Taylor polynomial-based constrained solver for fuel-optimal low-thrust trajectory optimisation

Abstract

This paper presents differential algebra-based differential dynamic programming (DADDy), a publicly available C++ framework for constrained, fuel-optimal low-thrust trajectory optimisation. The method uses differential algebra (DA) for two purposes: automatic differentiation and high-order Taylor expansions of the dynamics. These expansions replace many expensive numerical propagations with polynomial evaluations, reducing computational effort while preserving solution quality. The framework combines two complementary modules. First, a differential dynamic programming (DDP)/iterative linear-quadratic regulator (iLQR) stage computes an almost-feasible trajectory from imperfect initial guesses. Second, a polynomial-accelerated Newton stage enforces full feasibility with fast local convergence. Equality and inequality constraints are handled through an augmented Lagrangian formulation, and a pseudo-Huber homotopy is used to improve robustness for fuel-optimal objectives. The solver is evaluated on benchmark transfers in Sun-centred, Earth-Moon, and Earth-centred dynamical environments. Across these cases, the most robust configuration (iLQRDyn) converged systematically and reduced run time by 70% (Sun-centred), 51-88% (Earth-Moon), and 41-55% (Earth-centred) relative to the corresponding baseline. When convergent, the DDP-based variants are faster still. Overall, the results show that DA-based acceleration can substantially improve practical efficiency while retaining robust convergence behaviour on the tested benchmark set.