Hamiltonian modelling of relative motion.

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.
• 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.