Eventos Anais de eventos
COBEM 2021
26th International Congress of Mechanical Engineering
SEMI-ANALYTIC FIRST-ORDER SOLUTION FOR OPTIMAL LOW-THRUST TRAJECTORIES AROUND MOON
Submission Author:
Luiz Arthur Gagg Filho , SP
Co-Authors:
Sandro da Silva Fernandes, Luiz Arthur Gagg Filho
Presenter: Sandro da Silva Fernandes
doi://10.26678/ABCM.COBEM2021.COB2021-0159
Abstract
This work considers the development of a semi-analytical first-order solution for computing optimal low-thrust limited power trajectories around Moon. Based on the study by Knezevic and Milani about the main terms of Kaula's development for the disturbing potential of the Moon as function of the orbital elements, it is assumed that for a preliminary mission analysis of a space vehicle propelled by a low-thrust engine this perturbing potential is described by the main three zonal harmonics $J_2$, $J_3$ and $J_4$. It is important to note that the second zonal harmonic is not as dominant with respect to the higher terms as it occurs for Earth; indeed, the difference in the order of magnitude of $J_2$ with respect to $J_3$ and $J_4$ is not so high. The optimization problem is formulated as a Mayer problem of optimal control with the state variables defined by the Cartesian elements (components of the position vector and the velocity vector) and by a consumption variable which describes the fuel spent during the maneuver. By applying Pontryagin Maximum Principle the optimal thrust acceleration and the maximum Hamiltonian function are determined. Then, a set of classical orbital elements is introduced as a new set of state variables by means of an intrinsic canonical transformation defined by the general solution of the canonical system described by the undisturbed part of the maximum Hamiltonian. This study does not include orbits with small eccentricities and/or inclinations, such that classical orbital elements are introduced. The first-order solution includes periodic terms concerning with the zonal harmonics and concerning with the optimal thrust acceleration. This solution is derived by means of Hori method, a perturbation method based on Lie series. On contrary of analytical theories for the motion of artificial satellites, the solution of the average canonical system is computed by numerical techniques; that is, the average canonical system is integrated numerically by means a Runge-Kutta-Fehlberg algorithm of 4-5 order. It is remarkable to mention that the solution is expressed in closed-form with no development in powers of eccentricity. Some numerical results show the influence of the zonal harmonics on the optimal trajectories.
Keywords
Low-thrust trajectories, zonal harmonics of gravitational field, Semi-analytic solutions
