Mathematical Model of Unit Vector Method (MMUVM) and Convergence of Its Iteration Method in Simplified Form (PUVM1)

The unit vector method (PUVM1) by which the observed data of artificial satellites are used to determine the perturbed initial orbits has been widely applied. In order to further perfect and improve this method, first, a mathematical model (MMUVM) corresponding to this method is constructed on the basis of measuring errors. In essence, the MMUVM is a nonlinear optimization problem. In the light of the MMUVM, simulated and observed data of multiple circuits are taken to form specific objective functions, which are then processed by means of a direct search method and with of a tri-diagonal quadratic interpolation model for solving optimization problems. The calculated results show that the optimization model MMUVM is right and reasonable and the adopted direct method is practical and effective. Secondly, the relation between PUVM1 and MMUVM is further clarified: PUVM1 is essentially a simplified form of the MMUVM, and the primary reason is found from mathematical principles, why it is that PUVM1 can only be applied to short-arc data within one circuit and not to long-arc or multi-circuit data. Finally, a preliminary theoretical analysis of the convergence of the iteration algorithm of PUVM1 is carried out and instancesof numerical verification given. It is pointed out that the iterative scheme of PUVM1 is conditionally convergent. This means that sometimes, even though the quasi-normalization equation is reasonable, the iteration diverges.