Dynamically Scaled Generalized Inversion for Asymptotic Stabilization of Underactuated Spacecraft Dynamics

Abstract

Abstract Novel concept of feedback linearization is introduced for smooth asymptotic stabilization of underactuated spacecraft equipped with one and two degrees of actuation. The concept is based on generalized inversion, and is aimed at asymptotically realizing a perturbation from the unrealizable feedback linearizing transformation. A desired stable second-order linear dynamics in a norm measure of the angular velocity components about the unactuated axes is prescribed. Evaluation of this dynamics along the vector field defined by the underactuated Euler's dynamical equations of angular motion results in a relation that is linear in the control variables. This relation is used to assess realizability of the desired unactuated dynamics, resulting in necessary and sufficient conditions for asymptotic stabilizability of underactuated spacecraft. Generalized inversion of the relation produces a control law that is composed of particular and auxiliary parts. The generalized inverse in the particular part is scaled by a dynamic factor that depends on the spacecraft angular velocity components about the spacecraft actuated axes, such that the generalized inverse converges uniformly to the standard Moore-Penrose generalized inverse as the transient response decays, resulting in asymptotic realization of the desired unactuated stable linear dynamics. The null-control vector in the auxiliary part of the control law is chosen for asymptotic stable perturbed feedback linearization of the actuated subsystem.