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, i
@q k , i
@ q k , 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
#
@
q i @
q k
@
q i @
@
q i @
q k
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
2
.
(7.10)
i
@ _
tr @ A i
@
q i @ D i
(7.11)
q k 5
@ _
q k
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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