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Advancing Averaged Primer Vector Theory with Bang-Bang Control and Eclipsing

Noah Lifset, Ryan P. Russell

TL;DR

The paper addresses the challenge of efficiently designing minimum-fuel, low-thrust, many-revolution trajectories by advancing averaged primer-vector dynamics to include eclipsing, bang-bang control, and perturbations. It introduces a multi-arc averaging framework that uses the Leibniz integral rule to naturally incorporate co-state jumps due to eclipsing, derives a switching-function-based bound that limits thrust arcs per revolution, and fixes a singularity in the averaged eclipsing constraint. Variational equations are provided for fast STM computation, enabling accurate targeting and optimization within an augmented state framework. Validation against unaveraged dynamics in 48-revolution and 486-revolution GTO→GEO transfers demonstrates the averaged model closely tracks the full dynamics while offering substantial computational efficiency, highlighting its practical impact for mission-design trade studies and continuation- or optimization workflows. The work thus enhances the applicability and reliability of averaged dynamics for complex, long-duration low-thrust missions.

Abstract

Low-thrust, many-revolution spacecraft trajectories are increasingly required for mission design due to the efficiency and reliability of electric propulsion technology. Primer vector theory using averaged dynamics is well suited for such applications, but is difficult to implement in a way that maintains both optimality and computational efficiency. An improved model is presented that combines advances from several past works into a general and practical formulation for minimum-fuel, perturbed Keplerian dynamics. The model maintains computational efficiency of dynamics averaging with optimal handling of the eclipsing constraint and bang-bang control through the use of the Leibniz integral rule for multi-arc averaging. A subtle, but important singularity arising from the averaged eclipsing constraint is identified and fixed. A maximum number of six switching function roots per revolution is established within the averaged dynamics. This new theoretical insight provides a practical upper-bound on the number of thrusting arcs required for any low-thrust optimization problem. Variational equations are provided for fast and accurate calculation of the state transition matrix for use in targeting and optimization. The dynamics include generic two-body perturbations and an expanded state to allow for sensitivity calculations with respect to launch date and flight time. A 48-revolution GTO to GEO transfer is used to directly compare optimal averaged and unaveraged trajectories. The capabilities of averaged dynamics are then demonstrated with an optimal 486-revolution GTO to GEO minimum fuel transfer.

Advancing Averaged Primer Vector Theory with Bang-Bang Control and Eclipsing

TL;DR

The paper addresses the challenge of efficiently designing minimum-fuel, low-thrust, many-revolution trajectories by advancing averaged primer-vector dynamics to include eclipsing, bang-bang control, and perturbations. It introduces a multi-arc averaging framework that uses the Leibniz integral rule to naturally incorporate co-state jumps due to eclipsing, derives a switching-function-based bound that limits thrust arcs per revolution, and fixes a singularity in the averaged eclipsing constraint. Variational equations are provided for fast STM computation, enabling accurate targeting and optimization within an augmented state framework. Validation against unaveraged dynamics in 48-revolution and 486-revolution GTO→GEO transfers demonstrates the averaged model closely tracks the full dynamics while offering substantial computational efficiency, highlighting its practical impact for mission-design trade studies and continuation- or optimization workflows. The work thus enhances the applicability and reliability of averaged dynamics for complex, long-duration low-thrust missions.

Abstract

Low-thrust, many-revolution spacecraft trajectories are increasingly required for mission design due to the efficiency and reliability of electric propulsion technology. Primer vector theory using averaged dynamics is well suited for such applications, but is difficult to implement in a way that maintains both optimality and computational efficiency. An improved model is presented that combines advances from several past works into a general and practical formulation for minimum-fuel, perturbed Keplerian dynamics. The model maintains computational efficiency of dynamics averaging with optimal handling of the eclipsing constraint and bang-bang control through the use of the Leibniz integral rule for multi-arc averaging. A subtle, but important singularity arising from the averaged eclipsing constraint is identified and fixed. A maximum number of six switching function roots per revolution is established within the averaged dynamics. This new theoretical insight provides a practical upper-bound on the number of thrusting arcs required for any low-thrust optimization problem. Variational equations are provided for fast and accurate calculation of the state transition matrix for use in targeting and optimization. The dynamics include generic two-body perturbations and an expanded state to allow for sensitivity calculations with respect to launch date and flight time. A 48-revolution GTO to GEO transfer is used to directly compare optimal averaged and unaveraged trajectories. The capabilities of averaged dynamics are then demonstrated with an optimal 486-revolution GTO to GEO minimum fuel transfer.
Paper Structure (12 sections, 43 equations, 16 figures, 5 tables, 1 algorithm)

This paper contains 12 sections, 43 equations, 16 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Optimal Control
  3. Averaged Dynamics
  4. Basic method
  5. Multi-arc averaging
  6. Eclipsing singularity
  7. Variational Equations
  8. Results
  9. Approximate Unaveraged Dynamics
  10. 48-Revolution GTO to GEO example
  11. 486-Revolution GTO to GEO example
  12. Conclusion

Figures (16)

  • Figure 1: Angles used to calculate eclipsing function in conical shadow model.
  • Figure 2: Switching function and switching polynomial over averaging period at (a: left) beginning of transfer, (b: middle) middle of transfer, and (c: right) end of transfer.
  • Figure 3: Diagram of multi-arc breakdown for a hypothetical spacecraft period.
  • Figure 4: Contributions to averaged co-state dynamics in a hypothetical multi-arc period. The equation shown is simplified with separated integral and Leibniz boundary terms.
  • Figure 5: Updated minimum thrust value inside eclipse.
  • ...and 11 more figures