Controllability of spacecraft attitude under magnetic actuation

In this paper, we apply nonlinear controllability theory to the attitude dynamics of a magnetically actuated spacecraft in a Keplerian orbit in the geomagnetic field. The variation of the geomagnetic field along the orbit makes the dynamical equations time varying in nature. Hence we first present sufficient conditions for accessibility, strong accessibility and controllability of a general time-varying system. We apply these sufficient conditions to show that the attitude dynamics of a spacecraft actuated by three magnetic actuators and subjected to a general time-varying magnetic field are strongly accessible if the magnetic field and its first two time derivatives are linearly independent at every instant. In addition, if the magnetic field is periodic in time, then the attitude dynamics of the spacecraft are controllable. These results are used to show that the attitude dynamics of a spacecraft actuated by three magnetic actuators in a closed Keplerian orbit in a of the geomagnetic field are strongly accessible and controllable if the orbital plane contains neither the geomagnetic equator nor the geomagnetic poles.