Inertia plays a crucial role in the quaternion-based rigid body dynamics, the associated mass matrix, however, presents singularity in the traditional representation. Recent researches demonstrated that the singularity can be avoided by adding an extra term into kinetic energy via a multiplier. Here, we propose a modified inertia representation through splitting the kinetic energy into two parts, where a square term of quaternion velocity, governed by an extra inertial parameter, is separated from the original expression. We further derive new numerical integration schemes in both Lagrange and Hamilton framework. Error estimation shows that the extra inertial parameter has a significant influence on the numerical error in discretization, and an iterative scheme of optimizing the extra inertial parameter to reduce the numerical error in simulation is proposed for quaternion-based rigid body dynamics. Numerical results demonstrate that the mean value of the three principal moments of inertia is a reasonable value of the extra inertia parameter which can impressively improve the accuracy in most cases, and the iterative scheme can further reduce the numerical error for numerical integration, taking the implementation in Lagrange's frame as an example.

References

1.
Goldstein
,
H.
,
Poole
,
C.
, and
Safko
,
J.
,
2002
,
Classical Mechanics
, 3rd ed.,
Addison Wesley
,
New York
.
2.
Nikravesh
,
P. E.
, and
Chung
,
I.
,
1982
, “
Application of Euler Parameters to the Dynamic Analysis of Three-Dimensional Constrained Mechanical Systems
,”
ASME J. Mech. Des.
,
104
(
4
), pp.
785
791
.
3.
Nikravesh
,
P. E.
,
Wehage
,
R. A.
, and
Kwon
,
O. K.
,
1985
, “
Euler Parameters in Computational Kinematics and Dynamics—Part 1
,”
ASME J. Mech. Des.
,
107
(
3
), pp.
358
365
.
4.
Nikravesh
,
P. E.
,
1988
,
Computer-Aided Analysis of Mechanical Systems
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
5.
Vadali
,
S. R.
,
1988
, “
On the Euler Parameter Constraint
,”
J. Astronaut. Sci.
,
36
(
3
), pp.
259
265
.
6.
Chou
,
J. C.
,
1992
, “
Quaternion Kinematic and Dynamic Differential Equations
,”
IEEE Trans. Rob. Autom.
,
8
(
1
), pp.
53
64
.
7.
Morton
,
H. S.
, Jr.
,
1993
, “
Hamiltonian and Lagrangian Formulations of Rigid-Body Rotational Dynamics Based on the Euler Parameters
,”
J. Astronaut. Sci.
,
41
(4), pp.
569
591
.
8.
Shivarama
,
R.
, and
Fahrenthold
,
E. P.
,
2004
, “
Hamilton's Equations With Euler Parameters for Rigid Body Dynamics Modeling
,”
ASME J. Dyn. Syst., Meas., Control
,
126
(
1
), pp.
124
130
.
9.
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
.
10.
Nielsen
,
M. B.
, and
Krenk
,
S.
,
2012
, “
Conservative Integration of Rigid Body Motion by Quaternion Parameters With Implicit Constraints
,”
Int. J. Numer. Methods Eng.
,
92
(
8
), pp.
734
752
.
11.
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
.
12.
Udwadia
,
F. E.
, and
Schutte
,
A. D.
,
2010
, “
An Alternative Derivation of the Quaternion Equations of Motion for Rigid-Body Rotational Dynamics
,”
ASME J. Appl. Mech.
,
77
(
4
), p.
044505
.
13.
O'Reilly
,
O. M.
, and
Varadi
,
P. C.
,
1999
, “
Hoberman's Sphere, Euler Parameters and Lagrange's Equations
,”
J. Elasticity
,
56
(
2
), pp.
171
180
.
14.
Möller
,
M.
, and
Glocker
,
C.
,
2012
, “
Rigid Body Dynamics With a Scalable Body, Quaternions and Perfect Constraints
,”
Multibody Syst. Dyn.
,
27
(
4
), pp.
437
454
.
15.
Sherif
,
K.
,
Nachbagauer
,
K.
, and
Steiner
,
W.
,
2015
, “
On the Rotational Equations of Motion in Rigid Body Dynamics When Using Euler Parameters
,”
Nonlinear Dyn.
,
81
(
1
), pp.
343
352
.
16.
Shabana
,
A. A.
,
2014
, “
Euler Parameters Kinetic Singularity
,”
Proc. Inst. Mech. Eng., Part K
,
228
(
3
), pp.
307
313
.
17.
Krenk
,
S.
,
2009
,
Non-Linear Modeling and Analysis of Solids and Structures
,
Cambridge University Press
, New York.
18.
Wendlandt
,
J. M.
, and
Marsden
,
J. E.
,
1997
, “
Mechanical Integrators Derived From a Discrete Variational Principle
,”
Physica D
,
106
(
3
), pp.
223
246
.
19.
Gonzalez
,
O.
,
2000
, “
Exact Energy and Momentum Conserving Algorithms for General Models in Nonlinear Elasticity
,”
Comput. Methods Appl. Mech. Eng.
,
190
(
13
), pp.
1763
1783
.
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