H.j. Woltring, Fax/tel +31.40.413 744

05-18-1992, 03:06 AM

Dear Biomch-L readers,

Following the comments from Jesus Dapena and Paolo de Leva the other week on

the ISB Draft for 3-D Kine(ma)tics Standardization & Terminology, I'd like to

join them with complimenting the work by Peter Cavanagh and his colleagues.

Standardization is not a purely scientific matter in which "objective, scien-

tific" norms prevail (in my book, they ought!), but also a pragmatic, democra-

tic endeavour in which existing custums out in the field (and their authors?)

look for rationalization, recognition, and acceptance, while users and vendors

who have already systems with particular conventions in use or on the market

may not be highly thrilled by the idea of having to change their ways. Thus,

the academic ideal of uniqueness, universatility, and regularity may have to

be sacrificed; I think that the authors of the current draft have made this

very clear.

Below are some comments on the published texts, ordered as the drafts them-

selves; I hope thus to contribute my 3 cents to the present debate (-: as I

read on UseNet these days, inflation has hollowed out the original proverb

which only claims 2 cents :-)

A. Terminology

It would be useful if the text be more specific on terminology: definition of

terms such as poses and displacements, positions and translations, attitudes

and rotations, Cardanic and Euler angles, finite and instantaneous helical,

screw, and rotation axes would be recommendable. This should take place under

a separate heading rather than at different places throughout the draft. Some

of the draft's headings are concerned with poses, some with displacements:

thus, Part 3 is called "Global Displacements", while Parts 4 and 5 are called

"Global Attitudes" and "Relative Attitudes", respectively.

I think that "Global" should be replaced by "Absolute", in line with common

parlance in the Biomechanics / Kinesiology field where "absolute movement"

stands for segment movement with respect to an external, usually inertial

coordinate frame, and where "relative movement" is tantamount to joint

movement.

B. Kinematics and Kinetics / Dynamics

While the report is entitled "Recommendations for Standardization in the

Reporting of Kinematic Data", it is also concerned with Kinetics / Dynamics.

For example, in Part 5 (Relative Attitudes), reference is made to decomposi-

tion ...

"about axes which can be anatomically meaningful at the sacrifice of

establishing a reference frame with non-orthogonal axes. As long as

forces and moments are not resolved along those non-orthogonal axes,

this does not present a problem".

It would be interesting to see how this can be acceptable for kinematics

if it is anathema for kinetics, in particular because these non-orthogonal

axes are not rigid with respect to each other while those of cartesian

coordinate frames are.

Furthermore, Part 2 is concerned with Segmental Centres of Mass, at which

local Reference Frames are to be defined. Masses, however, are not relevant

in pure kinematics.

Finally, Part 6 (Joint Moments) provides some tentative recommendations which

leave open about what axes moments should be decomposed: if not about the axes

of a "Joint Coordinate System" (see below), should this be about the global /

inertial axes, about the proximal segment's axes, or about the distal segment's

axes? Patterns of moments will look quite different under these alternative

representations. Some authors have even proposed to decompose forces and mo-

ments along the generally oblique axes of a Cardanic convention, thus imposing

the increased correlations for Cardanic angles as compared to `helical angles'

also on joint forces and moments; fortunately, the ISB Committe does not advo-

cate this.

C. Definition of a Global Reference Frame

It might be useful to point out that the positive Z-axis in aeronautics points

downward. The choice of (Xg,Yg) in the sagittal plane on behalf of planar

studies seems to indicate a predisposition towards sagittal gait studies,

rather than frontal analysis as, in e.g., various posture, sports and ergo-

nomics studies.

D. Definition of Local Reference Frames

The draft proposes that local segment reference frames should have their

origin in the segment's centre of gravity, and that the Y-axis should be

proximally oriented (in the oblong direction), while the Z-axis should be

oriented towards the right of the subject (when standing in the neutral,

anatomical pose, i.e., erect, arms along the body with the head, palms

and feet directed forward). In this pose, the local and global coordinate

systems are parallel *if* the local Y-axis points upward which is not the

case for the head and upper trunk. A better convention would seem to be

that all local Y-axes should point upward when the subject is in the neutral,

anatomical pose.

E. The 4 x 4 Matrix Notation Model

The use of this model implies a particular approach for representing segment

positions and translations, as it implies the conventional rigid-body model

Xg = Rgl Xl + Pgl

where Xg is the position vector in global (inertial) coordinates of a point

on a body segment, Xl its position vector in local (segment-fixed) coordina-

tes, Rgl the orthonormal attitude matrix whose elements are the cosines of the

angles between corresponding coordinate axes of the two coordinate systems,

and Pgl the position vector of the local coordinate system's origin in the

global coordinate system. For joint poses and displacements, the local system

becomes the distal system, and the global system becomes the proximal system.

This model is to be seen only as an unambiguous formulation of the proposed

standards: it is emphasized in the draft's preamble that investigators should

be free to collect and process data at their own discretion, and the draft is

merely concerned with unifying the way that results should be presented (in

the literature, in clinical reports?).

With the chosen XYZ axes, the rotation sequence example given in Part 4 on

Global Attitudes does not correspond with the conventional Cardanic/Eulerian

sequence in which the ab/adduction axis corresponds to the floating axis:

the Y-axis corresponds to axial rotation of a segment when in the neutral or

anatomical pose. Furthermore, neither the attitude matrices of the head and

upper trunk nor those of the neck and of the "joint" between upper and lower

trunk are equal to the identity matrix in this pose. They would if all Y-axes

would point upwards in this pose as proposed above.

F. Relative Attitudes (Joint angles)

Based on the quoted work of a number of highly respected, North American

biomechanicians during the 70-ties and 80-ties, the draft currently opts

for a Cardanic/Eulerian sequence with one "rotation" axis imbedded in the

proximal segment (coincident with planar flexion/extension), one imbedded

in the distal segment (coincident with planar endo/exorotation), and one

normal to these two, the floating axis (identified with ab/adduction).

The example provided is for the knee joint.

Before making this choice, the draft recommends:

"Joint coordinate systems (which might better be called Joint Rotation

Conventions) defined for each joint individually. This system allows

sequence independent rotations about axes which can be anatomically

meaningful ..."

after which a choice is made for the "most well known example of such systems

[ as ] developed for the knee by Grood & Suntay (1983) and Chao (1986)"; see

also Chao (1980) for an even earlier presentation. Actually, figure 3 in the

Draft proposes to decompose positions/displacements along the generally oblique

axes of an (electrogoniometric) linkage system that mechanizes the proposed

Joint Rotation Convention; this seems at variance with the position definition

(Pgl) stated above.

As discussed - perhaps ad nauseam? - on the Biomch-L forum during the previous

2 months and at even earlier occasions (see the Biomch-L archives for February

and March 1990 and for March and April 1992), the Cardanic/Eulerian convention

has certain problems. In the present context, I fail to see how the arbitrary

choice of some imbedded and floating axes "can be anatomically meaningful" if

no anatomists are being referenced. Others have used terms like "physiological

angles" and "orthopaedic angles" ...

Next, the Committee kindly refers to my own work:

"Woltring (1990, 1991) and others have supported the use of helical axes

for the description of joint motion since it avoids some problems - such

as gimbal lock - inherent in Euler angle representations. More recently,

Woltring suggests the use of: "an attitude vector standard". At present,

we are not proposing a standard for this approach as debate continues on

its clinical application. Should there be a widespread support for such

a representation we would certainly consider a recommendation for standar-

ization of a helical axis representation."

I sincerely hope that "helical axes" will never be used as standards for *

describing joint poses (i.e., positions and attitudes), while I do feel *

that they are useful for describing joint displacements (movement, motion, *

translation, rotation) through instantaneous kinematics: velocities, acce- *

lerations, 3-D Centres of Rotation. As an approximation to Instantaneous *

Helical Axes between sampling times in digital measurement and processing *

systems, also the "Finite Helical Axis" or FHA is a usefuul tool. *

However, recent authors have used this so-called Finite Helical Axis for

describing joint *poses* rather than *displacements*, in which a current pose

is described in terms of a rotation about and translation along a directed

line in space through which this pose can be formally attained from the

reference pose. They typically assess the amount of rotation theta about and

translation d along the FHA, the unit direction vector N (i.e., N'N = 1, where

' denotes transposition) of the FHA, and the piercing point Q of the FHA with

some reference plane. Subsequently, the angles between N and the reference

frame's coordinate axes are considered, in combination with the total rotation

angle theta.

A serious problem with this approach is that the direction of N becomes highly

sensitive to measurement errors and coordinate system definition uncertainties

(even though its length remains 1) if theta becomes small (Woltring et al, JoB

1985). Also Q becomes ill-determined in that case, as is the displacement d

which is equal to the projection onto N of the displacement of some reference

point P on the moving object. Only the angle theta remains well-determined --

for a diligently chosen measurement configuration.

Thus (confining myself to attitudes only) plotting theta and N over time

results in very noisy components of N whenever theta approaches 0. Comparing

these 4 `dependent' entities with 3 `independent' Cardanic/Eulerian angles is

not easy, as recently confirmed by Small (1992; cf. her recent posting onto

Biomch-L). Note that in gimbal lock, the Cardanic/Eulerian approach has only

two degrees of freedom, as one degree of freedom is lost due to a mathematical

*and* physical/biomechanical singularity caused by the two imbedded axes

becoming parallel. Close to gimbal lock, Cardanic/Eulerian angles are

similarly noisy as N in the helical case when theta is close to 0.

In my own work, I have, therefore, proposed to use the product THETA = theta N

as an attitude "vector" whose components can be meaningfully interpreted as

angular quantifiers, and which are always regular given a suitable measurement

set-up. If the total "rotation" angle theta is small (

Following the comments from Jesus Dapena and Paolo de Leva the other week on

the ISB Draft for 3-D Kine(ma)tics Standardization & Terminology, I'd like to

join them with complimenting the work by Peter Cavanagh and his colleagues.

Standardization is not a purely scientific matter in which "objective, scien-

tific" norms prevail (in my book, they ought!), but also a pragmatic, democra-

tic endeavour in which existing custums out in the field (and their authors?)

look for rationalization, recognition, and acceptance, while users and vendors

who have already systems with particular conventions in use or on the market

may not be highly thrilled by the idea of having to change their ways. Thus,

the academic ideal of uniqueness, universatility, and regularity may have to

be sacrificed; I think that the authors of the current draft have made this

very clear.

Below are some comments on the published texts, ordered as the drafts them-

selves; I hope thus to contribute my 3 cents to the present debate (-: as I

read on UseNet these days, inflation has hollowed out the original proverb

which only claims 2 cents :-)

A. Terminology

It would be useful if the text be more specific on terminology: definition of

terms such as poses and displacements, positions and translations, attitudes

and rotations, Cardanic and Euler angles, finite and instantaneous helical,

screw, and rotation axes would be recommendable. This should take place under

a separate heading rather than at different places throughout the draft. Some

of the draft's headings are concerned with poses, some with displacements:

thus, Part 3 is called "Global Displacements", while Parts 4 and 5 are called

"Global Attitudes" and "Relative Attitudes", respectively.

I think that "Global" should be replaced by "Absolute", in line with common

parlance in the Biomechanics / Kinesiology field where "absolute movement"

stands for segment movement with respect to an external, usually inertial

coordinate frame, and where "relative movement" is tantamount to joint

movement.

B. Kinematics and Kinetics / Dynamics

While the report is entitled "Recommendations for Standardization in the

Reporting of Kinematic Data", it is also concerned with Kinetics / Dynamics.

For example, in Part 5 (Relative Attitudes), reference is made to decomposi-

tion ...

"about axes which can be anatomically meaningful at the sacrifice of

establishing a reference frame with non-orthogonal axes. As long as

forces and moments are not resolved along those non-orthogonal axes,

this does not present a problem".

It would be interesting to see how this can be acceptable for kinematics

if it is anathema for kinetics, in particular because these non-orthogonal

axes are not rigid with respect to each other while those of cartesian

coordinate frames are.

Furthermore, Part 2 is concerned with Segmental Centres of Mass, at which

local Reference Frames are to be defined. Masses, however, are not relevant

in pure kinematics.

Finally, Part 6 (Joint Moments) provides some tentative recommendations which

leave open about what axes moments should be decomposed: if not about the axes

of a "Joint Coordinate System" (see below), should this be about the global /

inertial axes, about the proximal segment's axes, or about the distal segment's

axes? Patterns of moments will look quite different under these alternative

representations. Some authors have even proposed to decompose forces and mo-

ments along the generally oblique axes of a Cardanic convention, thus imposing

the increased correlations for Cardanic angles as compared to `helical angles'

also on joint forces and moments; fortunately, the ISB Committe does not advo-

cate this.

C. Definition of a Global Reference Frame

It might be useful to point out that the positive Z-axis in aeronautics points

downward. The choice of (Xg,Yg) in the sagittal plane on behalf of planar

studies seems to indicate a predisposition towards sagittal gait studies,

rather than frontal analysis as, in e.g., various posture, sports and ergo-

nomics studies.

D. Definition of Local Reference Frames

The draft proposes that local segment reference frames should have their

origin in the segment's centre of gravity, and that the Y-axis should be

proximally oriented (in the oblong direction), while the Z-axis should be

oriented towards the right of the subject (when standing in the neutral,

anatomical pose, i.e., erect, arms along the body with the head, palms

and feet directed forward). In this pose, the local and global coordinate

systems are parallel *if* the local Y-axis points upward which is not the

case for the head and upper trunk. A better convention would seem to be

that all local Y-axes should point upward when the subject is in the neutral,

anatomical pose.

E. The 4 x 4 Matrix Notation Model

The use of this model implies a particular approach for representing segment

positions and translations, as it implies the conventional rigid-body model

Xg = Rgl Xl + Pgl

where Xg is the position vector in global (inertial) coordinates of a point

on a body segment, Xl its position vector in local (segment-fixed) coordina-

tes, Rgl the orthonormal attitude matrix whose elements are the cosines of the

angles between corresponding coordinate axes of the two coordinate systems,

and Pgl the position vector of the local coordinate system's origin in the

global coordinate system. For joint poses and displacements, the local system

becomes the distal system, and the global system becomes the proximal system.

This model is to be seen only as an unambiguous formulation of the proposed

standards: it is emphasized in the draft's preamble that investigators should

be free to collect and process data at their own discretion, and the draft is

merely concerned with unifying the way that results should be presented (in

the literature, in clinical reports?).

With the chosen XYZ axes, the rotation sequence example given in Part 4 on

Global Attitudes does not correspond with the conventional Cardanic/Eulerian

sequence in which the ab/adduction axis corresponds to the floating axis:

the Y-axis corresponds to axial rotation of a segment when in the neutral or

anatomical pose. Furthermore, neither the attitude matrices of the head and

upper trunk nor those of the neck and of the "joint" between upper and lower

trunk are equal to the identity matrix in this pose. They would if all Y-axes

would point upwards in this pose as proposed above.

F. Relative Attitudes (Joint angles)

Based on the quoted work of a number of highly respected, North American

biomechanicians during the 70-ties and 80-ties, the draft currently opts

for a Cardanic/Eulerian sequence with one "rotation" axis imbedded in the

proximal segment (coincident with planar flexion/extension), one imbedded

in the distal segment (coincident with planar endo/exorotation), and one

normal to these two, the floating axis (identified with ab/adduction).

The example provided is for the knee joint.

Before making this choice, the draft recommends:

"Joint coordinate systems (which might better be called Joint Rotation

Conventions) defined for each joint individually. This system allows

sequence independent rotations about axes which can be anatomically

meaningful ..."

after which a choice is made for the "most well known example of such systems

[ as ] developed for the knee by Grood & Suntay (1983) and Chao (1986)"; see

also Chao (1980) for an even earlier presentation. Actually, figure 3 in the

Draft proposes to decompose positions/displacements along the generally oblique

axes of an (electrogoniometric) linkage system that mechanizes the proposed

Joint Rotation Convention; this seems at variance with the position definition

(Pgl) stated above.

As discussed - perhaps ad nauseam? - on the Biomch-L forum during the previous

2 months and at even earlier occasions (see the Biomch-L archives for February

and March 1990 and for March and April 1992), the Cardanic/Eulerian convention

has certain problems. In the present context, I fail to see how the arbitrary

choice of some imbedded and floating axes "can be anatomically meaningful" if

no anatomists are being referenced. Others have used terms like "physiological

angles" and "orthopaedic angles" ...

Next, the Committee kindly refers to my own work:

"Woltring (1990, 1991) and others have supported the use of helical axes

for the description of joint motion since it avoids some problems - such

as gimbal lock - inherent in Euler angle representations. More recently,

Woltring suggests the use of: "an attitude vector standard". At present,

we are not proposing a standard for this approach as debate continues on

its clinical application. Should there be a widespread support for such

a representation we would certainly consider a recommendation for standar-

ization of a helical axis representation."

I sincerely hope that "helical axes" will never be used as standards for *

describing joint poses (i.e., positions and attitudes), while I do feel *

that they are useful for describing joint displacements (movement, motion, *

translation, rotation) through instantaneous kinematics: velocities, acce- *

lerations, 3-D Centres of Rotation. As an approximation to Instantaneous *

Helical Axes between sampling times in digital measurement and processing *

systems, also the "Finite Helical Axis" or FHA is a usefuul tool. *

However, recent authors have used this so-called Finite Helical Axis for

describing joint *poses* rather than *displacements*, in which a current pose

is described in terms of a rotation about and translation along a directed

line in space through which this pose can be formally attained from the

reference pose. They typically assess the amount of rotation theta about and

translation d along the FHA, the unit direction vector N (i.e., N'N = 1, where

' denotes transposition) of the FHA, and the piercing point Q of the FHA with

some reference plane. Subsequently, the angles between N and the reference

frame's coordinate axes are considered, in combination with the total rotation

angle theta.

A serious problem with this approach is that the direction of N becomes highly

sensitive to measurement errors and coordinate system definition uncertainties

(even though its length remains 1) if theta becomes small (Woltring et al, JoB

1985). Also Q becomes ill-determined in that case, as is the displacement d

which is equal to the projection onto N of the displacement of some reference

point P on the moving object. Only the angle theta remains well-determined --

for a diligently chosen measurement configuration.

Thus (confining myself to attitudes only) plotting theta and N over time

results in very noisy components of N whenever theta approaches 0. Comparing

these 4 `dependent' entities with 3 `independent' Cardanic/Eulerian angles is

not easy, as recently confirmed by Small (1992; cf. her recent posting onto

Biomch-L). Note that in gimbal lock, the Cardanic/Eulerian approach has only

two degrees of freedom, as one degree of freedom is lost due to a mathematical

*and* physical/biomechanical singularity caused by the two imbedded axes

becoming parallel. Close to gimbal lock, Cardanic/Eulerian angles are

similarly noisy as N in the helical case when theta is close to 0.

In my own work, I have, therefore, proposed to use the product THETA = theta N

as an attitude "vector" whose components can be meaningfully interpreted as

angular quantifiers, and which are always regular given a suitable measurement

set-up. If the total "rotation" angle theta is small (