M.Sc Student | Beigelman Igor |
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Subject | Graph-Theory-Based Optimal Impulsive Formation keeping and Collocation |

Department | Department of Aerospace Engineering |

Supervisor | Professor Pinchas Gurfil |

Full Thesis text |

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 *l ^{2}*-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 *J _{2}*
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.