Satellite Orbit and Ephemeris Determination using Inter Satellite Links

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