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
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
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
=
Search WWH ::




Custom Search