Powered Descent Guidance via First-Order Optimization with Expansive Projection

TL;DR

This work tackles PDG under nonconvex thrust bounds and pointing constraints by proposing ExpProj, a first-order method that directly handles nonconvex sets rather than relying on LCvx or Taylor-based approximations. ExpProj combines orthogonal projections onto nonconvex sets with an ADMM framework to compute feasible, fuel-efficient trajectories across variable time-of-flight scenarios. Numerical results show that ExpProj matches LCvx in the lossless, optimal time-of-flight case and outperforms it when time-of-flight is nonoptimal, by avoiding infeasibility and reducing linearization error. Indoor flight tests validate real-time applicability on embedded hardware and support practical deployment for diverse planetary soft-landing missions.

Abstract

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.

Figures (9)

• Figure 1: Orthogonal projections onto a nonconvex set $\mathcal{X}$. Observe that the projection onto nonconvex sets can be expansive.
• Figure 2: Orthogonal projection onto the surface of the second-order cone defined by $\mathcal{C}_1$. Note that the set is nonconvex and that the projection from region ① (exterior region) is nonexpansive, while the projection from region ③ (interior region) can be expansive.
• Figure 3: Orthogonal projection onto $\mathcal{C}_2$ (shaded area). Note that the set is nonconvex and that the projection from region ① (below $\mathcal{C}_2$) is nonexpansive, while the projection from region ③ (above $\mathcal{C}_2$) can be expansive.
• Figure 4: Relative residuals attained by using the Newton-Raphson method for the instances derived from ExProj. Note that the computational process exhibits exponential convergence, typically converging within a minimal number of iterations.
• Figure 5: Optimization results for the scenario of optimal time of flight case ($t_f=t^*_f$). It is observed that the convexification is lossless in this case, resulting in both LCvx and ExProj algorithms yielding nearly identical optimal solutions. However, note that in the latter phase of the thrust profile, the LCvx solution marginally underutilizes the maximum allowed thrust, rendering it slightly suboptimal.