Herman J. Woltring

02-19-1990, 07:01 AM

THE CURIOUS INCIDENT OF THE VECTORIAL TRIBE

It is rumoured that there once was a tribe of Indians who believed that

arrows are vectors. To shoot a deer due northeast, they did not aim an arrow

in the northeasterly direction; they sent two arrows simultaneously, one due

north and one due east, relying on the powerful resultant of the two arrows

to kill the deer.

Skeptical scientists have doubted the truth of this rumour, pointing out

that not the slightest trace of the tribe has ever been found. But the complete

disappearance of the tribe through starvation is precisely what one would ex-

pect under the circumstances; and since the theory that the tribe existed con-

firms two such diverse things as the NONVECTORIAL BEHAVIOR OF ARROWS and the

DARWINIAN PRINCIPLE OF NATURAL SELECTION, it is surely not a theory to be

dismissed lightly.

A. Banesh Hoffmann, About Vectors. Prentice-Hall, New Jersey 1966;

Dover, New York 1975, pp. 11-12.

Dear Biomch-L readers,

Ed Grood's reply of last Saturday is to be commended, and I'll address his

points more-or-less in the (time, first-last, top-bottom) sequence provided

by him.

1. `Current position':

While the (finite) helical axis has the usuful property of defining total dis-

placement (translation and rotation) between two orientations, where one can be

a reference orientation, and the other a current or actual orientation, this

does not preclude its utility for other purposes. The beguiling analogy between

rigid-body and point displacements will be discussed below.

I don't think that I proposed to decompose helical movement into s i x compo-

nents, with three orthogonal translation/position, and three orthogonal transla-

tion/attitude components. In the context of the current debate, my main focus

is on attitude/rotation parametrization as apparent from the original title. I

agree that position description is somewhat awkward under the helical approach

(as defined by the position of the helical axis, i.e., of some point on it re-

quiring 2 d.o.f.'s, and of the amount of translation along the helical axis when

moving from the reference to the current orientation, requiring 1 more d.o.f.),

but I disagree that the helical decomposition is incorrect for joint (or seg-

ment) attitude/rotation. While the G&S convention is useful, I would disagree

that it is the best (or, for truely 3-D joint rotation as in hip and shoulder,

even a good convention). Actually, position/translation description is also

awkward under the G&S approach since gimbal-lock affects both the rotational

a n d the translational parametrizations in that model... As said in the 14

Feb posting, gimbal-lock does not exist under the helical convention, and Ed

has not yet commented that point.

2. `Misconceptions and verbal ambiguities':

It seems that the confusion (including the wording in my 14 Feb posting) exists

between two different kinds of `sequence effects':

(a) Sequences in a `geometrical, top-down, or left-right' sense, as

in the matrix product Ri(PHIi)*Rj(PHIj)*Rk(PHIk)

or in the vector sum Xi( Di )+Xj( Dj )+Xk( Dk ),

(b) Sequences in a temporal sense, where the variables PHI. and D. in the above

expressions are changed from their reference values (zero) to the final

values that describe the current or actual orientation.

Different sequences under (a) are

Ri(PHIi)*Rj(PHIj)*Rk(PHIk) and Rk(PHIk)*Ri(PHIi)*Rj(PHIj), or

Xi( Di )+Xj( Dj )+Xk( Dk ) and Xk( Dk )+Xi( Di )+Xj( Dj ).

Now, the important point is that matrix multiplication is non-commutative, that

is, the resulting matrix products are generally n o t identical, while vector

addition i s commutative, that is, the vector sums are always identical.

One particular t i m e - sequence in the sense of (b) is:

Start: Ri( 0 )*Rj( 0 )*Rk( 0 ), Xi( 0)+Xj( 0)+Xk( 0)

1st displacement: Ri(PHIi)*Rj( 0 )*Rk( 0 ), Xi(Di)+Xj( 0)+Xk( 0)

2nd displacement: Ri(PHIi)*Rj(PHIj)*Rk( 0 ), Xi(Di)+Xj(Dj)+Xk( 0)

3rd displacement: Ri(PHIi)*Rj(PHIj)*Rk(PHIk), Xi(Di)+Xj(Dj)+Xk(Dk)

and a different t i m e - sequence is:

Start: Ri( 0 )*Rj( 0 )*Rk( 0 ), Xi( 0)+Xj( 0)+Xk( 0)

1st displacement: Ri( 0 )*Rj( 0 )*Rk(PHIk), Xi( 0)+Xj( 0)+Xk(Dk)

2nd displacement: Ri( 0 )*Rj(PHIj)*Rk(PHIk), Xi( 0)+Xj(Dj)+Xk(Dk)

3rd displacement: Ri(PHIi)*Rj(PHIj)*Rk(PHIk), Xi(Di)+Xj(Dj)+Xk(Dk)

Here, initial and final orientations are identical, buth the displacement

occurs (or is thought to occur) along a different p a t h.

The G&S approach imposes one particular sequence in the sense of (a), and

then proceeds by declaring it sequence-independent in the sense of (b). The

standard handbooks, however, interpret the term s e q u e n c e in the sense

of (a)... [The many Dutch readers on the list will undoubtedly recall the

recent parliamentory debate on `Social Innovation', where a highly ranking

civil servant gave four completely different explanations of the meaning of

that term.]

I believe that the major motivation underlying the G&S approach -- or, for that

matter, of any approach in terms of Cardanic/Eulerian rotations -- is to try

and mimick the p a t h properties of vectorial movement. If I decompose a

position or translation in terms of components X, Y, and Z, I may reach the

position (X,Y,Z)' from the reference position (0,0,0)' by sequence-independent

steps in the (b) sense above, but a l s o by sequence-independent steps in

the (a) sense above. For rotations, this is n o t the case, as I can only

have a sequence-independence in the (b) sense above. In the linkage-terminology

of the G&S approach: if the linkage is made to merely allow translations along

the linkage's axes, but no rotations about them, the choice of which axes are

to be the imbedded ones and which are to be the floating ones is irrelevant,

and has no influence on the amount of translation along each (permutated) axis.

For rotations, this sequence-independence in the (a) sense does not apply.

While it may be attractive to have an attitude parametrization that has the

physical property of path description, this property is in no way necessary

for the attitude parametrization goal. In fact, the imposition of this desire

entails a number of awkward side-effects such as gimbal-lock and Codman's para-

dox. Using a simple, while not exact analogon: while I may elect to travel

from Eindhoven to Cincinnati by first moving south (or north, via the North

Pole) until I reach the equator, thence west (or east) until I am due south of

Cincinnati, and finally north (or south, via the South Pole) until I can join

Ed for a drink, this does not imply that this is an elegant way to describe

Ed's position with respect to me.

The helical decomposition results in three orthogonal components that generally

do n o t have path description properties. If two of the three components are

zero, however, the single remaining non-zero term has a path describing nature.

While this may be regrettable, I am really not interested in thinking how to get

from some neutral hip or shoulder attitude to a complex, possibly pathological

one, in terms of elementary rotations about some anatomically or technically

defined co-ordinate axes, but only in a unique, well-behaved, minimally singular

parametrization; I think that the helical decomposition meets these requirements

quite well.

Current clinical practise is more-or-less in agreement on how to define pure

flexion-extension, pure ab-adduction, and pure endo-exorotation; such agreement

does, at this time, not exist when these elementary (`planar') rotations occur

simultaneously. Thus, we have the freedom to choose that particular convention

which corresponds with the already established, special cases (the G&S approach,

all other Cardanic conventions, and the helical approach meet that condition),

a n d which is best behaved in the complex, 3-D case.

Summarizing:

Both the G&S and the helical approach exhibit three `independent' components

except for the singular cases of gimbal-lock; in the helical approach, the com-

ponents are mutually orthogonal but do not have trajectory properties, and the

three axes are symmmetrical with respect to each other, without preferred `body-

fixed' and `floating' axes; in the G&S approach, the opposite is the case. I

believe that the balance is in favour of the proposed, helical standard, but it

is up to the relevant community to accept or reject such a proposal.

Herman J. Woltring ,

Eindhoven, The Netherlands.

It is rumoured that there once was a tribe of Indians who believed that

arrows are vectors. To shoot a deer due northeast, they did not aim an arrow

in the northeasterly direction; they sent two arrows simultaneously, one due

north and one due east, relying on the powerful resultant of the two arrows

to kill the deer.

Skeptical scientists have doubted the truth of this rumour, pointing out

that not the slightest trace of the tribe has ever been found. But the complete

disappearance of the tribe through starvation is precisely what one would ex-

pect under the circumstances; and since the theory that the tribe existed con-

firms two such diverse things as the NONVECTORIAL BEHAVIOR OF ARROWS and the

DARWINIAN PRINCIPLE OF NATURAL SELECTION, it is surely not a theory to be

dismissed lightly.

A. Banesh Hoffmann, About Vectors. Prentice-Hall, New Jersey 1966;

Dover, New York 1975, pp. 11-12.

Dear Biomch-L readers,

Ed Grood's reply of last Saturday is to be commended, and I'll address his

points more-or-less in the (time, first-last, top-bottom) sequence provided

by him.

1. `Current position':

While the (finite) helical axis has the usuful property of defining total dis-

placement (translation and rotation) between two orientations, where one can be

a reference orientation, and the other a current or actual orientation, this

does not preclude its utility for other purposes. The beguiling analogy between

rigid-body and point displacements will be discussed below.

I don't think that I proposed to decompose helical movement into s i x compo-

nents, with three orthogonal translation/position, and three orthogonal transla-

tion/attitude components. In the context of the current debate, my main focus

is on attitude/rotation parametrization as apparent from the original title. I

agree that position description is somewhat awkward under the helical approach

(as defined by the position of the helical axis, i.e., of some point on it re-

quiring 2 d.o.f.'s, and of the amount of translation along the helical axis when

moving from the reference to the current orientation, requiring 1 more d.o.f.),

but I disagree that the helical decomposition is incorrect for joint (or seg-

ment) attitude/rotation. While the G&S convention is useful, I would disagree

that it is the best (or, for truely 3-D joint rotation as in hip and shoulder,

even a good convention). Actually, position/translation description is also

awkward under the G&S approach since gimbal-lock affects both the rotational

a n d the translational parametrizations in that model... As said in the 14

Feb posting, gimbal-lock does not exist under the helical convention, and Ed

has not yet commented that point.

2. `Misconceptions and verbal ambiguities':

It seems that the confusion (including the wording in my 14 Feb posting) exists

between two different kinds of `sequence effects':

(a) Sequences in a `geometrical, top-down, or left-right' sense, as

in the matrix product Ri(PHIi)*Rj(PHIj)*Rk(PHIk)

or in the vector sum Xi( Di )+Xj( Dj )+Xk( Dk ),

(b) Sequences in a temporal sense, where the variables PHI. and D. in the above

expressions are changed from their reference values (zero) to the final

values that describe the current or actual orientation.

Different sequences under (a) are

Ri(PHIi)*Rj(PHIj)*Rk(PHIk) and Rk(PHIk)*Ri(PHIi)*Rj(PHIj), or

Xi( Di )+Xj( Dj )+Xk( Dk ) and Xk( Dk )+Xi( Di )+Xj( Dj ).

Now, the important point is that matrix multiplication is non-commutative, that

is, the resulting matrix products are generally n o t identical, while vector

addition i s commutative, that is, the vector sums are always identical.

One particular t i m e - sequence in the sense of (b) is:

Start: Ri( 0 )*Rj( 0 )*Rk( 0 ), Xi( 0)+Xj( 0)+Xk( 0)

1st displacement: Ri(PHIi)*Rj( 0 )*Rk( 0 ), Xi(Di)+Xj( 0)+Xk( 0)

2nd displacement: Ri(PHIi)*Rj(PHIj)*Rk( 0 ), Xi(Di)+Xj(Dj)+Xk( 0)

3rd displacement: Ri(PHIi)*Rj(PHIj)*Rk(PHIk), Xi(Di)+Xj(Dj)+Xk(Dk)

and a different t i m e - sequence is:

Start: Ri( 0 )*Rj( 0 )*Rk( 0 ), Xi( 0)+Xj( 0)+Xk( 0)

1st displacement: Ri( 0 )*Rj( 0 )*Rk(PHIk), Xi( 0)+Xj( 0)+Xk(Dk)

2nd displacement: Ri( 0 )*Rj(PHIj)*Rk(PHIk), Xi( 0)+Xj(Dj)+Xk(Dk)

3rd displacement: Ri(PHIi)*Rj(PHIj)*Rk(PHIk), Xi(Di)+Xj(Dj)+Xk(Dk)

Here, initial and final orientations are identical, buth the displacement

occurs (or is thought to occur) along a different p a t h.

The G&S approach imposes one particular sequence in the sense of (a), and

then proceeds by declaring it sequence-independent in the sense of (b). The

standard handbooks, however, interpret the term s e q u e n c e in the sense

of (a)... [The many Dutch readers on the list will undoubtedly recall the

recent parliamentory debate on `Social Innovation', where a highly ranking

civil servant gave four completely different explanations of the meaning of

that term.]

I believe that the major motivation underlying the G&S approach -- or, for that

matter, of any approach in terms of Cardanic/Eulerian rotations -- is to try

and mimick the p a t h properties of vectorial movement. If I decompose a

position or translation in terms of components X, Y, and Z, I may reach the

position (X,Y,Z)' from the reference position (0,0,0)' by sequence-independent

steps in the (b) sense above, but a l s o by sequence-independent steps in

the (a) sense above. For rotations, this is n o t the case, as I can only

have a sequence-independence in the (b) sense above. In the linkage-terminology

of the G&S approach: if the linkage is made to merely allow translations along

the linkage's axes, but no rotations about them, the choice of which axes are

to be the imbedded ones and which are to be the floating ones is irrelevant,

and has no influence on the amount of translation along each (permutated) axis.

For rotations, this sequence-independence in the (a) sense does not apply.

While it may be attractive to have an attitude parametrization that has the

physical property of path description, this property is in no way necessary

for the attitude parametrization goal. In fact, the imposition of this desire

entails a number of awkward side-effects such as gimbal-lock and Codman's para-

dox. Using a simple, while not exact analogon: while I may elect to travel

from Eindhoven to Cincinnati by first moving south (or north, via the North

Pole) until I reach the equator, thence west (or east) until I am due south of

Cincinnati, and finally north (or south, via the South Pole) until I can join

Ed for a drink, this does not imply that this is an elegant way to describe

Ed's position with respect to me.

The helical decomposition results in three orthogonal components that generally

do n o t have path description properties. If two of the three components are

zero, however, the single remaining non-zero term has a path describing nature.

While this may be regrettable, I am really not interested in thinking how to get

from some neutral hip or shoulder attitude to a complex, possibly pathological

one, in terms of elementary rotations about some anatomically or technically

defined co-ordinate axes, but only in a unique, well-behaved, minimally singular

parametrization; I think that the helical decomposition meets these requirements

quite well.

Current clinical practise is more-or-less in agreement on how to define pure

flexion-extension, pure ab-adduction, and pure endo-exorotation; such agreement

does, at this time, not exist when these elementary (`planar') rotations occur

simultaneously. Thus, we have the freedom to choose that particular convention

which corresponds with the already established, special cases (the G&S approach,

all other Cardanic conventions, and the helical approach meet that condition),

a n d which is best behaved in the complex, 3-D case.

Summarizing:

Both the G&S and the helical approach exhibit three `independent' components

except for the singular cases of gimbal-lock; in the helical approach, the com-

ponents are mutually orthogonal but do not have trajectory properties, and the

three axes are symmmetrical with respect to each other, without preferred `body-

fixed' and `floating' axes; in the G&S approach, the opposite is the case. I

believe that the balance is in favour of the proposed, helical standard, but it

is up to the relevant community to accept or reject such a proposal.

Herman J. Woltring ,

Eindhoven, The Netherlands.