Optimum Low Thrust Elliptic Orbit Transfer Using Numerical Averaging

Low-thrust electric propulsion is increasingly being used for spacecraft missions primarily due to its high propellant efficiency. Since analytical solutions for general low-thrust transfers are not available, a simple and fast method for low-thrust trajectory optimization is of great value for preliminary mission planning. However, few low-thrust trajectory tools are appropriate for preliminary mission design studies. The method presented in this paper provides quick and accurate solutions for a wide range of transfers by using numerical orbital averaging to improve solution convergence and include orbital perturbations. Thus allowing preliminary trajectories to be obtained for transfers which involve many revolutions about the primary body. This method considers minimum fuel transfers using first order averaging to obtain the fuel optimum rates of change of the equinoctial orbital elements in terms of each other and the Lagrange multipliers. Constraints on thrust and power, as well as minimum periapsis, are implemented and the equations are averaged numerically using a Gaussian quadrature. The use of numerical averaging allows for more complex orbital perturbations to be added without great difficulty. Orbital perturbations due to solar radiation pressure, atmospheric drag, a non-spherical central body, and third body gravitational effects have been included. These perturbations have not been considered by previous methods using analytical averaging. Thrust limitations due to shadowing have also been considered in this study. To allow for faster convergence of a wider range of problems, the solution to a transfer which minimizes the square of the thrust magnitude is used as a preliminary guess for the minimum fuel problem. Thus, this method can be quickly applied to many different types of transfers which may include various perturbations. Results from this model are shown to provide a reduction in propellant mass required over previous minimum fuel solutions. Minimum time transfers are also solved and compared to minimum fuel.