Unified kinematic framework for a non-nominal Euler axis/angle rotation

In this paper, kinematics equations of attitude parameters are derived for cases where the Euler rotation theorem cannot be applied and the single ro- tation that takes an initial reference frame to a target reference frame cannot be performed. It is the case when the nominal rotation is not allowed along a prescribed direction in space, namely an “underactuation” direction. As a matter of fact, a non-nominal maneuver planning scheme, expressed in terms of Euler axis/angle parameters, is admitted for the minimization of the align- ment error between the target and the attainable attitude. The derivation of kinematic equations, describing the time evolution of non-nominal rota- tion parameters, is performed by means of rigorous algebraic manipulations within a unified framework, where the underactuation direction is prescribed in either the moving frame or the target frame.