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Herman J. Woltring
04-08-1992, 08:57 AM
Dear Biomch-L readers,

Since last month's debate on attitude representation, Prof. Jorge Angeles of
McGill University (now on sabbatical at the T.U. of Munich, Dept. of Mechanics
B) kindly mentioned some recent references. The following papers are worth
reading:

(1) Simon L. Altmann (Oxford): Hamilton, Rodrigues, and the Quaternion
Scandal. Mathematics Magazine 62(1989/12)5, 291-308.

(2) Kerry W. Spring (McGill): Euler Parameters and the Use of Quaternion
Algebra in the Manipulation of Finite Rotations: a Review.
Mechanism and Machine Theory 21(1986/5)5, 365-373.

In (1), Altmann quotes Rodrigues' formula exactly as given in Rodrigues'
1840 paper, except that he introduced vector notation, which was nonexistent
in Rodrigues' time:

cos c/2 = cos a/2 cos b/2 - sin a/2 sin b/2 l.m
sin c/2 n = sin a/2 cos b/2 l + cos a/2 sin b/2 m + sin a/2 sin b/2 l x m

If we devide the second equation by the first on each side, and do some trivial
divisions on the right side, we obtain the formula

tan c/2 n = [ tan a/2 l + tan b/2 m + tan a/2 tan b/2 l x m ] /
[ 1 - tan a/2 tan b/2 l.m ]

as presented by Giovanni Legnani in his Biomch-L posting when substituting
his A, B, and C notation for what Spring calls "Rodrigues parameters (Gibbs
vector)".

In (2), the attitude matrix is expressed in exponential form as

Q(theta N) = exp(A{theta N})

where A{.} is the skew-symmetric matrix mentioned in previous postings, N the
unit direction vector, and theta the amount of rotation about N. Even though
the product theta N occurs here directly, Spring does not consider defining
the attitude vector THETA = theta N, merely quoting Stuelpnagel [SIAM Rev. 6,
422-430 (1964)] when he writes:

Euler angles are most commonly employed even though they are neither
invariant, nor well-behaved for free-body rotations (a non-singular
mapping requires a set of at least 4 parameters [10]). Apparently,
many authors, such as Stuelpnagel [10], give special significance to
Euler angles calling them physically significant in themselves. While
this is true for [spinning -- HJW] top dynamics and gimbaled gyros (and
Euler angles are well suited to the study of these topics), Euler angles
have no more physical significance than do Euler parameters [i.e., nor-
malized quaternions {sin theta/2 N', cos theta/2} -- HJW] for free-body
rotations. For small rotations, both the 1-2-3 Euler angles and the
vector portion of the Euler parameters may be interpreted as pitch, roll
and yaw. It is unfortunate that familiarity with Euler angles gained
through the study of top dynamics and related topics has led to their
applications to general rotational movement.
Of the common methods of representing rotations, only Euler parameters
and Cayley-Klein parameters are well behaved for arbitrary rotations.

Could the attitude "vector" THETA which is non-singular and yet consists
of only 3 independent parameters possibly become the common method for
representing rotations and attitudes?

HJW

P.S. Altmann's paper is a very tongue-in-cheek one, giving much background
on both Hamilton and Rodrigues. If you cannot locate it, his paper follows
closely the material in the first two chapters of Altmann's book Rotations,
Quaternions, and Double groups (Clarendon Press, Oxford 1986) which I have
not seen yet.