Real-time trajectory optimization for powered planetary landings based on analytical shooting equations

Traditionally, numerical trajectory integration for shooting equation calculation and iterations for shooting with randomly guessed initial solutions deteriorate the real-time performance of indirect methods for on-board applications. In this study, the indirect method is improved to achieve real-time trajectory optimization of fuel-optimal powered planetary landings with the help of analytical shooting equation derivations and a practical homotopy technique. Specifically, the contributions of this paper are threefold. First, the analytical expressions for shooting equation calculation are derived to replace the traditional time-consuming trajectory integration. Consequently, the computational efficiency is significantly improved. Second, the original three-dimensional landing problem is connected with a simplified one-dimensional problem that only involves the vertical dynamics, and its analytical solution is obtained based on Pontryagin’s minimum principle. Third, starting with the analytical solution, the accurate solution of the original landing problem can be obtained through an adaptive homotopy process. Simulation results of Earth landing scenarios are given to substantiate the effectiveness of the proposed techniques and illustrate that the developed method can obtain a fuel-optimal landing trajectory in 5 ms with 100% success rate.