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Hamiltonian modelling of relative motion.

N Jeremy Kasdin1, Pini Gurfil

  • 1Mechanical and Aerospace Engineering Department, Princeton University, Princeton, NJ 08544, USA. jkasdin,pgurfil@princeton.edu

Annals of the New York Academy of Sciences
|June 29, 2004
PubMed
Summary

This study introduces a Hamiltonian method for modeling spacecraft relative motion, yielding closed-form solutions. This approach simplifies complex dynamics, including orbital perturbations, using novel epicyclic elements.

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Area of Science:

  • Astrodynamics
  • Celestial Mechanics
  • Spacecraft Dynamics

Background:

  • Modeling relative spacecraft motion is crucial for rendezvous, proximity operations, and formation flying.
  • Existing models often struggle with high-order terms and orbital perturbations, limiting analytical solutions.
  • A need exists for robust methods that provide closed-form solutions for complex relative motion dynamics.

Purpose of the Study:

  • To develop a Hamiltonian approach for modeling relative spacecraft motion.
  • To derive canonical coordinates for relative state-space dynamics.
  • To obtain closed-form solutions for relative motion, incorporating high-order terms and perturbations.

Main Methods:

  • Derivation of canonical coordinates for relative state-space dynamics.

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  • Partitioning the Hamiltonian into linear and high-order terms.
  • Solving Hamilton-Jacobi equations for the linear part and introducing epicyclic elements.
  • Incorporating perturbations using a variation of parameters procedure.

    • Main Results:

    • A Hamiltonian formulation for relative spacecraft motion is presented.
    • Closed-form solutions are obtained for J(2-) and J(4-)invariant orbits.
    • Periodic high-order unperturbed relative motion solutions are derived using relative motion elements only.
    • The method effectively models high-order terms and orbital perturbations like Earth's oblateness.

    Conclusions:

    • The Hamiltonian approach provides a powerful framework for analyzing relative spacecraft motion.
    • Epicyclic elements offer new constants for understanding relative motion dynamics.
    • The derived closed-form solutions enhance the predictability and analysis of complex orbital scenarios.
    • This methodology facilitates accurate modeling for advanced space missions.