HYBRID TRAJECTORY OPTIMIZATION USING GENETIC ALGORITHMS

ABSTRACT A genetic algorithm is used together with calculus of variations to optimize an interplanetary trajectory for the Bryson-Ho Earth-to-Mars orbit transfer problem. The global search properties of genetic algorithm combine with the local search capabilities of calculus of variations to produce solutions that are superior to those generated with the calculus of variations alone, and these solutions require less user interaction than previously possible. The genetic algorithm is not hampered by ill-behaved gradients and is relatively insensitive to problems with a small radius of convergence, allowing it to optimize trajectories for which solutions had not yet been obtained. The use of the calculus of variations within the genetic algorithm optimization routine increased the precision of the final solution to levels uncommon for a genetic algorithm. 1. INTRODUCTION Techniques for optimizing spaceflight trajectories problems have become increasingly important as pressure to reduce the costs of space missions has increased. Both direct and indirect methods, known as “hill-climbing” methods, have been used to optimize space trajectories; however, for some scenarios, convergence to optimal solution is time-consuming, tedious, and sometimes not even possible. Direct methods that solve for controls to optimize the objective function directly, often via a gradient-based search, suffer from two major drawbacks. First, because search direction is ultimately driven by the local value of the gradient vector, the solution can converge on local, rather than global, minima, resulting in a final solution that is not globally optimal and cannot be further optimized [1]. Second, the optimal solution often has a small radius of convergence, requiring that the guesses for the initial parameters be close to the optimal answer [2]. Indirect methods, such as the calculus of variations, obtain optimal results by solving for the costates of a related boundary value problem and not for the controls directly. Although indirect methods are generally more likely to find a true, rather than local, optimum both direct and indirect approaches share many of the same drawbacks, most notably a small radius of convergence [3]. The “hill-climbing” methods exploit all local information in an efficient way, provided that certain conditions are fulfilled and, in particular, that the function to be minimized is “well-conditioned” in the neighborhood of the unique optimum [4]. Such a high level of exploitation requires a lot of local information to be known (gradient and, sometimes, Hessian matrix): the more intensive the exploitation, the stronger the need of specialized information about the function to be minimized. Moreover, if the basic requirements are not satisfied, the reliability of the “hill-climbing” methods is greatly jeopardized. Therefore, it is vital to choose initial parameter values intelligently; failure to do so will either dramatically increase the required computation or preclude obtaining a solution entirely. When indirect methods are used, where the optimization parameters are generally not related to the trajectory in an intuitive or straightforward manner, there may not be knowledge of the parameter bounds or their sensitivity. A common strategy to improve initial parameter selection uses previously optimized parameter values from a similar problem as an initial guess [1]. If no closely related solutions exist, initial values are found by optimization of an entire series of intermediate problems relating the new scenario to one with a known solution, a procedure known as homotopy analysis [5]. In recent years many techniques have been suggested for the avoidance of these shortcomings. A survey of these methods can be found in [6]. Evolutionary algorithms (EAs) are the best known. The usefulness of the genetic algorithms (GAs), well known of them, for solving impulsive trajectories is well documented [7, 8, 9]. The purpose of this study was to investigate the GA's effectiveness at determining a near optimal trajectory.