Geometric methods have become increasingly accepted in computational multibody system (MBS) dynamics. This includes the kinematic and dynamic modeling as well as the time integration of the equations of motion. In particular, the observation that rigid body motions form a Lie group motivated the application of Lie group integration schemes, such as the Munthe-Kaas method. Also established vector space integration schemes tailored for structural and MBS dynamics were adopted to the Lie group setting, such as the generalized α integration method. Common to all is the use of coordinate mappings on the Lie group $SE(3)$ of Euclidean motions. In terms of canonical coordinates (screw coordinates), this is the exponential mapping. Rigid body velocities (twists) are determined by its right-trivialized differential, denoted dexp. These concepts have, however, not yet been discussed in compact and concise form, which is the contribution of this paper with particular focus on the computational aspects. Rigid body motions can also be represented by dual quaternions, that form the Lie group $Sp̂(1)$, and the corresponding dynamics formulations have recently found a renewed attention. The relevant coordinate mappings for dual quaternions are presented and related to the $SE(3)$ representation. This relation gives rise to a novel closed form of the dexp mapping on $SE(3)$. In addition to the canonical parameterization via the exponential mapping, the noncanonical parameterization via the Cayley mapping is presented.

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