[Paper Review] Towards a High Fidelity Direct Transcription Method for Optimisation of Low-Thrust Trajectories

Chit Hong Yam, Dario Izzo+1 more|arXiv (Cornell University)|Apr 26, 2010

Spacecraft Dynamics and Control10 references19 citations

TL;DR

This paper proposes a high-fidelity direct transcription method for low-thrust trajectory optimization by replacing impulsive burns with continuous thrust propagation and introducing a Sundman transformation to enable on-the-fly mesh adaptation. The method improves trajectory accuracy and optimality by using Taylor integration for efficient numerical propagation and dynamically refining segment durations based on radial distance, resulting in higher final mass and better dynamical fidelity compared to traditional Sims-Flanagan approaches.

ABSTRACT

We build upon some new ideas in direct transcription methods developed within the Advanced Concepts Team to introduce two improvements to the Sims-Flanagan transcription for low-thrust trajectories. The obtained new algorithm is able to produce an operational trajectory accounting for the real spacecraft dynamics and adapting the segment duration on-line improving the final trajectory optimality.

Motivation & Objective

• To improve the accuracy of the Sims-Flanagan direct transcription method for low-thrust trajectory optimization by replacing impulsive maneuvers with continuous thrust arcs.
• To reduce computational overhead in continuous thrust propagation by employing Taylor integration instead of Runge-Kutta-Fehlberg schemes.
• To enable on-the-fly time-mesh adaptation during optimization by applying the Sundman transformation, which clusters segments near the central body where dynamics are faster.
• To develop a unified trajectory optimization framework suitable for both preliminary and operational mission design phases.
• To validate the method’s superiority in final mass and dynamical fidelity through a Mercury-bound mission case study.

Proposed method

• Replace impulsive $Δ V$ maneuvers in the Sims-Flanagan model with continuous thrust arcs, where thrust magnitude and direction are optimized as constant values per segment.
• Use numerical integration of the full two-body equations of motion with thrust, including mass propagation via the rocket equation, to model real spacecraft dynamics.
• Implement Taylor integration methods to reduce computational cost compared to standard Runge-Kutta-Fehlberg schemes, improving efficiency by nearly an order of magnitude.
• Apply the Sundman transformation to reparameterize time, making the independent variable $ s $ such that $ dt/ds = r $, leading to denser segment distribution near the central body.
• Introduce an additional constraint to match mission duration via $ T_f - T_0 = \int_{s_0}^{s_f} r \, ds $, enabling dynamic mesh adaptation without increasing problem dimensionality.
• Optimize both trajectory and mesh parameters simultaneously using sequential quadratic programming (SQP) via the SNOPT solver, maintaining low-dimensional problem structure.

Experimental results

Research questions

• RQ1 Can continuous thrust modeling improve the fidelity of low-thrust trajectory optimization compared to impulsive $Δ V$ transcription?
• RQ2 How does Taylor integration compare to Runge-Kutta-Fehlberg in terms of computational efficiency for continuous thrust propagation?
• RQ3 Can the Sundman transformation enable automatic, on-line mesh adaptation in direct transcription methods without increasing problem complexity?
• RQ4 Does adaptive meshing based on radial distance lead to higher final spacecraft mass and improved trajectory optimality?
• RQ5 Can the proposed method serve effectively across both preliminary and operational mission design phases?

Key findings

• The continuous thrust time-space method with Taylor integration achieved higher dynamical fidelity than the impulsive model, with a 3.2% higher final mass in the Mercury mission example.
• The $ s $-space method with Sundman transformation automatically concentrated segment intervals near Mercury, where orbital speed is highest, improving resolution in critical regions.
• Final mass in the $ s $-space method was higher than in the time-space method (3.2% improvement), demonstrating the benefit of adaptive meshing for control and fuel efficiency.
• The use of Taylor integration reduced computational cost by nearly an order of magnitude compared to Runge-Kutta-Fehlberg, making continuous thrust propagation feasible for optimization.
• The proposed method successfully balanced high fidelity and computational efficiency, enabling accurate trajectory design suitable for operational use after adding perturbations.
• The framework demonstrated versatility by supporting both preliminary design (low-dimensional) and operational design (high-fidelity) phases within a single unified approach.