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Research Papers

On the Numerical Influences of Inertia Representation for Rigid Body Dynamics in Terms of Unit Quaternion

[+] Author and Article Information
Xiaoming Xu

State Key Laboratory of Structural Analysis of
Industrial Equipment,
Department of Engineering Mechanics,
Dalian University of Technology,
Dalian 116023, China
e-mail: xxm020201@163.com

Wanxie Zhong

State Key Laboratory of Structural Analysis of
Industrial Equipment,
Department of Engineering Mechanics,
Dalian University of Technology,
Dalian 116023, China

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 7, 2015; final manuscript received March 12, 2016; published online March 29, 2016. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 83(6), 061006 (Mar 29, 2016) (11 pages) Paper No: JAM-15-1541; doi: 10.1115/1.4033031 History: Received October 07, 2015; Revised March 12, 2016

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.

Copyright © 2016 by ASME
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Figures

Grahic Jump Location
Fig. 4

Comparison of the variational integrator with γ=0, motion of mass center: (a) the x-componentand (b) the z -component. α=0 (), α=γm (), α=104 (), analytical (–––), and time step η=0.007.

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Fig. 2

Comparison of the variational integrator with γ=0: (a) relative energy error, (b) relative angular momentum error of l3 -component. α=0 (), α=γm (), α=104 (), and time step η=0.005.

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Fig. 3

Comparison of the variational integrator with γ=0: (a) normalization constraint and (b) orthogonality constraint. α=0 (), α=γm (), α=104 (), and time step η=0.005.

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Fig. 5

Comparison of the variational integrator with α=0 : (a) energy error with time step η=0.01 and (b) the energy error curve with γ increasing, γ=γm (), γ=γh (), ΔHm=−2.76×10−5, and ΔHm=−6.34×10−5

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Fig. 6

Comparison of the variational integrator: the maximum of periodic energy error with time step increasing. [γ, α]=[0, 0] (), [γ, α]=[0, 104] (), and [γ, α]=[γm, 0] ().

Grahic Jump Location
Fig. 7

Comparison of the variational integrator, motion of mass center: (a) the x-componentand (b) the z -component. [γ, α]=[0, 0] (), [γ, α]=[0, 104] (), [γ, α]=[γm, 0] (), analytical (–––), and time step η=0.007.

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Fig. 8

The energy optimization for variational integrator: (a) relative energy error and (b) the z -component of the mass center. γ=γh (), γ=γm (), γ=γopt-e (), analytical (–––), and time step η=0.007.

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Fig. 9

The phase optimization for variational integrator: (a) the phase errors of reference points—t0=4.350, tm=4.319, and th=4.375 and (b) the z -component of the mass center—γ=γh (), γ=γm (), γ=γopt-p (), analytical (–––), and time step η=0.007

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Fig. 10

Comparison of the energy–momentum scheme: (a) the x -component, (b) the z-component.[γ, α*]=[∞, 0] (), [γ, α*]=[γm, 0] (), [γ, α*]=[γm, 104] (), analytical (–––), and time step η=0.01.

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Fig. 11

The phase optimization for energy–momentum scheme: (a) the phase errors of reference points— t0=2.755, tm=2.770, and th=2.685 and (b) comparison of the z -component of the mass center. γ=γh (), γ=γm (), γ=γopt-p (), analytical (–––), and time step η=0.007.

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Fig. 12

Comparison of the energy–momentum scheme: the energy error curve with γ−1 increasing

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Fig. 13

Comparison of the variational integrator for regular precession, motion of mass center: (a) the x -component and (b) the z -component. γ=0 (), γ=γm (), γ=γh (), γ=3.715 (), analytical (–––), and time step η=0.007.

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Fig. 14

Comparison of the energy–momentum scheme for regular precession, motion of mass center: (a) the x -component and (b) the z -component. γ=∞ (), γ=γm (), γ=I1 (), analytical (–––), and time step η=0.007.

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