Dear BiomchL readers:
I enjoyed reading the debate concerning 'errors in the body segmental
parameters (BSP)' last week. Especially, Brian Davis' last posting was
instructional for me in that the discussion was quantitative (practical) as
well as analytic. I think it was a good demonstration showing how to deal with
'propagation of error' in calculation of a quantity (e.g., moment, M), which
is a function of more than one variable. Whenever such a function involves the
same variable more than once, some of the errors or uncertainties may cancel
themselves. This effect sometimes is called 'compensating errors'.
Neglecting compensating errors may result in overestimation of the final
uncertainty. At this point, those who are familiar with this matter may
want to skip the part (1) to the part (2) of this posting.
(1) Such overestimation can be avoided using the concept of 'partial
derivatives' in computing uncertainty in a functional quantity. Lets
suppose that x, y, ..., z are measured with uncertainties dx, dy, ..., dz
and the measured values are used to compute the function, q(x,y,...,z). Then
dq = @q/@xdx + @q/@ydy + ...... + @q/@zdz, [1]
~
where d and @ are symbols for uncertainty and partial derivative,
respectively. Note the symbol of 'approximation' instead of 'equality' in
[1]. Disregarding _independence_ and _randomness_ of uncertainties in the
variables of q, the uncertainty in q is never larger than the ordinary sum of
each term in [1], i.e.,
dq < @q/@xdx + @q/@ydy + ...... + @q/@zdz. [2]

However, if the uncertainties in the variables of q are _independent_ and
_random_, then the uncertainty in q can be given:
dq = sqrt of { [(@q/@x)dx]**2 + ...... + [(@q/@z)dz]**2 }. [3]
This is the equation that you saw in Brian Davis' posting last week.
(2) I would like to somewhat extend Brian Davis' argument over the
effects of uncertainties in BSP and location of joint to calculation of
moment, M. However, I'd like to simplify the inverse dynamics problem
itself as much as he did without hurting the error issue. Only the difference
I make here is replacing a GRF with a distal joint force, D. This allows
reaction forces on the segment (near or at two joints) to be comparable. I
think, however, it is legitimate to keep a net moment at the distal joint zero
in this problem. Then an appropriate diagram of the problem should look
something like below:
ma /\ P /
 /\ /
 alfa(@)  /
__________________________/
//\  CM M (@:ccw or positive
/   direction)
/  \/ mg
/ D
Now the net torque about the estimated center of mass (CM) of the segment of
interest is given:
I*alfa = D*cL + P*(1c)L + M, [4]
where c is the ratio of the distance between the distal joint and the
segmental CM to the segmental length, L. The c is one of BSP's, which appear
in [4], besides moment of inertia about the segmental CM, I. + The I (=mr**2)
does not depend explicitly on the c. The r is the radius of gyration of the
segment. Furthermore the c does not affect angular acceleration, alfa, as long
as the CM is along on the line of the segment of interest. Thus the term,
I*alfa, is regarded as constant with respect to c. Then changes in moment
estimation to small changes in L and c are given:
@M/@L = Dc  P(1c) and
@M/@c = (D+P)L,
respectively. Let e(L) and e(c) be errors in estimation of moment due to
uncertainties in L and c, respectively. ++ Let k be a 'given' fractional
uncertainty both in L and c. Then
e(L) = kLcD  KL(1c)P and
e(c) = kcL(D+P).
Taking the ratio of e(L) to e(c),
e(L)/e(c) = [ D  {(1c)/c}P ] / (D+P).
+++ Therefore,
e(L) < e(c) if sign(D) = sign(P),
e(L) > e(c) if sign(D) = sign(P), and
e(L) = e(c) if either D > P,
~
where .4 < c < .6 for major segments of human body.
(3) + Exclusion of the term, I*alfa, in computing e(L) and e(c) depends
on how the r is obtained. If the estimation of the r is based on
measuremnt, then the I*alfa can be treated independently of c. However, if
the estimation of the r is based on an analytical relationship with the c,
then partial derivative of the I*alfa should be taken with respect to the c.
++ I used the same k for fractional uncertainties in L and c with a clear
reason. The point of the late debate was the _sensitivities_ in estimation
of moment, M, to variation of L and c, respectively. Variations of L and c
are uncertainties, i.e., dL and dc, around their best estimated or measured
values. In practice, however, dc/c doesn't have to be the same as dL/L.
Therefore, the corresponding uncertainty in estimating or measuring each
variable (e.g., dc) should be used in calculating the net contribution of
each variable (e.g., @M/@cdc) to the final error, dM. Which one is larger
between dL and dc or how large they are is another matter to be asked in
addition to the sensitivity of dM (e.g., @M/@c) to uncertainty in each
variable.
+++ What I showed above is simply listing all possible cases under different
conditions. I don't have enough variety of data to take examples for each
condintion. Brian Davis suggested that the 2nd condition is where movemnts
would normally occur. I wonder which of those conditions is pertinent as
near the toeoff, especially, in kicking, and near the touchdown. I
believe there are many who are familiar with such data among the readers.
Resposes to this would be appreciated.
Thanks for reading along this far.
Regards,
Seogyong Lee
Department of Biomechanics
University of Maryland
College Park, MD 20742
phone: (301) 4052572
email: sL44@umail.umd.edu
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