Optimal Trajectory Planning for Multiphase Lunar Landing∗

Lunar landing problem is formulated as an optimal control problem and has been solved by Legendre Pseudospectral method. The landing problem is split into various stages to take into account various mission constraints. Based on these constraints, the problem is split into three stages-Rough Braking, Attitude Hold and Retargeting. Legendre Pseudospectral method is used for solving this multiphase lunar landing problem. The idea of this method is to discretize the trajectory optimization as a nonlinear programming problem. Legendre Pseudospectral method was applied to approximate the state and the state differential equations as algebraic constraints. Optimal solutions of the above problem are those control variables which dynamically satisfy all mission constraints. The cost function considered is minimum fuel satisfying all mission constraints envisaged. The method guarantees a minimum fuel solution which satisfies terminal mission constraints in each stage as part of the formulation. The mission constraints considered in studying this problem are desired altitude, attitude and velocity while reaching the desired landing site from a specified perilune height. At touchdown, the lander orientation should be vertical favoring landing leg of the Lander to touch the Moons surface as desired. The commanded acceleration should be within the limit of the main engine thruster capability (saturation limit) which enforces maximum and minimum thrust constraints.