Advanced

Coordinate representations for rigid parts in multibody dynamics

Afzali-Far, B. LU and Lidström, P. LU (2016) In Mathematics and Mechanics of Solids 21(8). p.990-1025
Abstract

The paper is concerned with coordinate representations for rigid parts in multibody dynamics. The discussion is based on the general theory of the dynamics of multibody systems under constraints. General configuration coordinates are introduced and the requirement of their regularity is discussed. The use of Euler angles, quaternions and linear coordinates, where quaternions and linear coordinates require constraint conditions, is analysed in detail. These coordinate systems are all shown to be regular. Equations of motion are formulated, using Lagrange's as well as Euler's equations, and they are supplemented by the appropriate constraint conditions in the cases of quaternions and linear coordinates. Mass matrices are derived, and in... (More)

The paper is concerned with coordinate representations for rigid parts in multibody dynamics. The discussion is based on the general theory of the dynamics of multibody systems under constraints. General configuration coordinates are introduced and the requirement of their regularity is discussed. The use of Euler angles, quaternions and linear coordinates, where quaternions and linear coordinates require constraint conditions, is analysed in detail. These coordinate systems are all shown to be regular. Equations of motion are formulated, using Lagrange's as well as Euler's equations, and they are supplemented by the appropriate constraint conditions in the cases of quaternions and linear coordinates. Mass matrices are derived, and in terms of the Euler angles the mass matrix components are products of trigonometric functions whereas in terms of quaternions the matrix components are quadratic polynomials. Using linear coordinates gives rise to a constant mass matrix. Thus, there is a decreasing degree of complexity, regarding mass matrix components, when going from Euler angles to linear coordinates. This is obtained at the expense of an increasing gross number of degrees of freedom and the necessary introduction of constraint conditions. The different equations of motion obtained are compared with respect to their structural complexity. In all representations the components of the angular velocity are explicitly calculated. This is not always the case in previous investigations of this subject. The paper also gives a new proof of the well-known relation between angular velocity and unit quaternions and their time derivative.

(Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
configuration coordinates, equations of motion, Euler angles, linear coordinates, Multibody dynamics, quaternions, rigid body, rotation tensor
in
Mathematics and Mechanics of Solids
volume
21
issue
8
pages
36 pages
publisher
SAGE Publications
external identifiers
  • scopus:84983028239
  • wos:000382845200005
ISSN
1741-3028
DOI
10.1177/1081286514546180
language
English
LU publication?
yes
id
3f169822-e9b6-4d0c-ac29-0afa8bdd1367 (old id 7863092)
date added to LUP
2016-01-26 14:39:39
date last changed
2017-01-01 08:17:51
@article{3f169822-e9b6-4d0c-ac29-0afa8bdd1367,
  abstract     = {<p>The paper is concerned with coordinate representations for rigid parts in multibody dynamics. The discussion is based on the general theory of the dynamics of multibody systems under constraints. General configuration coordinates are introduced and the requirement of their regularity is discussed. The use of Euler angles, quaternions and linear coordinates, where quaternions and linear coordinates require constraint conditions, is analysed in detail. These coordinate systems are all shown to be regular. Equations of motion are formulated, using Lagrange's as well as Euler's equations, and they are supplemented by the appropriate constraint conditions in the cases of quaternions and linear coordinates. Mass matrices are derived, and in terms of the Euler angles the mass matrix components are products of trigonometric functions whereas in terms of quaternions the matrix components are quadratic polynomials. Using linear coordinates gives rise to a constant mass matrix. Thus, there is a decreasing degree of complexity, regarding mass matrix components, when going from Euler angles to linear coordinates. This is obtained at the expense of an increasing gross number of degrees of freedom and the necessary introduction of constraint conditions. The different equations of motion obtained are compared with respect to their structural complexity. In all representations the components of the angular velocity are explicitly calculated. This is not always the case in previous investigations of this subject. The paper also gives a new proof of the well-known relation between angular velocity and unit quaternions and their time derivative.</p>},
  author       = {Afzali-Far, B. and Lidström, P.},
  issn         = {1741-3028},
  keyword      = {configuration coordinates,equations of motion,Euler angles,linear coordinates,Multibody dynamics,quaternions,rigid body,rotation tensor},
  language     = {eng},
  month        = {09},
  number       = {8},
  pages        = {990--1025},
  publisher    = {SAGE Publications},
  series       = {Mathematics and Mechanics of Solids},
  title        = {Coordinate representations for rigid parts in multibody dynamics},
  url          = {http://dx.doi.org/10.1177/1081286514546180},
  volume       = {21},
  year         = {2016},
}