Reduced transfer equations of ball-and-socket joint elements incorporated with Euler parameters

The reduced multibody system transfer matrix method is a completely recursive method utilizing joint coordinates and applicable for evaluating the generalized accelerations of a multibody system at any given moment, provided that the generalized coordinates and velocities are known. For an open-loop multi-rigid-body system, the generalized coordinates of the system are composed of the generalized relative coordinates of the joint elements. Typically, for each joint element, the dimension of the generalized relative coordinates is equal to its relative motion degrees of freedom, leading to minimum dimension of the generalized coordinates of the system, which is equal to the degrees of freedom of the system. However, this may result in singularity for a ball-and-socket joint element when evaluating its generalized accelerations if any triad of Euler angles is utilized as its generalized relative coordinates. The tetrad Euler parameters are an alternative to Euler angles to resolve such a singular problem, which is a common practice in the dynamics approaches using body coordinates as generalized coordinates; nevertheless, it has not been observed in the completely recursive methods with joint coordinates. In this paper, the self-constraint equations of Euler parameters are taken into account to establish the corresponding reduced transfer equations characterized by a symmetric generalized inertial matrix, which are in completely recursive form. Fundamental numerical stability analyses are conducted via condition numbers of corresponding matrices, demonstrating that employing Euler parameters to describe the relative kinematics of a ball-and-socket joint element enhances numerical stability compared to Euler angles.