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This paper introduces a first-order method for solving optimal powereddescent guidance (PDG) problems, that directly handles the nonconvexconstraints associated with the maximum and minimum thrust bounds with varyingmass and the pointing angle constraints on thrust vectors. This issue has beenconventionally circumvented via lossless convexification (LCvx), which lifts anonconvex feasible set to a higher-dimensional convex set, and via linearapproximation of another nonconvex feasible set defined by exponentialfunctions. However, this approach sometimes results in an infeasible solutionwhen the solution obtained from the higher-dimensional space is projected backto the original space, especially when the problem involves a nonoptimal timeof flight. Additionally, the Taylor series approximation introduces anapproximation error that grows with both flight time and deviation from thereference trajectory. In this paper, we introduce a first-order approach thatmakes use of orthogonal projections onto nonconvex sets, allowing expansiveprojection (ExProj). We show that 1) this approach produces a feasible solutionwith better performance even for the nonoptimal time of flight cases for whichconventional techniques fail and 2) the proposed method compensates for thelinearization error that arises from Taylor series approximation. We claim thatthe proposed approach offers more flexibility in generating feasibletrajectories for a wide variety of planetary soft landing problems.

A First-Order Method with Expansive Projection for Optimal Powered Descent Guidance