Biomedical Engineering Reference
In-Depth Information
y
x
z
FIGURE 7.12
Foot ground penetration conditions.
lower bound and upper bound to the neutral angle (the natural angle at rest)
instead of eliminating this DOF from the skeleton model.
7.9.3.1.2
Strength limits
Each joint torque is also bounded by its physical strength limits. These limits are
obtained from the strength experiments as presented in Chapter 6. Note that the
percent max of torque that is acting as a limit is a single point on the surface
depicted in Figures 6.8 and 6.9. Note that these surfaces must be represented into
a parametric equation in order to be used in the inequality constraint as follows:
L
U
τ
#
τðtÞ
#
τ
;
0
t
T
(7.28)
#
#
L
are the lower torque limits and
U
the upper limits.
where
τ
τ
7.9.3.1.3
Ground penetration
Walking is characterized with unilateral contact between the foot and ground as
shown in
Figure 7.12
. While the foot contacts the ground, the height and velocity
of the contacting points (circles) are zero. In contrast, the height of other points
(triangles) on the foot is greater than zero.
Therefore, the ground penetration constraints are formulated as follows:
y
i
ðtÞ
5
0
;
x
i
ðtÞ
5
_
0
;
y
i
ðtÞ
5
_
0
;
_
z
i
ðtÞ
5
0
;
i
AΩ
(7.29)
y
i
ðtÞ
$
ε;
i
2 Ω;
0
t
T
#
#
where
ε
is a small positive number and
Ω
is the set of contacting points.
7.9.3.1.4
Dynamic balance
The dynamic balance is achieved by forcing the ZMP to remain within the foot sup-
port polygon (FSP) as depicted in
Figure 7.13
,where
is a vector along the bound-
ary of the FSP and r is the position vector from a vertex of the FSP to the ZMP.
The ZMP constraint is mathematically expressed as follows:
Γ
ð
r
i
3
Γ
i
Þ
U
n
y
#
0
;
i
1
; ...;
4
(7.30)
5
where n
y
is the unit vector along the y-axis.
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