STABILIZATION OF UNDERACTUATED SPACECRAFT DYNAMICS VIA SINGULARLY PERTURBED FEEDBACK LINEARIZATION

The concept of control by feedback linearization is extended to underactuated spacecraft dynamics, by singularly perturbing the non-realizable linear system resulting from the feedback linearizing transformation, and by utilizing a controls coefficient generalized inversion design methodology. The underactuated Euler's system of dynamical equations for the spacecraft is partitioned into actuated and unactuated subsystems, and a function of the angular velocities about the unactuated body axes is used to prescribe a desired second-order dynamics for the unactuated subsystem. The evaluation of this dynamics along the trajectories defined by the underactuated Euler's dynamical equations results in a relation that is pointwise-linear in the control variables. A condition is derived based on this relation to assess the realizability of the desired unactuated dynamics by checking the rank of the involved controls coefficient Jacobian. The control variables are solved for by utilizing the Moore-Penrose generalized inverse of the controls coefficient, resulting in a control law that is composed of particular and auxiliary parts. The particular part works to realize the desired linear unactuated dynamics, and the pseudo-control vector in the auxiliary part is chosen to yield a singularly perturbed feedback linearization for the actuated subsystem. The singularity avoidance problem that is related to using the controls coefficient generalized inverse is solved by modifying the generalized inverse definition in the vicinity of the origin, and the large control effort reduction problem is solved by deactivating the particular part of the control law whenever the angular velocities about the actuated axes become high. The control law yields global Lyapanov stability of the origin and arbitrarily small uniform ultimate bounds for the closed loop trajectories when the spacecraft is equipped with two independent gas jet actuators. We show that singularly perturbed feedback linearization is not applicable when the degree of actuation is one.