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View Full Version : Angle(s) et Axe d'Euler (1980/1984)

Herman J. Woltring
01-14-1992, 07:15 AM

In my posting of 8 October, I referred to Paul Volkmann in Germany who, about
one century ago, allegedly introduced the helical angles for the first time.
While this rumour has not yet been confirmed (see the ATTJOB TEX manuscript
mentioned earlier today on this list), a few hours ago I received a highly
interesting fax from one of our readers, Giovanni Legnani of the University
of Brescia in Italy.

It is a copy from section 1.4.4 "Angle et Axe d'Euler [6]" (note the singular
form "angle") in Alain Liegeois, LES ROBOTS, Vol. 7, Analyses des Performances
et C.A.O., Hermes Publishing (France) 1984. My translation of the most rele-
vant parts is as follows:

Another non-redundant and representative way for defining the rotation
is of the form N f(theta) which provides a vector in R3. The simplest
form is f(theta) = theta, resulting in the rotation vector

[Vrx] [nx theta]
Vr = [Vry] = [ny theta] (1-42)
[Vrz] [nz theta]

Except for the trivial case theta = 0 where N is indetermined, on can
derive from this
2 2 2
theta = +/- SQRT(Vrx + Vry + Vrz),
(1-43)
nx = Vrx/theta, ny = Vry/theta, nz = Vrz/theta

which define Euler's angle and axis.
The vector Vr can be calculated from the rotation matrix R since ...

( ... formulae omitted, similar to those in Spoor & Veldpaus,
Journal of Biomechanics 13(1980)4, 391-393 ... )

As regards the instantaneous rotation velocity of the rigid object,
this is simply

o
omega = N theta (1-47)

The reference [6] in the paragraph's title is:

[6] J.-C. Latombe, J. Mazer, D'efinition d'un langage de programmation
pour la robotique (L.M.). Rapport de Recherche No. RR 197, Labora-
toire de Math'ematiques Appliqu'ees et Informatique, Grenoble, March
1980.

At the present time, this is the earliest source for what (lacking a better
term) I have called `helical angles'. Note that the last equation presupposes
that N is constant; the ATTJOB TEX manuscript contains a more general formula
describing the case that both N and theta in the product

THETA = N theta

are time-varying. The book's next section, 1.4.5 "Angles d'Euler" (plural!)
seems to discuss the classical Cardanic/Eulerian angles -- I merely have its
first three lines.

Anyone in the readership who knows of an older reference in the published or
gray literature?

With kind regards,

Herman J. Woltring, Eindhoven/NL, FAX +31.40.413 744