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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 (