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 rotation 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 alignment error between the target and the attainable attitude. The derivation of kinematic equations, describing the time evolution of non-nominal rotation 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.