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Section 2.1 and calculated in [12, 13]. In order to visualize the domain of attraction,
we show the level sets of
V
in most cases and a level surface plus three orthogonal
slices of the level surface that pass through the origin for the SO(3) case. In each
figure we also plot the sign of
V
at each point.
Subsection 2.3.2 then uses the results of Section 2.3.1 to show experimental re-
sults for a variety of the combinations of the degrees of freedom in the controlled
environments. We conduct 50 trials for each type of robot motion by going to the
edge of the domain of attraction as determined in Section 2.3.1.
2.3.1
Analysis of the Domain of Attraction
To find the domain of attraction experimentally, we perform the following sequence
for each type of robot motion:
1. gather images on an evenly space grid of the camera workspace;
2. compute the value of the Lyapunov function,
V
, at each position;
3. compute the control input,
u
, at each position as described in Section 2.2;
4. compute the gradient of the Lyapunov function,
∇
V
, using a finite central dif-
ference approximation at each position;
5. compute the time derivative of the Lyapunov function using the chain rule
V
=
∇
V
x
=
∇
V
u
;
and
(2.14)
6. search for the level set of V which is homeomorphic to
S
n
with the largest inte-
rior and for which
V
<
0 for every point on the interior except the goal position.
By the theory of Lyapunov stability[14], we can conclude that the volume described
by the interior of the level set of the Lyapunov function as described above is the
domain of attraction for the asymptotically stable system.
2.3.1.1
2D
Figure 3(b) shows the analysis of the domain of attraction for the goal scene and
kernels shown in Figure 3(a). The chosen kernels are placed around the star object
and provide for a domain of attraction that approaches the joint limits of the robot
for planar motions. Although not seen in this figure, kernel selection often results
in a Lyapunov bowl that is not symmetrically shaped. The positive definite matrix
P
from (2.5) can be engineered to change the shape of the Lyapunov function and
provide faster convergence in all directions.
In contrast, Figures 3(c) and 3(d) show how a poor selection of kernels unin-
tentionally resulted in a second subset of the domain of the Lyapunov function to
which the system would converge. Similar to the successful set of kernels, these
kernels were placed at different locations around the star object near the center of
the goal image. The combination of scene and kernels resulted in an invariant region
for which the robot motion was stable to some point, but not asymptotically stable
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