Rapid lunar soft-landing trajectory optimization by a legendre pseudospectral method

Determining how to seek the optimal soft-landing trajectory of the lunar probe from the lunar satellite orbit, so that it is able to safely and efficiently reach the lunar surface, involves the solutions of an optimal trajectory design and control problem. In this paper, the problem of rapid Lunar soft-landing trajectory optimization solved by using a Legendre pseudospectral method was studied. Firstly, some new hypotheses were introduced according to the distinguished features of the lunar and soft-landing trajectory. Based on these hypotheses, the set of dynamics and kinematics equations of the Lunar soft-landing was simplified in order to reduce the computation load, and the rationality of the simplification was analyzed. Then, the soft-landing trajectory optimization problem was transformed into a constrained parameter optimization problem by using a Legendre pseudospectral method, and a combination of nonlinear programming methods (Method of Multipliers and BFGS algorithm) was used to solve the transformed problem, that is, Method of Multipliers was applied to deal with the constraints and transform a constrained parameter optimization problem into a unconstrained one solved by the BFGS. Finally, we gave a numerical simulation example, in which the minimum principle of Pontryagin was used to verify the optimal condition of the optimal trajectory. Simulation results show the methodology and algorithms were able to generate a feasible soft-landing trajectory quickly on a desktop computer.