This paper introduces a first-order method for solving optimal powered descent guidance (PDG) problems, that directly handles the nonconvex constraints associated with the maximum and minimum thrust bounds with varying mass and the pointing angle constraints on thrust vectors. This issue has been conventionally circumvented via lossless convexification (LCvx), which lifts a nonconvex feasible set to a higher-dimensional convex set, and via linear approximation of another nonconvex feasible set defined by exponential functions. However, this approach sometimes results in an infeasible solution when the solution obtained from the higher-dimensional space is projected back to the original space, especially when the problem involves a nonoptimal time of flight. Additionally, the Taylor series approximation introduces an approximation error that grows with both flight time and deviation from the reference trajectory. In this paper, we introduce a first-order approach that makes use of orthogonal projections onto nonconvex sets, allowing expansive projection (ExProj). We show that 1) this approach produces a feasible solution with better performance even for the nonoptimal time of flight cases for which conventional techniques fail to generate achievable trajectories and 2) the proposed method compensates for the linearization error that arises from Taylor series approximation, thus generating a superior guidance solution with less fuel consumption. We provide numerical examples featuring quantitative assessments to elucidate the effectiveness of the proposed methodology, particularly in terms of fuel consumption and flight time. Our analysis substantiates the assertion that the proposed approach affords enhanced flexibility in devising viable trajectories for a diverse array of planetary soft landing scenarios.

翻译：本文提出一种求解最优动力下降制导（PDG）问题的一阶方法，直接处理与变质量情况下最大/最小推力边界及推力矢量指向角约束相关的非凸约束。传统上通过无损凸化（LCvx）方法规避该问题，该方法将非凸可行集提升至更高维的凸集，同时采用泰勒级数线性逼近另一由指数函数定义的非凸可行集。然而，当从高维空间获得的解被投影回原始空间时，特别是在涉及非最优飞行时间的问题中，该方法有时会导致不可行解。此外，泰勒级数逼近产生的近似误差会随飞行时间和偏离参考轨迹的程度而增大。本文引入一种利用非凸集正交投影的一阶方法，实现扩张投影（ExProj）。研究证明：（1）即使在传统技术无法生成可执行轨迹的非最优飞行时间场景中，该方法仍能产生性能更优的可行解；（2）所提方法能补偿泰勒级数逼近产生的线性化误差，从而生成燃料消耗更低的优越制导解。我们通过定量评估的数值算例阐明所提方法的有效性，特别是在燃料消耗与飞行时间方面的表现。分析结果证实：所提方法为多类行星软着陆场景的可行轨迹设计提供了更强的灵活性。