3-dimensional analytical solution for lunar descent scheme

lunar and mars landing is a crucial part of any exploration mission. its importance is manifested by the considerable interest in the past couple of decades. solution of lunar descent can be either numerical or analytical. numerical solutions are iterative and computationally demanding and some of them might not be suitable for on-board implementation. on the other hand, analytical solutions are not complex and are much more attractive for the same purposes. all the present analytical solutions schemes are two dimensional considering only altitude and down range to describe the reference trajectories of a lunar descent. the main contribution of this paper is the development of 3-dimensional analytical solution for the reference trajectory which is crucial for a precise landing of a lunar spacecraft. in this investigation cross range distance is incorporated for complete representation. comparisons are made by simulated responses between numerical and analytical solutions. detailed mathematical derivations are presented in this paper as well.