Rotation matrices, which are three-by-three orthonormal matrices with determinant equal to plus one, constitute the special orthogonal group of rigid-body rotations, denoted SO(3). Owing to the three-by-three nature of rotation matrices plus their orthonormality constraint, parameterizations are often used in favor of rotation matrices for computations and derivations. For example, Euler angles and Rodrigues parameters are common three-parameter unconstrained parameterizations, while unit-length quaternions are a popular four-parameter constrained parameterization. In this paper various identities associated with the parameterization of SO(3) are considered. In particular, we present six identities, three related to unconstrained parameterizations and three related to constrained parameterizations. We also discuss rotation matrix perturbations. The utility of these identities is highlighted when deriving the motion equations of a rigid body using Lagrange's equation. We also use them to examine some issues associated with spacecraft attitude determination.

References

1.
Hughes
,
P. C.
,
2004
,
Spacecraft Attitude Dynamics
,
Dover Publications
,
New York
.
2.
Chaturvedi
,
N. A.
,
Sanyal
,
A. K.
, and
McClamroch
,
N.-H.
,
2011
, “
Rigid-Body Attitude Control—Using Rotation Matrices for Continuous, Singularity-Free Control Laws
,”
IEEE Control Syst. Mag.
,
31
(
3
), pp.
30
51
.10.1109/MCS.2011.940459
3.
Crassidis
,
J. L.
, and
Junkins
,
J. L.
,
2011
,
Optimal Estimation of Dynamic Systems
, 2nd ed.,
CRC Press
,
New York
.
4.
Shuster
,
M. D.
,
1993
, “
A Survey of Attitude Representations
,”
J. Astronaut. Sci.
,
41
(
4
), pp.
439
517
.
5.
Murray
,
R. N.
,
Zexiang
,
L.
, and
Sastry
,
S. S.
,
1994
,
A Mathematical Introduction to Robotic Manipulation
,
CRC Press
, Ann Arbor, MI.
6.
Barfoot
,
T. D.
,
Forbes
,
J. R.
, and
Furgale
,
P. T.
,
2011
, “
Pose Estimation Using Linearized Rotations and Quaternion Algebra
,”
Acta Astronaut.
,
68
(
1–2
), pp.
101
112
.10.1016/j.actaastro.2010.06.049
7.
Meirovitch
,
L.
, and
Stemple
,
T.
,
1995
, “
Hybrid Equations of Motion for Flexible Multibody Systems Using Quasicoordinates
,”
AIAA J. Guid., Control, Dyn.
,
18
(
4
), pp.
678
688
.10.2514/3.21447
8.
D'Eleuterio
,
G. M. T.
, and
Barfoot
,
T. D.
,
2007
, “
A Discrete Quasicoordinate Formulation for the Dynamics of Elastic Bodies
,”
ASME J. Appl. Mech.
,
74
(
2
), pp.
231
239
.10.1115/1.2189873
9.
Kane
,
T. R.
, and
Levinson
,
D. A.
,
1980
, “
Formulation of Equations of Motion for Complex Spacecraft
,”
J. Guid. Control
,
3
(
2
), pp.
99
112
.10.2514/3.55956
10.
Schaub
,
H.-P.
, and
Junkins
,
J. L.
,
2003
,
Analytical Mechanics of Space Systems
,
AIAA Education Series
, Reston, VA.
11.
Whittaker
,
E. J.
,
1937
.
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies
, 4th ed.,
Dover Publications
,
New York
, pp.
43
44
.
12.
Meirovitch
,
L.
,
1991
, “
Hybrid State Equations of Motion for Flexible Bodies in Terms of Quasi-Coordinates
,”
AIAA J. Guid. Control
,
14
(
5
), pp.
1008
1013
.10.2514/3.20743
13.
Hurtado
,
J. E.
,
2003
, “
Hamel Coefficients for the Rotational Motion of a Rigid Body
,”
The AAS John L. Junkins Astrodynamics Conference
,
College Station, TX
, May 23–24, Paper No. AAS 03-283.
14.
Hurtado
,
J. E.
, and
Sinclair
,
A. J.
,
2004
, “
Hamel Coefficients for the Rotational Motion of an N-Dimensional Rigid Body
,”
Proc. R. Soc. London, Ser. A
,
460
(
2052
), pp.
3613
3630
.10.1098/rspa.2004.1320
15.
Luenberger
,
D. G.
, and
Ye
,
Y.
,
2008
,
Linear and Nonlinear Programming
, 3rd ed.,
Springer
,
New York
, pp.
323
325
.
16.
Bernstein
,
D. S.
,
2009
,
Matrix Mathematics
, 2nd ed.,
Princeton University
,
Princeton, NJ
, p.
103
.
17.
Boltzmann
,
L.
,
1902
, “
Uber die Form der Lagrange'schen Gleichungen fur Nichtholonome, Generalisierte Koordinaten
,”
Sitzungsber. Math.-Naturwiss. Kl. Kais. Akademieder Wiss.
,
61
, pp.
1603
1614
.
18.
Hamel
,
G.
,
1904
, “
Die Lagrange-Eulerschen Gleichungen der Mechanik
,”
Z. für Math. Phys.
,
50
, pp.
1
57
.
19.
Greenwood
,
D. T.
,
1977
,
Classical Dynamics
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
20.
Shuster
,
M. D.
, and
Oh
,
S. D.
,
1981
, “
Three-Axis Attitude Determination From Vector Observations
,”
AIAA J. Guid. Control Dyn.
,
4
(
1
), pp.
70
77
.10.2514/3.19717
21.
Shuster
,
M. D.
,
1990
, “
Kalman Filtering of Spacecraft Attitude and the QUEST Model
,”
J. Astronaut. Sci.
,
38
(
3
), pp.
377
393
.
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