Biomedical Engineering Reference
In-Depth Information
8
<
I i @ C i
@ _
T i 1 1 @ D i 1 1
@ _
ðk
q k 1
#
q k
@
D i
@ _
(4.46)
q k 5
:
T i 1 1 @
D i 1 1
@ _
ðk .
q k
8
<
C i
@ €
I i @
T i 1 1 @
D i 1 1
@ €
ðk
q k 1
#
q k
@
D i
@
(4.47)
q k 5
T i 1 1 @
:
D i 1 1
@ €
ðk .
q k
8
<
0
ðk
#
@
T i 1 1
@
T i 1 1 @
E i 1 1
@
E i 1 1 1
ðk
i
1 Þ
5
1
@
E i
q k
q k
(4.48)
q k 5
@
:
T i 1 1 @
E i 1 1
@
ðk . i 1
1 Þ
q k
8
<
ðk
0
#
@
F i 1 1 1 T i 1 1 @
T i 1 1
@
F i 1 1
@
ðk
i
1 Þ
5
1
@
F i
q k
q k
(4.49)
q k 5
:
@
T i 1 1 @
F i 1 1
@
ðk
i
1 Þ
.
1
q k
@ G i
@
0
(4.50)
q k 5
8
<
:
0
1
2 A i
2 A i
2 A i
@
q k D i 1 @
A i
@
q i @
D i
@
@
G i @
A i 2 1
@
@
A 2
g T
f T
tr
q k E i 2
q k F i 2
z 0
ðk
#
@
q i @
@
q k
@
q i @
@
q i @
q k
i
@
0
1
q k 5
tr @ A i
@
q i @ D i
g T @ A i
@
q i @ E i
f T @ A i
@
q i @ F i
@
A 2
ðk
q k 2
.
@
q k
@
@
q k
(4.51)
i
@ _
tr @ A i
@
q i @ D i
(4.52)
q k 5
@ _
q k
i
@ €
tr @
A i
@
q i @
D i
@ €
(4.53)
q k 5
q k
Thus, the gradients of torque with respect to state variables are obtained
through Equations (4.51 4.53) . It is essential to have closed-form expressions
for
the gradients as
this
facilitates
fast computation of
the optimization
algorithm.
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