Controllability of nonlinear time-varying systems: applications to spacecraft attitude control using magnetic actuation

Nonlinear controllability theory is applied to the time-varying attitude dynamics of a magnetically actuated spacecraft in a Keplerian orbit in the geomagnetic field. First, sufficient conditions for accessibility, strong accessibility and controllability of a general time-varying system are presented. These conditions involve application of Lie-algebraic rank conditions to the autonomous extended system obtained by augmenting the state of the original time-varying system by the time variable, and require the rank conditions to be checked only on the complement of a finite union of level sets of a finite number of smooth functions. At each point of each level set, it is sufficient to verify escape conditions involving Lie derivatives of the functions defining the level sets along linear combinations over smooth functions of vector fields in the accessibility algebra. These sufficient conditions are used 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 time derivative 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 nonrotating dipole approximation of the geomagnetic field are strongly accessible and controllable if the orbital plane does not coincide with the geomagnetic equatorial plane.