Biomedical Engineering Reference
In-Depth Information
8
<
I
i
@
C
i
@ _
T
i
1
1
@
D
i
1
1
@ _
ðk
iÞ
q
k
1
#
q
k
@
D
i
@ _
(4.46)
q
k
5
:
T
i
1
1
@
D
i
1
1
@ _
ðk
.
iÞ
q
k
8
<
C
i
@ €
I
i
@
T
i
1
1
@
D
i
1
1
@ €
ðk
iÞ
q
k
1
#
q
k
@
D
i
@
€
(4.47)
q
k
5
T
i
1
1
@
:
D
i
1
1
@ €
ðk
.
iÞ
q
k
8
<
0
ðk
iÞ
#
@
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
iÞ
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
iÞ
#
@
q
i
@
@
q
k
@
q
i
@
@
q
i
@
q
k
@τ
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
iÞ
q
k
2
.
@
q
k
@
@
q
k
(4.51)
@τ
i
@ _
tr
@
A
i
@
q
i
@
D
i
(4.52)
q
k
5
@ _
q
k
@τ
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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