M.Sc Thesis

M.Sc StudentMeltzer Daniel
SubjectStationkeeping about the Collinear Equilibrium Points in the
Restricted Three-Body Problem
DepartmentDepartment of Aerospace Engineering
Supervisor PROF. Pinchas Gurfil
Full Thesis text - in Hebrew Full thesis text - Hebrew Version


The utilization of the collinear libration points of the restricted three-body problem (R3BP) for space mission trajectories has been intensively investigated over the past few decades. In this work, calculation of a periodic halo-like orbit is accomplished by formulating the restricted three-body dynamics as a control problem. This approach is applied on the dynamical model of Earth-Moon system. This work considers four dynamical models of which, one is for the basic circular restricted three-body problem and three other models consider secondary effects such as: oblateness of the primaries, the Earth and the Moon, eccentricity of the Moon’s orbit about the Earth and the gravitational effect of the Sun on the motion of spacecraft in Earth-Moon system. For each of the considered models the position of the collinear libration points is calculated.

Linearization about the libration points yields dynamics equations about the collinear libration points. Using linear systems theory for constant linear systems and Floquet’s theory for a periodically varying linear systems it is found that motion about each one of the collinear libration points is unstable.

A continuous acceleration control term is introduced into the state-space dynamics

and use the pole-assignment technique to find linear periodic reference trajectories.

In periodically varying system a generalized version of pole-assignment technique is required.

For the calculation of a non-linear solution to the restricted three-body problem, a disturbance-accommodating control scheme adopted coupled with an extended restricted three-body model, which incorporates effects such as oblateness of the primaries, eccentricity of the secondary orbit and the gravitational effect of the Sun.

The non-linear terms are modeled as persistent disturbances that are output from a second order linear system. Then, a LQR control scheme is used to track the libration point reference trajectory under the nonlinear term periodic disturbances.

Simulation experiments of a generalized model were conducted. This model include non-linear equations of motion for a restricted three body problem in an Earth-Moon-spacecraft system with elliptic orbit of the Moon, gravitational effect of the Sun and oblateness of the Earth and the Moon. These experiments show that small amounts of propellant are required. The requirement of control effort is in order from few to ten’s of meters per second per year and control acceleration of micrometer per square second make this trajectory control method suitable for a realization with an electric propulsion system such as Hall thruster.