An analytical solution of fuel-near-optimal trajectory for three-dimensional planetary landings

Abstract

The existing solutions for three-dimensional fuel-optimal trajectory generation are inadequate in terms of real-time performance and convergence property, which are vital to successful and reliable powered planetary landings. In this study, the fast generation of three-dimensional fuel-near-optimal landing trajectories is achieved based on problem decomposition and analytical iterative techniques. Specifically, the following three contributions are emphasized. First, the original three-dimensional (3D) thrust-constrained powered planetary landing problem is decoupled into three analytically solvable one-dimensional (1D) sub-problems: one along the vertical direction, and two along the horizon direction. These three 1D problems are re-coupled together by six to-be-determined coordination coefficients. Second, an analytical iteration algorithm is developed to determine these six coordination coefficients and to ensure the satisfaction of thrust magnitude and direction constraints. Third, three necessary proofs are provided to validate the guaranteed convergence of the proposed analytical iteration algorithm. Simulation results of various planetary landing missions on Earth demonstrate that the proposed analytical algorithm can achieve a significant fast generation of 3D fuel-near-optimal landing trajectories with acceptable optimality (2.6% more fuel on average) and guaranteed convergence rate (100%) for planetary landings.