The Vectorial Parameterization of Rotation

The parameterization of rotation is the subject of continuous research and development in many theoretical and applied fields of mechanics, such as rigid body, structural, and multibody dynamics, robotics, spacecraft attitude dynamics, navigation, image processing, and so on. This paper introduces the vectorial parameterization of rotation, a class of parameterization techniques encompassing many formulations independently developed to date for the analysis of rotational motion. The exponential map of rotation, the Rodrigues, Cayley, Gibbs, Wiener, and Milenkovic parameterization all are special cases of the vectorial parameterization. This generalization parameterization sheds additional light on the fundamental properties of these techniques, pointing out the similarities in their formal structure and showing their inter-relationships. Although presented in a compact manner, all of the formulae needed for a complete implementation of the vectorial parameterization of rotation are included in this paper.

[1]  Gabriela Koreisová,et al.  Scientific Papers , 1911, Nature.

[2]  Thomas Freud. Wiener,et al.  Theoretical analysis of gimballess inertial reference equipment using delta-modulated instruments , 1962 .

[3]  J. Stuelpnagel.  On the Parametrization of the Three-Dimensional Rotation Group , 1964 .

[4]  William Rowan Hamilton,et al.  Elements of Quaternions , 1969 .

[5]  R. M. Bowen,et al.  Introduction to vectors and tensors, Vol 1: linear and multilinear algebra , 1976 .

[6]  Stanley W. Shepperd,et al.  Quaternion from Rotation Matrix , 1978 .

[7]  R. A. Spurrier.  Comment on " Singularity-Free Extraction of a Quaternion from a Direction-Cosine Matrix" , 1978 .

[8]  J. Argyris.  An excursion into large rotations , 1982 .

[9]  Roger A. Wehage.  Quaternions and Euler Parameters — A Brief Exposition , 1984 .

[10]  K. Spring.  Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: A review , 1986 .

[11]  M. Géradin,et al.  Kinematics and dynamics of rigid and flexible mechanisms using finite elements and quaternion algebra , 1988 .

[12]  Satya N. Atluri,et al.  Primal and mixed forms of Hamiltons's principle for constrained rigid body systems: numerical studies , 1991 .

[13]  J. C. Simo,et al.  Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum , 1991 .

[14]  M. Shuster.  A survey of attitude representation , 1993 .

[15]  J. C. Simo,et al.  A new energy and momentum conserving algorithm for the non‐linear dynamics of shells , 1994 .

[16]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[17]  J. C. Simo,et al.  Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms , 1995 .

[18]  J. Junkins,et al.  Stereographic Orientation Parameters for Attitude Dynamics: A Generalization of the Rodrigues Parameters , 1996 .

[19]  J. Junkins,et al.  HIGHER-ORDER CAYLEY TRANSFORMS WITH APPLICATIONS TO ATTITUDE REPRESENTATIONS , 1997 .

[20]  A. Ibrahimbegovic.  On the choice of finite rotation parameters , 1997 .

[21]  F. Pfister,et al.  Bernoulli Numbers and Rotational Kinematics , 1998 .

[22]  Carlo L. Bottasso,et al.  Integrating Finite Rotations , 1998 .

[23]  O. Bauchau,et al.  On the design of energy preserving and decaying schemes for flexible, nonlinear multi-body systems , 1999 .

[24]  M. Borri,et al.  On Representations and Parameterizations of Motion , 2000 .

[25]  M. Borri,et al.  A Novel Momentum-Preserving Energy-Decaying Algorithm for Finite Element Multibody Procedures , 2000 .

[26]  M. Borri,et al.  Integration of elastic multibody systems by invariant conserving/dissipating algorithms. II. Numerical schemes and applications , 2001 .

[27]  M. Borri,et al.  Integration of elastic multibody systems by invariant conserving/dissipating algorithms. I. Formulation , 2001 .

[28]  M. Borri,et al.  Geometric invariance , 2002 .

[29]  Olivier A. Bauchau,et al.  Robust integration schemes for flexible multibody systems , 2003 .