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Figure 14.3 shows the result of a successful experiment if there is no joint limit
for the robot. Figure 14.3(a-b) show the exponential decay of the error and the
velocity of the robot, respectively. Though minor variations may arise, this 2D be-
havior is similar for all the successful experiments reported in this section. Figure
14.4 depicts the joint position of six axes of the robot.
Then we set the joint limits of q 6 to be smaller than 0.19. Therefore, if no par-
ticular strategy is considered to avoid joint limits, the visual task fails. Figure 14.5
shows the block diagram considering the joint limit. We performed a set of exper-
iments using the cost function defined previously with various values of the coef-
ficient
κ
(
κ
=0.1,
κ
=1,
κ
=100). During the experiments for these different values,
if
=0.1), the motion generated by the main task in the direction of
the joint limits is not compensated enough by the secondary task, and then the joint
limits cannot be avoided effectively. Figure 14.6(a) shows that q 6 exceeds the upper
limit of the joint. However, if
κ
is too small (
κ
=100), it may result in too large ve-
locities variance in the first seconds as shown in Figure 14.6(c). Tuning is therefore
performed based on trial and error. This solution is not good.
The last plot (auto) on Figure 14.6(d) depicts the results obtained using the ap-
proach proposed where
κ
is too high (
κ
κ
with respect to the time is shown in Figure 14.7. It illustrates that the joint approach
the joint limit and then move away from it.
κ
is automatically computed by (14.40). The change of
14.4
Gradient Projection Method for Occlusion Avoidance
In this section, we describe how gradient projection methods can be used to prevent
the occlusion of target features by static objects during the camera motion [11]. Be-
low, we adopt a cost function proposed in [10], and develop the necessary methods
for exploiting this cost function in a gradient projection framework.
14.4.1
Basic Algorithm Design
We assume here that image processing methods are available to detect the presence
of a potential occluding object in the image. Given that the potential occlusion can
be detected, let d = d x + d y be the distance in the image from a feature point to
the contour of the occluding object, and let x a be the point of the occluding object
that is the closest to the target in the image. The cost function V occ is defined in the
image space, so that it is maximal when d is 0, and nearly 0 when d is high. This is
achieved by the cost function
V occ = e β d 2
.
(14.41)
is arbitrary and can be used to tune the effect of the avoidance
control law. The gradient in the image space is obtained by a simple calculation:
The parameter
β
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