Biomedical Engineering Reference
In-Depth Information
T
T
local frame
k
; z
0
5
0010
for a revolute joint and z
0
5
0000
for a prismatic joint; and finally,
δ
ik
is Kronecker delta.
The first term in the torque expression (equation of motion) is the inertia and
Coriolis torque; the second term is the torque due to gravity load; the third term
is the torque due to external force; and the fourth term represents the torque due
to the external moment.
7.5.2
Sensitivity analysis
In order to calculate the sensitivity, we must first find expressions for the deriva-
tives of various quantities. The derivatives,
@τ
i
@q
k
,
@τ
i
@ q
k
,
@τ
i
(
i
1to
n
;
k
1to
n
),
5
5
@ q
k
can be evaluated for the articulated spatial human mechanical system in a recursive
way using the foregoing recursive Lagrangian dynamics formulation:
8
<
tr
@
2
A
i
2
A
i
2
A
i
q
k
D
i
1
@
A
i
@
q
i
@
D
i
@
@
@
q
k
F
i
2
G
i
@
A
i
2
1
@
2
g
T
q
k
E
i
2
f
T
z
0
ðk
iÞ
#
@
q
i
@
q
k
@
q
i
@
@
q
i
@
q
k
@τ
i
@
tr
@
q
k
5
:
@q
i
@
A
i
D
i
@q
k
g
T
@
@q
i
@
A
i
E
i
@q
k
2
f
T
@
@q
i
@
A
i
F
i
@q
k
ðk
iÞ
2
.
(7.10)
@τ
i
@ _
tr
@
A
i
@
q
i
@
D
i
(7.11)
q
k
5
@ _
q
k
@τ
i
@
€
tr
@
A
i
@
q
i
@
D
i
@
€
(7.12)
q
k
5
q
k
A more detailed treatment on the derivation of sensitivity equations can be
found in some of the published work on sensitivity (
Xiang et al., 2008
).
7.5.3
Mass and inertia property
Predictive dynamics uses forward kinematics to transfer the motion from the ori-
gin towards the end-effector along the branch as shown in
Figure 7.3
. This pro-
cess involves only state variables and geometrical parameters. However,
backward dynamics propagates forces from the end-effector to the origin. Mass
and inertia property of
the links are taken into consideration for dynamic
analysis.
In
Figure 7.3
, joint (
k
) and joint (
k
1) for which
mass and inertia properties are defined in the local coordinate
z
i
1
3
. The links
between coordinates
z
i
1
3
and
z
i
1
2
, and
z
i
1
2
and
z
i
1
1
have zero link length, and
zero mass and inertia, so that the force is correctly transferred back through
z
i
1
3
,
z
i
1
2
, and
z
i
1
1
for the joint (
k
).
1
1) are connected by link (
j
1
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