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View Full Version : Re: Quaternions vs. Euler angles



Arnim Henze
04-23-2001, 07:41 PM
Chris,
as Ton already pointed out, there is an extensive explanation of the use
of quaternions (in their "euler parameter" representation) in "Computer
Aided Kinematics and Dynamics of Mechanical Systems", by E.J. Haug. In
this Book Haug introduces the theory the multi-body-simulation software
DADS is based on.
>From the theoretical point of view euler parameters
e0 = cos(A/2)
e1 = Ux*sin(A/2)
e2 = Uy*sin(A/2)
e3 = Uz*sin(A/2)
are an excellent tool to describe orientations and numerically solve
equations of motion of mechanical multi-body-systems, since there is no
gimbal lock, they are well defined on the unit-sphere (any orientation
of a rigid body can be uniquely/continuesly described by the set of (Ux,
Uy, Uz, A).
Whereas there is no way of integrating angular velocity w to obtain
orientation (since it's not integrable) one may integrate the time-
derivative of euler parameters to calculate e0(t), e1(t), e2(t), e3(t).

On the other hand euler parameters cannot be used to describe multiple
revolutions (uniqueness only on the unit-sphere) which is not too much
of a problem in biomechanics, since no joint range in biological systems
is greater. A problem with euler parameters is though, that it's almost
impossible to set up 3D-torque elements (e.g. for spherical joints).
Except for the trivial case where the joint torque always acts along
(Ux,Uy,Uz) it is very hard to describe a certain experimental behavior
with an euler parameter torque element. I tried this when I wanted to
limit the range of motion for a spherical joint but finally gave up,
since it was simply impossible (at least for me) to find the equations
using euler parameters. So I ended up using angles again.

The interpretaton of euler parameters is simple as long as one looks at
orientation only. But as soon as torque elements or experimental data
are to be described, interpretation seems to be impossible.

Arnim Henze.
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Institut f"ur Astronomie und Astrophysik
Arnim Henze Abt. Computational Physics - Biomechanik -

Universit"at T"ubingen Tel.: ++49 7071 29 78654
Auf der Morgenstelle 10 Fax : ++49 7071 29 5889
D-72076 T"ubingen, Germany email: henze@tat.physik.uni-tuebingen.de
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