It is well known that use of quaternions in dynamic modeling of rigid bodies can avoid the singularity due to Euler rotations. This paper shows that the dynamic response of a rigid body modeled by quaternions may become unbounded when a torque is applied to the body. A theorem is derived, relating the singularity to the axes of the rotation and applied torque, and to the degrees of freedom of the body in rotation. To avoid such singularity, a method of equivalent couples is proposed.

References

References
1.
Gray
,
A.
, 1918,
A Treatise on Gyrostatics and Rotational Motion
,
Macmillan
,
London
.
2.
Pio
,
R. L.
, 1966, “
Euler Angle Transformations
,”
IEEE Trans. Autom. Control.
11
(
4
), pp.
707
715
.
3.
Stuelpnagel
,
J.
, 1964, “
On the Parameterization of the Three-Dimensional Rotation Group
,”
SIAM Rev.
6
(
4
), pp.
422
430
.
4.
Hamilton
,
W. R.
, 1853,
Lectures on Quaternions
,
Hodges and Smith
,
Dublin
.
5.
Mitchell
,
E. E. L.
, and
Rogers
,
A. E.
, 1965, “
Quaternion Parameters in Simulation of Spinning Rigid Body
,”
Simulation.
4
(
6
), pp.
390
396
.
6.
Bar-Itzhack
,
I. Y.
, and
Oshman
,
Y.
, 1985, “
Attitude Determination from Vector Observations Quaternion Estimation
,”
IEEE Trans. Aerosp. Electron. Syst.
21
(
1
), pp.
128
136
.
7.
Taylor
,
R. H.
, 1979, “
Planning and Execution of Straight Line Manipulator Trajectories
,”
IBM J. Res. Dev.
23
(
4
), pp.
424
436
.
8.
Gu
,
Y. L.
, 1988, “
Analysis of Orientation Representations by Lie Algebra in Robotics
,”
Proceedings of the 1988 IEEE International Conference on Robotics and Automation (Cat. No. 88CH2555-1
), Vol.
2
, pp.
874
879
.
9.
Kavan
,
L.
,
Collins
,
S.
,
Zara
,
J.
, and
O’Sullivan
,
C.
, 2008, “
Geometric Skinning with Approximate Dual Quaternion Blending
,”
ACM Trans. Graphics.
27
(
4
), pp.
105, 1
23
10.
Yamamoto
,
H.
, and
Aoshima
,
N.
, 2002, “
Three Dimensional Signal Processing based on Quaternion
,”
SICE 2002. Proceedings of the 41st SICE Annual Conference (Cat. No. 02TH8648.
), Vol.
3
, pp.
1420
1424
.
11.
Markey
,
B. R.
, and
Walmsley
,
S. H.
, 1982, “
Quaternion Formulation of Lattice Dynamics of Molecular Crystals
,”
Chem. Phys. Lett.
92
(
3
), pp.
257
261
.
12.
Miller
,
T. F.
III
,
Eleftheriou
,
M.
,
Pattnaik
,
P.
,
Ndirango
,
A.
,
Newns
,
D.
, and
Martyna
,
G. J.
, 2002, “
Symplectic Quaternion Scheme for Biophysical Molecular Dynamics
,”
J. Chem. Phys.
116
(
20
), pp.
8649
8659
.
13.
Kuipers
,
J. B.
, 1999,
Quaternions and Rotation Sequences
,
Princeton University Press
,
New Jersey
.
14.
Shuster
,
D. M.
, 1993, “
A Survey of Attitude Representations
,”
J. Astronaut. Sci.
41
(
4
), pp.
439
517
. Available at http://www.dimnp.unipi.it/gabiccini-m/RAR/___RotationsSurvey.pdf.
15.
Shabana
,
A. A.
, 2005,
Dynamics of Multibody Systems
,
Cambridge University Press
,
New York
.
16.
Howland
,
R. A.
, 2006,
Intermediate Dynamics: A linear Algebraic Approach
,
Springer Science + Business Media, Inc.
,
New York
.
17.
Nikravesh
,
P. E.
, 1998,
Computer-Aided Analysis of Mechanical Systems
,
Prentice-Hall, Englewood Cliffs
,
NJ
.
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