The parametrization of a rigid-body rotation is a classical subject in rigid-body dynamics. Euler angles, the rotation matrix and quaternions are the most common representations. However, Euler angles are known to be prone to singularities, besides not being frame-invariant. The full 3 × 3 rotation matrix conveys all the motion information, but poses the problem of an excessive number of parameters, nine, to represent a transformation that entails only three independent parameters. Quaternions are singularity-free, and thus, ideal to study rigid-body kinematics. However, quaternions, comprising four components, are subject to one scalar constraint, which has to be included in the mathematical model of rigid-body dynamics. The outcome is that the use of quaternions imposes one algebraic constraint, even in the case of mechanically unconstrained systems. An alternative parametrization is proposed here, that (a) comprises only three independent parameters; (b) is fairly robust to representation singularities; and (c) satisfies the quaternion scalar constraint intrinsically. To illustrate the concept, a simple, yet nontrivial case study is included. This is a mechanical system composed of a rigid, toroidal wheel rolling without slipping or skidding on a horizontal surface. The simulation algorithm based on the proposed parametrization and fundamentally on quaternions, together with the invariant relations between the quaternion rate of change and the angular velocity, is capable of reproducing the falling of the wheel under deterministic initial conditions and a random disturbance acting on the tilting axis. Finally, a comparative study is included, on the numerical conditioning of the parametrization proposed here and that based on Euler angles. Ours shows as broader well-conditional region than Euler angles offer. Moreover, the two parametrizations exhibit an outstanding complementarity: where the conditioning of one degenerates, the other comes to rescue.

## References

1.
Stuelpnagel
,
J.
,
1964
, “
On the Parameterization of the Three-Dimensional Rotation Group
,”
SIAM Rev.
,
6
(
4
), pp.
422
430
.10.1137/1006093
2.
Pina
,
E.
,
1983
, “
A New Parameterization of the Rotation Matrix
,”
Am. J. Phys.
,
51
(
4
), pp.
375
379
.10.1119/1.13253
3.
Tilma
,
T.
, and
Sudarshan
,
E.
,
2002
, “
Generalized Euler Angle Parameterization for SU(N)
,”
J. Phys. A: Math. Gen.
,
35
(
48
), p.
10467
.10.1088/0305-4470/35/48/316
4.
Tilma
,
T.
, and
Sudarshan
,
E.
,
2004
, “
Generalized Euler Angle Parameterization for U(N) With Applications to SU(N) Coset Volume Measures
,”
J. Geom. Phys.
,
52
(
3
), pp.
263
283
.10.1016/j.geomphys.2004.03.003
5.
Dewey
,
J.
,
Helman
,
M.
,
Knott
,
S.
,
Turco
,
E.
, and
Hutton
,
D.
,
1989
, “
Kinematics of the Western Mediterranean
,”
Geol. Soc., London, Special Pub.
,
45
(
1
), pp.
265
283
.10.1144/GSL.SP.1989.045.01.15
6.
Colombet
,
P.
,
Robinson
,
J.
,
Christel
,
P.
,
Franceschi
,
J.
, and
Djian
,
P.
,
2007
, “
Using Navigation to Measure Rotation Kinematics During ACL Reconstruction
,”
Clin. Orthop. Related Res.
,
454
(
1
), pp.
59
65
.10.1097/BLO.0b013e31802baf56
7.
Danielson
,
D. A.
, and
Hodges
,
D. H.
,
1987
, “
Nonlinear Beam Kinematics by Decomposition of the Rotation Tensor
,”
ASME J. Appl. Mech.
,
54
(
2
), pp.
258
262
.10.1115/1.3173004
8.
Ishii
,
T.
,
Mukai
,
Y.
,
Hosono
,
N.
,
Sakaura
,
H.
,
Nakajima
,
Y.
,
Sato
,
Y.
,
Sugamoto
,
K.
, and
Yoshikawa
,
H.
,
2004
, “
Kinematics of the Upper Cervical Spine in Rotation: In Vivo Three-Dimensional Analysis
,”
Spine
,
29
(
7
), pp.
E139
E144
.10.1097/01.BRS.0000116998.55056.3C
9.
Baruh
,
H.
,
1999
,
Analytical Dynamics
,
WCB/McGraw-Hill
,
Boston
.
10.
Pio
,
R. L.
,
1966
, “
Euler Angle Transformations
,”
IEEE Trans. Autom. Control
,
11
(
4
), pp.
707
715
.10.1109/TAC.1966.1098430
11.
Evans
,
D. J.
,
1977
, “
On the Representation of Orientation Space
,”
Mol. Phys.
,
34
(
2
), pp.
317
325
.10.1080/00268977700101751
12.
Burger
,
H.
,
1995
, “
Use of Euler-Rotation Angles for Generating Antenna Patterns
,”
IEEE Trans. Antennas Propag.
,
37
(
2
), pp.
56
63
.10.1109/74.382344
13.
Horn
,
A.
,
1954
, “
Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix
,”
Am. J. Math.
,
76
(
3
), pp.
620
630
.10.2307/2372705
14.
Raptis
,
I. A.
,
Valavanis
,
K. P.
, and
Moreno
,
W. A.
,
2011
, “
A Novel Nonlinear Backstepping Controller Design for Helicopters Using the Rotation Matrix
,”
IEEE Trans. Control Syst. Technol.
,
19
(
2
), pp.
465
473
.10.1109/TCST.2010.2042450
15.
Rusydi
,
M. I.
,
Okamoto
,
T.
,
Ito
,
S.
, and
Sasaki
,
M.
,
2014
, “
Rotation Matrix to Operate a Robot Manipulator for 2D Analog Tracking Objects Using Electrooculography
,”
Robotics
,
3
(
3
), pp.
289
309
.10.3390/robotics3030289
16.
Zhang
,
Y.
, and
Xiao
,
D.
,
2014
, “
An Image Encryption Scheme Based on Rotation Matrix Bit-Level Permutation and Block Diffusion
,”
Commun. Nonlinear Sci. Num. Simul.
,
19
(
1
), pp.
74
82
.10.1016/j.cnsns.2013.06.031
17.
Shoemake
,
K.
,
1985
, “
Animating Rotation With Quaternion Curves
,”
12th Annual Conference on Computer Graphics and Interactive Techniques
(
SIGGRAPH'85
),
San Francisco, CA
, July 22–26, pp.
245
254
.10.1145/325334.325242
18.
Bar-Itzhack
,
I. Y.
,
2000
, “
New Method for Extracting the Quaternion From a Rotation Matrix
,”
J. Guid., Control, Dyn.
,
23
(
6
), pp.
1085
1087
.10.2514/2.4654
19.
Lekkas
,
A.
, and
Fossen
,
T. I.
,
2013
, “
A Quaternion-Based LOS Guidance Scheme for Path Following of AUVs
,”
9th IFAC Conference on Control Applications in Marine Systems
, Osaka, Japan, Sept. 17–20, pp.
245
250
.10.3182/20130918-4-JP-3022.00070
20.
Wu
,
S.
,
Wu
,
G.
,
Tan
,
S.
, and
Wu
,
Z.
,
2013
, “
Quaternion-Based Adaptive Terminal Sliding Mode Control for Spacecraft Attitude Tracking
,”
10th IEEE International Conference on Control and Automation
(
ICCA
), Hangzhou, China, June 12–14, pp.
913
917
.10.1109/ICCA.2013.6565120
21.
Argyle
,
M. E.
,
Beach
,
J. M.
,
Beard
,
R. W.
,
McLain
,
T. W.
, and
Morris
,
S.
,
2014
, “
Quaternion Based Attitude Error for a Tailsitter in Hover Flight
,”
American Control Conference
(
ACC
), Portland, OR, June 4–6, pp.
1396
1401
.10.1109/ACC.2014.6859324
22.
Fister
,
I.
,
Yang
,
X.
,
Brest
,
J.
, and
Fister
,
I.
, Jr.
,
2013
, “
Modified Firefly Algorithm Using Quaternion Representation
,”
Expert Syst. Appl.
,
40
(
18
), pp.
7220
7230
.10.1016/j.eswa.2013.06.070
23.
Terze
,
Z.
,
Muller
,
A.
, and
Zlatar
,
D.
,
2014
, “
Redundancy-Free Integration of Rotational Quaternions in Minimal Form
,”
ASME
Paper No. DETC2014-35118.10.1115/DETC2014-35118
24.
Angeles
,
J.
,
2014
,
Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms
, 4th ed.,
Springer
,
New York
.
25.
Synge
,
J. L.
,
1960
, “
Classical Dynamics
,”
Principles of Classical Mechanics and Field Theory
,
Springer
,
Berlin
, pp.
1
225
.
26.
Akcoglu
,
M. A.
,
Bartha
,
P. F.
, and
Ha
,
D. M.
,
2011
,
Analysis in Vector Spaces
,
Wiley
,
Hoboken, NJ
.
27.
Ascher
,
U. M.
, and
Petzold
,
L. R.
,
1998
,
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
, Vol.
61
,
SIAM
,
Toronto
.
28.
Golub
,
G. H.
, and
Van Loan
,
C. F.
,
2012
,
Matrix Computations
, Vol.
3
,
Johns Hopkins University Press
,
Baltimore, MD
.
29.
Angeles
,
J.
, and
Lee
,
S. K.
,
1988
, “
The Formulation of Dynamical Equations of Holonomic Mechanical Systems Using a Natural Orthogonal Complement
,”
ASME J. Appl. Mech.
,
55
(
1
), pp.
243
244
.10.1115/1.3173642