Information Technology Reference
In-Depth Information
q
min
li
=
q
min
+
ρ
q
(14.32)
i
q
max
li
=
q
max
−
ρ
,
q
(14.33)
i
<
ρ
<
.
.
in which 0
0
5 is a tuning parameter (in [4], the value
ρ
= 0
1 is typically
used).
The cost function used to avoid joint limits
V
jl
is given by
n
i
q
i
,
V
jl
(
q
)=
1
2
i
=1
δ
(14.34)
where
⎧
⎨
q
min
l
i
q
min
l
i
q
i
−
,
if
q
i
<
q
max
l
i
q
max
l
i
δ
i
=
q
i
−
,
if
q
i
>
.
(14.35)
⎩
0
,
else
The gradient of the cost function is easily obtained as
⎧
⎨
q
i
−
q
max
l
i
Δ
q
i
q
max
l
i
,
q
i
>
q
min
l
i
g
(
q
)=
q
i
−
.
(14.36)
q
min
l
i
,
q
i
<
⎩
Δ
q
i
0
,
else
To apply this method in an image-based visual servoing (IBVS) approach [2, 3],
it is necessary to relate changes in the image to joint velocities. To this end, IBVS
control laws are typically expressed in the operational space (
i.e.
, in the camera
frame), and then computed in the joint space using the robot inverse Jacobian. IBVS
can be augmented to include the avoidance of joint limits by directly expressing the
control law in the joint space, since manipulator joint limits are defined in this space.
The subtask functions
e
i
used are computed from visual features:
s
∗
i
e
i
=
s
i
−
(14.37)
where
s
i
is the current value of the visual features for subtask
e
i
and
s
∗
i
their desired
value. The interaction matrix
L
s
i
related to
s
i
is defined so that
s
i
=
L
s
i
ξ
,where
ξ
is the instantaneous camera velocity [8]. The interaction matrix
L
s
i
and the task
Jacobian
J
i
are linked by the relation:
J
i
=
L
s
i
MJ
q
,
(14.38)
where the matrix
J
q
denotes the robot Jacobian (
r
=
J
q
q
)and
M
is the matrix that
relates the variation of the camera velocity
ξ
to the variation of the chosen camera
pose parametrization
r
(
ξ
=
M r
)[8].
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