Optimal low-thrust transfers between coplanar orbits with small eccentricities

Abstract

In this paper, a first-order analytical solution, which includes the short periodic terms, for the problem of optimal time-fixed low-thrust limited-power transfers (no rendezvous), in an inverse-square force field, between coplanar orbits with small eccentricities is obtained through canonical transformation theory. Short periodic terms are eliminated from the maximum Hamiltonian, expressed in non-singular orbital elements, through an infinitesimal canonical transformation built through the Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton-Jacobi equation through the separation of variables technique. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition.

Spacecraft trajectory optimization: A review of models, objectives, approaches and solutions

“…Notable researches are dedicated to analytical approaches in spacecraft trajectory optimization. For example, Fernandez [123] developed a complete first-order analytical approach for the problem of optimal low-thrust limitedpower transfers in an inverse-square force field between coplanar orbits with small eccentricities. The presented approach eliminates the singularity for circular orbits and can be applied for time-fixed transfers between coplanar orbits with small eccentricities.…”

“…Reference Description Typical transfer da Silva et al [123] A complete first-order analytical solution, which includes the short periodic terms in an inverse-square force field, between coplanar orbits with small eccentricities is presented.…”

Analytical solutions for optimal low-thrust limited-power transfers between coplanar orbits

“…A first order analytical solution for the system of differential equations governed by the maximum Hamiltonian, which includes short periodic terms, can be derived applying canonical transformation theory, as described in [2] for transfers between elliptical orbits and in [1] for transfers between orbits with small eccentricities.…”

“…Several researchers have obtained numerical and analytical solutions for several maneuvers involving specific initial and final orbits and specific thrust profiles. In the analytical studies, 1 sandro@ita.br 2 fchagas.carvalho@gmail.com averaging techniques and perturbation methods are applied and analytical solutions of the averaged equations, as well as first order solutions which include short periodic terms, are obtained [1][2][3][4][5].…”

“…The authors have presented a complete analytical solution, which includes the short periodic terms, for the problem of optimal low-thrust limited-power transfers between arbitrary elliptical coplanar orbits [2] and for transfers between orbits with small eccentricities [1] by using a canonical approach based on canonical transformation theory, including the Hamilton-Jacobi theory and a perturbation method based on Lie series. Classical orbital elements are used for describing the analytical solution in the case of transfers between elliptical orbits, and, a suitable set of nonsingular elements are used in the case of transfers between orbits with small eccentricities.…”