Stochastic method of investigating perturbed motion of a space vehicle in the atmosphere

When space vehicles (SVs) are moving in the atmosphere detailed accounting of the influence of external perturbation factors is quite difficult: they are either unknown or have a complex nature. This necessitates development of methods to solve the problems of controlling SVs in conditions of uncertainty. A stochastic method of investigating perturbed paths of space vehicles in the atmosphere has been elaborated. The method is based on the mathematical theory of continuous Markovian processes. The principle of reduction from deterministic description of space vehicle motion to the stochastic one is applied. A SV is supposed to be in a limited number of states that are determined by a previously chosen algorithm of phase space discretization. Differential Kolmogorov equations for determining the probabilities of a SV being in each of the states and relationships for the calculation of mathematical expectations of SV space phases are set up. Relationships for calculating the rates of SV transition from one state to another are developed and validated. Ways of reducing the duration of the processes of calculating SV flight paths are proposed. Analytical relationships for determining the values of phase coordinates are formulated. Numerical results of calculating flight paths in a wide range of boundary conditions of SV design characteristics are given. The possibility of applying the proposed method for the analysis of SV motion in the atmosphere in conditions of conditions of flight indeterminacy is shown.