Biomedical Engineering Reference
In-Depth Information
where A 0 5 1 and B 0 5 C 0 5 0. Then, the global position, velocity, and accelera-
tion of a point in the Cartesian coordinate system can be calculated using the
following formulas:
0 r j 5
0
0
A j r j ;
r j 5
B j r j ;
r j 5
C j r j
(8.4)
where 0 r j and r j are global and local augmented coordinates, respectively. It is
noted that the repeated index is not summed in the above equations or the equa-
tions that follow.
8.2.2 Backward recursive dynamics
We proceed to develop the backward recursion for the dynamic analysis, which is
accomplished by defining a 4
4 transformation matrix D i and 4
1 transforma-
3
3
tion vectors E i , F i , and G i as follows.
Given
f k 5 k f x
0
k f y
k f z
the
external
force
and
the moment
h k 5 k h x k h y k h z 0 for the link k , defined in the global coordinate system, the
joint actuation torques
τ i for i
n to 1 are computed as:
5
tr @
A i
@
g T @
A i
@
f k @
A i
@
G i A i 2 1 z 0
τ i 5
q i D i
q i E i 2
q i F i 2
(8.5)
2
I i C i 1
D i 5
T i 1 1 D i 1 1
(8.6)
i r i 1 T i 1 1 E i 1 1
E i 5 m i
(8.7)
k r f δ ik 1
F i 5
T i 1 1 F i 1 1
(8.8)
h k δ ik 1
G i 5
G i 1 1
(8.9)
where D n 1 1 5
0; I i is the inertia matrix for link i ; m i
is the mass of link i ; g is the gravity vector;
0 and E n 1 1 5
F n 1 1 5
G n 1 1 5
i r i is the location of center of mass
k r f
of link i in the local frame i ;
is the position of the external force in the local
T for a revolute joint; and
frame k ; z 0 5 ½ 0010
δ ik is Kronecker delta.
Where the first term in the equations of motion is the inertia and Coriolis tor-
que, 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.
8.2.3 Sensitivity analysis
This is a highly nonlinear programming problem, and optimization lends itself
well to calculating reasonable solutions. Accurate sensitivity is a key factor for
efficiently achieving an optimal solution for a gradient-based optimization algo-
rithm, such as the sequential quadratic programming (SQP) method.
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