Relative Orbit Determination for Unconnected Spacecraft Within a Constellation

Open AccessEngineering NotesRelative Orbit Determination for Unconnected Spacecraft Within a ConstellationTong Qin, Dong Qiao and Malcolm MacdonaldTong QinTsinghua University, 100084 Beijing, People’s Republic of China*Postdoctoral Researcher, Department Precision Instrument; .Search more papers by this author, QiaoBeijing Institute Technology, 100081 China†Professor, School Aerospace Engineering; (Corresponding Author).Search author MacdonaldUniversity Strathclyde, Glasgow, Scotland G1 1XJ, United Kingdom‡Professor, Electronic Electrical authorPublished Online:21 Dec 2020https://doi.org/10.2514/1.G005424SectionsPDFPDF Plus ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail AboutI. IntroductionLarge constellations spacecraft are set become increasingly common. determination (OD) can be nontrivial task large constellation operator, efficient operation such is crucial challenge. This Note aims at addressing one critical issue constellations, that is, the relative OD. Relative OD provides basic state information subsequent control operation. In addition, determining orbits used aid certain scientific operational objectives, as gravity recovery in [1] Earth observation [2].A significant volume work [3–7] has focused on two spacecraft; essentially an estimation problem based intersatellite ranging measurements. The prior [8] shown measurements not enough determine all orbit elements. Hill Born [9,10] detail obtainable elements using range measurements, full element only obtained through consideration multibody dynamics. Qin et al. [11] further analyzed sensitivity orbital geometric configurations, showing performance some special configurations degraded, making underivable.To best authors’ knowledge, no considered between nondirectly connected spacecraft. challenge unconnected needs addressed so every obtain its orbit, completely derived. Without direct processed neither nor simple linear problem.To realize whole first proposes indirect method spherical geometry. A link established via intermediate target according [11]. then developed trigonometry proposed various different number intermediated Then, extension perturbed dynamics discussed.II. Indirect Method Based Spherical GeometryA. ModelsConsider SA SB in, respectively, OA OB with following Keplerian elements: EA=[aA,eA,iA,?A,?A,fA](1)EB=[aB,eB,iB,?B,?B,fB](2)where aA aB semimajor axes, eA eB eccentricities, iA iB true anomalies, ?A ?B longitude ascending nodes, ?A ?B argument periapsides, fA fB respectively. According [11], measurement most axis, eccentricity, orientation, anomalies orbits, given XA,B=[aA,eA,fA,aB,eB,fB,?,?A,?B]T(3)where ?, ?A, ?B three describing orientation projected celestial sphere, Fig. 1. angle ? planes, whereas ?A angular distances from periapsis intersections orbits. positive direction motion.The expressed elements, ?=cos?1(cos(iA)cos(iB)+sin(iA)sin(iB)cos(??))(4)?A=tan?1(sin(??)sin(iA)cot(iB)?cos(iA)cos(??))??A(5)?B=tan?1(sin(??)?sin(iB)cot(iA)+cos(iB)cos(??))??B(6)where ??=?2??1 difference nodes.Fig. 1 elliptic onto sphere.B. MethodsIn than spacecraft, it reasonable consider may linked other might due curvature Earth, limitations hardware, or any reasons. conclusion longer applicable Therefore, presented.Consider SA, SB, SC corresponding OA, OB, OC. projections sphere 2. Crosslink also SC. connected. Denote ?i,j great circles i j, ?ii,j distance projection periapse intersection j. vector: XA,C=[aA,eA,fA,aC,eC,fC,?A,C,?AA,C,?BB,C]T(7)Since aA, eA, determined aC, eC, fC SC, ?A,C, ?AA,C, ?CA,C.Fig. 2 Schematic sphere.Based law cosines angles, ?A,C 3 ?A,C=cos?1(cos(?A,B)cos(?B,C)?sin(?A,B)sin(?B,C)cos(?BA,B??BB,C))(8)Obtaining calculated sines, sin(?BA,B??BB,C)sin(?A,C)=sin(?AA,B??AA,C)sin(?B,C)=sin(?CA,C??CB,C)sin(?A,B)(9)?AA,C=?AA,B?sin?1(sin(?BA,B??BB,C)sin(?A,C)sin(?B,C))(10)?CA,C=?CB,C+sin?1(sin(?BA,B??BB,C)sin(?A,C)sin(?A,B))(11)Fig. All potentials triangular.Equations (8–11) give process calculate orientations OC 3. However, constructed Thus, solutions unique. Considering possible triangular, there total eight types geometrical 3.The computation equations Oc under Table 1, accompanied classification criteria.In ?cos=cos(?A,B)cos(?B,C)(12)?sin=sin(?A,B)sin(?B,C)cos(?BA,B??BB,C)(13)??A=sin?1(sin(?BA,B??BB,C)sin(?A,C)sin(?B,C))(14)and ??C=sin?1(sin(?BA,B??BB,C)sin(?A,C)sin(?A,B))(15)It potential lead four groups equations. Cases 8 have same do cases 7, 6, 4 5. It therefore whether inputs constrain solution sets into uniqueness. Consider 4. 4a, symmetric about OB. 4b, 4c, both those Because symmetry, Figs. ?A,B, ?B,C, ?AA,B, ?BA,B, ?BB,C, ?CB,C, which means But geometry 2.Considering above cases, symmetry cannot limit An priori value each needed distinguish configuration. Usually, roughly knows either ground station onboard integration. rough help accurate method.Fig. Three having 2.Note geometry-based method. authors illustrates quite change configuration.III. Sensitivity Analysis Geometric ConfigurationsA. For Two Connected SpacecraftIn was coplanar, circular, fewer results summarized Tables general refer non-coplanar, nonsymmetric, noncircular orbits.It seen coplanar configuration causes unobservability planes. circular anomaly distance. There even severe configurations. As takes inputs; hence, analysis performed reveal spacecraft.B. Spacecraft1. One Intermediate SpacecraftWithout loss generality, taken example analysis. Eq. (10), ?BA,B??BB,C ?A,C. requirements these parameters non-coplanar nonsymmetric nonsymmetric.For data, planes does exist When ?A,B=?B,C,?BA,B=?BB,C(16)Therefore, ?A,C=0(17)Equation (17) indicates that, though angles if point significance designing crosslink multiple maximum long crosslinks, determined.In where ?A,C?0, ?AA,C ?CA,C depends ?AA,B ?CB,C obtained. detail, 3, thus (12). If fA??AA,B obtained, fA??AA,C solved.In ?A,C=0, satisfies conditions ?A,B=?B,C ?BA,B=?BB,C. right equality (11), we get ?AA,B??AA,C=?CB,C??CA,C. determined, requires ?AA,C??CA,C calculated. them combination 5.Comparing advantages. availability (fA??AA,C)?(fC??CA,C) orbit. weakens and, when there’s third better determined.2. More SpacecraftWhen required simultaneously. Assume SD performing SD, need ?A,D=cos?1(cos(?A,C)cos(?C,D)+sin(?A,C)sin(?C,D)cos(?CC,D??CA,C))(18)sin(?CC,D??CA,C)sin(?A,D)=sin(?DA,D??DC,D)sin(?A,C)=sin(?AA,D??AA,C)sin(?C,D)(19)?AA,D=?AA,C+sin?1(sin(?CC,D??CA,C)sin(?A,D)sin(?C,D))(20)?DA,D=?DC,D+sin?1(sin(?CC,D??CA,C)sin(?A,D)sin(?A,C))(21)To ?A,D, still ?C,D ?CC,D (or fC??CC,D circular) fC??CA,C C combining circular), ?CA,C??CC,D solved. ?A,C=0. Thus (18) simplified ?A,D=?C,D(22)The ?A,D solvable. requirement Similar case ?AA,D ?DA,D related eccentricities detained omitted here.The 6. Compared 5, identical By parity reasoning, illustrating feasibility method.IV. Extension Non-Keplerian OrbitsThe two-body difficult, unobservable. rotation along axes inertial coordinate system [8]. investigated. Perturbed would improve observability, observable. once realized dynamics, they easily. Next, technique non-Keplerian nonspherical discussed.Considering J2 perturbation, acceleration rotationally z axis system. Using node (inclination periapsis) ?? express model X=[aA,eA,iA,?A,fA,aB,eB,iB,?B,fB,??](23)The drift rate (23) referred [12]. used, by: ?=?rB?TBArA?(24)where TBA transformation matrix reference rB rA are, their own systems, position vectors rA=[rAcos(fA),rAsin(fA),0]T(25)rB=[rBcos(fB),rBsin(fB),0]T(26)rA=aA(1?eA2)1+eAcos(fA)(27)rB=aB(1?eB2)1+eBcos(fB)(28)TBA=[cos(?B)?sin(?B)0sin(?B)cos(?B)0001][1000cos(?)sin(?)0?sin(?)cos(?)]?[cos(?A)sin(?A)0?sin(?A)cos(?A)0001](29)Use (24) estimate states (23). After obtaining easier way, ??A,C=??A,B+??B,Cor??A,C=??A,B???B,C(30)An above, situations Sec. II. numerical verification part V.C.V. SimulationAn Earth-centered simulation applied verify theoretical integrated fourth-order Runge–Kutta effect weakening studied simulations Then simulated.A. Validation MethodThe six elliptical values listed denoting ith Si Oi. P1 P10 10 points. Si+1 i=1, 2, 4, constellation, maintains line-of-sight neighboring always available.The O1 O3, O4, O5, O6 validate initial knowledge errors inclination, node, periapsis, km, 0.01, 0.1°, 0.1°. Intersatellite accuracy m estimated computed triangles 6.Fig. 5 Projection spherical.Figures 6a–6d correspond sequential over time conducted 7. real converge, indicating succeeds without link.Fig. 6 pairs orbits.Fig. 7 Errors Promotion SensitivityThe approach surface among move orbit; 8.The O2 8. leads error fails converge. Involving taking accurately approach.If zero, S1 S2 will reduce determined. With O3 S3, 9.Fig. Error methods.Fig. 9 Comparison method.C. considering perturbation. converge near Obtaining (30).Fig. measurements.VI. ConclusionsVia Given respect sets, make Of note configuration, method, observable.AcknowledgmentThis paper sponsored National Natural Science Foundation China (Grant No. 51827806). References Mao X., Visser P. van den IJssel J., “Absolute CHAMP/GRACE Constellation,” Advances Space Research, Vol. 63, 12, 2019, pp. 3816–3834. https://doi.org/10.1016/j.asr.2019.02.030 CrossrefGoogle Scholar[2] Lee S. Mortari D., “Design Constellations Observation Links,” Journal Guidance, Control, Dynamics, 2016, 1–9. https://doi.org/10.2514/1.G001710 Google Scholar[3] Abusali P., Tapley B. D. Schutz E., “Autonomous Navigation Global Positioning System Satellites Cross-Link Measurements,” 21, 1998, 321–327. https://doi.org/10.2514/2.4238 LinkGoogle Scholar[4] Psiaki M. 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Lafontaine “Linearized Dynamics Formation Flying J2-Perturbed Elliptical Orbit,” Guidance Control 1649–1658. https://doi.org/10.2514/1.29438 ScholarTablesTable Computation configurationsCaseGeometric criteria?A,C?AA,C?CA,C1?AA,B>?AA,C, ?BA,B>?BB,C, ?CB,C>?CA,C?A,C=cos?1(?cos+?sin)?AA,C=?AA,B???A?CA,C=?CB,C???C2?AA,B>?AA,C, ?BA,B?CA,C?A,C=cos?1(?cos+?sin)?AA,C=?AA,B+??A?CA,C=?CB,C+??C3?AA,B?CA,C?A,C=cos?1(?cos??sin)?AA,C=?AA,B+??A?CA,C=?CB,C???C4?AA,B?CA,C?A,C=cos?1(?cos??sin)?AA,C=?AA,B???A?CA,C=?CB,C+??C5?AA,B>?AA,C, ?CB,C?AA,C, ?CB,C?AA,C, ?CB,C?AA,C, ?CB,C>?CA,C?A,C=cos?1(?cos+?sin)?AA,C=?AA,B???A?CA,C=?CB,C???CFigure 4c?AA,B>?AA,C, ?CB,C<?CA,C?A,C=cos?1(?cos??sin)?AA,C=?AA,B???A?CA,C=?CB,C+??CTable Obtainable configurationsCaseGeometry descriptionObtainable elements1General configurationaA, aB, eB, fA, fB, ?A,B2Non-coplanar orbitaA, fA??AA,B, ?A,B3Two orbitsaA, fB??BA,B, ?A,B4Two phaseaA+aB, eA+eB, fA+fB, ?A+?B, ?A,B5Two fA+fB??A??B, ?A,BTable descriptionShape size1Two ?AA,B??BA,B2One fA??A+?B3Two fA?fB4The ?AA,B??BA,B5The orbiteA, aA?aB, c1(aA+aB)+2c2(fA?fB??AA,B+?BA,B)Table configurationsRelative OCRelative elementsNon-coplanar orbits?A,C, ?CA,CNon-coplanar OC?A,C, fA??AA,C, OA?A,C, fC??CA,CNon-coplanar fC??CA,CCoplanar ?AA,C??CA,CCoplanar (fA??AA,C)?(fC??CA,C)Table ODRelative orbits?A,D, ?AA,D, ?DA,DNon-coplanar OD?A,D, fA??AA,D, OA?A,D, fD??DA,DNon-coplanar fD??DA,DCoplanar ?AA,D??DA,DCoplanar (fA??AA,D)?(fD??DA,D)Table constellationSpacecrafta (km)ei (deg)? (deg)? (deg)f (deg)S126,5600.360201020S226,5600.360801080S326,5600.36014010140S426,5600.36020010200S526,5600.36026010260S626,5600.36032010320Table orbitSpacecrafta (deg)S126,5600.360201020S226,5600.360201080S326,5600.3608010140 Previous article Next FiguresReferencesRelatedDetails What's Popular Volume 44, Number 3March 2021 CrossmarkInformationCopyright © 2020 American Aeronautics Astronautics, Inc. rights reserved. requests copying permission reprint should submitted CCC www.copyright.com; employ eISSN 1533-3884 initiate your request. See AIAA Rights Permissions www.aiaa.org/randp. TopicsCelestial MechanicsPlanetsSpace Systems VehiclesSpacecraftsSpace TechnologyKepler's Laws Planetary MotionPlanetary ExplorationAstronomySatellite TechnologyUncrewed SpacecraftSatellite-Based ServicesSatellite Broadcasting KeywordsSpacecraftsConstellationsOrbit DeterminationSensitivity AnalysisTarget SpacecraftPeriapsisEarthNon OrbitSpacecraft ControlMultibody DynamicsAcknowledgmentThis Received5 June 2020Accepted18 November 2020Published online21 December