Biomedical Engineering Reference
In-Depth Information
k r f
link i in the local frame i ;
is position of the external force in the local frame
T
T
k ; z 0 5 ½ 0010
for a revolute joint and z 0 5 ½ 0000
for a pris-
matic joint.
δ ik is Kronecker delta.
The first term in torque expression is the inertia and Coriolis torque, the second
term denotes the torque of force due to gravity, the third term is the torque due to
external force, and the fourth term represents the torque due to external moment.
4.5.3 Sensitivity analysis
The derivatives, i
@
q k , i
, i
@
( i
1to n ; k
1to n ), can be evaluated for a
5
5
@ _
q k
q k
mechanical system in a recursive way using the foregoing recursive Lagrangian
dynamics formulation.
4.5.4 Kinematics sensitivity analysis
For a given point, sensitivity of position, velocity, and acceleration with respect
to state variables relates to transformation matrices A, B, and C:
8
<
@
A i 2 1
@
T i
ðk
,
q k
@
A i
@
A i 2 1 @
T i
(4.38)
q k 5
:
ðk
5
@
q k
0
ðk
.
<
@
B i 2 1
@
T i 1 @
A i 2 1
@
@
T i
@
q i _
q i
ðk
,
q k
q k
@
B i
2 T i
@
(4.39)
q k 5
B i 2 1 @
T i
@q k 1
A n 2 1 @
@
:
q k _
q i
ðk
5
0
ðk
.
8
<
@
B i 2 1
@ _
T i
ðk
,
q k
@
B i
@ _
A i 2 1 @ T i
@
(4.40)
q k 5
:
ðk
5
q k
0
ðk
.
8
<
2 T i
@
@ C i 2 1
@
2 @ B i 2 1
@
@ T i
@
q i 1 @ A i 2 1
@
@
q i 1 @ A i 2 1
@
@ T i
@
T i 1
q i _
_
q i
q i
ðk
,
q i
q k
q k
q k
q k
@
C i
@
2 T i
@
3 T i
@
2 T i
@
q k 5
C i 2 1 @
T i
@q k 1
2B i 2 1 @
A i 2 1 @
A i 2 1 @
:
q i 1
q k _
q i 1
q k _
q k
q i
ðk
5
0
ðk
.
(4.41)
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