Satellite-to-satellite orbit determination using minimum, discrete range and range-rate data only

The problem of determining the orbit of an unknown satellite from another satellite given minimum range and range-rate observations at discrete times is considered. The problem considered is, for concreteness, a geocentric satellite observing another geocentric satellite. The orbit determination goal is for simplicity, the calculation of position and velocity at one of the observation times. It is shown that the unknown position and velocity at the chosen epoch can be obtained by solving a set of six nonlinear algebraic equations in which the position and velocity at other times are computed by using f and g functions. The resulting system of six nonlinear algebraic equations is then solved by iteration with a modified Newton's method. Several sophisticated methods were applied and a hybrid method due to Powell was selected from these. Results obtained from application of the method to a number of examples are examined. Rules for constructing the initial guesses that reduce computation and improve the probability of convergence to the true solution are presented.