Almost-global exponential state-feedback stabilization of an underactuated rigid-body in 3D

The classical third-order model describing the rotation of a rigid body in 3D is here considered, with two torques being applied as controls around two of the three main rotation inertial axes. By using a new method, that we call `Quadratic Immersion´, we show that the global stabilization can be attained with a static state-feedback - i.e. a static loop making the zero a `practical´ globally asymptotically stable point for the closed-loop system. Here `practical´ adds up to a slight modification of the usual notion of asymptotic stability, where the point is required to lie in just the closure of the closed-loop system domain - and in particular not to be an equilibrium of the closed-loop system - and the attraction set is given by IR3 unless a zero-measure set (`almost global´-stability). Such stabilization is exponential, with rate and max overlength that can be arbitrarily and simultaneously set in advance by the designer. Numerical simulations of the method are reported, showing the exponential-stabilizing behavior of the closed loop system in various situations.