New Angles-only Algorithms for Initial Orbit Determination

Transcription

1 New Angles-nly Algrithms fr Initial Orbit Determinatin Gim J. Der Abstract DerAstrdynamics The extractin f range frm ptical sensr bservatin data has always been difficult, expensive and smetimes unreliable. This paper presents three new angles-nly algrithms t help mitigate thse prblems. First, the crrect range r rt f the Gauss r Laplace eighth-degree plynmial equatin is cnsistently deduced withut guesswrk fr any bject in any rbit regime. Secnd, the Keplerian slutin is analytically extended t include perturbatins fr fast and accurate Initial Orbit Determinatin (IOD). Using simulated and real data, with r withut angular measurement errrs, ten numerical examples are presented fr cmparisn. Mre than a hundred Gecentric and Helicentric test cases have been independently cnfirmed by third parties. These analytic angles-nly algrithms imprve data cllectin efficiency, catalg maintenance, uncrrelated target catalging, and ultimately cntributing t real-time Space Situatin Awareness at a greatly reduced cst. Cpyright 011 DerAstrdynamics 1

2 1. Intrductin Initial rbit determinatin is required fr newly detected unknwn bjects, such as a new freign launch, an Uncrrelated Target (UCT) r a piece f tracked but un-catalged space debris. IOD, which requires nly sparse data, cmputes a starting state vectr r nminal rbit. Differential Crrectin (DC), which requires dense data, prcesses systems and statistical measurement errrs t imprve a nminal rbit. Dense data cllectin is time cnsuming, sensr resurce limited, and is nt cmpatible with autmatic and real-time prcessing. Even at these prhibitive csts, dense data IOD can still fail t initiate tracks, the detected new bjects will remain as UCT s and uncatalged. As such, they pse a higher threat t space asset than the catalgued bjects. In this paper, unless stated therwise, angles-nly algrithm means angles-nly IOD algrithm. Deep space bject bservatins are typically angles-nly based. The classical angles-nly prblem as described in [1] t [1], uses three sets f angles t slve fr a -Bdy (Keplerian) rbit f an bject. All classical methds, which slve fr the general slutin, unfrtunately have tw majr weaknesses: range guessing t start and -Bdy slutins t finish. Range guessing is due t ignring r nt slving the crrect real rt f the Gauss and Laplace eighth-degree plynmial equatins r the Equatins fr shrt. The - Bdy slutins are due t the difficulty f adding perturbatins analytically. Even if the crrect initial -Bdy rbit is fund, it is ften nt accurate enugh t predict a future acquisitin f a Near-Earth bject after 4 hurs. IOD and DC prcesses have been successfully applied t GEO and Deep Space bjects [ t 4] when a priri knwledge f the slw mving and high altitude bject is assumed. Clearly, when the bject is cmpletely unknwn, even with the help f pwerful cmputers, the classical angles-nly methds have been difficult t perfrm IOD. The new angles-nly algrithms remve these weaknesses by range slving and perturbed rbits. Mdern ptical sensrs [5] have been prviding accurate angles data since 003. Hwever, they fail t cnnect with prper time span r angular separatin t prduce reasnably accurate IOD rbits. Prper time span r angular separatin guidelines fr Gecentric and Helicentric IODs are depicted in Figure 1 and summarized as fllws: 1. Time and angular spacing: Three sets f angles at reasnably separated time spans f abut 10 minutes (Gecentric) and days r mnths (Helicentric) are required. A prper angular separatin is when the variatins f the angular values fr tw pints differ by at least 0.05 degrees.. Multi-revlutins: The Gauss r Laplace algrithm is frmulated fr the three data pints within ne rbit at least fr the first-pass. If the hurs- r days-apart data pints are indeed within an arc f ne rbit, then a gd slutin may still be fund, but there is n guarantee. 3. Angular measurement accuracy: Angles data cllected by an peratinal ptical sensr [5] with a Field-f-View reslutin f abut arc-secnds ( degrees) prvides accurate angles inputs fr the Gauss, Laplace and Duble-r angles-nly algrithms. Errr-free inputs imply that all the six angles are accurate with five r mre significant digits as described in the Numerical Example Sectin. Cpyright 011 DerAstrdynamics

3 Nrmal bject angular spacing within an arc f the rbit Unknwn Orbit bject at t 3 bject at t r 3 x = r bject at t 1 Unknwn Orbit 1 revlutin Imprper Obs data spacing: 1. Clsely spaced data (Time span t clse) bject at t 3 r 3 r r 1 bject t clse at t and t 1 Observer r 1 * 3 sets f angle t i i i i = 1,, 3,.. * 1, and/r 3 sets f Observer crdinates Unknwn Orbit revlutins r bject in first rev at t and t 1 r 1 Prper Obs Angular data spacing: (1) Gecentric: 10 minutes apart r > 0.05 degrees (max: 90 minutes) () Helicentric: 30 arc minutes apart (max: t be determined) bject in secnd rev at t 3 Figure 1. Guidelines fr time and angular spacing f angles data r 3. Multi-revlutins data (Time span t far apart) Assuming that prper angles data is given, each crrect psitive real rt f the Equatins is the initial range estimate at the secnd time pint f the unknwn rbit as shwn in Figure 1. Theretically there are eight rts and can be cmputed by a standard rt slver. Charlier applied the Descartes Rule f Signs [0] and [1], t the Laplace Equatin fr Helicentric bjects in 1910 indicating at mst three psitive real rts. Hwever, the ten numerical examples f this paper shw that sme f Charlier s theretical assertins are incrrect in practice, especially fr Gecentric applicatins. Selecting the crrect rt can be very difficult, as indicated by Vallad [7]. Withut guesswrk, the new Gauss and Laplace algrithms slve fr the crrect psitive real rt f the Equatins by cmbining Descartes Rule f Signs and the Gauss gemetric methd. All previus papers may use the same cmbinatin, but they start with guessing and/r finish with multiple rts. The algrithms f this paper always prvide ne crrect rt. The new Duble-r algrithm takes advantage f the range slving slutin f Gauss r Laplace t cnstruct tw Cartesian psitin vectrs ( r 1 and r 3 ), which in turn transfrms the angles-nly prblem t a Lambert prblem. Algrithms that invlve range guessing, range hypthesis, partial derivative evaluatins, high-rder matrix inversins, sensitive iterative methds and unpredictable cnvergence are avided. Cpyright 011 DerAstrdynamics 3

4 . The Crrect Rt f the Equatins f Gauss and Laplace This sectin presents a straightfrward methd t find the crrect rt f the Equatins. Since the frmulatins leading t the Equatins can be fund in many textbks, they will nt be repeated here. Althugh the Gauss and Laplace frmulatins are different, the resulting equatins have the same frm and can be expressed as: f ( x ) = x a x b x c 0 (1) where x r r ( t) is the magnitude f the psitin vectr at t. The cefficients a, b, and c, which can be cmputed frm the given inputs t the angles-nly prblem, are different in numerical values between thse f Gauss and Laplace. Theretically, if the input angles are errr-free, then these three cefficients are determined accurately, and the resulting range estimate f Equatin (1) is within 99% f the real range at t..1 Descartes Rule f Signs Descartes Rule f Signs [6] states that: If f ( x ) is a plynmial with nn-zer real cefficients, the number f psitive rts f Equatin (1), cannt exceed the number f changes f signs f the numerical cefficients f f ( x ). If the bservatin r the angles data is real, the range x must be psitive. There are nly fur terms in Equatin (1), and the cefficient c is always negative. Therefre, at mst three changes f signs can ccur, implying at mst three psitive real rts. If the cefficients indicate ne psitive real rt, the crrect value f x can be fund withut guesswrk as in Sectin.3, and the Gauss gemetric methd described in Sectin. may be skipped Transfrm: f(x) = x a x b x c = 0 r = R + bject at t 4 t: f( ) = sin M sin ( m = 0 M and m are cmputed frm a, b and c Gauss: Determine the crrect rt r > R > r Earth and < (180 ) < 90 Observer L r Unknwn Orbit r sin = = sin ( + R sin R Earth r = ECI psitin vectr t bject R = ECI psitin vectr t bserver = range vectr frm bserver L = Line f sight vectr frm bserver Figure. Gauss gemetric methd Cpyright 011 DerAstrdynamics 4

5 . Gauss Gemetric Methd As described in [1], Gauss transfrmed Equatin (1) t: 4 f ( ) = sin M sin (m ) = 0 () where M and m are cnstants cmputed frm the angles-nly inputs, and the unknwn is the angle between the psitin vectrs and r as shwn in Figure. Theretically, any visible bject must lie abve the grund bserver s hrizn, implying that and The Gauss gemetric methd als uses the Law f Sines as: r R = = (3) sin sin ( ) sin t relate the magnitudes f the three psitin vectrs, r,, and R, after is fund. The crrect rt that crrespnds t the real unknwn rbit must satisfy: r Earth R r (4) 0 (180 ) 90 Fr an rbiting space sensr f Gecentric IOD (and Helicentric IOD in Celestial Mechanics), at mst tw ther rts may exist, but they are assciated with 90. By nt allwing the line f sight t see thrugh the Earth, ne f the tw pssible rts can ften be eliminated. Other intuitive relatins may be needed..3 A Simple Iterative Methd fr Equatins (1) and () Suppsing the search is fr an Earth satellite r an rbital debris bject by a grund sensr, the starting magnitude f the Cartesian r Earth Centered Inertial (ECI) psitin vectr is x1 r ( t) r Earth fr Equatin (1). The iterative methd, which assume n a priri knwledge, may end at any desired range limit, say 100,000 km fr Gecentric IOD with a step f 500 km. Since the altitude f 90% f the Space bjects arund the Earth are abut,000 km r less, the slutin f a Near Earth bject can be fund in a few steps. If there are three psitive real rts, Gauss gemetric methd is required. The starting value f is zer fr Equatin (), and the maximum value is 90 fr grund sensrs. The cnfusin f multiple rts begins with 0 180, and applies nly t rbiting space sensr and Helicentric IODs. If the evaluatin f is separated between 0 90 and , finding the crrect and then x f Equatin (1) is intuitive. 3. A New Duble-r Algrithm Tw arbitrary starting values and a nn-rbust iterative methd (Newtn-Raphsn) are prblematic fr the traditinal Duble-r algrithms. Even thugh range guessing is replaced by range slving, frcing their target functins t cnvergence is tricky when high-rder partial derivatives, multi-dimensinal matrix inversins and sensitive iterative methds are invlved. The new duble-r algrithm avids these prblems. Cpyright 011 DerAstrdynamics 5

6 Unknwn Orbit t 3 r 3 Three psitins nminal Unknwn Orbit t Index at t, j = 1,, 3 3 j = 1 Nine angular variatins frm the nminal Predicted psitin at t frm i = j = 1 Nine angular variatins ij frm the nminal i = 1,, 3 j = 1,, 3 R Observer r r 1 t Three 1 psitins r ij nminal r 1 3 i = 1 bject at t Three 1 psitins Index at t, i = 1,, 3 1 Example: Three psitins at tw end pints R Target functin = min { } ij Figure 3. Range slving and Lambert targeting f the new Duble-r algrithm With the tw reasnably accurate initial range estimates at t 1 and t 3 prvided by range slving, Lambert targeting with analytic perturbed trajectries [7] and [8], becmes pssible. As shwn in the left f Figure 3, the tw initial range estimates f Gauss r Laplace at t 1 and t 3 can be varied, and multiple Lambert slutins between the end pints 1 and 3 can be easily cmputed. Fr example: chsing i = 1,,3 and j = 1,,3 at respectively t 1 and t 3 requires 9 Lambert slutins. In the right f Figure 3, the simple target functin is ij, the angle between the given nminal line-f-sight vectr and r the ECI psitin vectr cmputed by Lambert targeting at t. The ptimum ranges, i and r j, at respectively t 1 and t 3 are thse assciated with the -dimensinal minimum ij. If the nminal is allwed t vary (assume measurement errr), a 3-dimensinal minimum can ften be fund smaller than the -dimensinal ne. The analytic ij perturbed r Kepler+ trajectries with ( J, J 3, J 4, Sun, Mn,.. ) can be replaced by thse frm numerical integratin, if higher accuracy instead f speed is desired. The variatins f prvide a view f the ptimum slutin. The cmparisn f the new ij Duble-r algrithms and a versin f Gding [18] is illustrated in Figure 4. Cpyright 011 DerAstrdynamics 6

7 Der range slving th Gauss 8 - degree plynmial equatin Unknwn Orbit Observer r 3 bject at t 3 x = r r 1 bject at t bject at t 1 Gding range guessing 1. r and r estimates frm range slving instead f range guessing 1 3. r, r and r are ptimized instead f assumed with n errrs r by Gibbs gives direct r retrgrade instead f direct and retrgrade 4. Analytic Kepler + perturbed slutin instead f -Bdy slutins Duble _ r (Der) Lambert + and Kepler + _ Duble r (Gding) Lambert and -Bdy 5. One (near SP) slutin versus Multiple (-bdy) slutins Figure 4. Duble-r algrithms cmparisn 4. Numerical Examples Fr Gecentric rbits, the Space Catalg prvides initial state vectrs fr knwn bjects at different rbit regimes. Fr Helicentric rbits, the state vectrs f knwn planetary bjects can be extracted frm the JPL DE405 ephemeris file and the Minr Planet file f Internatinal Astrnmical Unin (IAU) fr almst any date. These accurate Gecentric and Helicentric state vectrs frm the errr-free reference IOD slutins fr all practical purpses. Hw accurate des the angles data fr any bject have t be such that Equatin (1) is useful? In the fllwing ten examples, the estimated ranges x s and cefficients a, b, and c f Equatin (1) frm bth the Gauss and Laplace algrithms are given fr bjects at different rbit regimes as well as rbits f different eccentricities, inclinatins and lcatins n the rbits. Range slving is successful, if (x, a, b, c) f each example satisfies f ( x ) = x a x b x c 0. Accurate input angles data cupled with prper time r angular spacing give accurate cefficients, which in turn prduce reasnable range estimates. Reasnable means that the estimated range is within five percent and usually ne percent f the desired range at t. Nte that, x is nrmalized with the Earth radius f km r AU f km. Cpyright 011 DerAstrdynamics 7

8 In practice, the mst likely surce f errrs f ptical sensrs and ptical telescpes is usually the angles bservatin data. Time errr is insignificant due t the use f atmic clcks in peratinal sensrs. The bserver crdinates r psitin vectr is usually accurate. If real bservatin data is nt available, then ther means f injecting errrs are needed fr rbustness evaluatins f the algrithms. Since errr-free angles data can be cmputed with the use f the Space Catalg and the JPL DE405 ephemeris file, angular measurement errrs can be intrduced systematically. On the ther hand, inaccurate angles data f asterids and cmets frm mst Astrnmy magazines are unable t prduce accurate cefficients f Equatin (1), and the resulting estimated ranges d nt match even within 90% percent f thse knwn reference slutins. The characteristics f angular errrs and estimated ranges f Equatin (1) are illustrated as fllws: Input angles data Number f accurate significant digits in the angles data Example: Right ascensin r Declinatin (degrees) Expected % f errrs Estimated range Errr-free 5 r mre r mre 1 r less excellent Marginal 3 r r 1.34 t 5 gd t pr Inaccurate r less 1. r less >> 5 unacceptable Mre than a hundred bjects and six hundred test cases were analyzed with and withut measurement errrs. The psitin and velcity vectrs cmputed by the new angles-nly algrithms are cmpared with knwn and unknwn answers. Ten examples fr Gecentric and Helicentric IODs were chsen and the estimated Gauss / Laplace parameters, x, eccentricity and inclinatin are summarized as fllws: Example Input errrs Estimated Gauss / Laplace Parameters (%) x= ECI - SCI range (km) Eccentricity Inclinatin (deg 1. Gecentric, MEO / / / 3.1. Gecentric, Mlniya / / / Gecentric, GEO / / / Gecentric, LEO / / / Gecentric, HEO / / / Gecentric, MEO / / / Gecentric, MEO Text / / / 39. Bk 8. Helicentric, Saturn / / /.5 9. Helicentric, Jupiter / / / Helicentric, Ceres / / / 8.0 In general, the slutins f Laplace in Examples 1 t 10 are nt as accurate as thse f Gauss. As indicated in [16], the Laplace algrithm requires tpcentric crrectin, while the Gauss algrithm handles tpcentric bservatin data naturally. Since the new algrithms are range independent, slutins are nt affected by input data frm multiple sensrs r bservers. In the fllwing examples, nly ne bserver is used. The results f the new algrithms are clred in red, and the desired answer in green in the Tables. Cpyright 011 DerAstrdynamics 8

9 Example 1. MEO, errr-free Descartes Rule f Signs indicates ne psitive real rt fr bth Gauss and Laplace, and can be fund as shwn in Sectin. The cnverged ECI psitin and velcity vectrs f the new Duble-r (Lambert targeting) algrithm are much clser t thse f the unknwn rbit than bth the Gauss and Laplace algrithms as shwn in Table 1. Grund Sensrs Latitude (deg) Lngitude (deg) Altitude (km) Sites 1,, Date (yr/m/dd/hr/min/sec) Right ascensin (deg) Declinatin (deg) 009/ 5/ 6/ 16/ 0/ / 5/ 6/ 16/ 10/ / 5/ 6/ 16/ 0/ Output cmparisn f the three new algrithms estimated Gauss cefficients (a = , b = 1.770, c = ) estimated Laplace cefficients (a = , b = 4.778, c = ) t t sec nds ECI psitin vectr (km) ECI velcity vectr (km/s) Gauss Laplace Duble-r (targeting) Answer Table 1. Cmparisn f slutins f the new algrithms fr MEO at t Example. Mlniya, errr-free Descartes Rule f Signs indicates ne psitive real rt fr bth Gauss and Laplace. All the slutins frm the Gauss, Laplace and Duble-r (targeting) algrithms are clse t the unknwn Mlniya rbit as shwn in Table. Grund Sensrs Latitude (deg) Lngitude (deg) Altitude (km) Sites 1,, Date (yr/m/dd/hr/min/sec) Right ascensin (deg) Declinatin (deg) 011/ 1/ 4/ 13/ 0/ / 1/ 4/ 13/ 5/ / 1/ 4/ 13/ 10/ Output cmparisn f the three new algrithms estimated Gauss cefficients (a = , b = , c = ) estimated Laplace cefficients (a = 1.004, b = , c = ) Cpyright 011 DerAstrdynamics 9

10 t t sec nds ECI psitin vectr (km) ECI velcity vectr (km/s) Gauss Laplace Duble-r (targeting) Answer Table. Cmparisn f slutins f the new algrithms fr Mlniya at t Example 3. GEO, errr-free Descartes Rule f Signs indicates three psitive real rts fr Gauss; tw with the LEO ranges and ne GEO range. Hwever, The Gauss gemetric methd deduced a range near GEO, and therefre the crrect range is that f the GEO. Grund Sensrs Latitude (deg) Lngitude (deg) Altitude (km) Sites 1,, Date (yr/m/dd/hr/min/sec) Right ascensin (deg) Declinatin (deg) 006/ 9/ 11/ 4/ 45/ / 9/ 11/ 4/ 50/ / 9/ 11/ 4/ 55/ Output cmparisn f the three new algrithms estimated Gauss cefficients (a = , b = , c = ) estimated Laplace cefficients (a = , b = , c = ) t t 300 sec nds ECI psitin vectr (km) ECI velcity vectr (km/s) 1 Gauss Laplace Duble-r (targeting) Answer Table 3. Cmparisn f slutins f the new algrithms fr GEO at t Example 4. LEO, with 0.1% angular errrs A piece f rcket bdy in LEO at a very high inclinatin and the additin f 0.1% angular measurement errrs did nt cause any prblem fr range slving. Laplace slutin is sensitive t angular errrs. Grund Sensrs Latitude (deg) Lngitude (deg) Altitude (km) Sites 1,, Date (yr/m/dd/hr/min/sec) Right ascensin (deg) Declinatin (deg) 011/ 1/ 18/ 10/ 9/ / 1/ 18/ 10/ 1/ / 1/ 18/ 10/ 14/ Cpyright 011 DerAstrdynamics 10

11 Output cmparisn f the three new algrithms: estimated Gauss cefficients (a = , b = , c = ) estimated Laplace cefficients (a = , b = , c = 0.71) t t 154 sec nds ECI psitin vectr (km) ECI velcity vectr (km/s) 1 Gauss Laplace Duble-r (targeting) Answer Table 4. Cmparisn f slutins f the new algrithms fr LEO at t Example 5. HEO, with 0.1% angular errrs An bject in HEO in a retrgrade rbit f inclinatin 15 degrees and the additin f 0.1% angular measurement errrs als did nt cause any prblem fr range slving. Grund Sensrs Latitude (deg) Lngitude (deg) Altitude (km) Sites 1,, Date (yr/m/dd/hr/min/sec) Right ascensin (deg) Declinatin (deg) 011/ 1/ 4/ 1/ 4/ / 1/ 4/ 1/ 47/ / 1/ 4/ 1/ 5/ Output cmparisn f the three new algrithms: estimated Gauss cefficients (a = , b = , c = ) estimated Laplace cefficients (a =.3968, b = 48.48, c = ) t t 300 sec nds ECI psitin vectr (km) ECI velcity vectr (km/s) 1 Gauss Laplace Duble-r (targeting) Answer Table 5. Cmparisn f slutins f the new algrithms fr HEO at t Example 6. MEO, with 0.1% angular errrs An MEO bject in a retrgrade rbit f inclinatin 105 degrees and the additin f 0.1% angular measurement errrs als did nt cause any prblem fr range slving. Grund Sensrs Latitude (deg) Lngitude (deg) Altitude (km) Sites 1,, Date (yr/m/dd/hr/min/sec) Right ascensin (deg) Declinatin (deg) 011/ 1/ 5/ 0/ 3/ / 1/ 5/ 0/ 8/ / 1/ 5/ 0/ 13/ Cpyright 011 DerAstrdynamics 11

12 Output cmparisn f the three new algrithms: estimated Gauss cefficients (a = , b =.593, c = ) estimated Laplace cefficients (a = , b = , c =.470) t t 300 sec nds ECI psitin vectr (km) ECI velcity vectr (km/s) 1 Gauss Laplace Duble-r (targeting) Answer Table 6. Cmparisn f slutins f the new algrithms fr LEO at t Example 7 MEO, with angular errrs Example 7 cmpares the traditinal and new Gauss, Laplace and Duble-r angles-nly algrithms fr a MEO bject. The traditinal algrithms prduce Keplerian slutins and thse f the new algrithms include J, J 3, J 4, Sun, Mn, and ther perturbatins when applicable. As expected, the Kepler+ initial rbits f the new algrithms are mre accurate fr the unknwn MEO bject. The Laplace slutins are prblematic due t the inaccurate derivatives f the line-f-sight vectrs and the lack f tpcentric crrectin. Grund Sensrs Latitude (deg) Lngitude (deg) Altitude (km) Sites 1,, Date (yr/m/dd/hr/min/sec) Right ascensin (deg) Declinatin (deg) 007/ 8/ 0/ 11/ 40/ / 8/ 0/ 11/ 50/ / 8/ 0/ 1/ 0/ Output cmparisn f the traditinal Gauss and the new Gauss algrithms: estimated Gauss cefficients (a = , b = , c = 7.804) t t 600 sec nds ECI psitin vectr (km) ECI velcity vectr (km/s) 1 Gauss (traditinal, knew answer) Gauss (traditinal, range guessing) Gauss (range slving) Answer Table 7a. Cmparisn f the traditinal and new Gauss algrithms at t Output cmparisn f the traditinal Laplace and the new Laplace algrithms: estimated Laplace cefficients (a = , b = , c = ) t t 600 sec nds ECI psitin vectr (km) ECI velcity vectr (km/s) 1 Laplace (traditinal, knew answer) Laplace (traditinal, range guessin Laplace (new range slving) Answer Table 7b. Cmparisn f the traditinal and new Laplace algrithms at t Cpyright 011 DerAstrdynamics 1

13 Output cmparisn f the traditinal Duble-r and the new Duble-r algrithms: t t 600 sec nds ECI psitin vectr (km) ECI velcity vectr (km/s) 1 Duble-r (traditinal, knew answer Duble-r (traditinal, range guessin Duble-r (new Lambert-targeting) Answer Table 7c. Cmparisn f the traditinal and new Duble-r algrithms at t Example 8. Saturn, errr-free The state vectrs f the Earth (bserver) and Saturn (unknwn bject) were btained frm the DE405 ephemeris file. Since DE405 data is sufficiently accurate, the cefficients f the Equatins are cmputed accurately and the resulting range estimates shw less than 0.01 percent f errr. Numerus test cases similar t this shw that the angles data generated frm state vectrs f DE405 can be cnsidered errr-free fr Helicentric IOD. Date (yr/m/dd/hr/min/sec) Sun Centered Inertial (SCI) psitin vectr f Earth (km) 01/ 1/ 5/ 0/ 0/ 0 01/ 1/ 15/ 0/ 0/ 0 01/ 1/ 5/ 0/ 0/ 0 Date (yr/m/dd/hr/min/sec) Right ascensin (deg) Declinatin (deg) 01/ 1/ 5/ 0/ 0/ 0 01/ 1/ 15/ 0/ 0/ 0 01/ 1/ 5/ 0/ 0/ 0 Output cmparisn f the three new algrithms: estimated Gauss cefficients (a = , b = , c = ) estimated Laplace cefficients (a = 9.599, b = , c = ) t =01/ SCI psitin vectr (km) SCI velcity vectr (km/s) 1/ 5/ 0/ 0/ 0 Gauss Laplace Duble-r (targeting) Answer Table 8. Cmparisn f slutins f the new algrithms fr Saturn at t Example 9. Jupiter, errrs in bth Earth psitin and measurement angles The state vectrs f the Earth (bserver) and Jupiter (unknwn bject) were btained frm the DE405 ephemeris file. Only the first fur digits f each cmpnent f the Earth psitin vectr were kept, and abut 0.1% angular measurement errrs were added t the Cpyright 011 DerAstrdynamics 13

14 given angles. Even thugh the injected errrs prduced substantial differences in the cefficients f the Equatins with respect t the errr-free set, the resulting slutins f Gauss and Duble-r are still reasnable. The Laplace slutin is pr due mre t inaccurate angles data than the lack f tpcentric crrectin. Date (yr/m/dd/hr/min/sec) Sun Centered Inertial (SCI) psitin vectr f Earth (km) 1990/ 1/ 13/ 0/ 0/ / 6/ / 0/ 0/ / 1/ 8/ 0/ 0/ 0 Date (yr/m/dd/hr/min/sec) Right ascensin (deg) Declinatin (deg) 1990/ 1/ 13/ 0/ 0/ / 6/ / 0/ 0/ / 1/ 8/ 0/ 0/ 0 Output cmparisn f the three new algrithms: estimated Gauss cefficients (a = , b = , c = ) estimated Laplace cefficients (a = , b = , c = ) desired DE405 cefficients (a =.869, b = , c = ) t =1990/ 6/ SCI psitin vectr (km) SCI velcity vectr (km/s) / 0/ 0/ 0 Gauss Laplace Duble-r (targeting) Answer Table 9. Cmparisn f slutins f the new algrithms fr Jupiter at t Example 10. Ceres, n errr in Earth psitin (DE405) and 0.1% errrs in angles The state vectrs f the Earth (bserver) were btained frm the DE405 ephemeris file. The angles data f the Asterid, Ceres, (unknwn bject) were btained frm the IAU Minr Planet file, and then abut 0.1% f the angles values were injected. Again, the resulting slutins f Gauss and Duble-r are reasnable, and that f Laplace is slightly wrse (n tpcentric crrectin) as cmpared t the reference r errr-free slutin. Date (yr/m/dd/hr/min/sec) Sun Centered Inertial (SCI) psitin vectr f Earth (km) 007/ 9/ 1/ 11/ 0/ 0 007/ 11/ 16/ 8/ 1/ 0 007/ 1/ 7/ 0/ 8/ 0 Date (yr/m/dd/hr/min/sec) Right ascensin (deg) Declinatin (deg) 007/ 9/ 1/ 11/ 0/ 0 007/ 11/ 16/ 8/ 1/ 0 007/ 1/ 7/ 0/ 8/ 0 Cpyright 011 DerAstrdynamics 14

15 Output cmparisn f the three new algrithms: estimated Gauss cefficients (a = , b = , c = ) estimated Laplace cefficients (a = , b =.634, c = 7.059) estimated reference cefficients (a = 8.687, b = 7.376, c = 1.653) t =007/ 11/ SCI psitin vectr (km) SCI velcity vectr (km/s) 16/ 8/ 1/ 0 Gauss Laplace Duble-r (targeting) Answer Table 10. Cmparisn f slutins f the new algrithms fr Ceres at t Cnclusins This paper presents three new angles-nly IOD algrithms that can cnsistently deduce the crrect range r rt f the Gauss r Laplace eighth-degree plynmial equatins withut guesswrk fr any unknwn bject in any rbit regime, and therefre the 00- year angles-nly prblem is slved. The -Bdy slutin is then analytically extended t include perturbatins fr speed and accuracy. These new angles-nly algrithms have been further imprved t meet mre accurate catalging and crrelatin requirements f the Space Based Space Surveillance and the Space Fence systems, while als prvide the needed algrithms fr multi-sensr multibject peratins f Space Situatinal Awareness. The new angles-nly algrithms allw nt just ptical sensrs and ptical telescpes t deduce accurate IOD rbits f any bject with sparse data, but reduce greatly the csts f data cllectin thrughput, sftware implementatin flexibility, and many ther grund and space peratins. References 1. Gauss, C.F. Thery f Mtin f the Heavenly Bdies Mving Abut the Sun in Cnic Sectins, 1809, (English translatin, Dver Phenix Editins).. F.R. Multn, An Intrductin t Celestial Mechanics, The Macmillan Cmpany, New Yrk, Escbal, P.R., Methds f Orbit Determinatin, Rbert E. Krieger Publishing Cmpany, Malabar, Flrida, USA, Herrick, S., Astrdynamics, Vl 1, Van Nstrand Reinhld, Lndn, Bate, R., and Mueller, D., and White, W., Fundamentals f Astrdynamics, Dver Publicatins, Inc., Trnt, Canada, Battin, R.H. An Intrductin t The Mathematics and Methds f Astrdynamics, American Institute f Aernautics and Astrnautics, Educatin Series, 1987 Cpyright 011 DerAstrdynamics 15

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