Trajectory Optimization of Lunar Soft Landing Using Differential Evolution

Velocity at landing should be within a safe limit, to make soft landing possible. For this the thrust steering angle should be controlled to optimize the fuel consumption. This paper deals with designing soft landing trajectory, from lunar parking orbit to the surface of Moon. To ensure vertical landing using minimum fuel, the landing trajectory design is done in four phases: Hohmann transfer phase, power descend phase, attitude maneuver and vertical descend phase. The fuel optimization is done in the power descend phase. Then the attitude of the probe is adjusted and the probe is made vertical during the attitude maneuver. In final vertical descend phase the steering angle is kept as 90 degree to land the probe vertically. Optimal soft landing trajectory is designed using indirect approach of optimization. The indirect approach uses, Pontryagin's minimum principle to transfer optimal control problem to Two Point Boundary Value (TPBV) problem, and to express the control variable as a function of costate variables. Then the state and costate equations are integrated at optimum control angle, with the initial costate found by the different optimization techniques, to get the optimal trajectory. Two optimization techniques are used in this paper: Control Random Search (CRS) and Differential Evolution (DE). The former optimization techniques are compared with optimal trajectory designed by indirect approach. The simulation results show that CRS has taken 10000 iterations in a simulation time of 120 minutes to converge while DE converged with 2000 iteration in 30 minutes of CPU time. Even though the velocity obtained after power descent phase and optimum landing mass are almost same in both technique, the time of convergence of CRS is four times that of DE. i.e., DE is found superior to CRS, since it converges faster and load taken by system is less. The problem of soft landing is extended to design a fuel optimal three dimensional trajectory using DE technique.