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Brian Davis
06-02-1994, 04:51 AM
Dear Biomechers

I enjoyed reading Zvi Ladin's posting on the network, since I think this
makes for an interesting discussion. He raised some points that I would
like to address below.

One such issue concerns the statement he made in his last paragraph
"any errors in the location of the COM would have just as grave
consequences as errors in the locations of the joint centers". At first
glance this statement appears to be correct, since the equations
of equilibrium do indeed contain terms that are the radii vectors from the
segmental COM to the joint centers. However, after spending the morning
thinking about it, I am still of the opinion that errors in the locations
of joint axes are MUCH MORE damaging than errors in the location of the
segment's COM. I will give my reasons in two ways; (i) a "logical"
approach, and (ii) an "analytical" approach.

(i). If one regards the foot, and for the moment assumes the GRF acts at
the 2nd metatarsal head (MTH), and the COM is midway between the 2nd MTH
and the ankle joint, then one can take moments about the COM and solve
for the (unknown) joint moment. This equation will involve the distance
between the 2nd MTH and COM (call this distance "A") as well as the
distance between the COM and the ankle (call this distance "B"). If one
makes a mistake in locating the COM, then either (a) A will be bigger and B
smaller, or (b) A will be smaller and B will be bigger. The point is that
in the equation, these errors will to some extent CANCEL each other.
However, if there is an error in the location of the GRF (say A is bigger),
then there is nothing in the equation that will cancel this. This is my
first argument to support my opinion that errors in either joint axis
location or location of GRF are more detrimental than errors in BSP
parameters. (The second argument is much longer!)

(ii) Analytical approach. Here I am going to try an do a complete error
analysis of a somewhat simplified case--a foot in contact with the ground.
(The foot does have an angular acceleration (alpha) and a vertical
acceleration (Vacc).) F is the magnitude of the GRF, H is the reaction
force at the ankle, mg is the weight of the foot (mass, m = 1.16kg,
g = 9.81m/s/s) and the moment at the ankle = M. "O" represents the COM and
"A" represnts the ankle joint. Fx, Ox and Ax are the x-coordinates of F, O
and A respectively. "I" is the moment of inertia of the foot (about the
COM). Some of these quantities are indicated in the sketch below.

| |
| |
| |
J \
_______ / A @ ) @ represents a clockwise moment
----^ o /|\ / of magnitude M at the ankle joint
(____T_______|____|___/
/|\ \|/ |H
|F mg
|

One can sum the vertical forces and take moments at O to obtain the
following equation;

M = -F*(Ox - Fx) + (m*Vacc + m*g - F)*(Ax - Ox) + I*alpha

In performing an uncertainty analysis, one needs to know the partial
derivatives of M with repect do each variable in the above equation:

dM/dF = -(Ox - Fx)
dM/dOx = -m*(Vacc + g)
dM/dFx = F
dM/dm = (Vacc + g)*(Ax - Ox)
dM/dVacc = m*(Ax - Ox)
dM/dg = m*(Ax - Ox)
dM/dAx = m*(Vacc + g) - F
dM/dI = alpha
dM/dalpha = I

One then needs to substitute actual values into the above equations and
multiply by the errors associated with each variable. Then square the
results, add up and then take the square root to find the overall
uncertainty. i.e., overall uncertainty equals;
square root of {([dM/dF]*error in F)^2 + ([dM/dOx]*error in Ox)^2 +.....}

I did this for typical data used in gait analysis (using SI units);
F = 830 N, Ox = 0.58 m, Fx = 0.5 m, m = 1.16 kg, Vacc = 5.54 m/s/s,
g = 9.81 m/s/s, Ax = 0.66 m, I = 0.0099 kg/m2, alpha = 36 r/s/s.

By substituting these values into the partial derivatives above, one can
get the following values: (next to each is an assumed error for each
variable)
error
dM/dF = -0.08 8.3 N (i.e., 1% of F)
dM/dOx = -17.81 0.01 m (i.e., 1 cm)
dM/dFx = 830 0.01 m "
dM/dm = 1.228 0.116 kg (i.e., 10% of mass)
dM/dVacc = 0.093 0.554 m/s/s (i.e., 10% of Vacc)
dM/dg = 0.093 0 m/s/s
dM/dAx = -812.2 0.01 m (i.e., 1 cm)
dM/dI = 37 0.00099 kg/m/m (i.e., 10% of I)
dM/dalpha = 0.0099 3.7 m/s/s (i.e., 10% of alpha)

The calculated moment at the ankle is 131 Nm with an overall uncertainty
of 11.634 Nm. Now for the "bottom line". If the error associated with Fx
is reduced to zero, then the overall uncertainty becomes 8.15 Nm. If the
error associated with Ax is reduced to zero, the overall uncertainty
becomes 8.33Nm. (Both 8.15 and 8.33 are better than 11.6.) However, if
the error associated with Ox is reduced to zero (perfect BSP data), the
overall uncertainty is 11.632 Nm. (Hardly different to 11.634Nm) So, this
tedious exercise has suggested that, for the data given above, COM location
is not nearly as important as the location of the external force or the
ankle joint center.

Another issue that I would like to mention concerns the inclusion or
exclusion of inertial components in the Inverse Dynamics Approach. I do
not want to give readers the impression that I think inertial components
should be excluded. Although I believe errors in either acceleration data
or BSP parameters are not that serious (compared to errors in location of
GRF and/or joint axes), I do not recommend that the dynamic terms
should be neglected. That would be like saying "I am going to CONSISTENTLY
overestimate (or underestimate) BSP and acceleration terms by 100%". This
scenario is different to the situation where one might have large
uncertainties in different variables--which results in some terms being
overestimated, and some underestimated. Thus, errors in joint moments will
only be linearly affected by errors in BSP parameters (e.g. limb masses) if
the estimates are consistently too large or consistently too small.

I hope this has added to what I consider an interesting question.

Regards,
Brian L. Davis, Ph.D.
Dept. Biomedical Engineering (Wb3)
Cleveland Clinic Foundation
9500 Euclid Avenue
Cleveland, Ohio 44195, U.S.A

E-Mail: davis@bme.ri.ccf.org

Ph: (216) 444-1055 (Work)
Fax:(216) 444-9198 (Work)