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

TL;DR

The paper presents DADDy, a DA-based solver for constrained, fuel-optimal low-thrust trajectory optimization. It couples a robust DDP/iLQR stage with a polynomial Newton polishing step, using differential algebra to perform automatic differentiation and high-order Taylor expansions of the dynamics, thereby accelerating repeated evaluations. Constraints are handled via an Augmented Lagrangian scheme, while a pseudo-Huber fuel-cost surrogate and homotopy guide the optimization toward fuel-optimal solutions; a DA-accelerated Newton method rapidly enforces full feasibility. Across Sun-centered, Earth–Moon CR3BP, and Earth-centered transfers, DADDy achieves comparable accuracy to state-of-the-art methods but with substantial runtime reductions (up to ~70%), aided by a dynamic polynomial forward pass and efficient linear-algebra updates. The results demonstrate a favorable robustness-efficiency trade-off and establish DA as a key enabler for fast, accurate second-order optimal control in astrodynamics, with the solver publicly available for broader use.

Abstract

This paper presents the differential algebra-based differential dynamic programming (DADDy) solver, a publicly available C++ framework for constrained, fuel-optimal low-thrust trajectory optimization. The method exploits differential algebra (DA) to perform automatic differentiation and provides high-order Taylor polynomial expansions of the dynamics. These expansions replace repeated numerical propagation with polynomial evaluations, significantly reducing computational cost while maintaining accuracy. The solver combines two complementary modules: a fast Differential Dynamic Programming or iterative Linear Quadratic Regulator (DDP/iLQR) scheme that generates an almost-feasible trajectory from arbitrary initial guesses, and a polynomial-based Newton solver that enforces full feasibility with quadratic convergence. The solver accommodates equality and inequality constraints efficiently, while a pseudo-Huber cost function and homotopy continuation enhance convergence robustness for fuel-optimal objectives. The performances of the DADDy solver are assessed through several benchmark cases, including Sun-centered, Earth-Moon, and Earth-centered transfers. Results show that the solver achieves accuracy comparable to state-of-the-art methods while providing substantial computational savings. The most robust configuration (iLQRDyn) converged in all cases, reducing run times by 70% for Sun-centered, 51-88% for Earth-Moon, and 41-55% for Earth-centered problems. When convergence is achieved, the DDP variant attains even faster solutions. These results demonstrate that DA enables a favorable trade-off between robustness and efficiency in second-order optimal control.

Figures (13)

• Figure 1: Summary flow chart of the DADDy solver.
• Figure 2: Solution to the Earth--Mars transfer.
• Figure 3: Earth--Mars transfer performance metrics for different values of $\varepsilon_{\rm DA}$.
• Figure 4: Earth-Mars transfer performance metrics.
• Figure 5: Proportion of dynamics approximations performed by the iLQRDyn method during the Earth--Mars transfer.