Giovanni Legnani. University Of Brescia

03-05-1990, 10:22 PM

Dear biomch-l subscriber,

Returning from a few days holiday I found

other papers on angles, screw axes and so on... It looks as if

everybody wants to show how good he is in math.... And although my

English and my Math are very bad, I also want to do the same...

Like many people, I have studied many different ways to express

rotations. A chapter of my doctoral dissertation (1986) was

devoted to this study. I do believe that there is NOT a BEST way

to represent the angular position of rigid bodies. Every method

has good points and bad points. Probably each method has been

established because it was useful in a practical application. For

example, I guess, that the Euler's angles have an interesting

meaning in Astronomic study about planets.

Any way I devoted a little time in studying the following sets of

"angular parameters":

1) Euler's angles

2) Cardan's angles (also known as Tait-Brians' angles)

3) Euler's parameters and "quaternions"

4) Rodriguez-Hamilton's parameters

5) Euler's axis and angle (also known as finite rotational axis)

6) a system based on Latitude and longitude

7) director cosines and rotational matrices

8) screw theory

Each of the methods, except 8), falls in one of the two following

groups:

a) a three independent parameters system

1) 2) 4) 5)

b) a more (>3) NOT-independent parameters system

3) 6) 7)

the screw theory describes either rotations and displacements and

from the rotational point of view is "similar" to method 5.

Every system which consists of 3 parameters has mathematical

singularities for a few particular values of its parameters. (i.e.

the Rodriguez parameters for TETA==90 degrees or the Euler angles

if the node axis is not defined, ....) and there are high

numerical errors when the parameters are closed to these

singularity points. Any way this is not a problem if one is sure

that in its problem he will never reach or approach these points.

Every system consisting in more than 3 parameters removes the

singularity but forces to use a set of NOT independent

"coordinates". The meaning of a set of coordinates of group 2 is

also generally less expressive or clear for Humans.

I do think that the choice of an angular parameter system can be

done taking into account many aspects as (for instance):

1) the meaning of each parameter in each practical application

2) the risk to fall into a singularity

3) the math simplicity

......

.... and last but not the least

999) the researcher's experience and beliefs...

I like to use a 4*4 matrix method based on the well known

Transformation matrices approach. I extended this method to full

kinematics (speed and acceleration) and dynamics (wrenches

(forces+torques), linear and angular momentum, inertial terms). I

think that, at least for computer applications, this method is

very convenient for its programming simplicity and because it has

not any math singularity. A 3*3 sub-matrix of a 4*4 matrix is the

well known rotational matrix that can be easily built from

whatever of the above coordinate systems. However, the inverse

transformation (i.e. from matrix to parameters) is, of course,

possible and easy only if we are not close to one of the math sin-

gularities of that parameters set.

Using this approach, my colleagues and I have been developing

SPACELIB, a computer library for the kinematic and dynamic analy-

sis of systems of rigid bodies. Using this library we have reali-

zed many computer programs for the study of robots and for human

body simulation (direct and inverse dynamics and direct and inver-

se kinematics).

Summarizing:

1) each method has good and bad points

2) I like matrices

3) I am interested in the exchange of papers, SHORT mails and

computer libraries.

If someone is interested in my topics, he can have a look at the

proceedings of the last Int. Congress of Biomechanics (LA 1989) or

its satellite meeting on computer simulation (DAVIS, CA 1989) or

write or mail to me directly.

I am looking forward to hearing from someone about angles,

matrices or grammar mistakes.

Yours Faithfully

Giovanni LEGNANI

Giovanni LEGNANI

University of Brescia

Mech. Eng. Dep.

Via Valotti 9

25060 MOMPIANO BS

ITALY

tel +39 30 3996.446

fax +39 30 303681

Returning from a few days holiday I found

other papers on angles, screw axes and so on... It looks as if

everybody wants to show how good he is in math.... And although my

English and my Math are very bad, I also want to do the same...

Like many people, I have studied many different ways to express

rotations. A chapter of my doctoral dissertation (1986) was

devoted to this study. I do believe that there is NOT a BEST way

to represent the angular position of rigid bodies. Every method

has good points and bad points. Probably each method has been

established because it was useful in a practical application. For

example, I guess, that the Euler's angles have an interesting

meaning in Astronomic study about planets.

Any way I devoted a little time in studying the following sets of

"angular parameters":

1) Euler's angles

2) Cardan's angles (also known as Tait-Brians' angles)

3) Euler's parameters and "quaternions"

4) Rodriguez-Hamilton's parameters

5) Euler's axis and angle (also known as finite rotational axis)

6) a system based on Latitude and longitude

7) director cosines and rotational matrices

8) screw theory

Each of the methods, except 8), falls in one of the two following

groups:

a) a three independent parameters system

1) 2) 4) 5)

b) a more (>3) NOT-independent parameters system

3) 6) 7)

the screw theory describes either rotations and displacements and

from the rotational point of view is "similar" to method 5.

Every system which consists of 3 parameters has mathematical

singularities for a few particular values of its parameters. (i.e.

the Rodriguez parameters for TETA==90 degrees or the Euler angles

if the node axis is not defined, ....) and there are high

numerical errors when the parameters are closed to these

singularity points. Any way this is not a problem if one is sure

that in its problem he will never reach or approach these points.

Every system consisting in more than 3 parameters removes the

singularity but forces to use a set of NOT independent

"coordinates". The meaning of a set of coordinates of group 2 is

also generally less expressive or clear for Humans.

I do think that the choice of an angular parameter system can be

done taking into account many aspects as (for instance):

1) the meaning of each parameter in each practical application

2) the risk to fall into a singularity

3) the math simplicity

......

.... and last but not the least

999) the researcher's experience and beliefs...

I like to use a 4*4 matrix method based on the well known

Transformation matrices approach. I extended this method to full

kinematics (speed and acceleration) and dynamics (wrenches

(forces+torques), linear and angular momentum, inertial terms). I

think that, at least for computer applications, this method is

very convenient for its programming simplicity and because it has

not any math singularity. A 3*3 sub-matrix of a 4*4 matrix is the

well known rotational matrix that can be easily built from

whatever of the above coordinate systems. However, the inverse

transformation (i.e. from matrix to parameters) is, of course,

possible and easy only if we are not close to one of the math sin-

gularities of that parameters set.

Using this approach, my colleagues and I have been developing

SPACELIB, a computer library for the kinematic and dynamic analy-

sis of systems of rigid bodies. Using this library we have reali-

zed many computer programs for the study of robots and for human

body simulation (direct and inverse dynamics and direct and inver-

se kinematics).

Summarizing:

1) each method has good and bad points

2) I like matrices

3) I am interested in the exchange of papers, SHORT mails and

computer libraries.

If someone is interested in my topics, he can have a look at the

proceedings of the last Int. Congress of Biomechanics (LA 1989) or

its satellite meeting on computer simulation (DAVIS, CA 1989) or

write or mail to me directly.

I am looking forward to hearing from someone about angles,

matrices or grammar mistakes.

Yours Faithfully

Giovanni LEGNANI

Giovanni LEGNANI

University of Brescia

Mech. Eng. Dep.

Via Valotti 9

25060 MOMPIANO BS

ITALY

tel +39 30 3996.446

fax +39 30 303681