First published online February 1, 2008
Journal of Experimental Biology 211, 467-481 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.008573
Angular momentum in human walking
Hugh Herr1,2,* and
Marko Popovic1
1 The MIT Media Laboratory, 20 Ames Street, Cambridge, MA 02139, USA
2 The Harvard-MIT Division of Health Sciences and Technology, 20 Ames Street,
Cambridge, MA 02139, USA
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Fig. 1. Human model and coordinate frame. The human model has 16 segments with 32
internal degrees of freedom. Using human morphological data from the
literature, mass is distributed throughout the model in a realistic manner.
The coordinate frame is oriented by the right-hand rule with the
z-axis directed vertically, the y-axis pointing in the
direction of the walking motion (anterior–posterior direction), and the
x-axis pointing to the right of the participant (medio-lateral
direction).
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Fig. 2. Centroidal moment pivot (CMP). The CMP is the point where the ground
reaction force would have to act to keep the horizontal component of the
whole-body angular momentum constant. When the moment about the center of mass
(CM) is zero (B), the CMP coincides with the center of pressure (CP). However,
when the CM moment is non-zero (A), the extent of separation between the CMP
and CP is equal to the magnitude of the horizontal component of moment about
the CM, divided by the normal component of the ground reaction force.
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Fig. 3. Whole-body angular momentum and moment. (A) A normalized angular momentum
for walking is plotted about three orthogonal directions versus
percentage gait cycle. The angular momentum is normalized by the product of
each participant's mass, CM height and self-selected gait speed (MVH;
see Table 1 for values). (B)
Normalized CM moment is plotted about three orthogonal directions
versus percentage gait cycle. Moment is normalized by the product of
each participant's weight and CM height (MGH). For both A and B, the
solid line is the mean normalized value, and the dashed lines are one standard
deviation about the mean (10 participants and seven walking trials per
participant). In addition, 0% and 100% gait cycles correspond to consecutive
heel strikes of the same foot.
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Fig. 4. Horizontal ground reaction force and CP predictions. (A,B) The horizontal
ground reaction forces in walking are plotted versus percentage gait
cycle in the medio-lateral (x) and anterior–posterior
(y) directions, respectively. The thick red line is the calculated
zero-moment force (see Eqns 9 and
10), and the thin blue line is
the force measured experimentally using force platforms. (C) Plotted are the
CP (blue dashed line), CMP (red solid line) and CM ground projection (green
dash-dotted line) trajectories and corresponding footprints. The two circles
on each line denote the transition from single to double support, and vice
versa. In all plots, only half the gait cycle is shown. Data span from
the middle of a single-support phase (0% gait cycle) to the middle of the next
single-support phase of the opposite limb (50% gait cycle). Data shown are for
one representative participant and trial (participant no. 1 in
Table 1).
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Fig. 5. The mean participant-dependent first PC about three spatial directions.
Here the participant-dependent first PCs were averaged across the 10 study
participants. Error bars are one standard deviation about the mean. The
abscissa numbers and human model segments are paired to the right of the
figure. In the anterior–posterior (y) direction, large
variations in the relative contribution of angular momentum are observed for
the pelvis and abdomen (segment 13), chest (segment 14) and head (segment 16)
[see large standard deviations in Fig. 5 for PC1 (y)].
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Fig. 6. The participant-independent PCs about three spatial directions. Plotted are
the PCs that when combined account for more than 90% of experimental data. The
abscissa numbers correspond to the same human model segments as defined in
Fig. 5. While only three PCs
are needed to explain 90% of the data in the sagittal and transverse planes,
four PCs are required in the coronal (x–z) plane.
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Fig. 7. Tuning coefficients. The mean values, over all participants and trials, of
the normalized tuning coefficients are plotted for three spatial directions.
The tuning coefficients correspond to the PCs shown in
Fig. 6. Here the normalized
tuning coefficients were computed using Eqn
17, and were obtained as gait percentage averages over all
participants and walking trials. To provide information on the variability of
each normalized tuning coefficient, a variability number
Bi(j) is assigned to each curve, where
j=1...3 (spatial directions) and i=1...N (model
segments or PCs). Each number was computed by first estimating the area
between plus and minus one standard deviation about the tuning coefficient
mean, and then dividing by the total area beneath the tuning coefficient mean
– from 0% to 100% gait cycle.
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Fig. 8. Hula-hoop body motions. Angular momentum, horizontal ground reaction force
and ground references points are plotted in the medio-lateral (x) and
anterior–posterior (y) directions. In this experiment a
participant rotated his hips while standing in double support, similar to how
one twirls a hula hoop, at an increasing and then decreasing rotational speed
for approximately 10 s (see A for one representative cycle). (B) The
horizontal components of normalized angular momentum are plotted
versus time. For ease of comparison with walking values shown in
Fig. 3A, the angular momentum
is normalized by the product of the participant's mass, CM height and
self-selected gait speed (MVH; see participant no. 1 in
Table 1). (C,D) The horizontal
ground reaction forces measured experimentally (thin blue line) are plotted
along with the calculated zero-moment forces (thick red line) versus
time for the same participant and trial as in B. Both experimental and
calculated zero-moment forces are normalized by the stiffness term,
Fz/zCM, and the radius of the ground
support base in the appropriate direction (see Eqns
9 and
10). In the medio-lateral
(x) direction, the radius was measured while standing in double
support, and was equal to one-half the distance from the lateral edge of the
right foot to the lateral edge of the left foot. In the
anterior–posterior (y) direction, the radius was equal to
one-half the participant's foot length. (E) Plotted are the CP (blue dashed
line), CMP (red solid line) and CM ground projection (green dash-dotted line)
trajectories and corresponding footprints. As in A, only one hula-hoop cycle
is shown from 7.2 to 8 s. The ground CM projection remains within the support
envelope while the CMP often falls outside the region.
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Fig. 9. Exaggerated walking gait. Angular momentum, horizontal ground reaction
force and ground reference points are plotted in the medio-lateral
(x) and anterior–posterior (y) directions. In this
experiment a participant walked with exaggerated leg protraction and
retraction movements, similar to a military marching gait, at a forward speed
of 1.3 m s–1 (see whole-body sketches above plots). (A,B) The
horizontal components of normalized angular momentum are plotted
versus percentage gait cycle. For ease of comparison with walking
values shown in Fig. 3A, the
angular momentum is normalized by the product of the participant's mass, CM
height, and self-selected gait speed (MVH; see participant no. 1 in
Table 1). Here 0% and 100% gait
cycles correspond to consecutive heel strikes of the same foot. (C,D) The
horizontal ground reaction forces measured experimentally (thin blue line) are
plotted along with the calculated zero-moment forces (thick red line)
versus percentage gait cycle for the same participant and trial as in
A and B. Here 0% to 50% gait cycle spans from the middle of a single-support
phase to the middle of the next single-support phase of the opposite limb. (E)
Plotted are the CP (blue dashed line), CMP (red solid line) and CM ground
projection (green dash-dotted line) trajectories and corresponding footprints.
As in C and D, only 50% of the gait cycle is shown. The two circles on each
line denote the transition from single to double support, and vice
versa. For this exaggerated gait, the CMP often falls outside the ground
support envelope.
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© The Company of Biologists Ltd 2008
