A First‐Order Analytical Theory for Optimal Low‐Thrust Limited‐Power Transfers between Arbitrary Elliptical Coplanar Orbits

Abstract

A complete first-order analytical solution, which includes the short periodic terms, for the problem of optimal low-thrust limited-power transfers between arbitrary elliptic coplanar orbits in a Newtonian central gravity field is obtained through canonical transformation theory. The optimization problem is formulated as a Mayer problem of optimal control theory with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin maximum principle and determining the maximum Hamiltonian, classical orbital elements are introduced through a Mathieu transformation. The short periodic terms are then eliminated from the maximum Hamiltonian through an infinitesimal canonical transformation built through Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton-Jacobi equation through separation of variables technique. For transfers between close orbits a simplified solution is straightforwardly derived by linearizing the new Hamiltonian and the generating function obtained through Hori method.

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.…”

Optimal low-thrust transfers between coplanar orbits with small eccentricities

“…For maneuvers with arbitrary duration, the periodic terms must be included (Edelbaum 1966;Da Silva Fernandes and Carvalho 2008). Following the algorithm the Hori method and An approximate expression for the fuel consumption along the optimal trajectory can be put in a compact form as (Da Silva Fernandes and Carvalho 2008),…”

“…Several researchers have obtained numerical and analytical solutions for several maneuvers involving specific initial and final orbits and specific thrust profiles (Edelbaum 1965(Edelbaum , 1966Marec et al 1980;Haissig et al 1993;Geffroy and Epenoy 1997;Sukhanov 2000;Kiforenko 2005;Bonnard et al 2006;Da Silva Fernandes and Carvalho 2008;Jamison and Coverstone 2010;Quarta and Mengali 2013). In the analytical studies, averaging techniques are applied and solutions of the averaged equations are obtained such that only secular behavior of the optimal solutions is discussed.…”