Convex Trajectory Optimization via Monomial Coordinates Transcription for Cislunar Rendezvous

TL;DR

This work introduces a nonlinear convex framework for fuel-optimal impulsive rendezvous near a reference orbit by employing overparameterized monomial coordinates and high-order Taylor expansions computed with differential algebra. The approach reformulates the OCP in monomial coordinates, enabling sequential convex programming to converge on feasible, near-optimal trajectories with significantly reduced real-time computational burden. It demonstrates robustness across fixed-time, constrained, and free-time rendezvous scenarios in the CR3BP, achieving small guidance and open-loop errors while maintaining comparable Delta-v to canonical SCP. The method is well-suited for autonomous onboard guidance in the cislunar domain, with demonstrated potential for hardware-limited implementations and future enhancements via alternative state representations and stochastic extensions.

Abstract

This paper proposes a nonlinear guidance algorithm for fuel-optimal impulsive trajectories for rendezvous operations close to a reference orbit. The approach involves overparameterized monomial coordinates and a high-order approximation of the dynamic flow precomputed using differential algebra, which eliminates the need for real-time integration. To address non-convexity in the monomial coordinate formulation of the guidance problem, sequential convex programming is applied. Using the methodology presented in this paper, repeatedly evaluating the nonlinear dynamics is not necessary, as in shooting or collocation methods. Instead, only the monomial equations require updating between iterations, drastically reducing computational burden. The proposed algorithm is tested in the circular restricted three-body problem framework with the target spacecraft on a near-rectilinear halo orbit. The results demonstrate stability, efficiency, and low computational demand while achieving minimal terminal guidance errors. Compared to linear methods, this nonlinear convex approach exhibits superior performance in open-loop propagation of impulsive maneuvers in cislunar space, particularly in terms of accuracy. These advantages make the algorithm an attractive candidate for autonomous onboard guidance for rendezvous operations in the cislunar domain.

Figures (21)

• Figure 1: Impulsive maneuver trajectory transcription via $\Psi_m$.
• Figure 2: Flowchart of the fixed-final time nonlinear convex impulsive guidance.
• Figure 3: Block diagram illustrating the closed-loop guidance.
• Figure 4: Flowchart of the free-final time nonlinear convex algorithm.
• Figure 5: Target's NRHO and time interval of the generated $\Psi_m$ for rendezvous scenario 1.