Acquisition of a Desired Pure-Spin Condition for a Magnetically Actuated Spacecraft

A rigorous proof of global exponential stability is derived for a magnetic control law that drives a rigid satellite toward a pure-spin condition around a prescribed principal axis of inertia with a desired angular rate. The proof represents an extension and a generalization of a method proposed by two of the authors of the present note for demonstrating global asymptotic stability for a B-dot-like control law that detumbles a spacecraft to rest by means of magnetic actuators only. The proof of stability in the case of acquisition of a non-zero desired angular rate pure spin state is derived in terms of robustness of the global exponential stability of a nominal system by means of generalized exponential asymptotic stability in variations (GEASV) tools. To this aim, the error dynamics equation is first derived in the classical form of a nominal system perturbed by a vanishing perturbation term. Then, after proving the generalized exponential stability for the nominal system, such result is extended to the perturbed system. As a further contribution, an approach for the choice of the control law gain is proposed to the present application, thus allowing to perform the acquisition of the desired pure-spin condition in quasi-minimum time from arbitrary initial tumbling conditions. Stability and performance of the approach are extensively tested by means of numerical simulation.