Perturbed feedback linearization of attitude dynamics

Abstract

The paper introduces a theoretical foundation of the perturbed feedback linearization methodology for realization of linear spacecraft attitude deviation dynamics. The approach is based on nonuniqueness representation of under- determined linear algebraic equations solution via nullspace parametrization. A prescribed stable linear second order time-invariant ordinary differential equation in a spacecraft attitude deviation norm measure is evaluated along the solution trajectories of the spacecraft equations of motion, yielding a linear relation in the control variables. Generalized inversion of the relation results in a control law that consists of particular and auxiliary parts. The particular part acts on the range space of the controls coefficient row vector, and it works to drive the spacecraft attitude variables in order to nullify the attitude deviation norm measure. The auxiliary part acts on the complementary orthogonal subspace, and provides the necessary spacecraft internal stability. The null-control vector in the auxiliary part is projected onto the controls coefficient nullspace by a null projection matrix, and is designed to yield perturbed feedback linearization of the spacecraft internal dynamics. The feedback control design utilizes the concept of damped generalized inverse to limit the growth of the Moore-Penrose generalized inverse. The control law yields globally uniformly ultimately bounded trajectory tracking errors, and it reveals a tradeoff between trajectory tracking accuracy and damped generalized inverse stability.