Information Technology Reference
In-Depth Information
If we choose the control input
x
(
t
))
d
w
T
∂
K
(
w
)
∂
P
T
(
u
=
−
I
(
w
,
ξ
−
ξ
0
)
(2.8)
x
I
the time derivative of
V
is given by
x
(
t
))
d
w
ξ
−
ξ
0
)
T
P
2
∂
K
(
w
)
∂
V
=
−
(
I
(
w
,
≤
0
.
(2.9)
x
I
by construction
and have shown a control input
u
that ensures its time derivative is negative semidef-
inite in
We now have a Lyapunov function that is positive definite in
ξ
. Asymptotic stability to the goal configuration, however, must be shown
in
x
(
t
) (assuming without loss of generality that
x
0
= 0). That is, we must show that
both
V
and
V
are positive definite and negative semidefinite in
x
(
t
), respectively.
To see this locally, we look at the first order Taylor series expansion of
ξ
ξ
about the
point
ξ
0
, as done for classical visual servoing in [5]:
ξ
0
+
∂ξ
∂
x
x
(
t
)+
O
(
x
2
)
ξ
=
0
=
∂ξ
∂
(2.10)
x
x
(
t
)+
O
(
x
2
)
ξ
−
ξ
0
=
J
x
(
t
)+
O
(
x
2
)
ξ
−
ξ
.
Then, inserting (2.10) into (2.5) and (2.6) we achieve the desired definiteness in
x
(
t
)
under certain conditions on the Jacobian matrix
J
:
V
=
1
2
x
(
t
)
T
J
T
PJ
x
(
t
)+
O
(
x
3
)
Q
=
J
T
PJ
,
,
(2.11)
and
V
=
x
(
t
)
T
QQ
T
x
(
t
)+
O
(
x
3
)
−
.
(2.12)
p
is full column rank, where
n
is the number of kernels
and
p
is the dimension of
x
(
t
),then
Q
will be a full rank
p
n
×
If the Jacobian matrix
J
∈
R
p
matrix. From (2.11),
(2.12), and a full rank assumption for Q, we can conclude that
V
is positive definite
and
V
is negative definite in some neighborhood of the goal with
V
= 0and
V
= 0
at the goal. The rank condition on Q also clearly sets forth a necessary condition on
the number of kernels required; namely that there must be at least as many kernel
measurements as there are degrees of freedom.
Although we have only shown the method for the simple 2D case, the reader can
infer how it extends to other motions given an appropriate change of coordinates
based on the relationship between camera motion and image transformation. Several
motions are worked out in great detail in our previous work [12, 11, 13]. These
include 2D translation, depth, roll, rigid body rotations, and some combinations of
these.
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