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On Singularity of Rigid-Body Dynamics Using Quaternion-Based Models

[+] Author and Article Information
Homin Choi

Department of Aerospace and Mechanical Engineering,University of Southern California,3650 McClintock Avenue, Room 430, Los Angeles, CA 90089-1453

Bingen Yang1

Department of Aerospace and Mechanical Engineering,University of Southern California,3650 McClintock Avenue, Room 430, Los Angeles, CA 90089-1453bingen@usc.edu

1

Corresponding author.

J. Appl. Mech 79(2), 024502 (Feb 09, 2012) (7 pages) doi:10.1115/1.4005575 History: Received November 10, 2010; Revised October 21, 2011; Posted February 01, 2012; Published February 09, 2012; Online February 09, 2012

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.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

A constrained rigid body

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Figure 2

A rotation sequence with Euler angles: (a) yawing ψ; (b) pitching φ; and (c) rolling θ

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Figure 3

Deformation of the springs

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Figure 4

The rotational axis of the body and the rotational axis of torque

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Figure 5

Free vibration (Case 1): (a) translation; (b) rotation in terms of quaternion components

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Figure 6

Free vibration (Case 1): rotation of the body in terms of Euler angles

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Figure 7

Forced vibration (Case 2): (a) translation; (b) rotation in terms of quaternion components

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Figure 8

Forced vibration (Case 2) with modified χk given in Eq. 28: (a) translation; (b) rotation in terms of quaternion

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