Continuation from a flat to a round Earth model in the coplanar orbit transfer problem

Abstract

In this article we focus on the problem of minimization of the fuel consumption for the coplanar orbit transfer problem. This problem is usually solved numerically by a shooting method, based on the application of the Pontryagin Maximum Principle, however the shooting method is known to be hard to initialize and the convergence is difficult to obtain due to discontinuities of the optimal control. Several methods are known in order to overcome that problem, however in this article we introduce a new approach based on the following preliminary remark. When considering a 2D flat Earth model with constant gravity, the optimal control problem of passing from an initial configuration to some final configuration by minimizing the fuel consumption can be very efficiently solved, and the solution leads to a very efficient algorithm. Based on that, we propose a continuous deformation from this flat Earth model to a modified flat Earth model that is diffeomorphic to the usual round Earth model. The resulting numerical continuation process thus provides a new way to solve the problem of minimization of the fuel consumption for the coplanar orbit transfer problem.

Optimal Control and Applications to Aerospace: Some Results and Challenges

“…Denoting by F the change of variables from Cartesian to cylindrical coordinates, one passes from the adjoint vector in Cartesian coordinates to cylindrical coordinates by applying the transpose of the inverse of the differential of F . This is indeed a general geometric result whose proof is provided in the appendix of [186].…”

“…In this section we sketch shortly an alternative approach to the strong thrust minimal consumption orbit transfer planification problem developed in [186], consisting of considering at first the problem for a flat model of the Earth with constant gravity, and then of introducing step by step (by continuation) the variable gravity and the curvature of the Earth, in order to end up with the true model. Of course, the fuel efficient orbit transfer of a satellite has been widely studied (see [187,2]), with many possible approaches.…”

“…The optimal control problem under consideration consists of steering the above system from a given initial configuration r(0) = r 0 , ϕ(0) = ϕ 0 , v(0) = v 0 , γ(0) = γ 0 , m(0) = m 0 , to some point of a specified final orbit r(t f ) = r f , v(t f ) = v f , γ(t f ) = γ f (other kinds of final conditions have been considered in [186]), by maximizing the final mass m(t f ). Note that the final time t f must be fixed in this problem, otherwise the optimal control problem would not have any solution (see [180,194]) since it is always better in terms of consumption to let the engine turn more around the planet with shorter thrust arcs.…”

“…It can even be solved explicitly, analytically, and the shooting method can be simplified in order to converge automatically and instantaneously, without any careful initialization (see [186] for details). In view of that it is tempting to try to pass continuously from this simple model to the initial one by acting on the gravity and on the curvature of the planet.…”

“…The details of the procedure as well as numerical simulations are provided in [186], and comparisons are led with classical direct methods.…”