Satellite phasing manoeuvre design has been widely addressed in literature in case of chemical propulsion (impulsive manoeuvre) with applications mainly to Low-Earth-Orbit (LEO) and geostationary (GEO) Earth observation and telecommunications satellite missions. Limited work can be found regarding low-thrust orbital phasing optimal design of satellites at medium altitudes (MEO), where mainly GNSS constellations operate. In this case, additional constraints, mission and/or platform-related (e.g. Earth-shadow eclipses, angular rates, Sun illumination restrictions, etc.), and constraints on orbital parameters (eccentricity in particular) - can further increase the optimization problem complexity as well as the computational effort.
This paper presents a straightforward methodology to address the problem of low-thrust orbital phasing of a MEO satellite in quasi-circular orbit under a set of constraints. More in detail, an analytical formulation has been developed to design a low-thrust phasing campaign under Sun illumination restrictions, with limited delta-V budget, and a time minimization objective. The design of the thrust control strategy is based on a geometrical approach aiming at minimizing thrust deviation with respect to velocity direction while satisfying the above-mentioned illumination constraint. Therefore, firstly, a geometric definition of the instantaneous minimum angle between thrust and velocity vector, taking into account the Sun illumination constraint, is provided. Afterwards, an expression of the average tangential acceleration due to low-thrust (over one orbit revolution) is formulated, by exploiting trigonometric identities and symmetry properties, and reduced to a closed-form analytical solution in terms of elliptic integrals of second kind. Leveraging the closed-form solution, the approximated minimum duration of the constrained, low-thrust phasing campaign is computed as a function of the spacecraft properties, available delta-V budget, total phasing angle and Sun elevation angle with respect to the satellite orbital plane (assumed constant as a first approximation).
Although the presented analytical formulation is based on the hypothesis of circular orbit, symmetry properties of the thrust control strategy with respect to the angular variable identifying instantaneous satellite position along the orbit allow to keep the eccentricity trend bounded when the thrust strategy is incorporated in a routine based on numerical propagation accounting for the osculating eccentricity evolution.
Finally, the analytical formulation results are validated against the ones obtained through numerical simulations. The benchmark used for validation of the presented analytical formulation is indeed a flexible routine, based on numerical propagation, implementing a refinement of the above-mentioned thrust control strategy, which further optimizes the eccentricity evolution trend and includes additional platform constraints, like limitation on the maximum satellite platform angular rate and need to use reduced thrust magnitude during Earth-shadow eclipses.
The matching between analytical and numerical results demonstrates that the derived formulation represents a good approximation of the problem dynamics while offering several advantages: it provides a computationally efficient approach for constrained low-thrust phasing in MEO, enabling rapid trade-off analyses supporting preliminary mission design stages and can be used to generate reliable initial guess solutions for more complex, numerical optimization algorithms.