A rapid method for calculating maximal and minimal inter-satellite distances

This paper proposes a method to rapidly and effectively calculate the maximal and minimal distances between a pair of satellites in which the leader is in an elliptic orbit. The principal idea of this method is simplifying the nonlinear squared distance function into a Taylor series with limited orders and further fitting the trigonometric functions in the derivative function of the simplified squared distance by piece-wise quadratic polynomials. By solving the zero-crossing points of the fitted quadratic curve, the critical points of the original nonlinear distance function are approximately determined. It turns out that the accuracy of the obtained solutions of the extreme distances depends on the number of intervals of the polynomial fitting. The bigger the number of intervals is, the better the accuracy. However, it is also noticed that the number of intervals is not necessary too big. For real applications a small value (e.g. 8) may be enough for the number of intervals. Besides, the method is apparently more effective for the small eccentricity cases. Finally, some simulations are further carried out to demonstrate the performances of this new method.