Biomedical Engineering Reference
In-Depth Information
For instance, a performance measure for the biosystem has been identified as
the sum of squares of all joint torques with coefficients p as in
Equation (5.16)
.
ð
T
X
n
τ
2
S
J
ð
q
; τ;
t
j
p
Þ
5
p
i
τ
i
dt
(5.16)
0
i
5
1
where
X
n
τ
p
i
5
1
and
p
i
$
0
(5.17)
i
5
1
The parameters p are determined by solving the inner optimization problem as
defined in
Equation (5.18)
so that the exact performance measure can be identified.
mi
p
ε
s
:
t
:
:
h
ð
p
Þ
#
0
min
q; τ
S
J
ð
q
;
τ
;
t
j
p
Þ
(5.18)
f
ð
q
;
q
;
q
;
t
Þ
5
0
s
:
t
:
:
τ
-
g
ðϒÞ
#
0
q
L
U
#
q
#
q
L
U
τ
#τ#τ
where h
ð
p
Þ
#
0 are the possible equality or inequality constraints on the para-
meters p satisfying the normalization and non-negativity conditions.
is the error
defined in
Equation (5.3)
. The process of identifying the unknown performance
measure is transformed to find the parameters p that will minimize the error
ε
ε
.
5.6
Constraints
Constraints are formulated based on the available information
about the biosys-
tem. In general, two types of constraints are included in this set: (i) boundary con-
ditions, and (ii) state response at some time points, q
ðt
j
Þ;
ϒ
, obtained from
either experiments or observations. In addition, boundary conditions consist of
time boundary, q
t
j
,
t
j
A
Γ
t
j
A
Ω
, where X
represents the global Cartesian coordinates that capture the geometrical environ-
ment for the biosystem. For example, given initial and final postures, a walking
task is performed to predict the walking motion between the two postures. The
initial and final postures are the time boundaries, and the ground is formulated as
a geometrical boundary. However, there are many options for state response con-
straints based on available information about the walking motion, such as transi-
tion posture between single support and double support phases, knee flexion
, and geometrical boundary, X
ð
q
t
j
Þ
,
t
j
A
Ω
,
Γ
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