unknown user

02-17-1990, 02:27 AM

Dear Biomch-L Readers,

I appreciate Dr. Herman Woltring's invitation to debate his proposal for

standardization of 3-D joint attitude representation. In this first response

I have three goals. These are: a) to outline my current position on Herman's

proposal; b) to address the common misconceptions about sequence dependency

of finite rotations, and c) to stimulate thought and discussion about the

description of particle displacements and how the principles used are applied

to the description of rigid body displacements.

My Current Position

Let me start with the area of agreement. I completely accept Dr. Woltring's

proposal for the use of the helical axis to describe joint attitude and

rotational displacement. The helical axis has the important characteristic

that the magnitude of the rotation and translation are independent of the

coordinate system chosen. The analogy to particle displacement is clear,

the magnitude of the total displacement vector is also independent of the

coordinate system chosen.

Now the area of disagreement. Herman has proposed the helical displacement

be decomposed into orthogonal components in either body segment's coordinate

system. This produces six components, three for translation and three for

rotation. I believe this is not fully satisfactory for describing joint

translation and incorrect for the joint rotation components. It is my

position that the proper joint rotation components are those describe by

Fred Suntay and I. My reasons for this will become apparent in the course of

the debate.

Misconceptions

Herman and many others refer to the rotations Fred and I described as being

an "ordered sequence of rotations". I would agree they are an ordered triple,

just like the orthogonal components of particle displacement are an ordered

triple. I disagree with the terminology "ordered sequence" because the

final displacement is not dependent upon the sequence the rotations are

performed. Am I missing some other meaning of this phrase?

There is a general, and incorrect, belief that finite three dimensional

rotations are sequence dependent. This is not surprising as almost every

text on the subject gives the example of a book rotated using two different

sequences resulting in two different final positions. This example is passed

along without any careful analysis of what is actually happening.

The basic problem with the book example is the three axes used are those of

an orthogonal coordinate system located in one of the body segments. While

such axes do define independent translational degrees-of-freedom (dof),

they do not define independent rotational dof. The independent rotational

degrees-of-freedom are those Herman referred to: a fixed axis in each body

segment and their common perpendicular. The orientation of the fixed axes

are chosen for convenience. This is similar to selecting an appropriate

orthogonal system for describing particle displacements.

To better understand the origin of the sequence dependency I will give a

similar example for particle displacement. It starts by first specifying

the displacements (x,y,z) without specifying the three independent dof.

Next, perform the displacements along the axes of any orthogonal

coordinate system and note its final location. Third change the orientation

of the orthogonal coordinate system. We still have three independent

dof but the directions have a different physical significance. Now

perform the three component displacements in any sequence. The final

position is clearly not the same as before. The problem is not that particle

displacements are sequence dependent, it's that we changed the independent

dof used.

Independent Rotational Degrees-of-Freedom

At the risk of being unnecessarily redundant I will again describe an

appropriate set of independent rotation axes. First locate two body fixed

axes, one in each body segment. These axes are selected so that rotation

about them is a motion of interest. The third rotation axis is the common

perpendicular to the two body fixed axes. The three angles which define the

orientation were described in the paper with Fred Suntay. Briefly, they

are:

1. The rotation about the common perpendicular axis is given by the

angle between the two body fixed axes.

2. The rotation about each body fixed axis is given by the angle

between the common perpendicular and a reference line located

in the same body as the fixed axis. It is convenient to

select the reference line so it is also perpendicular to the

body fixed axis. In this way the body fixed axes are normal

to the plane which contains both the reference line and the

common perpendicular axis.

In closing this first round of the debate I will state the primary reasons

for using the system we proposed as the components of the helical rotation.

1. They are independent components.

2. They add (in a screw sense) to the total helical rotation.

3. They correspond to common clinical descriptions of joint

rotation.

4. They are easy to compute from the rotation matrix and have a

well defined mathematical relationship with the total helical

rotation.

Edward S. Grood

Cincinnati, Ohio, USA

I appreciate Dr. Herman Woltring's invitation to debate his proposal for

standardization of 3-D joint attitude representation. In this first response

I have three goals. These are: a) to outline my current position on Herman's

proposal; b) to address the common misconceptions about sequence dependency

of finite rotations, and c) to stimulate thought and discussion about the

description of particle displacements and how the principles used are applied

to the description of rigid body displacements.

My Current Position

Let me start with the area of agreement. I completely accept Dr. Woltring's

proposal for the use of the helical axis to describe joint attitude and

rotational displacement. The helical axis has the important characteristic

that the magnitude of the rotation and translation are independent of the

coordinate system chosen. The analogy to particle displacement is clear,

the magnitude of the total displacement vector is also independent of the

coordinate system chosen.

Now the area of disagreement. Herman has proposed the helical displacement

be decomposed into orthogonal components in either body segment's coordinate

system. This produces six components, three for translation and three for

rotation. I believe this is not fully satisfactory for describing joint

translation and incorrect for the joint rotation components. It is my

position that the proper joint rotation components are those describe by

Fred Suntay and I. My reasons for this will become apparent in the course of

the debate.

Misconceptions

Herman and many others refer to the rotations Fred and I described as being

an "ordered sequence of rotations". I would agree they are an ordered triple,

just like the orthogonal components of particle displacement are an ordered

triple. I disagree with the terminology "ordered sequence" because the

final displacement is not dependent upon the sequence the rotations are

performed. Am I missing some other meaning of this phrase?

There is a general, and incorrect, belief that finite three dimensional

rotations are sequence dependent. This is not surprising as almost every

text on the subject gives the example of a book rotated using two different

sequences resulting in two different final positions. This example is passed

along without any careful analysis of what is actually happening.

The basic problem with the book example is the three axes used are those of

an orthogonal coordinate system located in one of the body segments. While

such axes do define independent translational degrees-of-freedom (dof),

they do not define independent rotational dof. The independent rotational

degrees-of-freedom are those Herman referred to: a fixed axis in each body

segment and their common perpendicular. The orientation of the fixed axes

are chosen for convenience. This is similar to selecting an appropriate

orthogonal system for describing particle displacements.

To better understand the origin of the sequence dependency I will give a

similar example for particle displacement. It starts by first specifying

the displacements (x,y,z) without specifying the three independent dof.

Next, perform the displacements along the axes of any orthogonal

coordinate system and note its final location. Third change the orientation

of the orthogonal coordinate system. We still have three independent

dof but the directions have a different physical significance. Now

perform the three component displacements in any sequence. The final

position is clearly not the same as before. The problem is not that particle

displacements are sequence dependent, it's that we changed the independent

dof used.

Independent Rotational Degrees-of-Freedom

At the risk of being unnecessarily redundant I will again describe an

appropriate set of independent rotation axes. First locate two body fixed

axes, one in each body segment. These axes are selected so that rotation

about them is a motion of interest. The third rotation axis is the common

perpendicular to the two body fixed axes. The three angles which define the

orientation were described in the paper with Fred Suntay. Briefly, they

are:

1. The rotation about the common perpendicular axis is given by the

angle between the two body fixed axes.

2. The rotation about each body fixed axis is given by the angle

between the common perpendicular and a reference line located

in the same body as the fixed axis. It is convenient to

select the reference line so it is also perpendicular to the

body fixed axis. In this way the body fixed axes are normal

to the plane which contains both the reference line and the

common perpendicular axis.

In closing this first round of the debate I will state the primary reasons

for using the system we proposed as the components of the helical rotation.

1. They are independent components.

2. They add (in a screw sense) to the total helical rotation.

3. They correspond to common clinical descriptions of joint

rotation.

4. They are easy to compute from the rotation matrix and have a

well defined mathematical relationship with the total helical

rotation.

Edward S. Grood

Cincinnati, Ohio, USA