### Citation:

Rita Neves and Joan Pau Sanchez Cuartielles. Gauss’ variational equations for low-thrust optimal control problems in low-energy regimes. Proceedings of the International Astronautical Congress 2018 (IAC ’18), 1-5 October 2018, Bremen, Germany.

### Abstract:

With the pursuit of increasingly innovative and complex space missions, the focus of the space industry
has been turning towards electric propulsion systems. Due to their high specific impulse - about ten times
that of a chemical engine - they provide large savings in propellant mass, decreasing the overall cost of
the mission. This proves to be essential for small low cost missions, such as interplanetary CubeSats, and
more ambitious endeavours such as asteroid retrieval or crewed missions to Mars.
Designing a low-thrust trajectory is a more complex task than doing so for a high-thrust one, since
computing the thrust sequence that minimizes the fuel spent requires a search over a huge and complex
design space. Setting up the optimal control problem generally requires a good first-guess solution, a fine
tuning of the parameters involved, and the definition of feasible bounds for the trajectory. In order to
converge to a solution, the problem settings are simplified as much as possible. This includes the dynamical
framework used, which often may not be sensitive enough to describe the low-energy trajectory regime
necessary for some of the mission examples mentioned above.
This abstract proposes a new set of equations of motion to solve the optimal control problem. These
are derived from the disturbing function of the previously studied Keplerian Map, formulated from the
Hamiltonian of the CR3BP. Its motion corresponds to the propagation of Gauss’s planetary equations
with both the disturbing potential of the CR3BP, and the accelerations of the electric engine. The
novelty of this formulation is that it describes a third-body motion in terms of the orbital elements that
define the osculating orbit of the spacecraft, in a barycentric coordinate system. This is advantageous in
several respects: first, low-thrust sub-optimal control laws can be easily generated and explored to find
a first guess solution near global optima. Second, bounds for the optimal control problem, as well as
the boundary values, can be easily defined, which allows for a much faster convergence. This dynamical
framework is accurate until very close to the sphere of influence of the perturbing body, and thus can
be efficiently used to target low-energy hyperbolic invariant manifold structures associated with periodic
orbits near it.
The paper presents the methodology as well as a full retrieval trajectory for asteroid 2018 AV2, a
small co-orbital asteroid that could be retrieved during its next Earth encounter in 2037.