M.Sc Thesis

M.Sc StudentBeigelman Igor
SubjectGraph-Theory-Based Optimal Impulsive Formation keeping and
DepartmentDepartment of Aerospace Engineering
Supervisor PROF. Pinchas Gurfil
Full Thesis textFull thesis text - English Version


We develop a spacecraft formation flying control algorithm using relative orbital element corrections, which represent the differences between the orbital element corrections of any two spacecraft in a formation. This formalism introduces an inherent freedom that is used for deriving a formationkeeping algorithm optimally balancing the fuel consumption among the formation members based on the impulsive Gauss Variational equations. The main idea is that formulating the problem of formationkeeping in terms of relative orbital element corrections leaves the final values of the orbital elements unconstrained, thus allowing the spacecraft to create a natural energy-balanced formation. The freedom rendered by this modeling is used to find optimal impulsive maneuvers minimizing the squared l2-norm of the velocity corrections vector, which can be used for formation initialization and control. The optimization is solved using the method of least squares. The optimal formationkeeping method is designed to accommodate the effects of oblateness and drag. Based on graph theory, it is shown that the spacecraft will naturally form a stable energy-balanced formation, and that the optimal formationkeeping strategy is invariant to the spanning tree.

As a particular case of impulsive formationkeeping an algorithm for collocation of geostationary satellites in the same geostationary slot is developed. Synchronous elements are used for linear approximation of the relative motion in order to detect optimal impulsive velocity corrections for the collocation problem. The minimum distance between satellites that guarantees collision-free motion was taken into account in order to find necessary and sufficient conditions for the relative eccentricity and inclination vectors guaranteeing safe collocation. This collocation algorithm was simulated while incorporating J2 perturbations, since it is a major perturbation affecting geostationary satellites.

The main contribution of this research is the development of a generic optimal impulsive maneuvers scheme for multiple spacecraft formation flying and formationkeeping. This algorithm does not requires complicated calculations, since it is based upon simple matrix operations a change in the number of satellites does not affect the structure of the proposed algorithm and the calculation time. Another unique feature of this method is the independence on mission type and position in space, since all orbital perturbations can be taken into account during the model definition.