Herman J. Woltring

02-14-1990, 06:43 AM

Dear Biomch-L readers,

The abstract below has been accepted for presentation at the previously announ-

ced, 4th International Biomechanics Seminar in Gothenburgh/Sweden on 26 and 27

April, 1990. In view of various comments about this proposal, both from Dr

Grood in Cincinnati/Ohio and elsewhere, Ed Grood has agreed to a debate on the

pro's and con's of various 3-D joint attitude parametrizations. For the sake

of clarity, this text deviates slightly from the accepted text.

Following earlier items on Biomch-L last year, I hope that this debate may be

one of many to follow. It is for this kind of activities that an email discus-

sion list can be much more efficient than, say, Letters to the Editors in formal

journals. This is not to say that such letters are not useful; rather, they

might become the result of more interactive debates, whether at conferences,

during laboratory visits, or from behind a terminal. Ed, it's up to you, now!

Herman J. Woltring

Eindhoven, The Netherlands.

3-D ATTITUDE REPRESENTATION : A NEW STANDARDIZATION PROPOSAL (1)

Herman J. Woltring, Eindhoven, The Netherlands

Grood & Suntay (Journal of Biomechanical Engineering 1983, 136) have proposed a

`sequence independent, oblique co-ordinate system' in which the current orienta-

tion (i.e., position and attitude) is thought to be reached from a predefined

reference orientation via an ordered sequence of rotations ijk of three elemen-

tary, helical displacements a b o u t (PHI.) and a l o n g (D.) the axes i,

j, and k of an electrogoniometric linkage system. The terminal axes i and k

are imbedded in the body segments comprising the joint, and they are identical

to prior selected, Cartesian co-ordinate axes defined in these segments; the

intermediate or `floating' axis j is normal to the two imbedded axes, and iden-

tical to the `line of nodes' in classical handbook descriptions of Euler/Cardan

angles.

Although t e m p o r a l sequence dependency of Cardan/Eulerian rotation

conventions is avoided in such predefined electrogoniometric systems, a similar

effect is now imposed by the g e o m e t r i c a l choice of imbedded and

floating axes. Thus, different numerical results may be obtained for current

joint attitudes (given identical segment co-ordinate systems), and adverse

effects such as g i m b a l - l o c k (i.e., for some PHIj, either the sum

or the difference of PHIi and PHIk is undefined) and C o d m a n ' s P a r a -

d o x (i.e., both {PHIi,PHIj,PHIk} and {PHIi+PI,PI-PHIj,PHIk+PI} (N.B.: -PI

works also) are valid solutions; cf. A.E. Codman, The Shoulder, Boston 1934)

continue to occur.

Instead of defining joint orientation or movement in terms of an ordered

sequence of three helical displacements, it seems more appropriate to view

the current orientation in terms of a s i n g l e helical displacement,

to be decomposed into orthogonal components in either body segment's co-ordi-

nate system which, apart from a sign inversion, appear to be identical. For

attitude representation (position representation is more complicated), one

can define an attitude `vector' THETA = theta * N, where N is the unit direction

vector about which the helical, scalar rotation theta occurs. This vector

THETA, while not a true vector as rotations are not additive, is symmetrical

in its three components and not affected by gimbal-lock or Codman's Paradox.

Unlike N, THETA is well-determined from noisy measurements even for small theta.

A further advantage is that the helical representation corresponds approximately

with the mean value of all valid, Cardanic representations once Codman's Paradox

is accounted for.

(1) This is a paper under the CAMARC project. CAMARC, for "Computer Aided

Movement Analysis in a Rehabilitation Context", is a project under the Advanced

Informatics in Medicine action of the Commission of the European Communities,

XIII-F/CEC), with academic, public-health, industrial, and independent partners

from Italy, France, U.K. and The Netherlands. Its scope is pre-competitive.

The abstract below has been accepted for presentation at the previously announ-

ced, 4th International Biomechanics Seminar in Gothenburgh/Sweden on 26 and 27

April, 1990. In view of various comments about this proposal, both from Dr

Grood in Cincinnati/Ohio and elsewhere, Ed Grood has agreed to a debate on the

pro's and con's of various 3-D joint attitude parametrizations. For the sake

of clarity, this text deviates slightly from the accepted text.

Following earlier items on Biomch-L last year, I hope that this debate may be

one of many to follow. It is for this kind of activities that an email discus-

sion list can be much more efficient than, say, Letters to the Editors in formal

journals. This is not to say that such letters are not useful; rather, they

might become the result of more interactive debates, whether at conferences,

during laboratory visits, or from behind a terminal. Ed, it's up to you, now!

Herman J. Woltring

Eindhoven, The Netherlands.

3-D ATTITUDE REPRESENTATION : A NEW STANDARDIZATION PROPOSAL (1)

Herman J. Woltring, Eindhoven, The Netherlands

Grood & Suntay (Journal of Biomechanical Engineering 1983, 136) have proposed a

`sequence independent, oblique co-ordinate system' in which the current orienta-

tion (i.e., position and attitude) is thought to be reached from a predefined

reference orientation via an ordered sequence of rotations ijk of three elemen-

tary, helical displacements a b o u t (PHI.) and a l o n g (D.) the axes i,

j, and k of an electrogoniometric linkage system. The terminal axes i and k

are imbedded in the body segments comprising the joint, and they are identical

to prior selected, Cartesian co-ordinate axes defined in these segments; the

intermediate or `floating' axis j is normal to the two imbedded axes, and iden-

tical to the `line of nodes' in classical handbook descriptions of Euler/Cardan

angles.

Although t e m p o r a l sequence dependency of Cardan/Eulerian rotation

conventions is avoided in such predefined electrogoniometric systems, a similar

effect is now imposed by the g e o m e t r i c a l choice of imbedded and

floating axes. Thus, different numerical results may be obtained for current

joint attitudes (given identical segment co-ordinate systems), and adverse

effects such as g i m b a l - l o c k (i.e., for some PHIj, either the sum

or the difference of PHIi and PHIk is undefined) and C o d m a n ' s P a r a -

d o x (i.e., both {PHIi,PHIj,PHIk} and {PHIi+PI,PI-PHIj,PHIk+PI} (N.B.: -PI

works also) are valid solutions; cf. A.E. Codman, The Shoulder, Boston 1934)

continue to occur.

Instead of defining joint orientation or movement in terms of an ordered

sequence of three helical displacements, it seems more appropriate to view

the current orientation in terms of a s i n g l e helical displacement,

to be decomposed into orthogonal components in either body segment's co-ordi-

nate system which, apart from a sign inversion, appear to be identical. For

attitude representation (position representation is more complicated), one

can define an attitude `vector' THETA = theta * N, where N is the unit direction

vector about which the helical, scalar rotation theta occurs. This vector

THETA, while not a true vector as rotations are not additive, is symmetrical

in its three components and not affected by gimbal-lock or Codman's Paradox.

Unlike N, THETA is well-determined from noisy measurements even for small theta.

A further advantage is that the helical representation corresponds approximately

with the mean value of all valid, Cardanic representations once Codman's Paradox

is accounted for.

(1) This is a paper under the CAMARC project. CAMARC, for "Computer Aided

Movement Analysis in a Rehabilitation Context", is a project under the Advanced

Informatics in Medicine action of the Commission of the European Communities,

XIII-F/CEC), with academic, public-health, industrial, and independent partners

from Italy, France, U.K. and The Netherlands. Its scope is pre-competitive.