Journal of Mechanical Science and Technology 23 (2009) 3239~3244 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-009-0915-1 Journal of Mechanical Science and Technology An analytical guidance law of planetary landing mission by minimizing the control effort expenditure † Hamed Hossein Afshari * , Alireza Basohbat Novinzadeh and Jafar Roshanian Department of Aerospace Engineering, K.N. Toosi University of Technology, Tehran, Iran (Manuscript Received August 4, 2008; Revised June 12, 2009; Accepted August 16, 2009) -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract An optimal trajectory design of a module for the planetary landing problem is achieved by minimizing the control ef- fort expenditure. Using the calculus of variations theorem, the control variable is expressed as a function of costate variables, and the problem is converted into a two-point boundary-value problem. To solve this problem, the perform- ance measure is approximated by employing a trigonometric series and subsequently, the optimal control and state trajectories are determined. To validate the accuracy of the proposed solution, a numerical method of the steepest de- scent is utilized. The main objective of this paper is to present a novel analytic guidance law of the planetary landing mission by optimizing the control effort expenditure. Finally, an example of a lunar landing mission is demonstrated to examine the results of this solution in practical situations. Keywords: Nonlinear optimal control; Analytic solution; Control effort expenditure -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction Researchers and engineers have not been as suc- cessful in dealing with nonlinear optimal control problems as they have been in solving linear optimi- zation problems in control. In general, the optimal formulations of nonlinear dynamic systems, either through dynamic programming or through a varia- tional approach, lead to nonlinear partial differential equations. The numerical solution of these equations when dealing with complex nonlinear systems is al- ways difficult, particularly for real-world physical problems. One optimal control solution of the nonlin- ear lunar landing mission is obtained either by a dy- namic programming approach or through its varia- tional formulation [1, 2]. Useful mathematical meth- ods specially for approximating the mathematical functions are presented in [3]. To create a closed-loop guidance policy of the satellite injection problem, Pourtakdoust and Novinzadeh presented a fuzzy algo- rithm that was augmented to the solution of the time- optimal guidance strategy [4]. Afshari et al. presented some analytic approaches in spacecraft guidance [5-7]. An optimal guidance law that minimized the com- manded acceleration in three dimensions was ob- tained by Souza [8]. Ramana has designed an optimal trajectory for soft landing on the moon by solving the boundary value equations through a numerical ap- proach named controlled random search [9]. Lee in- vestigated on the optimal trajectory and the feedback linearization control of a re-entry vehicle during the terminal-area energy management (TAEM) phase [10]. Employing the nonlinear function approach, an improved model-based predictive control of vehicle trajectory has been developed [11]. This paper fo- cuses on a novel solution to design the optimal con- trol for the nonlinear problem of planetary landing mission by optimizing the control effort expenditure. To obtain an analytical solution, a set of state- dependent nondimensional variables is introduced. †This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin * Corresponding author. Tel.: +98 93 6669 1025, Fax.: +98 21 77991045 E-mail address: h.h.afshari@gmail.com © KSME & Springer 2009