Evolved Non-Keplerian Spacecraft Trajectories for Near–Earth Orbital Maneuvers D.W. Hinckley Jr. a* D.L. Hitt a† M.J. Eppstein b ‡ a School of Engineering b Department of Computer Science University of Vermont Burlington, VT 05465 USA In this paper we use Differential Evolution (DE), with best-evolved results refined using a Nelder-Mead optimization, to solve complex problems in orbital mechanics relevant to low Earth orbits (LEO) and within the Earth-Moon system. A class of Lambert problems is examined to evaluate the performance and robustness of this evolutionary approach to orbit optimization. We evolve impulsive initial velocity vectors giving rise to intercept trajecto- ries that take a spacecraft from given initial positions to specified target positions. We seek to minimize final positional error subject to time-of-flight and/or energy (fuel) constraints. We first validate that the method can recover known analytical solutions obtainable with the assumption of Keplerian motion. We then apply the method to more complex and re- alistic non-Keplerian problems incorporating trajectory perturbations arising in LEO due to the Earth’s oblateness and rarefied atmospheric drag. Finally, a rendezvous trajectory from LEO to the L4 Lagrange point is computed. The viable trajectories obtained for these challenging problems suggest the robustness of our computational approach for real-world orbital trajectory design in LEO situations where no analytical solution exists. I. Introduction The planning of orbital maneuvers and/or trajectories for spacecraft represents a design optimization problem that is associated with multiple engineering constraints (e.g., time of flight, fuel consumption, and positional accuracy). Aside from the inherently nonlinear equations of classical orbital motion, modern problems of practical interest are further complicated by various sources of perturbations such as planetary oblateness, atmospheric drag for low Earth orbits (LEO), and solar radiation pressure among others. With the emergence of satellite formation-flying mission concepts, additional constraints are often required in order to achieve satisfactory performance. For example, the satellite formation topology may be required to sat- isfy a specified criterion during a finite portion of the orbit for the purposes of a coordinated measurements. NASA’s Magnetospheric Multi-Scale Mission (MMS) provides an excellent example of such constraints (see mms.gsfc.nasa.gov) . The MMS mission consists of four satellites that need to be in a tetrahedral arrange- ment during the region of measurement performance; this region is defined by a symmetric range of anomaly about apogee. Owing to the multiple objectives and system complexity, analytical approaches to trajectory optimiza- tion are generally not available and numerical optimization is required. To this end, various evolutionary approaches for trajectory optimization have been explored over the past decade. Cacciatore & Toglia 2 con- sidered minimum fuel orbital trajectories resulting from a finite series of impulsive thrusts using a genetic algorithm (GA). Lee et al. 19 also used GAs to evolve orbital elements (semi-major axis, eccentricity, inclina- tion) instead of an initial trajectory velocity. As such, their approach was necessarily limited to the idealized * Mechanical Engineering Graduate Student, AIAA Student Member † Professor of Mechanical Engineering, AIAA Associate Fellow, and corresponding author: dhitt@uvm.edu ‡ Professor of Computer Science 1 of 19 American Institute of Aeronautics and Astronautics