  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.
