Biomedical Engineering Reference
In-Depth Information
(A)
(B)
M
A
F
A
M
A
F
A
A
F
I
M
I
M
M
C
A
F
F
C
G
FZMP/FRI
G
ZMP=COP
P (COP)
P'
FIGURE 7.7
Balanced and unbalanced states: (A) balanced state, (B) unbalanced state.
variables, external loads, and gravity loads as well as the GRF, the real joint
torques are recovered from the equations of motion using
Equation (7.5)
.
7.7
Zero-Moment point (ZMP)
The ZMP approach is a well-established bipedal dynamic balance criterion that
has been widely used in the fields of robotics and biomechanics (
Vukobratovic
and Borovac, 2004
). The ZMP is the point on the ground where the resultant tan-
gential moments of the active forces are zero, as illustrated in
Figure 7.7
.
Here, the forces are distinguished into two categories: active forces and pas-
sive forces. Active forces include inertia, Coriolis, gravity, and external forces.
Passive forces are the GRFs. The balanced and unbalanced states of the human
system are illustrated in
Figure 7.7
, where point A denotes the ankle, point C is
the center of mass of foot, G is foot gravity force, F
A
and M
A
are resultant force
and moment of body parts excluding the contacting foot, F and M are the resul-
tant GRFs, and F
I
and M
I
are the inertia forces of the foot due to rotation. FZMP
represents fictitious ZMP, and FRI is foot rotation indicator.
The position of ZMP can be calculated using the conditions:
M
z
5
0 and
M
x
5
0(
z
is the walking direction,
x
is the lateral direction, and
y
is the vertical
direction; refer to
Figure 7.4
).
P
nlink
i
5
1
J
i
θ
iz
1
ðm
i
ð
2
€
y
i
1
gÞx
i
1
m
i
€
x
i
y
i
2
f
iy
x
i
2
f
ix
y
i
1
h
iz
Þ
x
zmp
5
(7.13a)
P
nlink
i
1
m
i
ð
2
€
y
i
1
gÞ
5
P
nlink
i
5
1
J
i
θ
ix
1
ðm
i
ð
2
€
y
i
1
gÞz
i
1
m
i
€
z
i
y
i
1
f
iy
z
i
2
f
iz
y
i
2
h
ix
Þ
z
zmp
5
(7.13b)
P
nlink
i
5
1
m
i
ð
2
€
y
i
1
gÞ
where
x
i
,
y
i
,
z
i
are the global coordinates of the center of mass for the link
i
;
m
i
is
the mass,
J
i
is the global inertia,
θ
i
is the global angular acceleration with link
i
;
f
i
and
h
i
are the external force and moment applied on link
i
;and
g
52
s
2
.
9
:
8062 m
=
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