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

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.

Copyright © 2017 by ASME
Topics: Rotation , Screws
Your Session has timed out. Please sign back in to continue.


Selig, J. M. , 2005, Geometric Fundamentals of Robotics (Monographs in Computer Science Series), Springer-Verlag, New York.
Shabana, A. A. , 2013, Dynamics of Multibody Systems, 4th ed., Cambridge University Press, New York.
Müller, A. , 2016, “ A Note on the Motion Representation and Configuration Update in Time Stepping Schemes for the Constrained Rigid Body,” BIT Numer. Math., 56(3), pp. 995–1015. [CrossRef]
Celledoni, E. , and Owren, B. , 1999, “ Lie Group Methods for Rigid Body Dynamics and Time Integration on Manifolds,” Comput. Methods Appl. Mech. Eng., 192(3–4), pp. 421–438. [CrossRef]
Iserles, A. , Munthe-Kaas, H. Z. , Nørsett, S. P. , and Zanna, A. , 2000, “ Lie-Group Methods,” Acta Numer., Vol. 9, pp. 215–365.
Krysl, P. , and Endres, L. , 2005, “ Explicit Newmark/Verlet Algorithm for Time Integration of the Rotational Dynamics of Rigid Bodies,” Int. J. Numer. Methods Eng., 62(15), pp. 2154–2177. [CrossRef]
Munthe-Kaas, H. , 1999, “ High Order Runge–Kutta Methods on Manifolds,” Appl. Numer. Math., 29(1), pp. 115–127. [CrossRef]
Owren, B. , and Marthinsen, A. , 1999, “ Runge–Kutta Methods Adapted to Manifolds and Based on Rigid Frames,” BIT, 39(1), pp. 116–142. [CrossRef]
Park, J. , and Chung, W. K. , 2005, “ Geometric Integration on Euclidean Group With Application to Articulated Multibody Systems,” IEEE Trans. Rob. Autom., 21(5), pp. 850–863. [CrossRef]
Simo, J. C. , and Wong, K. K. , 1991, “ Unconditionally Stable Algorithms for Rigid Body Dynamics That Exactly Preserve Energy and Momentum,” Int. J. Numer. Methods Eng., 31(1), pp. 19–52. [CrossRef]
Terze, Z. , Müller, A. , and Zlatar, D. , 2014, “ Lie-Group Integration Method for Constrained Multibody Systems in State Space,” Multibody Syst. Dyn., 34(3), pp. 275–305. [CrossRef]
Terze, Z. , Müller, A. , and Zlatar, D. , 2015, “ An Angular Momentum and Energy Conserving Lie-Group Integration Scheme for Rigid Body Rotational Dynamics Originating From Störmer-Verlet Algorithm,” ASME J. Comput. Nonlinear Dyn., 10(5), p. 051005. [CrossRef]
Darboux, G. , 1887, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitesimal, Vol. 4, Gautiers-Villars, Paris, France.
Sattinger, D. H. , and Weaver, O. L. , 1993, Lie Groups and Algebras With Applications to Physics, Geometry and Mechanics, Springer, New York.
Marsden, J. , 1997, Introduction to Mechanics and Symmetry, Springer-Verlag, New York.
Brüls, O. , Cardona, A. , and Arnold, M. , 2012, “ Lie Group Generalized-Alpha Time Integration of Constrained Flexible Multibody Systems,” Mech. Mach. Theory, 48, pp. 121–137. [CrossRef]
Müller, A. , and Terze, Z. , 2014, “ The Significance of the Configuration Space Lie Group for the Constraint Satisfaction in Numerical Time Integration of Multibody Systems,” Mech. Mach. Theory, 82, pp. 173–202. [CrossRef]
Altmann, S. L. , 1986, Rotations, Quaternions, and Double Groups, Oxford University Press, New York.
Borri, M. , Mello, F. , and Atluri, S. N. , 1990, “ Variational Approaches for Dynamics and Time-Finite-Elements: Numerical Studies,” Comput. Mech., 7(1), pp. 49–76. [CrossRef]
McCarthy, J. M. , 1990, Introduction to Theoretical Kinematics, The MIT Press, Cambridge, MA.
Géradin, M. , and Rixen, D. , 1995, “ Parametrization of Finite Rotations in Computational Dynamics: A Review,” Rev. Eur. Élém., 4(5–6), pp. 497–553.
Veldkamp, G. R. , 1976, “ On the Use of Dual Numbers, Vectors and Matrices in Instantaneous, Spatial Kinematics,” Mech. Mach. Theory, 11(2), pp. 141–156. [CrossRef]
Murray, R. M. , Li, Z. , and Sastry, S. S. , 1993, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL.
Helgason, S. , 1978, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, San Diego, CA.
Betsch, P. , and Siebert, R. , 2009, “ Rigid Body Dynamics in Terms of Quaternions: Hamiltonian Formulation and Conserving Numerical Integration,” Int. J. Numer. Methods Eng., 79(4), pp. 444–473. [CrossRef]
Siminovitch, D. , 1997, “ Rotations in NMR: Part I. Euler–Rodrigues Parameter and Quaternions,” Concepts Magn. Reson., 9(3), pp. 149–171. [CrossRef]
Müller, A. , 2010, “ Group Theoretical Approaches to Vector Parameterization of Rotations,” J. Geom. Symmetry Phys., 19, pp. 43–72.
Tsiotras, P. , Junkins, J. , and Schaub, H. , 1997, “ Higher Order Cayley Transforms With Applications to Attitude Representations,” J. Guid., Control, Dyn., 20(3), pp. 528–534. [CrossRef]
Milenkovic, V. , 1982, “ Coordinates Suitable for Angular Motion Synthesis in Robots,” Robot VI Conference, Detroit, MI, Society of Manufacturing Engineers, Dearborn, MI, Mar. 2–4, pp. 407–420.
Bauchau, O. A. , and Trainelli, L. , 2003, “ The Vectorial Parameterization of Rotation,” Nonlinear Dyn., 32(1), pp. 71–92. [CrossRef]
Bauchau, O. A. , 2011, Flexible Multibody Dynamics, Springer, Dortrecht, Heidelberg, New York.
Schaub, H. , and Junkins, J. L. , 1995, “ Stereographic Orientation Parameters for Attitude Dynamics: A Generalization of the Rodrigues Parameters,” J. Aeronautical Sci., 44(1), pp. 1–19.
Magnus, W. , 1954, “ On the Exponential Solution of Differential Equations for a Linear Operator,” Commun. Pure Appl. Math., 7(4), pp. 649–673. [CrossRef]
Ibrahimbegović, A. , Frey, F. , and Kožar, I. , 1995, “ Computational Aspects of Vector-Like Parametrization of Three-Dimensional Finite Rotations,” Int. J. Numer. Methods Eng., 38(21), pp. 3653–3673. [CrossRef]
Andrle, M. S. , and Crassidis, J. L. , 2013, “ Geometric Integration of Quaternions,” J. Guid., Control, Dyn., 36(6), pp. 1762–1767. [CrossRef]
Terze, Z. , Müller, A. , and Zlatar, D. , 2016, “ Singularity-Free Time Integration of Rotational Quaternions Using Non-Redundant Ordinary Differential Equations,” Multibody Syst. Dyn., 37(3), pp. 1–25.
Selig, J. M. , 2007, “ Cayley Maps for SE(3),” 12th IFToMM World Congress, Besancon, France, June 18–21.
Brodsky, V. , and Shoham, M. , 1999, “ Dual Numbers Representation of Rigid Body Dynamics,” Mech. Mach. Theory, 34(5), pp. 693–718. [CrossRef]
Cohen, A. , and Shoham, M. , 2015, “ Application of Hyper-Dual Numbers to Multibody Kinematics,” ASME J. Mech. Rob., 8(1), p. 011015.
Dimentberg, F. M. , 1965, The Screw Calculus and Its Application in Mechanics, Nauka, Moscow (Clearinghouse for Federal and Scientific Technical Information), Russia.
Rico Martinez, J. M. , and Duffy, J. , 1993, “ The Principle of Transference: History, Statement and Proof,” Mech. Mach. Theory, 28(1), pp. 165–177. [CrossRef]
Rooney, J. , 1975, “ On the Principle of Transference,” Fourth World Congress on the Theory of Machines and Mechanisms, Newcastle upon Tyne, England, Sept., pp. 1089–1094.
Husty, M. , and Schröcker, H.-P. , 2010, “ Algebraic Geometry and Kinematics,” Nonlinear Computational Geometry, (The IMA Volumes in Mathematics and Its Applications, Vol. 151), Springer, New York, pp. 85–107.
Bauchau, O. A. , and Choi, J. Y. , 2003, “ The Vector Parameterization of Motion,” Nonlinear Dyn., 33(2), pp. 165–188. [CrossRef]


Grahic Jump Location
Fig. 1

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



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In