Satellite Orbit and Ephemeris Determination using Inter Satellite Links

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Orbit ComputationInter Satellite LinksThe classical Runge-Kutta algorithm is of 4 th order and has 4 stages. The stage numberindicates, how often the right hand function f(x, t) has to be evaluated. The four derivatives k 1through k 4 are computed the following way:kkkk1234= f( x )= fEq. 4.2-4i⎛= f ⎜ x⎝⎛= f ⎜ x⎝( x + h ⋅ k )iiih ⎞+ ⋅ k1⎟2 ⎠h ⎞+ ⋅ k2 ⎟2 ⎠3With these, the new state vector can be obtained by( k1+ 2 ⋅ k2+ 2 ⋅ k3k4)hxi + 1= xi+ ⋅+6Eq. 4.2-5The step width h can be varied easily to minimise degradation due to round of errors. For analgorithm of order n, the error is of order n+1.Multistep procedures use the last n state vectors to obtain the state at time k+1. The Adams-Bashford algorithm, indicated in Eq. 4.2-6 is called predictor, because it uses the past functionevaluations to compute the present state. If the f k 's are stored, only one function evaluation pertime interval h is required, regardless of the order.xn−1k+ 1= xk+ h∑βifk−ii=0Eq. 4.2-6The coefficients β i are determined by the order of the algorithm, as indicated in Table 4-1. Adrawback of the prediction algorithm are round off errors due to large coefficients at highorders. It is therefore often combined with a so called corrector algorithm (Adams-Moulton),using a predicted state at time k+1 to evaluate the right hand function.xn −1*k + 1= xk+ h ⋅β−1⋅ fk+ 1+ h∑βifk−ii=0Eq. 4.2-7The coefficients β* are determined by the order of the procedure and are indicated in Table4-2.The combined predictor-corrector-algorithm leads to satisfactory results, comparable with aRunge-Kutta procedure of the same order. It requires only 2 function evaluation per timeinterval h, regardless of the order. In practice, only the combined predictor-correctoralgorithm is used.The following equations describe explicitly the algorithm for a 4 th order Adams-Bashford-Moulton numerical integration procedure.Page 24R. Wolf
Inter Satellite LinksOrbit Computation h Eq. 4.2-8pk+1= xk+ ⋅ ( 55 ⋅ f ( xk) − 59 ⋅ f ( xk−1) + 37 ⋅ f ( xk−2) − 9 ⋅ f ( xk−3))24 h xk+1= xk+ ⋅ ( 9 ⋅ f ( pk+1) + 19 ⋅ f ( xk) − 5 ⋅ f ( xk−1) + f ( xk−2))24The following tables summarise the coefficients for Adams-Bashford predictor and thecorresponding Adams-Moulton corrector up to 8 th order. Note, that the error of a n th orderalgorithm is also of (n+1) th order, similar to the Runge-Kutta type algorithms.i 0 1 2 3 4 5 6 7β 1i1β 2i3/2 -1/2β 3i23/12 -16/12 5/12β 4i55/24 -59/24 37/24 -9/24β 5i1901/720 -1387/360 109/30 -637/360 251/720β 6i4277/1440 -2641/480 4991/720 -3649/720 959/480 -95/288β 7i198721/60480β 8i16083/4480 -1152169/120960-18637/ 2520 235183/20160242653/13440-10754/ 945 135713/20160-296053/134402102243/120960-5603/2520 19087/ 60480-115747/13440Table 4-1 Coefficients of the Adams-Bashford Algorithm32863/ 13440 -5257/ 17280i -1 0 1 2 3 4 5 6β ∗ 1i 1/21/2β ∗ 2i5/12 2/3 -1/12β ∗ 3i9/24 19/24 -5/24 1/24β ∗ 4i251/720 323/360 -11/30 53/360 -19/720β ∗ 5i95/288 1427/1440 -133/240 241/720 -173/1440 3/160β ∗ 6i19087/ 60480 2713/2520 -15487/20160586/945 -6737/ 20160 263/2520 -863/60480β ∗ 7i5257/17280 139849/120960-4511/4480 123133/120960-88547/1209601537/4480 -11351/120960275/24192Table 4-2 Coefficients of the Adams-Moulton AlgorithmR. Wolf Page 25
Page 1 and 2: Satellite Orbit and EphemerisDeterm
Page 3 and 4: Inter Satellite LinksAbstractGlobal
Page 5 and 6: Inter Satellite LinksTable of Conte
Page 7 and 8: Inter Satellite Links6.2.2 IGSO WAL
Page 9 and 10: Inter Satellite LinksList of Figure
Page 11 and 12: Inter Satellite LinksFigure 6-38 Tr
Page 13 and 14: Inter Satellite LinksList of Tables
Page 15 and 16: Inter Satellite LinksJ2Earths Oblat
Page 17 and 18: Inter Satellite LinksIntroduction1
Page 19 and 20: Inter Satellite LinksIntroductionGE
Page 21 and 22: Inter Satellite LinksISL Observatio
Page 23 and 24: Inter Satellite LinksState Estimati
Page 26 and 27: State EstimationInter Satellite Lin
Page 36 and 37: Orbit ComputationInter Satellite Li
Page 86 and 87: Software DescriptionInter Satellite
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Software DescriptionInter Satellite
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Simulations and ResultsInter Satell
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Satellite Orbit and Ephemeris Determination using Inter Satellite Links

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