalgorithm to compute coverage statisticsto (such as the Earth’s orbit), where gravitation is answer the revisit question: howlong do I have much moreuniform. In this case, a rigid forma- to wait until the next satellite can seeme? In this tion might be held together for some time be- instance, the green area providesa revisit timeof cause the orbitchanges so slowly. less than 30 minutes, the yellow area providesa The second exoticconcept is a quasihalo con- rwisit time of lessthan one hour, andthe orange stellation (Figure 10) around the L1 Lagrange area providesa revisit time greaterthan one hour. point. LI is a point between the sun and the The instruments do notview the black region at Earth, roughly 1.5 million km away from Earth all. Note theresulting map’s complex geometry. along the sun-Earth line (where EartMsungrav- itation balances with the rotational force to produce a relative equilibrium).A particle placedat Exotic constellations L1 willremain there forever, ifno perturbations Two exoticconstellation concepts areunder con- are introduced. But L1 is unstable, and the re- sideration for various missions. The first con- gion around it in the phase space is chaotic. cept is formation flying. The idea is to put a There are 3D quasihalo orbits around L1 that group of satellites in orbit around Earth and wind around a torus6J These orbits provide the force them to fly in a geometric pattern such as a dynamics to control a constellation near L1 fly- triangle. In the case of loose formations, this is ing information. reasonable. The polygon formation is not re- The design of such constellations requires an quired to be rigid and might flex and change understanding of dynamical systems theory, par- shape as the orbits evolve. Recall that on a ticularly of invariant manifolds. A great deal of sphere, two great circles must alwaysintersect at work isrequired to understand the dynamics of the antipodes. This fact forces two circular or- these orbits and howto control them. In partic-

66 COMPUTING IN SCIENCE & ENGINEERING ular, continuous thrusting is required for their control, perhaps with ion engines.This requires an understanding of optimal control theoryin the three-body problem-currently an active area of research and development.

but are incomplete. The problem of multiple satellites has not been investigated. Although satellite constellations have been around for some time, the recent interest in the satcom industry has given the subject new life because so much is at stake. But despitethe fact Figure 10. A quasihalo constellation near11. that this is a multibillion-dollar industry just waiting to take off, the efforts expended in the area of mission analysis and designare dispro- figures. Lastly,I thank Ralph Roncoli forhis in- portionately small. Considering how much sightful review. cheaper simulation and analysis are compared with attempts to fix complex system problems References after launch, this is incomprehensible. 1. M. Lo and P. Estabrook, "The NUONCE Engine for LEO Net- This problem isa result of the past successof works,"Pmc. Fourth lnt'l Mobile Satellite Conf., 1995, pp. 193-197. the satellite industry, where so much has been 2. J.C.Walker, Circular Orbit Patterns Providing Whole Earth Cover- done with seemingly so little investment into age, Tech. Report 70211, Royal Aircraft Establishment, 1970. mission design and trajectory analysis. The 3. R. Lu&K, "Satellite Networks for Continuous Zonal Coverage," growth of technology has created much more American Rocket SOL I., Vol. 31, Feb. 1961, pp. 179-1 84. complex instruments and even more complex 4. W. Adams and L. Rider, "Circular Polar Constellations Providing mission requirements unimagined in the 1950s. Continuous Single or Multiple Coverage above a Specific Lati- tude," /. Astronautical Science, Vol. 35, No. 2, Apr. june 1987, Despite the incredible advances incomputation pp. 155-192. hardware and software, at the end of the 20th 5. M. Lo, "The Invariant Measure for the Satellite Ground Station century, we have reacheda plateau wherethe old View Period Problem," Hamiltonion Systems with Three or More trajectory technology no longer adequately Degrees offreedom, Kluwer Academic Publishers, Dordrecht, The serves modern mission requirements. New Netherlands, 1998. 6. C. Comez et al., "Lissajous Trajectories Around Halo Orbits," methods, new paradigmssuch as dynamicalsys- Tech. Report AAS 97-106, AAS/AIAA Space Flight Mechanics tems theory, chaotic orbits, ergodic theory, and Meeting, Huntsville, Ala., Feb. 1997. so forth, will provide new capabilitiesto respond 7. K. Howell and 6. Barden, "Formation Flying in the Vicinity of Li- to new challenges.9 bration Point Orbits," Tech. ReportAAS 98-169, AAS/AIAA Space Flight Mechanics Meeting, Monterey, Calif., 1998. A&"gment% This article presents work carried out at the Martin W. Lo is a member of the Navigation and Jet Propulsion-Laboratory, California Institute Flight Mechanics Sectionat the JetPropulsion Labora- of Technology under a contract with NASA. I tory. Lo received his PhD from Cornell University and thank the JPL Telecommunications and MissionBS from Caltech. His technical interests include trajec- Operations Directorate, the Center forSpace tory design and dynamical systems theory. Contact Mission Architecture and Design, and Dr. him at Jet Propulsion Laboratory, 4800 Oak Grove Dr., William Johnson for supportof the work pre- Pasadena, Calif. 91 109; [email protected]. sented inthis article. Kathleen Howell and Brian Barden providedthe quasihalo orbit. S.D. Ross, G. Hockney, andB. Kwan providedmany of the

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