Real-time control of optimal low-thrust transfer to the Sun–Earth L 1 halo orbit in the bicircular four-body problem $ Majid Salmani n , Christof B ¨ uskens Center of Industrial Mathematics, Bremen University, Bibliothekstrasse 1, 28359 Bremen, Germany article info Article history: Received 18 February 2011 Received in revised form 9 May 2011 Accepted 12 May 2011 Available online 20 July 2011 Keywords: Optimal control Sensitivity analysis Real-time control Bicircular four-body problem Halo orbit transfer Correction mission abstract In this article, after describing a procedure to construct trajectories for a spacecraft in the four-body model, a method to correct the trajectory violations is presented. To construct the trajectories, periodic orbits as the solutions of the three-body problem are used. On the other hand, the bicircular model based on the Sun–Earth rotating frame governs the dynamics of the spacecraft and other bodies. A periodic orbit around the first libration-point L 1 is the destination of the mission which is one of the equilibrium points in the Sun–Earth/Moon three-body problem. In the way to reach such a far destination, there are a lot of disturbances such as solar radiation and winds that make the plans untrustworthy. However, the solar radiation pressure is considered in the system dynamics. To prevail over these difficulties, considering the whole transfer problem as an optimal control problem makes the designer to be able to correct the unavoidable violations from the pre-designed trajectory and strategies. The optimal control problem is solved by a direct method, transcribing it into a nonlinear programming problem. This transcription gives an unperturbed optimal trajectory and its sensitivities with respect perturbations. Modeling these perturbations as parameters embedded in a parametric optimal control problem, one can take advantage of the parametric sensitivity analysis of nonlinear programming problem to recalculate the optimal trajectory with a very smaller amount of computation costs. This is obtained by evaluating a first-order Taylor expansion of the perturbed solution in an iterative process which is aimed to achieve an admissible solution. At the end, the numerical results show the applicability of the presented method. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction The optimal control theory has been used for solving a wide range of technical problems in the last decades. One of these problems is the orbital transfer problem. The transfer considered in this article starts from a Low Earth Orbit (LEO), a geocentric orbit used as a temporary parking orbit, to a three-dimensional non-geocentric orbit which is actually a periodic orbit around the libration point L 1 in the Sun–Earth/Moon three-body system. The libration point L 1 is an unstable libration point between Sun and Earth at almost 1.5 million kilometers away from the Earth in the direction of the Sun [1,2]. Indeed, because of very small gravitational disturbance forces, the periodic orbits around libration points have been identified as convenient locations for positioning spacecrafts in a mission with high performance requirements. Because of convenience of positioning and station- keeping in periodic orbits, transfers to these periodic orbits around libration points have been widely studied in space science and astrodynamics [1–6]. One of the properties of the periodic orbits around L 1 is that they are located in the vicinity of the Earth and the Moon under Contents lists available at 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.2011.05.017 $ This work has been supported by the scholarship in Scientific Computing in Engineering group. n Corresponding author. E-mail address: salmani@math.uni-bremen.de (M. Salmani). Acta Astronautica 69 (2011) 882–891