A Geometrical Approach to the Design of Multi-Gravity Assist Trajectories

Gravity-assisted trajectories take advantage of the gravitational pull provided by celestial bodies. By momentum exchange, the velocity of the spacecraft is modified. In this way, propellant is saved and high-Δv targets can be reached. The preliminary design of gravity-assisted trajectories relies on two-body dynamics, patched conics and impulsive maneuvers. Furthermore, the engine of all calculations is a Lambert solver employed in combination with an optimizer that searches for the event dates (departure, arrival, and intermediate encounters) that minimize fuel consumption, time of flight and/or other mission parameters. This approach is certainly robust, but may be computationally heavy. In this contribution, we propose a geometrical approach to the problem of designing trajectories with a given itinerary. Discrete sets of conic sections connecting the orbits (assumed circular and coplanar) of the planets of a selected itinerary are generated and then matched on the basis of flyby dynamics. The matching can be done either by imposing that the gravity assist is unpowered or by allowing an instantaneous velocity variation at the pericenter of the flyby hyperbola. The time matching with respect to the actual planetary configuration is carried out a posteriori and is only executed on the subset of solutions that satisfy any additional requirements set by the user. Multiple solutions are found for the complete itinerary. The individual portions of the trajectory are pre-computed, hence the algorithm does not resort to the solution of Lambert's problem. The method naturally aims at providing optimal, minimum-Δv trajectories. It is applicable to the design of interplanetary transfers as well as paths within multi-moon systems.