Probabilistic Trajectory Design Via Approximate Gaussian Mixture Steering

TL;DR

The paper tackles stochastic spacecraft trajectory design under uncertainty by formulating a distribution steering problem that drives an uncertain state from an initial pdf to a final pdf while minimizing control effort. It introduces a Gaussian mixture uncertainty framework to propagate non-Gaussian state distributions through nonlinear dynamics, using a node-based affine feedback control with a Gaussian collapse to enable tractable chance constraints. The method yields a nonlinear, nonconvex program solved with SNOPT and is evaluated on a finite-thrust Earth-to-Mars transfer, comparing single-Gaussian and Gaussian mixture configurations. Monte Carlo analysis shows GM-based designs achieve higher terminal constraint satisfaction (83% to 99%) and lower average $\Delta V$ costs, demonstrating improved robustness to non-Gaussian uncertainty and suggesting practical advantages for interplanetary mission design.

Abstract

A method is presented to solve a stochastic, nonlinear optimal control problem representative of spacecraft trajectory design under uncertainty. The problem is reformulated as a chance constrained nonlinear program, or what is known as a distribution steering problem. Typical distribution steering problems rely on the underlying uncertainties to be Gaussian distributions. This work expands on previous developments by embedding Gaussian mixture distributions into the formulation to better handle the uncertainty propagation and chance constraints involved. The method is applied to a finite-thrust Earth-to-Mars transfer problem. Evaluation via Monte Carlo analysis shows a greater satisfaction of constraints under non-Gaussian distributions of the state and a statistically lower cost.

Figures (18)

• Figure 1: GM approximation of standard normal using $\bm{L = 3}$ splitting library. The standard Gaussian is given by the red line, the individual GM components are shown in gray, and the overall GM is shown in black.
• Figure 2: Flow chart of propagation through dynamics and control update with associated equations. Recursion is established by $\bm{t_{k-1} \leftarrow t_{k_+}}$.
• Figure 3: $\bm{L=1}$ optimized means (yellow), covariance ellipses (black), and $\bm{\Delta V}$ vectors (purple).
• Figure 4: $\bm{L=1}$ terminal desired covariance (green), optimized covariance (black), and mean (yellow).
• Figure 5: $\bm{L=27}$ optimized means (yellow), covariance ellipses (black), and $\bm{\Delta V}$ vectors (purple).