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Low Thrust Trajectory Design Using A Semi-Analytic Approach

Madhusudan Vijayakumar, Ossama Abdelkhalik

TL;DR

This work introduces a semi-analytic framework for designing low-thrust trajectories by extending the Clohessy–Wiltshire Hill's equations to a 3D, linearized form with a constant thrust magnitude. Out-of-plane motion is solved independently via Laplace transforms, while in-plane motion is coupled yet admits closed-form solutions, enabling rapid generation of trajectory guesses. The trajectory is partitioned into segments, with NLP adjustments to thrust directions and timing, providing efficient initial guesses for both orbit insertion and rendezvous tasks and enabling validation against high-fidelity propagations. Case studies demonstrate the method’s versatility across 2D/3D transfers, including Earth-centric and interplanetary missions, and underscore its practicality as a fast, reliable precursor to more detailed optimization tools.

Abstract

Space missions that use low-thrust propulsion technology are becoming increasingly popular since they utilize propellant more efficiently and thus reduce mission costs. However, optimizing continuous-thrust trajectories is complex, time-consuming, and extremely sensitive to initial guesses. Hence, generating approximate trajectories that can be used as reliable initial guesses in trajectory generators is essential. This paper presents a semi-analytic approach for designing planar and three-dimensional trajectories using Hills equations. The spacecraft is assumed to be acted upon by a constant thrust acceleration magnitude. The proposed equations are employed in a Nonlinear Programming Problem (NLP) solver to obtain the thrust directions. Their applicability is tested for various design scenarios like orbit raising, orbit insertion, and rendezvous. The trajectory solutions are then validated as initial guesses in high-fidelity optimal control tools. The usefulness of this method lies in the preliminary stages of low-thrust mission design, where speed and reliability are key.

Low Thrust Trajectory Design Using A Semi-Analytic Approach

TL;DR

This work introduces a semi-analytic framework for designing low-thrust trajectories by extending the Clohessy–Wiltshire Hill's equations to a 3D, linearized form with a constant thrust magnitude. Out-of-plane motion is solved independently via Laplace transforms, while in-plane motion is coupled yet admits closed-form solutions, enabling rapid generation of trajectory guesses. The trajectory is partitioned into segments, with NLP adjustments to thrust directions and timing, providing efficient initial guesses for both orbit insertion and rendezvous tasks and enabling validation against high-fidelity propagations. Case studies demonstrate the method’s versatility across 2D/3D transfers, including Earth-centric and interplanetary missions, and underscore its practicality as a fast, reliable precursor to more detailed optimization tools.

Abstract

Space missions that use low-thrust propulsion technology are becoming increasingly popular since they utilize propellant more efficiently and thus reduce mission costs. However, optimizing continuous-thrust trajectories is complex, time-consuming, and extremely sensitive to initial guesses. Hence, generating approximate trajectories that can be used as reliable initial guesses in trajectory generators is essential. This paper presents a semi-analytic approach for designing planar and three-dimensional trajectories using Hills equations. The spacecraft is assumed to be acted upon by a constant thrust acceleration magnitude. The proposed equations are employed in a Nonlinear Programming Problem (NLP) solver to obtain the thrust directions. Their applicability is tested for various design scenarios like orbit raising, orbit insertion, and rendezvous. The trajectory solutions are then validated as initial guesses in high-fidelity optimal control tools. The usefulness of this method lies in the preliminary stages of low-thrust mission design, where speed and reliability are key.
Paper Structure (10 sections, 17 equations, 18 figures, 8 tables)

This paper contains 10 sections, 17 equations, 18 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Low Thrust Analytic Approximation of Hill's Equations
  3. Trajectory Design
  4. Case Studies
  5. Case 1: Coplanar Transfer LEO-to-GEO (Orbit Insertion)
  6. Case 2: Coplanar Transfer LEO-to-MEO (Rendezvous)
  7. Case 3: 3D Transfer LEO-to-GEO (Orbit Insertion)
  8. Case 4: 3D Transfer Earth-to-Mars (Rendezvous)
  9. Conclusion
  10. APPENDIX-A

Figures (18)

  • Figure 1: Coordinate Frames
  • Figure 2: Hills Frame
  • Figure 3: Hill's vs Keplerian Motion - Circular
  • Figure 4: Hill's vs Keplerian Motion - Eccentric
  • Figure 5: Case 1: Trajectory (ECI Frame)
  • ...and 13 more figures