Performance analysis of the vectorized high order expansions method in the accurate landing problem of reusable boosters

This paper addresses the design of the terminal phase maneuver of the booster landing in the presence of initial deviations and uncertainties. The primary purpose of this study is to evaluate the effectiveness of the vectorized high order expansions method to extract commands and near-optimal trajectory, while the accurate satisfaction of the terminal conditions is of great importance. After reviewing high order expansions methods, the Vectorized High Order Expansions method is briefly discussed. Then, the solution for the booster landing problem is extracted up to the third order. To study the solution quality in the presence of uncertainties, mass, density, and gravity are assumed to be constant in extracting the 3rd order solution. The assumptions however are eliminated in the simulations, and the effectiveness of high order expansions solution is studied subjectively and numerically in the presence of initial deviations and uncertainties. After implementing simulations and considering different assumptions, it will be shown that the 3rd order solution holds a desirable quality in accurately satisfying final conditions in presence of uncertainties and initial deviations and provides far better results than the 1st order solution. Moreover, in this paper, a novel method is proposed for online updating, and the performance and effect of this modification are studied using simulations. Reviewing the results of the 3rd order solution combined with the online update strategy, it becomes clear that the implementation of the online update significantly reduces landing point errors and improves the performance of the high order method. Furthermore, the study compares the performance of this strategy with the well-known state-dependent Riccati equation method to better evaluate the effectiveness of the proposed approach.