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Advances in Cislunar Periodic Solutions via Taylor Polynomial Maps

Mohammed Atallah, Simone Servadio

Abstract

In this paper, novel approaches are developed to explore the dynamics of motion in periodic orbits near libration points in cislunar space using the Differential Algebra (DA) framework. The Circular Restricted Three-Body Problem (CR3BP) models the motion, with initial states derived numerically via differential correction. Periodic orbit families are computed using the Pseudo-Arclength Continuation (PAC) method and fitted. Two newly developed polynomial regression models (PRMs) express initial states as functions of predefined parameters and are used in the DA framework to evaluate propagated states. The initial states, expressed via PRM, are propagated in the DA framework using the fourth-order Runge-Kutta (RK4) method. The resultant polynomials of both PRM and DA are employed to develop a control law that shows significantly reduced control effort compared to the traditional tracking control law, demonstrating their potential for cislunar space applications, particularly those requiring computationally inexpensive low-energy transfers.

Advances in Cislunar Periodic Solutions via Taylor Polynomial Maps

Abstract

In this paper, novel approaches are developed to explore the dynamics of motion in periodic orbits near libration points in cislunar space using the Differential Algebra (DA) framework. The Circular Restricted Three-Body Problem (CR3BP) models the motion, with initial states derived numerically via differential correction. Periodic orbit families are computed using the Pseudo-Arclength Continuation (PAC) method and fitted. Two newly developed polynomial regression models (PRMs) express initial states as functions of predefined parameters and are used in the DA framework to evaluate propagated states. The initial states, expressed via PRM, are propagated in the DA framework using the fourth-order Runge-Kutta (RK4) method. The resultant polynomials of both PRM and DA are employed to develop a control law that shows significantly reduced control effort compared to the traditional tracking control law, demonstrating their potential for cislunar space applications, particularly those requiring computationally inexpensive low-energy transfers.
Paper Structure (23 sections, 17 equations, 19 figures)

This paper contains 23 sections, 17 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Mathematical Model of the CR3BP
  3. Periodic Orbits Near $\mathcal{L}_1$ and $\mathcal{L}_2$
  4. Linearized Equations of Motion
  5. Differential Correction for Computing Initial States
  6. Computing Members across the Orbit Families
  7. LO Families
  8. HO and NRHO Families
  9. Overview of Differential Algebra
  10. Polynomial Regression Model
  11. Global Polynomial Regression Model
  12. Local Polynomial Regression Model
  13. Periodic Orbit Representation Using Differential Algebra
  14. Propagation with Respect to Normalized Time
  15. Numerical Simulations
  16. ...and 8 more sections

Figures (19)

  • Figure 1: The members of the LO Families.
  • Figure 2: The members of the HO families.
  • Figure 3: HO and NRHO families near L1 and L2.
  • Figure 4: Implementing operations on real numbers and functions by transforming them to floating point numbers or DA series.
  • Figure 5: Propagating the states in the DA framework.
  • ...and 14 more figures