Gabriel Baud-bovy

10-01-1995, 12:57 AM

A couple of years ago, there was a very interesting discussion=20

about Eulerian-Cardanic angles and helical angles in this

group. The main contributions are summarized in one of the Biomech-L=20

file ANGLES3D.TOPIC.

In this discussion, it was pointed out that there are no less that 12 ways=

=20

to define Eulerian-Cardanic angles (see also Woltring's papers). Formulas

to extract these angles from a rotation matrix and to translate between=20

the various conventions were worked out (see also Woltring's PRP routines).

Unfortunately, the "longitude-latitude-twist" or "azimuth-elevation-twist"

angles were not discussed in great details.

In my dictionnary, these angles are defined in the following way:

Let (x,y,z) and (X,Y,Z) be two frames of reference sharing the same

origin O. Let (Om) be the orthogonal projection of axis Z on=20

the (xOy) plan. Longitude (or azimuth) is defined by the=20

angle (xOm) and latitude (or elevation) by the angle (zOZ). =20

Thus (x,y,z) will become (X,Y,Z) after the following 3 rotations:

1) Rz(longitude) -> x and (Om) are aligned

2) Ry(latitude) -> z and Z are aligned

3) Rz(twist)

Of course, some parts of this definition are arbitrary. For example,

one could:

1) choose a different plane to project Z axis.

2) choose to project a different axis than Z axis.

Each choice will yield different angles and different axis of

rotation. Nevertheless, all these possibilities share the same features:

1) The first rotation has the affect to align one axis of (x,y,z)

with the orthogonal projection of one axis of (X,Y,Z).

2) First and last rotation are around the same axis.

My questions are the following:

1) How to define precisely these parameters? How many different

conventions respecting the two above features are there? How

to name these angles?

2) For each convention, how to extract the corresponding angles=20

from a rotation matrix?=20

3) How to go from one convention to another?

4) Reference paper, book, software?

Note: I don't think it is interesting to discuss which convention is the=

best

but if somebody has a strong opinion ...

A summary of the replies will be posted. Sorry for my poor english.

Gabriel Baud-Bovy

=09

Gabriel Baud-Bovy

Universit=E9 de Gen=E8ve, FAPSE

9, route de Drize

1227 Carouge - Switzerland

about Eulerian-Cardanic angles and helical angles in this

group. The main contributions are summarized in one of the Biomech-L=20

file ANGLES3D.TOPIC.

In this discussion, it was pointed out that there are no less that 12 ways=

=20

to define Eulerian-Cardanic angles (see also Woltring's papers). Formulas

to extract these angles from a rotation matrix and to translate between=20

the various conventions were worked out (see also Woltring's PRP routines).

Unfortunately, the "longitude-latitude-twist" or "azimuth-elevation-twist"

angles were not discussed in great details.

In my dictionnary, these angles are defined in the following way:

Let (x,y,z) and (X,Y,Z) be two frames of reference sharing the same

origin O. Let (Om) be the orthogonal projection of axis Z on=20

the (xOy) plan. Longitude (or azimuth) is defined by the=20

angle (xOm) and latitude (or elevation) by the angle (zOZ). =20

Thus (x,y,z) will become (X,Y,Z) after the following 3 rotations:

1) Rz(longitude) -> x and (Om) are aligned

2) Ry(latitude) -> z and Z are aligned

3) Rz(twist)

Of course, some parts of this definition are arbitrary. For example,

one could:

1) choose a different plane to project Z axis.

2) choose to project a different axis than Z axis.

Each choice will yield different angles and different axis of

rotation. Nevertheless, all these possibilities share the same features:

1) The first rotation has the affect to align one axis of (x,y,z)

with the orthogonal projection of one axis of (X,Y,Z).

2) First and last rotation are around the same axis.

My questions are the following:

1) How to define precisely these parameters? How many different

conventions respecting the two above features are there? How

to name these angles?

2) For each convention, how to extract the corresponding angles=20

from a rotation matrix?=20

3) How to go from one convention to another?

4) Reference paper, book, software?

Note: I don't think it is interesting to discuss which convention is the=

best

but if somebody has a strong opinion ...

A summary of the replies will be posted. Sorry for my poor english.

Gabriel Baud-Bovy

=09

Gabriel Baud-Bovy

Universit=E9 de Gen=E8ve, FAPSE

9, route de Drize

1227 Carouge - Switzerland