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The parameter that sets the amplitude of the control law due to the secondary task
is very important. In (14.31), the secondary task may be insufficient to avoid a joint
limit if
will result in some overshoot
in the effector velocity. One solution, proposed in [4], is to select the most critical
axis and to compute automatically the minimum value of
κ
is too small. Furthermore, if it is too large,
κ
to stop any motion on
this axis. More precisely, a critical axis is defined as an axis
i
such that
q
i
is between
its joint limits and its activation threshold, and such that
q
i
approaches its joint limit
due of the effect of the primary task. The effect of the primary task can be computed
by performing a prediction step. Assuming that the robot is located in
q
(
t
), without
considering a secondary task, the predicted position
q
(
t
+ 1) is given by
κ
q
(
t
+ 1)=
q
(
t
)+
q
Δ
t
.
(14.39)
From the set of all axes in the critical area such that
q
i
moves toward its joint
limit (
i.e.
,
q
i
(
t
+ 1) is nearer to the corresponding joint limit than
q
i
(
t
)), the axis for
which
q
k
(
t
+ 1) is the closest to its joint limits is selected. Then
κ
is computed to
stop any motion on this axis (
i.e.
,
q
k
(
t
+ 1)
−
q
k
(
t
)=0). The constraint
Δ
q
k
= 0
leads to compute
κ
as
(
J
1
e
1
)
k
κ =
(
P
1
g
(
q
))
k
,
(14.40)
where
k
is the number of the closest axis from the joint limit and (
.
)
k
is the
k
th value
of the vector.
14.3.2
Simulation Results
The whole of our simulations were conducted using Matlab
R
and the Machine
Vision Toolbox [7] and Robotics Toolbox [6]. For each simulation, feature points
consist of two points in 3D. This leaves the system with multiple DOF beyond that
required to achieve the primary task (
i.e.
, the robot is redundant with respect to the
given task).
Images were projected using a simulated camera with a focal length of 8
10
−
3
m. The u-coordinate (pix) and v-coordinate (pix) of the principal point are 256 and
256. The horizontal pixel pitch and vertical pixel pitch (pix/m) are 80000 and 80000.
The simulations use a 6 DOF eye-in-hand PUMA 560 robot. The standard PUMA
robot device module (PUM) controls a 5 or 6 DOF PUMA-type robot. A PUMA
robot usually consists of six revolute axes. The first three axes roughly correspond
to a torso, shoulder, and elbow rotation. The last three axes implement a standard
roll, pitch, roll wrist.
We assume that the camera velocity is the same as the end effector velocity (
M
=
I
). The estimate of depth of IBVS uses the depth value for the goal configuration.
Considering only the main task, the block diagram used by Simulink
R
×
is shown in
Figure 14.1.
Examples of the initial robot position and the initial and goal positions of feature
points can be seen in Figure 14.2.
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