On the Numerical Violation of the Mathematical Constraint when using Unit Quaternions as Orientation Parametrization in Multibody Systems

Multibody systems composed by interconnected bodies incorporate constraints, either coming from the explicit formulation of kinematic joints or resulting from the parametrization of the orientation of the bodies by dependent coordinates, e.g., Euler parameters as a special choice of quaternions. Since the four Euler parameters are over-determined for the three degrees of freedom for the rotation of a body, a mathematical constraint has to be satisfied. This means that the unit length constraint is enforced explicitly by means of an algebraic constraint. The problem of numerical violation of such mathematical constraints concerning Euler parameterization is discussed within the present work.