On one Approach to the Solution of the Lambert Problem using the Decompositional Method of Modal Control

Abstract

A new approach to the solution of the Lambert’s problem in spaceflight mechanics is proposed for elliptical orbits. The system of four transcendental algebraic equations is solved using the method of modal synthesis which is based on multilevel decomposition of discrete dynamic system and applied to solve the problem of identification of parameters of discrete system by a state observer. The solution algorithm is as follows: conditional and identification discrete models (systems) are built for the specified system of equations; initial values of estimates are given; initial conditions in the equations of residuals are formed. Using the method of modal synthesis, the problem of search for control of the auxiliary system is solved, as a result of which the matrix of state observer feedback coefficients is calculated. This matrix is used to predict the state vector and to obtain refined estimates --- parameters of the planar orbit. A numerical example of the Lambert’s problem solution using the proposed algorithm is given. In essence, an approach to the solution of nonlinear algebraic systems of the fourth order, which can be extended to systems of any observable order, is proposed. The peculiarity of the proposed algorithm is that the convergence of the iterative process of finding a solution can have a different "adjustable" speed using the control law