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Research Papers

# Coordinate Mappings for Rigid Body Motions

[+] Author and Article Information
Andreas Müller

Institute of Robotics,
Johannes Kepler University,
Linz 4040, Austria
e-mail: a.mueller@jku.at

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 5, 2016; final manuscript received August 30, 2016; published online December 2, 2016. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 12(2), 021010 (Dec 02, 2016) (10 pages) Paper No: CND-16-1320; doi: 10.1115/1.4034730 History: Received July 05, 2016; Revised August 30, 2016

## Abstract

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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Topics: Rotation , Screws

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## Figures

Fig. 1

(a) Rodrigues parameters c are obtained from Euler parameters via gnomonic projection, and (b) modified Rodrigues parameters a via stereographic projection

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