Extension of finite perturbative elements for multi-revolution, low-thrust propulsion transfer optimisation

This paper presents an extension of the analytical solution for perturbed Keplerian motion of a spacecraft under the effect of a low-thrust action (Zuiani et al., Acta Astronautica, 2011). The new formulation will include the possibility for treating two different thrusting modes, i.e. with a fixed thrust direction either in a rotating or in an inertial frame. Moreover the contribution of the J2 effect is also included in the analytical formulae. It will be shown that this approach allows for the fast computation of long, many revolution spirals while maintaining adequate accuracy, and it is able to include the combined actions of different perturbations. The proposed approach will also be applied to the case of a spacecraft with a low-thrust engine, which is injected into a Geostationary Transfer Orbit and will subsequently use its on-board propulsion to transfer to a final circular orbit around the Earth. The completion of the whole transfer might require several spirals and this makes the use of a full numerical propagation prohibitive on a sequential machine. In the proposed method, the thrusting pattern, duration and start of each thrusting arc, is defined through a parameterised function. The spiral is then propagated with the above-mentioned analytical approximation. A direct optimisation approach is then used to optimise these control parameters in order to minimise the propellant cost of the transfer, given a fixed transfer time and a set of
boundary conditions.