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.

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.