Herman J. Woltring

03-08-1990, 10:18 PM

Dear Biomch-L readers,

The number of (lengthy) responses on the current joint `angles' debate has been

rather small, both posted and emailed to me, so I agree with Ed Grood that it is

time to put things to an end with this reply. Of course, other subscribers are

free to continue, but the major philibusters should, perhaps, exercise some con-

straint.

* * * 1. Dwight Meglan's posting (Wed, 28 Feb 90 11:49 EDT) * * *

I am delighted that someone took the trouble of processing real joint data and

to show the different graphical results; time permitting, this is precisely what

I have been planning to do. Regretfully, Dwight's abstract as published in

the ORS '90 Proceedings on p. 558 (D.A. Meglan, J. Pisciotta, N. Berme and

S.R. Simon, Effective Use of Non-Sagittal Plane Joint Angles in Clinical Gait

Analysis) merely refers to what were called `different Euler angle systems: 1)

The fixed xyz axis system [Inman, V. et al., Human Walking, Williams & Wilkins,

1981], and 2) the floating xyz axis system (or Joint Coordinate System) [Chao,

E.Y., J. Biomech. 13:989-1006, 1980; Grood, E.S. & Suntay, W.J., J. Biomech.

Eng., 104:126-144, 1983]'. Thus, helical `angles' were not anticipated at the

time of abstract submission. On the other hand, co-ordinate system changes were

discussed, and Dwight's point that these should also be taken into account is

quite appropriate. However, this does not mean that the current joint `angle'

debate is superfluous since the differences in calculated angles under different

conventions can be quite dramatic, other things being equal.

Actually, Dwight's fixed xyz Euler angles are, in my mind, not at all what are

usually seen as Euler angles, i.e., some (well defined) sequence in the Carda-

nic/Eulerian sense that we have been debating on this list, but angles between

projections in fixed laboratory coordinate planes (XY, YZ, ZX) of spatial lines

(e.g., longitudinal limb axes). Since these angles are even less well-behaved

than Cardanic angles, I decided not to discuss them in the current debate.

Of course, the problem with abstracts submitted to large conferences is that

they may be obsolete or incomplete (because of new results) once the conference

takes place, and this may have been the case with the OSU study. Fortunately,

most conference organizers have the flexibility to accept such changes.

That I make this seemingly unfriendly remark has a real purpose: when I quoted

the debate between Alice and Humpty Dumpty, this was not merely a (poor?) joke,

but an indication of what seems to happen perpetually in the present debate.

Words are used in slightly different meanings, with all the concomitant, Babylo-

nian confusion. Before the Iron Curtain became torn, there were big signs once

one entered Eastern from Western Germany saying "You are now entering the Ger-

man Democratic Republic" -- which I, because of a western bias, just thought

to have left. Jim Andrew's Letter to the Editor [J. Biomech. 1984, 155-158],

referred to by Ed Grood in his last posting, claims that Ed's use of the term

`co-ordinate system' is unconventional, and that Ed, in fact, merely described a

fictitious linkage system. In a similar fashion, I think that Ed has reinter-

preted the term `sequence independence' in an unusual and unnecessarily confu-

sing way.

While Dwight's `shotgun' approach is to be commended, his explanations and con-

clusions worry me to some extent. When he says `It is important to note, how-

ever, that the finite screws describe the joint motion as a single rotation

about an axis in space which is exactly what the joint motion is, not a sequence

of ordered rotations', I believe that he is quite mistaken. Again, the helical

convention merely proposes to describe a current or actual joint attitude AS IF

it is attained from the reference attitude via a single, helical motion about

some directed line in space, it does NOT claim that this, in fact, occurs.

What one can do, though, is to view the movement, at each moment in time, as an

i n s t a n t a n e o u s rotation about plus translation along some directed

line in space. Now we talk about the Instantaneous Helical Axis which is some-

thing quite different. At each time instant, the `amount' of movement is defi-

ned by the instantaneous rotation velocity about and translation velocity along

this IHA, while the `mode' of the movement is defined by the position of (some

point on) the IHA and the unit direction vector of the IHA.

Dwight suggests that one should decompose the helical angle `vector' into com-

ponents along the generally oblique axes of a Cardanic linkage system like the

one preferred by Ed Grood. It may be that many orthopaedists are now familiar

with this Cardanic convention (but I believe that the non-Eulerian, `projected

angles' in San Diego are well understood there), but that is not a valid reason

to stick to them, especially if there are serious disadvantages. Good surgeons

are keen on learning new things once they believe that it will help them in

their work.

Jim Andrews mentions three key arguments for any joint angle definition: they

should explain the movement (or orientation) easily, they should not exhibit

singularities, and they should be easy to calculate. The first argument ap-

plies, in my mind, both to helical and Cardanic angles, with a preference for

the former since I think that sequence effects are more difficult to explain

than the notion of orthogonally decomposing the helical `vector'. The second

argument is in favour of the helical approach, where it might be useful to

note that close to gimbal-lock, certain differential displacements of a joint

or body will hardly be reflected by the chosen joint angle `co-ordinates',

whereas other differential displacements of the same magnitude will result in

very strong changes in these angles; this makes visual interpretation of angular

graphs rather difficult. The third argument is obsolete with current computa-

tional facilities.

Instead, the advantages of maximizing orthogonality should be clear to anyone

who wishes to describe joint angulation and to relate it to forces and moments

which are true vectors, commonly decomposed in (truely) independent, orthogonal

components.

Under various Cardanic conventions, strong flexion, abduction, and endorotation

under one convention become about the same flexion, but adduction and/or exo-

rotation under another one, while no strong differences are observable when all

angles are small. Thus, different p a t t e r n s of joint angles are to be

expected necessitating agreement between investigators (and their institutions?)

in order to allow valid comparisons. Since the `helical convention' more-or-less

provides the mean values of all possible Cardanic conventions, this might be an

additional argument in its favour, despite the generally non-physical nature of

its component angles.

* * * 2. Ed Grood's posting of Sat 3 Mar 90 14:29 EST * * *

I appreciate Ed's attempt to clearly define his key considerations, which should

make it easy to follow the debate.

a. "Independent" in the non-statistical, mathematical sense. Agreed, but in-

dependence is better if it applies also in the statistical, mathematical sense

with orthogonal (uncorrelated) components. Besides, some of Ed's examples for

his 6 generalized co-ordinates do not reflect independence in his use of the

term: the direction cosines of a position or direction `vector' are not, since

their squares add up to unity. In an earlier email note or posting (off my

head, there's too much paper on this debate already), Ed suggested using the

length and two of the three direction cosines as independent variables; this,

in my mind, is extremely unelegant and at variance with Jim Andrew's first two

conditions.

b. I agree that both Eulerian/Cardanic and helical angles are independent in the

above sense, but close to gimbal lock, they are quite differently behaved. This

may not be a problem in level, straight gait analysis (the current paradigm),

but it certainly is a problem in complex, sportive movements. When Ed claims

that generalized co-ordinates have trajectory properties, he reinterprets a

word from its intended meaning (for which I am the guilty one if I have been

insufficiently clear in my words): certainly, I thought in terms of a physical

path about and along the axes of Ed's linkage system, not in terms of some

abstract, mathematical space whose parameters have the dimension of angles,

behave as angles in certain special cases, but which are, in general, not real,

physically identifiable angles. ["It looks like a duck, walks like a duck,

quacks like a duck, so it must be a ..."]. Similarly, I did not think in terms

of the continuous time-dependent movement that our joints exhibit, but about how

to `optimally' describe a given attitude at some specific time only.

c. Ed's argument on `different sets of independent co-ordinates' comes back on

what I stated above with Dwight Meglan. Ed defines `sequence' in an unusual,

and, in my mind, unnecessarily confusing way. If I first (i.e., proximally)

translate x along the X-axis, then (distally) y along the (displaced) Y-axis

of a given, Cartesian co-ordinate system, I wind up in the same position as

when I had first (proximally) translated y along the Y-axis and then (distally)

x along the (displaced) X-axis. For rotations, however, different attitudes

are attained. Furthermore, the distinction between temporal and geometrical se-

quences is not used with particle displacements because it is not needed there.

When Ed says that this terminology `... is unnecessary when discussing rotatio-

nal displacements. We only need to talk about specifying the set of co-ordina-

tes to be used for a particular problem and whether such co-ordinates are com-

mutative in the ordinary sense', I do not know whether he refers to Cartesian

co-ordinates or linkage co-ordinates (when are they the same?) and to what kind

of `ordinary sequence' he refers to.

d. `Woltring angles'. While I feel honoured to see my name attached to gene-

rally non-existing angles (at least in real, physical space), I would prefer to

stick to the name `helical angles'. I maintain my position that, for attitude

description purposes, there is no need that these angles be generally identifi-

able with some specific physical angles; their orthogonality is the more impor-

tant property. Jim Andrews made some related remarks on this point.

e. At the present time, there is no clinically well-accepted convention for 3-D

joint attitude parametrization. Various orthopaedic surgeons accept what their

engineers tell them, but if these engineers cannot agree amongst themselves ...

f. Cardanic/Eulerian angles fail close to and at gimbal-lock, in both direct

and inverse dynamics.

g. Ed's angles are `commutative' and `additive' in his definition of the terms,

at the expense of a generally non-Cardanic, oblique `co-ordinate system'.

h. There is no need to advocate the use of any proposed joint convention for

joint ATTITUDE quantification because of its properties to explain certain

peculiarities of particular conventions.

Ed, let's wait and see what the (electronic) community has to say; may-be, its

contribution will eventually make a published debate useful.

* * * 3. Dr Legnani's posting of Tue 6 Mar 90 11:22 N * * *

Dr Legnani has given a nice summary of various attitude parametrization methods;

in the present debate (human interpretation of joint angle graphs), his `3 para-

meter models' are the relevant ones.

His claim that `(e)very system which consists of three parameters has mathema-

tical singularities for a few particular values of its parameters' does (unless

I am mistaken) not always hold true. If they were, the covariance matrix for

these parameters should become unbounded when the singular points are approach-

ed. This is not the case when calculating the covariance matrix for the `vec-

tor' THETA (with 0 .le. theta .le. pi) using the relation THETA = theta N and

formulae (20) and (21) in a paper on finite helical axes and centres of rotation

in the Journal of Biomechanics 1985, p. 382. Working out the various partial

derivatives and matrix products yields

COV(THETA) = k [ {theta^2/(1 - cos(theta))} (I - NN') + 2 cos(theta)^2 NN' ]

which, for theta --> 0, reduces to 2 k I,

where I is the identity matrix, and k the variance of incremental disturbances

on the attitude matrix in arbitrary directions; see the quoted paper for further

details where k is a function of isotropic measurement noise per co-ordinate

axis and of an isotropic landmark distribution in photogrammetric rigid-body

reconstruction.

If theta approaches 2 k pi, with k integer and non-zero, the covariance matrix

becomes unbounded, but this is irrelevant in the present debate where arbitrary

attitudes can be represented for 0 .le. theta .le. pi; cf. the planar case where

the unit direction vector N is replaced by a + or - sign, with theta periodic in

2 pi both in the 2-D and 3-D cases.

While I have no proof, I believe that the `helical vector' is well behaved also

for other situations than the isotropic case referred to above, as long as the

landmark distribution and the noise are non-pathological. Proving this might be

a nice challenge for a mathematically oriented MSc or PhD thesis ?

Herman J. Woltring

Eindhoven, The Netherlands

The number of (lengthy) responses on the current joint `angles' debate has been

rather small, both posted and emailed to me, so I agree with Ed Grood that it is

time to put things to an end with this reply. Of course, other subscribers are

free to continue, but the major philibusters should, perhaps, exercise some con-

straint.

* * * 1. Dwight Meglan's posting (Wed, 28 Feb 90 11:49 EDT) * * *

I am delighted that someone took the trouble of processing real joint data and

to show the different graphical results; time permitting, this is precisely what

I have been planning to do. Regretfully, Dwight's abstract as published in

the ORS '90 Proceedings on p. 558 (D.A. Meglan, J. Pisciotta, N. Berme and

S.R. Simon, Effective Use of Non-Sagittal Plane Joint Angles in Clinical Gait

Analysis) merely refers to what were called `different Euler angle systems: 1)

The fixed xyz axis system [Inman, V. et al., Human Walking, Williams & Wilkins,

1981], and 2) the floating xyz axis system (or Joint Coordinate System) [Chao,

E.Y., J. Biomech. 13:989-1006, 1980; Grood, E.S. & Suntay, W.J., J. Biomech.

Eng., 104:126-144, 1983]'. Thus, helical `angles' were not anticipated at the

time of abstract submission. On the other hand, co-ordinate system changes were

discussed, and Dwight's point that these should also be taken into account is

quite appropriate. However, this does not mean that the current joint `angle'

debate is superfluous since the differences in calculated angles under different

conventions can be quite dramatic, other things being equal.

Actually, Dwight's fixed xyz Euler angles are, in my mind, not at all what are

usually seen as Euler angles, i.e., some (well defined) sequence in the Carda-

nic/Eulerian sense that we have been debating on this list, but angles between

projections in fixed laboratory coordinate planes (XY, YZ, ZX) of spatial lines

(e.g., longitudinal limb axes). Since these angles are even less well-behaved

than Cardanic angles, I decided not to discuss them in the current debate.

Of course, the problem with abstracts submitted to large conferences is that

they may be obsolete or incomplete (because of new results) once the conference

takes place, and this may have been the case with the OSU study. Fortunately,

most conference organizers have the flexibility to accept such changes.

That I make this seemingly unfriendly remark has a real purpose: when I quoted

the debate between Alice and Humpty Dumpty, this was not merely a (poor?) joke,

but an indication of what seems to happen perpetually in the present debate.

Words are used in slightly different meanings, with all the concomitant, Babylo-

nian confusion. Before the Iron Curtain became torn, there were big signs once

one entered Eastern from Western Germany saying "You are now entering the Ger-

man Democratic Republic" -- which I, because of a western bias, just thought

to have left. Jim Andrew's Letter to the Editor [J. Biomech. 1984, 155-158],

referred to by Ed Grood in his last posting, claims that Ed's use of the term

`co-ordinate system' is unconventional, and that Ed, in fact, merely described a

fictitious linkage system. In a similar fashion, I think that Ed has reinter-

preted the term `sequence independence' in an unusual and unnecessarily confu-

sing way.

While Dwight's `shotgun' approach is to be commended, his explanations and con-

clusions worry me to some extent. When he says `It is important to note, how-

ever, that the finite screws describe the joint motion as a single rotation

about an axis in space which is exactly what the joint motion is, not a sequence

of ordered rotations', I believe that he is quite mistaken. Again, the helical

convention merely proposes to describe a current or actual joint attitude AS IF

it is attained from the reference attitude via a single, helical motion about

some directed line in space, it does NOT claim that this, in fact, occurs.

What one can do, though, is to view the movement, at each moment in time, as an

i n s t a n t a n e o u s rotation about plus translation along some directed

line in space. Now we talk about the Instantaneous Helical Axis which is some-

thing quite different. At each time instant, the `amount' of movement is defi-

ned by the instantaneous rotation velocity about and translation velocity along

this IHA, while the `mode' of the movement is defined by the position of (some

point on) the IHA and the unit direction vector of the IHA.

Dwight suggests that one should decompose the helical angle `vector' into com-

ponents along the generally oblique axes of a Cardanic linkage system like the

one preferred by Ed Grood. It may be that many orthopaedists are now familiar

with this Cardanic convention (but I believe that the non-Eulerian, `projected

angles' in San Diego are well understood there), but that is not a valid reason

to stick to them, especially if there are serious disadvantages. Good surgeons

are keen on learning new things once they believe that it will help them in

their work.

Jim Andrews mentions three key arguments for any joint angle definition: they

should explain the movement (or orientation) easily, they should not exhibit

singularities, and they should be easy to calculate. The first argument ap-

plies, in my mind, both to helical and Cardanic angles, with a preference for

the former since I think that sequence effects are more difficult to explain

than the notion of orthogonally decomposing the helical `vector'. The second

argument is in favour of the helical approach, where it might be useful to

note that close to gimbal-lock, certain differential displacements of a joint

or body will hardly be reflected by the chosen joint angle `co-ordinates',

whereas other differential displacements of the same magnitude will result in

very strong changes in these angles; this makes visual interpretation of angular

graphs rather difficult. The third argument is obsolete with current computa-

tional facilities.

Instead, the advantages of maximizing orthogonality should be clear to anyone

who wishes to describe joint angulation and to relate it to forces and moments

which are true vectors, commonly decomposed in (truely) independent, orthogonal

components.

Under various Cardanic conventions, strong flexion, abduction, and endorotation

under one convention become about the same flexion, but adduction and/or exo-

rotation under another one, while no strong differences are observable when all

angles are small. Thus, different p a t t e r n s of joint angles are to be

expected necessitating agreement between investigators (and their institutions?)

in order to allow valid comparisons. Since the `helical convention' more-or-less

provides the mean values of all possible Cardanic conventions, this might be an

additional argument in its favour, despite the generally non-physical nature of

its component angles.

* * * 2. Ed Grood's posting of Sat 3 Mar 90 14:29 EST * * *

I appreciate Ed's attempt to clearly define his key considerations, which should

make it easy to follow the debate.

a. "Independent" in the non-statistical, mathematical sense. Agreed, but in-

dependence is better if it applies also in the statistical, mathematical sense

with orthogonal (uncorrelated) components. Besides, some of Ed's examples for

his 6 generalized co-ordinates do not reflect independence in his use of the

term: the direction cosines of a position or direction `vector' are not, since

their squares add up to unity. In an earlier email note or posting (off my

head, there's too much paper on this debate already), Ed suggested using the

length and two of the three direction cosines as independent variables; this,

in my mind, is extremely unelegant and at variance with Jim Andrew's first two

conditions.

b. I agree that both Eulerian/Cardanic and helical angles are independent in the

above sense, but close to gimbal lock, they are quite differently behaved. This

may not be a problem in level, straight gait analysis (the current paradigm),

but it certainly is a problem in complex, sportive movements. When Ed claims

that generalized co-ordinates have trajectory properties, he reinterprets a

word from its intended meaning (for which I am the guilty one if I have been

insufficiently clear in my words): certainly, I thought in terms of a physical

path about and along the axes of Ed's linkage system, not in terms of some

abstract, mathematical space whose parameters have the dimension of angles,

behave as angles in certain special cases, but which are, in general, not real,

physically identifiable angles. ["It looks like a duck, walks like a duck,

quacks like a duck, so it must be a ..."]. Similarly, I did not think in terms

of the continuous time-dependent movement that our joints exhibit, but about how

to `optimally' describe a given attitude at some specific time only.

c. Ed's argument on `different sets of independent co-ordinates' comes back on

what I stated above with Dwight Meglan. Ed defines `sequence' in an unusual,

and, in my mind, unnecessarily confusing way. If I first (i.e., proximally)

translate x along the X-axis, then (distally) y along the (displaced) Y-axis

of a given, Cartesian co-ordinate system, I wind up in the same position as

when I had first (proximally) translated y along the Y-axis and then (distally)

x along the (displaced) X-axis. For rotations, however, different attitudes

are attained. Furthermore, the distinction between temporal and geometrical se-

quences is not used with particle displacements because it is not needed there.

When Ed says that this terminology `... is unnecessary when discussing rotatio-

nal displacements. We only need to talk about specifying the set of co-ordina-

tes to be used for a particular problem and whether such co-ordinates are com-

mutative in the ordinary sense', I do not know whether he refers to Cartesian

co-ordinates or linkage co-ordinates (when are they the same?) and to what kind

of `ordinary sequence' he refers to.

d. `Woltring angles'. While I feel honoured to see my name attached to gene-

rally non-existing angles (at least in real, physical space), I would prefer to

stick to the name `helical angles'. I maintain my position that, for attitude

description purposes, there is no need that these angles be generally identifi-

able with some specific physical angles; their orthogonality is the more impor-

tant property. Jim Andrews made some related remarks on this point.

e. At the present time, there is no clinically well-accepted convention for 3-D

joint attitude parametrization. Various orthopaedic surgeons accept what their

engineers tell them, but if these engineers cannot agree amongst themselves ...

f. Cardanic/Eulerian angles fail close to and at gimbal-lock, in both direct

and inverse dynamics.

g. Ed's angles are `commutative' and `additive' in his definition of the terms,

at the expense of a generally non-Cardanic, oblique `co-ordinate system'.

h. There is no need to advocate the use of any proposed joint convention for

joint ATTITUDE quantification because of its properties to explain certain

peculiarities of particular conventions.

Ed, let's wait and see what the (electronic) community has to say; may-be, its

contribution will eventually make a published debate useful.

* * * 3. Dr Legnani's posting of Tue 6 Mar 90 11:22 N * * *

Dr Legnani has given a nice summary of various attitude parametrization methods;

in the present debate (human interpretation of joint angle graphs), his `3 para-

meter models' are the relevant ones.

His claim that `(e)very system which consists of three parameters has mathema-

tical singularities for a few particular values of its parameters' does (unless

I am mistaken) not always hold true. If they were, the covariance matrix for

these parameters should become unbounded when the singular points are approach-

ed. This is not the case when calculating the covariance matrix for the `vec-

tor' THETA (with 0 .le. theta .le. pi) using the relation THETA = theta N and

formulae (20) and (21) in a paper on finite helical axes and centres of rotation

in the Journal of Biomechanics 1985, p. 382. Working out the various partial

derivatives and matrix products yields

COV(THETA) = k [ {theta^2/(1 - cos(theta))} (I - NN') + 2 cos(theta)^2 NN' ]

which, for theta --> 0, reduces to 2 k I,

where I is the identity matrix, and k the variance of incremental disturbances

on the attitude matrix in arbitrary directions; see the quoted paper for further

details where k is a function of isotropic measurement noise per co-ordinate

axis and of an isotropic landmark distribution in photogrammetric rigid-body

reconstruction.

If theta approaches 2 k pi, with k integer and non-zero, the covariance matrix

becomes unbounded, but this is irrelevant in the present debate where arbitrary

attitudes can be represented for 0 .le. theta .le. pi; cf. the planar case where

the unit direction vector N is replaced by a + or - sign, with theta periodic in

2 pi both in the 2-D and 3-D cases.

While I have no proof, I believe that the `helical vector' is well behaved also

for other situations than the isotropic case referred to above, as long as the

landmark distribution and the noise are non-pathological. Proving this might be

a nice challenge for a mathematically oriented MSc or PhD thesis ?

Herman J. Woltring

Eindhoven, The Netherlands