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0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0.15 0.1 0.05
0
0.05
0.1
0.15
(a)
(b)
Fig. 2.4
An analysis of the domain of attraction for depth motion: (a) the goal image and the
contours of the kernels used; and (b) the plot of
V
. The arrow indicate the value of
V
at each
depth
function of the camera depth along the optical axis for which
V
was negative every-
where except the goal location and zero at the goal, as seen in Figure 2.4.
2.3.1.3
SO(3)
Figure 2.5 shows the analysis of the domain of attraction for the set of kernels shown
in Figure 2.6. To determine the domain of attraction, we determined all the orienta-
tions of the robot for which our empirical computation of the time derivative of the
Lyapunov function was positive. We then start at the level surface
V
=
c
for
c
very
large and slowly decrease
c
until arriving at a level surface where
V
0forevery
point in its interior except at the goal. The volume defined by the interior of the level
surface defines the domain of attraction for this 3 degrees of freedom system. Figure
5(a) shows this level surface and nearby points where
V
<
0. Figures 5(b), 5(c) and
5(d) show orthogonal slices of the Lyapunov function through the origin. It becomes
much easier to visualize the magnitude and shape of the domain of attraction using
these slices, taking into account that these may give a inaccurate representation for
strangely shaped level surfaces of
V
.
Figure 2.6 shows the kernels used for determining the domain of attraction and
for the experiments shown later. It is often useful to show the kernel overlayed on
top of the image at the goal position to give an idea of how the kernel-projected
values are generated from the image at the goal. It is important to note that a change
in the kernels, whether location or covariance for our Gaussian kernels, carries with
it an implicit change in the domain of attraction.
>
2.3.2
Experimental Results
To validate the domains of attraction determined above, we conducted 50 trials for
each of the degrees of freedom analyzed in Section 2.3.1. We randomly set the
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