Optimization of Low-Thrust Trajectories for Formation Flying with Parallel Multiple Shooting

Abstract

In this work the problem of optimization of low-thrust space trajectory is addressed. The problem can be stated as the solution of an optimal control problem in which an objective function related to controls is minimized satisfying a series of constraints on the trajectory which are both difierential and algebraic. The problem has been faced transcribing the difierential constraints with a parallel multiple shooting transcription method into a NLP problem which has been solved with an interior point method. The method that has been developed is particularly suited for the solution of problems in which the trajectory is constrained with a great number of inequalities both on states and controls. As an example of such kind of problem, the method has been applied to the design of reconflguration maneuvers for spacecrafts ∞ying in formation, where the collision avoidance issue leads to the imposition of a large number of inequalities on states. In the fleld of space trajectory design the optimization of the resources is vital, as in an environment where the refueling is not possible and it is necessary to reduce the overall weight of the satellites, the amount of propellant needed can highly in∞uence the operative life of the satellites. For this reason LowThrust propulsion is becoming quite popular as it is characterized by high value of speciflc impulse, which allows to reduce the amount of propellant needed for a mission. The use of such kind of continuous thrust propulsion will lead to the necessity to flnd the optimal continuous control law. The problem can be stated as the solution of an optimal control problem in which an objective function related to controls is minimized satisfying a series of constraints on the trajectory which are both difierential and algebraic. Moreover the emerging problems in space mission lead to trajectories which are constrained with a great number of inequalities both on states and controls. An example of such problem is formation ∞ying trajectory design where the collision avoidance issue leads to the imposition of a large number of inequalities on states. In this work an optimization methods for the solution of problems with an high number of inequality constraints will be shown. The numerical solution of the optimal control problem which is at the base of the trajectory optimization is usually based on the idea of discretizing the continuous problem using a transcription technique into a discrete problem which can be faced with the techniques developed for parameter optimization which are usually based on the Newton method. 1 Following this paradigm, the problem is conceptually divided in two parts, a transcription of the difierential optimal control problem in a parameter optimization problem and the solution of that discrete problem. For what concerns the transcription of the optimal control problem in a Non Linear Programming (NLP) problem, a direct transcription method based on parallel multiple shooting techniques has been developed. It allows to exploit the advantages of parallel computing reducing at the same time the dimension of the resulting parameters optimization problem. The parameters optimization deriving from the transcription of the optimal control problem with the multiple shooting techniques is characterized both by nonlinear constraints and nonlinear objective function, leading to a NLP problem. The solution of this NLP problem is usually based on Sequential Quadratic Programming (SQP) methods associated with active set approach for active inequalities identiflcation. 2 In