Satellite Orbit and Ephemeris Determination using Inter Satellite Links
Satellite Orbit and Ephemeris Determination using Inter Satellite Links Satellite Orbit and Ephemeris Determination using Inter Satellite Links
State EstimationInter Satellite LinksThe greater flexibility of the extended filter if compared to the linearized filter is an advantageas well as a disadvantage. Good measurements presumed, the extended filter stays closer tothe true state than the linearized, but it can be corrupted easily by biased measurements.In practice, a mixed structure would be applied to the orbit estimation problem. The estimatederrors are only fed back into the orbit propagator, if they are assumed to be known preciseenough. The "feed back" criterion could be for examplexσxwith> cTreshold,1 < cTreshold< 10Eq. 3.5-10where σ x is the square root of the variance, obtain from the Kalman filter covariance matrix.In other words this would mean, the trajectory is corrected only if the uncertainty of the erroris several times lower than the error itself.Page 18R. Wolf
Inter Satellite LinksOrbit Computation4 ORBIT COMPUTATIONThe computation of a satellite orbit can be done using different approaches:• The analytical solution, where orbits are treated as conical sections (Kepler orbits)• The numerical integration of the equations of motion, described by a (more or less)accurate force model.Satellite orbiting in the relative vicinity of the earth are subject to a lot of disturbing forces,thus only the numeric integration approach leads to satisfactory result. An accurate orbitpropagator is required not only for simulation purpose, but also for state prediction in the orbitestimation process, where differential corrections are applied to a reference trajectory. Thelonger the processed orbit arc, the more accurate the force model has to be.4.1 Analytical SolutionThe analytical solution is obtained by neglecting all acting forces but the central force. This isalso known as the restricted two body problem, which has first been solved by JohannesKepler. Starting with Newton's law of gravity about the attraction of two masses A and BFAB( x)= G ⋅ mAmB⋅x( x − x ) 3AA− xBBEq. 4.1-1and assuming one mass to be negligible if compared to the other and building the sum ofkinetic and potential energy leads to the Keplerian equations, where satellite orbits are treatedas conical sections. Depending on whether the sum of kinetic and potential energy is positive,negative or zero determines the type of conical section.v 22GM GM− = − Ellipser 2aEq. 4.1-2v 2 GM− = 0 Parabola2 rv 2 GM GM− = Hyperbola2 r 2awith GM Gravitation constant times mass of central bodyv Velocity of point massr Distance of point massa Major semiaxis of conical sectionEq. 4.1-3Eq. 4.1-4R. Wolf Page 19
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State Estimation<strong>Inter</strong> <strong>Satellite</strong> <strong>Links</strong>The greater flexibility of the extended filter if compared to the linearized filter is an advantageas well as a disadvantage. Good measurements presumed, the extended filter stays closer tothe true state than the linearized, but it can be corrupted easily by biased measurements.In practice, a mixed structure would be applied to the orbit estimation problem. The estimatederrors are only fed back into the orbit propagator, if they are assumed to be known preciseenough. The "feed back" criterion could be for examplexσxwith> cTreshold,1 < cTreshold< 10Eq. 3.5-10where σ x is the square root of the variance, obtain from the Kalman filter covariance matrix.In other words this would mean, the trajectory is corrected only if the uncertainty of the erroris several times lower than the error itself.Page 18R. Wolf
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