View Full Version : About vectors and matrices: round #2

Herman J. Woltring
02-19-1990, 08:01 AM

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.


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.