Fuel-optimal and Energy-optimal guidance schemes for lunar soft landing at a desired location

Abstract

Two guidance schemes (i) fuel-optimal (ii) energy-optimal to realize soft landing at a desired location on the moon are developed using the optimal control laws. The optimal control laws are obtained by solving a two-point boundary value problem formulated based on Pontryagin’s principle. The guidance laws, adapted from the optimal control laws, are obtained as a function of unknown co-state variables. Differential Transformation (DT) technique is employed to determine the unknown co-states at each time instant of landing trajectory using the information on the current vehicle state, target landing site (loaded on-board apriori) and the time-to-go. The numerical integration of co-state dynamics is avoided due to the DT based approach. The time-to-go, a critical parameter for any guidance scheme, is computed and updated real time using a simple strategy which uses the current and end states. The simple strategy for time-to-go works well even when the shape of the trajectory is nonlinear. Extensive analysis is carried out to evaluate and compare the proposed guidance schemes. Further, the proposed schemes are compared with other popular guidance schemes. The DT based proposed schemes help quantify the landing masses for fuel-optimal and energy-optimal objectives. Other features of the proposed schemes are that they do not assume constant gravity field and independent of reference trajectory.