Sub-Optimal Fast Fourier Series Approximation for Initial Trajectory Design

TL;DR

This work extends the Finite Fourier Series Shape-Based trajectory design to generate sub-optimal yet feasible initial trajectories by introducing a time-of-flight penalty in the NLP objective. The approach preserves the FFS SB representation of the state and leverages an EoM residual to balance trajectory fidelity with shorter flight times, exploring orbit raising and rendezvous scenarios. Through comparisons with GPOPS-II optimal control solutions, the study demonstrates that penalized ToF trajectories can substantially reduce ToF while maintaining physical feasibility, albeit with trade-offs in EoM accuracy and thrust profiles. The results highlight the method's potential as a fast, initialization-friendly alternative to direct solvers, capable of producing near-optimal results in complex dynamical campaigns and guiding subsequent optimization steps.

Abstract

The Finite Fourier Series (FFS) Shape-Based (SB) trajectory approximation method has been used to rapidly generate initial trajectories that satisfy the dynamics, trajectory boundary conditions, and limitation on maximum thrust acceleration. The FFS SB approach solves a nonlinear programming problem (NLP) in searching for feasible trajectories. This paper extends the development of the FFS SB approach to generate sub optimal solutions. Specifically, the objective function of the NLP problem is modified to include also a measure for the time of flight. Numerical results presented in this paper show several solutions that differ from those of the original FFS SB ones. The sub-optimal trajectories generated using a time of flight minimization are shown to be physically feasible trajectories and potential candidates for direct solvers.

Figures (14)

• Figure 1: FFS Trajectory: Orbit Raising
• Figure 2: Thrust Acceleration: Orbit Raising
• Figure 3: Optimal EoM vs. Weight
• Figure 4: Optimal ToF vs Weight
• Figure 5: FFS Trajectories: Orbit Raising