Semi-Analytic Electrodynamic Tether Guidance Using Rotating Dipole Model of Earth's Magnetic Field

An electrodynamic tether provides a propellantless means of propulsion for spacecraft in Low Earth Orbit. Several proposed applications of the electrodynamic tether require a general change in the orbit, such as rendezvous for space surveillance or debris mitigation. This paper derives a current control law for electrodynamic tethers to achieve a specific general orbital change using a tilted, rotating dipole model of Earth's magnetic field. The control law is developed using a modified, semi-analytic derivation of the Gaussian Variation of Parameters equations. The semi-analytic method is used to solve the problem of integrating the perturbation equations with respect to time and true anomaly. A unique method was developed to account for the rotation of the magnetic field by taking advantage of the different time scales between Earth's rotation and the orbital motion of the spacecraft. Approximating the position of the dipole as fixed for a single orbital period allows for the integration of the Gaussian equations. Summing the changes in the orbital elements for each orbit over the entire maneuver provides a semi-analytic method that accounts for the magnetic field rotation. This semi-analytic method allows for rapid calculation of the open-loop guidance law while allowing for the added complexity of a tilted, rotating dipole as well as the changing orbital elements of the spacecraft during the maneuver. The solution is more accurate than the fixed dipole model while maintaining greater computing efficiency than numeric solutions.