Particle swarm optimization applied to impulsive orbital transfers Mauro Pontani a,n , Bruce A. Conway b a Scuola di Ingegneria Aerospaziale, University of Rome ‘‘La Sapienza’’, Rome, Italy b Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA article info Article history: Received 24 March 2011 Received in revised form 19 August 2011 Accepted 17 September 2011 Available online 12 January 2012 Keywords: Swarming theory Heuristic optimization methods Globally optimal orbital transfers abstract The particle swarm optimization (PSO) technique is a population-based stochastic method developed in recent years and successfully applied in several fields of research. It mimics the unpredictable motion of bird flocks while searching for food, with the intent of determining the optimal values of the unknown parameters of the problem under consideration. At the end of the process, the best particle (i.e. the best solution with reference to the objective function) is expected to contain the globally optimal values of the unknown parameters. The central idea underlying the method is contained in the formula for velocity updating. This formula includes three terms with stochastic weights. This research applies the particle swarm optimization algorithm to the problem of optimizing impulsive orbital transfers. More specifically, the following problems are considered and solved with the PSO algorithm: (i) determination of the globally optimal two- and three-impulse transfer trajectories between two coplanar circular orbits; (ii) determination of the optimal transfer between two coplanar, elliptic orbits with arbitrary orientation; (iii) determination of the optimal two-impulse transfer between two circular, non-coplanar orbits; (iv) determination of the globally optimal two-impulse transfer between two non-coplanar elliptic orbits. Despite its intuitiveness and simplicity, the particle swarm optimization method proves to be capable of effectively solving the orbital transfer problems of interest with great numerical accuracy. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction A variety of numerical solution techniques have been applied to the problem of space trajectory optimization. For several decades, direct and indirect deterministic methods have been used to locate the optimal solution for the problems of interest. Indirect optimization methods apply the necessary conditions for optimality, which arise from the calculus of variations. Direct methods convert an optimal control problem into a nonlinear programming problem, which involve a (usually large) number of parameters to be optimized. Due to their theoretical foundations, these two approaches exhibit specific advantages and disadvantages, which are extensively dealt with in the scientific literature. The implementation of deterministic techniques can require considerable analytical developments, as well as the defini- tion of specific routines for finding some mathematical entities that are involved in the solution process (e.g., the Jacobian matrix of the objective function). However, the two intrinsic features of deterministic methods that represent their main limitations are: (i) the need of a starting guess; (ii) the locality of results. In fact, both direct and indirect methods are local in nature, in the sense that the optimal solution they are able to detect depends on the available first attempt guess ‘‘solution’’. The numerical result found by deterministic methods usually lies in the neighborhood of the guess. With regard to convergence, in general direct techniques are more robust, because they are occasionally capable of converging to the desired result even in the presence of a poor guess. Conversely, indirect methods are more numerically accurate and do not need any Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/actaastro Acta Astronautica 0094-5765/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2011.09.007 n Corresponding author. E-mail addresses: mauro.pontani@uniroma1.it (M. Pontani), bconway@uiuc.edu (B.A. Conway). Acta Astronautica 74 (2012) 141–155