Learning Optimal Control and Dynamical Structure of Global Trajectory Search Problems with Diffusion Models

TL;DR

The paper addresses parameterized global trajectory search in the Circular Restricted Three-Body Problem (CR3BP) by learning structured optimal-control solutions conditioned on problem parameters. It extends the AmorGS framework with diffusion probabilistic models to capture solution structures such as bang-bang thrust profiles and invariant-manifold basins, enabling fast, high-quality initial guesses for offline optimization and accelerating online design. Two CR3BP problems are studied: a hybrid minimum-fuel/minimum-time objective and a variable-terminal-boundary problem tied to energy-dependent halo manifolds. Results show diffusion-based initial guesses substantially improve feasibility and optimality rates and reduce solving times, while accurately reflecting parameter-driven shifts in solution distributions and manifold basins. The work demonstrates a data-driven pathway to leverage dynamical structures for rapid, robust cislunar mission design with quantified generalization to unseen parameter values.

Abstract

Spacecraft trajectory design is a global search problem, where previous work has revealed specific solution structures that can be captured with data-driven methods. This paper explores two global search problems in the circular restricted three-body problem: hybrid cost function of minimum fuel/time-of-flight and transfers to energy-dependent invariant manifolds. These problems display a fundamental structure either in the optimal control profile or the use of dynamical structures. We build on our prior generative machine learning framework to apply diffusion models to learn the conditional probability distribution of the search problem and analyze the model's capability to capture these structures.

Figures (11)

• Figure 1: Hyperplane structure existed in locally optimal solutions to a cislunar transfer in CR3BP li2023amortized. The initial/final coast time and shooting time of the spacecraft are presented in orange (Groundtruth solution) and blue (AmorGS prediction). The maximum allowable thrust $0.15$N and $0.85$N are unseen in the AmorGS training dataset.
• Figure 2: Projection onto the Earth-Moon orbital plane of the GTO spiral (gray line), low thrust trajectory solution (black), the initial manifold arc that was targeted (magenta), and the final manifold arc that is targeted (green). Optimal solutions to a discretized grid of the stable manifold in the variable space $t_1$ and $t_2$ are shown in the background, illustrating how the initial guess solution evolves to an insertion region of the invariant manifold that minimizes fuel usage, revealing multiple basins in the invariant manifolds. Examples of the basins are highlighted in blue.
• Figure 3: Workflow of the Amortized Global Search (AmorGS) Framework.
• Figure 4: The average time of flight and final fuel mass are visualized for fixed objective function weights $\alpha$ from the training dataset. Each point is at the center of an ellipse, representing the corresponding covariance matrix based on one standard deviation. The samples from the diffusion model (DM) for three $\alpha$ values not included in the training data are shown as triangles, with the corresponding covariance ellipses highlighted.
• Figure 5: The throttle profiles for the minimum time ($\alpha=0$) and minimum fuel ($\alpha=1$) cost function are shown as a color map, visualizing the density of 1,500 solutions from the training data. The range of the throttle is split into fixed intervals, and the percentage of solutions within each interval is shown as a colored rectangle for each segment.