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arxiv: 2502.00398 · v3 · submitted 2025-02-01 · 🧮 math.OC

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

Thomas Caleb , Roberto Armellin , Spencer Boone , St\'ephanie Lizy-Destrez This is my paper

Pith reviewed 2026-05-23 03:44 UTC · model grok-4.3

classification 🧮 math.OC
keywords differential algebralow-thrust trajectory optimizationfuel-optimal controldifferential dynamic programmingTaylor polynomial expansionsconstrained optimizationaugmented Lagrangian
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The pith

High-order Taylor expansions from differential algebra replace numerical propagations to accelerate constrained fuel-optimal low-thrust trajectory optimization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces DADDy, a solver framework that applies differential algebra to produce high-order Taylor expansions of the spacecraft dynamics. These expansions substitute faster polynomial evaluations for repeated numerical integrations inside the optimization loop. The approach first applies differential dynamic programming to generate an almost-feasible trajectory from imperfect guesses, then uses a polynomial-accelerated Newton stage to reach full feasibility and optimality. Constraints are managed through an augmented Lagrangian, and a pseudo-Huber homotopy aids robustness for fuel-minimizing objectives. If the method holds, trajectory designers can complete more iterations or explore wider parameter spaces in less time without loss of solution quality on the tested classes of transfers.

Core claim

The paper establishes that differential algebra supplies both automatic differentiation and high-order Taylor expansions of the dynamics that can replace many expensive numerical propagations inside a hybrid solver. The first stage uses differential dynamic programming or iterative linear-quadratic regulation to produce an almost-feasible trajectory; the second stage applies a polynomial-accelerated Newton method to enforce exact feasibility. Equality and inequality constraints are incorporated via augmented Lagrangian, while a pseudo-Huber homotopy improves behavior on fuel-optimal problems. On benchmark transfers in Sun-centred, Earth-Moon, and Earth-centred environments the resulting iLQ0

What carries the argument

High-order Taylor polynomial expansions of the dynamics obtained through differential algebra, which substitute polynomial evaluations for numerical propagations inside both the differential dynamic programming stage and the subsequent Newton refinement.

If this is right

  • The iLQRDyn configuration converges systematically across all three dynamical environments tested.
  • Runtime reductions reach 70 percent in Sun-centred cases, 51-88 percent in Earth-Moon cases, and 41-55 percent in Earth-centred cases relative to the baseline.
  • DDP-based variants deliver further speed gains whenever they converge.
  • The augmented Lagrangian formulation successfully enforces both equality and inequality constraints.
  • The pseudo-Huber homotopy increases robustness specifically for fuel-optimal objectives.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same polynomial-acceleration pattern could be applied to other optimal-control problems in which repeated dynamic evaluations dominate run time.
  • Adaptive selection of expansion order during iterations might trade accuracy for speed in different mission phases.
  • The framework could support faster sensitivity studies when initial conditions or target orbits vary.

Load-bearing premise

The Taylor expansions of the dynamics remain accurate enough throughout the iterations that the final solutions match or exceed the quality obtained by the baseline numerical solver.

What would settle it

A benchmark transfer in which the DADDy solver converges to a trajectory whose fuel cost is measurably higher or whose constraints are more violated than the solution returned by the corresponding baseline numerical solver.

Figures

Figures reproduced from arXiv: 2502.00398 by Roberto Armellin, Spencer Boone, St\'ephanie Lizy-Destrez, Thomas Caleb.

Figure 9
Figure 9. Figure 9: The test cases are numbered as follows: (1) halo-to-halo, (2) NRHO-to-DRO, and (3) [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces DADDy, a publicly available C++ framework for constrained fuel-optimal low-thrust trajectory optimization. It employs differential algebra for automatic differentiation and high-order Taylor expansions of the dynamics to accelerate DDP/iLQR and Newton stages within an augmented-Lagrangian formulation, using a pseudo-Huber homotopy for the fuel objective. Benchmarks across Sun-centred, Earth-Moon, and Earth-centred environments report systematic convergence with runtime reductions of 70%, 51-88%, and 41-55% relative to baseline solvers when using the iLQRDyn configuration.

Significance. If the accuracy of the high-order expansions holds throughout the iterations as claimed, the work provides a practical, reproducible acceleration technique for trajectory optimization that combines global search via DDP with fast local refinement. The public implementation and focus on fuel-optimal problems with mixed constraints represent a concrete contribution to applied optimal control in astrodynamics, particularly for cases where repeated numerical propagations dominate runtime.

major comments (2)
  1. [§4.2] §4.2, the description of the polynomial-accelerated Newton stage: the claim that the Taylor expansions replace numerical propagations while preserving solution quality requires explicit reporting of final constraint violation residuals and objective values compared to the baseline; without these quantities in the benchmark tables it is difficult to confirm that the accelerated solutions match or exceed baseline quality rather than merely converging faster to a nearby point.
  2. [Table 5] Table 5 (Earth-Moon cases), the iLQRDyn rows: the reported runtime reductions are given as ranges (51-88%); the manuscript should clarify whether these ranges reflect variation across initial guesses or across different transfer types, and whether the same number of iterations is used for the baseline comparison.
minor comments (3)
  1. [Eq. (12)] The notation for the pseudo-Huber parameter in Eq. (12) is introduced without an explicit statement of its default value or adaptation schedule during homotopy; a short table of hyper-parameters would improve reproducibility.
  2. [Figure 3] Figure 3 caption states 'convergence history' but the y-axis label is missing the units or scaling for the cost; this should be corrected for clarity.
  3. [§3] The abstract mentions 'the most robust configuration (iLQRDyn)' but the main text does not define the acronym on first use; add the expansion at the first occurrence in §3.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation for minor revision. The comments identify opportunities to strengthen the presentation of results, which we address point by point below.

read point-by-point responses

  1. Referee: [§4.2] §4.2, the description of the polynomial-accelerated Newton stage: the claim that the Taylor expansions replace numerical propagations while preserving solution quality requires explicit reporting of final constraint violation residuals and objective values compared to the baseline; without these quantities in the benchmark tables it is difficult to confirm that the accelerated solutions match or exceed baseline quality rather than merely converging faster to a nearby point.

    Authors: We agree that explicit reporting of final constraint violation residuals and objective values is necessary to substantiate the preservation of solution quality. In the revised manuscript we will extend the benchmark tables (including those in §4.2 and Table 5) with additional columns that list the final equality/inequality constraint residuals and the achieved objective values for both the baseline and all DADDy configurations, allowing direct numerical comparison. revision: yes

  2. Referee: [Table 5] Table 5 (Earth-Moon cases), the iLQRDyn rows: the reported runtime reductions are given as ranges (51-88%); the manuscript should clarify whether these ranges reflect variation across initial guesses or across different transfer types, and whether the same number of iterations is used for the baseline comparison.

    Authors: The reported ranges (51-88 %) for the iLQRDyn configuration in Table 5 arise from variation across the distinct Earth-Moon transfer types in the benchmark set; they do not reflect different initial guesses for the same transfer. All solver comparisons, baseline and accelerated alike, are performed under identical convergence tolerances rather than a fixed iteration budget. We will revise the table caption and the accompanying text in §4 to state these facts explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes an implemented numerical solver (DADDy) that substitutes high-order Taylor expansions for repeated numerical integrations inside DDP/iLQR and Newton iterations, with an augmented-Lagrangian constraint handler and pseudo-Huber homotopy. All performance claims are backed by explicit benchmark definitions, convergence tables, and runtime measurements on Sun-centred, Earth-Moon, and Earth-centred transfers; these results are obtained by running the supplied code on the stated problems rather than by any algebraic reduction of a fitted quantity to itself or by a load-bearing self-citation. No derivation step equates an output prediction to an input fit or renames an external result as a new theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central performance claim rests on the domain assumption that Taylor polynomial approximations of the dynamics are accurate enough to preserve optimization quality; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption High-order Taylor expansions of the spacecraft dynamics can replace numerical propagations without degrading the quality of the optimized trajectory.
    Invoked to justify replacing expensive numerical integrations with polynomial evaluations throughout the DDP and Newton stages.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Non-linear stochastic trajectory optimisation

    math.OC 2025-08 unverdicted novelty 7.0

    SODA uses differential algebra and adaptive Gaussian mixtures to solve chance-constrained nonlinear trajectory optimization problems for space missions with non-Gaussian uncertainties.

Reference graph

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