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Practically all anthropomorphic models assume that human body
segments are rigid, have clearly defined intersegmental
boundaries, and possess an invariant endo-structure. All of
these assumptions are convenient for modeling purposes but
are basically incorrect: First, body segments change their
exo- and endomorphologies under the influence of external
(inertial) and internal (muscular and passive viscoelastic)
forces substantially, sometimes even dramatically (such as the
thorax during extensive breathing, and the "wobbling" of soft
tissue structures during running). Secondly, segment boundaries
are by no means fixed but vary as a segment moves through its
range of motion. In this process, mass particles are transferred
from one segment to the adjacent one and vice versa. Other
contributors to changing segment boundaries are non-stationary
joint axes of rotation. Thirdly, the constantly changing volume
of body fluids (mainly blood) within a segment also
substantially alters the segment's mass distribution.
In a 1980-publication, one of us (H. Hatze, Journal of Bio-
mechanics 13, pp. 833-843) described all of these phenomena
and estimated their influence on the accuracy of the computed
parameter values to be no more than 6%. This estimate may,
however, be completely wrong. To our knowledge, there exists no
reliable study at present which would allow us to give a more
accurate estimate. However, we may not be aware of pertinent
studies that appeared recently, such as the apparently important
MS thesis of Ori Sarfati on which, according to Zvi Ladin, the
recent publication by Ori Sarfati and Zvi Ladin of the video-
based system for the inertial parameter determination (J. of
Biomechanics 26(8), 1011-1016, 1993) is based. (Strangely
enough, however, Sarfati and Ladin did not find it worth
quoting Sarfati's MS thesis in the reference list of their own
publication, although this MS thesis is allegedly fundamental
to this publication. Under these circumstances, it is
obviously difficult for others to discover such hidden
references).
Considering the above arguments, one wonders to what extent
precision in determining segmental parameter values ( masses,
principal moments of inertia, mass centroid coordinates, etc.)
is really of relevance. At present, this question is difficult
to answer. It is certainly not of much use to the biomechanics
researcher or to the clinician when error bounds of 2% are
quoted for parameter determinations of artificial uniform test
bodies, as is the case with a recently announced system. The
user is not interested in errors that pertain to unrealistic
uniform test bodies. He wants to know what accuracy he can
expect in determining inertial parameter values of real (and
much more complex) body segments, the densities and shapes of
which are by no means uniform but vary across the cross-section
of the segments as well as along their longitudinal axes,
facts, which all have to be taken into account in an appropriate
anthropomorphic model for inertial parameter determination.
Reasonable estimates of error bounds relating to realistic
techniques for the determination of inertial segment parameter
values exist. The question remains what relevance the
stipulation of such error bounds might have in view of the
fuzziness of intersegmental boundaries and the inaccuracies
introduced by the apparent non-rigidity of real body segments.
This is the topic on which we would like to initiate a
discussion on this forum.
Sincerely
Herbert Hatze and Arnold Baca
________________________
In a recent posting, H. Hatze and A.Baca suggested a "discussion forum on
the accuracy required in determining human body segment parameters"
(BSP's). I would like to address this issue in the context of using the
Inverse Dynamic Approach (IDA) to calculate joint moments. (In this
approach, kinematic, Ground Reaction Forces (GRF's) and BSP data are all
utilized.) The question that has interested me (and others) for some time
is, "How important are uncertainties in each type of data set--e.g., does
an error of 1% in GRF data degrade the accuracy of hip moments as much as
say a 10% error in BSP data?
If one uses an approach described by Cappozzo, Leo and Pedotti (Equation
17 in their article in J. Biomechanics, v.8, 307-320, 1975), one can assess
the contribution of different errors on the overall uncertainty of a
calculated value for a joint moment. I have used the sagittal plane data
described in Winter's 1979 book on gait and, as a starting point, assumed
the following uncertainties;
1. BSP and acceleration data: 10% of nominal values
2. GRF data: 1% of nominal value (based on forceplate specifications)
3. Position of resultant GRF vector: 6.3 mm (based on Bobbert and
Schamhardt's article in J. Biomechanics, v23, 7, 705-710, 1990)
4. Coordinates of joints: 10mm
The results of this exercise showed that uncertainties in BSP and
acceleration data had little effect on the overall uncertainty of the hip
moment during the stance phase of gait, accounting for less than 2% of the
total uncertainty. Errors in Center of Pressure (CoP) location, and joint
axes had much greater ramifications, accounting for over 64% of the
uncertainty. Percentage-wise, the contributions of BSP and
accelaration errors became greater during swing (accounting for up to 50%
of the average uncertainty), but this must be seen in light of the fact
that moments during swing are extremely small. (The only people who
placed great importance on the moments during the swing phase were Braune
and Fischer who at the turn of the century were testing the theory of the
Weber brothers that the leg acted as a pendulum!)
The above statements are directly in line with the findings of Capozzo, Leo
and Pedotti who stated "Errors in determining the position of the
structure, with respect to the vector of ground reactions, are major
determinants of the errors on the muscular moments".
In summary, one could make a 50% error in one's estimates of BSP
parameters and hardly effect the accuracy of calculated values for joint
moments. So, in the context of human gait, it makes little sense in
striving to get perfect BSP data. It makes more sense to obtain reliable,
accurate data for joint centers of rotation and GRF data.
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)
____________________________
In recent postings Brian Davis and Paul Devita discussed the role of
the inertial parameters in the determination of joint moments, suggesting
that even large errors in the inertial parameters will lead to relatively
small errors in the calculated moments. I would like to address two
issues that are related to this discussion - the overall magnitude of
the inertial loads (i.e. forces and moments that arise from the motion
of the body segments), and the role of the inertial parameters in the
determination of the inertial loads.
In a study that Ge Wu (who is now at Penn State) and I conducted a few
years ago, we used the kinematometers (devices that combine position
markers, linear accelerometers and angular velocity sensors) to monitor the
kinematics of the lower limb during physical activities ranging from
slow walking to jumping. We then compared the magnitudes of inertial
and static components of the joint loads (forces and moments). The
results were presented last year, during the Second International
Symposium on 3-D Analysis of Human Movement (Poitiers, France), and
in summary we found that the inertial effects are largest in the
transverse plane and that they increase in magnitude in the proximal
direction and with an increase in the speed of the activity.
We compared the ratios of the maximum inertial forces to the maximum
static forces (i.e. the forces needed to maintain equilibrium).
In SLOW WALKING the maximum inertial forces for the ankle were on the
order of 3-6% (of the maximum static forces) in the vertical direction,
16% in A-P direction and 60-80% in the M-L direction.
The corresponding values for the hip were 20% in the vertical direction,
100-110% in the A-P direction and 200-300% in the M-L direction.
As the speed of the activity increased, we saw the expected increase in
the inertial effects, leading to inertial forces that were either HIGHER
than or similar to the static components in the A-P and M-L directions,
and are on the order of 10-20% in the vertical direction. These results
would be linearly affected by errors in the segmental masses.
The corresponding values for the moments were generally smaller, however
even there we saw values that were on the order of 20-40% in the medial
direction for running.
As both Devita and Davis suggest, the results of the moments will
be very sensitive to errors in the locations of the joints, however,
since the accepted practice of the inverse dynamic solution of the moment
equations invovles conducting the calculations in a Body Coordinate
System centered at the segmental center of mass, any errors in the location
of the COM would have just as grave consequences as errors in the locations
of the joint centers (since the equations contain terms that are the radii
vectors from the segmental COM to the joint centers).
References:
Ge Wu and Zvi Ladin. The effect of inertial load pm human joint force
and moment during locomotion. Proceedings of the Second International
Symposium on Three-Dimensional Analysis of Human Movement, 6/30-7/3/1993,
Poitiets, France, pp 106-107.
Zvi Ladin
Biomedical Engineering Department
Boston University
e-mail: ZL@buenga.bu.edu
________________________
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)
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