Three-Dimensional Trajectory Optimization for soft lunar landing considering landing constraints^*

It is the goal for future lunar missions to seek and design an optimal soft-landing trajectory of the lunar probe from the lunar satellite orbit to the lunar surface. This study aims at the optimization of three-dimensional trajectory method able to achieve soft precision landing at the desired site with a minimum fuel. The trajectory optimization model is constrained by the use of two types of sets: the positions and velocities of the lander. The Pontryagain maximum principle (PMP) is used to transform the nonlinear optimal control into a two-point boundary value problem(TPBVP) in which the backward integration method is applied to compute the initial value of conjugate variables. The trajectory generation can be formulated as a time-invariant set of equations by a monotonically increasing state variable because the thrust control angles are obtained by conjugate equations. The robustness and optimality for three-dimensional trajectory is shown in a numerical simulation. The results of the new optimization method satisfy the fuel consumption condition with permissible tolerances.