Biomedical Engineering Reference
In-Depth Information
subject to the initial conditions
y(
0
)
=
0
,
y(
0
)
˙
=
0
,
z(
0
)
=
0
,
z(
0
)
˙
=
V,
(5.90)
find the optimal control
u
s
=
u
s
0
(t)
such that
=
u
s
{
J
2
(u
s
)
|
≤
D
}
.
J
2
(u
s
0
)
min
J
1
(u
s
)
(5.91)
Based on the optimal time history of the absolute acceleration of the seat
pan, the desired control force
F(t)
is
m
d
ξ
dc
F(t)
=
(m
s
+
m
d
) u
0
s
(t)
+
−
(m
s
+
m
d
) g.
(5.92)
5.4.2
Solution Procedure
General Description
≤
t
≤
T
sim
will be used to calculate the optimal control. The simulation time
T
sim
can be estimated as
An iterative search over a finite time interval 0
2
D
V
+
T
sim
≥
τ,
(5.93)
where
V
is the impact velocity,
τ
is the duration of the crash deceleration
pulse, and
D
is the maximum magnitude of the displacement relative to
the base that is allowed for the seat pan. The right-hand side of Eq. (5.93)
is the time required for the center of mass of the entire system (the seat
and the occupant) to come to a complete stop, provided that the force
F
is
constant and the maximum displacement of the center of mass relative to
the base is equal to
D
.
Divide the time interval 0
≤
t
≤
T
sim
into
n
subintervals by the points
t
i
,
i
=
1
, ..., n
−
1, such that
0
=
t
0
<t
1
<
···
<t
n
−
1
<t
n
=
T
sim
.
(5.94)
Approximate
u
s
(t)
by a piecewise linear function
u
(k)
u
(k
−
1
)
−
s
s
u
s
(t)
=
u
(k
−
1
)
+
(t
−
t
k
−
1
)
s
t
k
−
t
k
−
1
(5.95)
for
t
k
−
1
≤
t
≤
t
k
,k
=
1
, ..., n,
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