Stabilization of Underactuated Spacecraft Dynamics via Singularly Perturbed Feedback Linearization

The concept of control by feedback linearization isextended to underactuated spacecrafts, by singularly perturbing thenonrealizable linear system resulting from the feedback linearizingtransformation, and by utilizing the control coefficient pseudoinversedesign methodology that has been recently developed. Theunderactuated Euler’s system of dynamical equations for thespacecraft is partitioned into actuated and unactuated subsystems, anda function of the angular velocities about the unactuated body axes isused to prescribe a desired second-order dynamics for the unactuatedsubsystem. The evaluation of this dynamics along the trajectoriesdefined by the underactuated Euler’s dynamical equations results in arelation that is pointwise-linear in the control variables, and acondition is derived based on this relation to assess the realizability ofthe desired unactuated dynamics by checking the rank of the involvedcontrol coefficient Jacobian. The control variables are solved byutilizing the Moore-Penrose pseudoinverse of the control coefficient,resulting in a control law that is composed of particular and auxiliaryparts. The particular part works to realize the desired linearunactuated dynamics, and the pseudo-control vector in the auxiliarypart is chosen to yield a singularly perturbed feedback linearizationfor the actuated subsystem. The singularity avoidance problem that isrelated to using the control coefficient pseudoinverse is solved bymodifying the pseudoinverse definition in the vicinity of thesingularity, and the large control effort reduction problem is solvedby deactivating the particular part of the control law whenever theangular velocities about the actuated axes become high. The controllaw yields a global asymptotic stability of the origin for the closedloop system when the spacecraft is equipped with two independentgas jet actuators. We show that perturbed feedback linearization is notapplicable when the degree of actuation is one.