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motion between views
{
B
}
and
{
C
}
is the same as the 3D motion between views
{
}
{
}
A
and
C
.
}
are collocated. That is, when
t
= 0, the differential of the mapping defined by (8.1)
is degenerate. Indeed, in this case the normal to the plane
n
is not observable. The
singularity of the parametrization does not affect the validity of the correspondence
H ≡
{
}
{
Remark 8.1.
The local parametrization given by (8.1) is singular when
A
and
B
SL
(3), however, it does mean that the parametrization (8.1) is very poorly con-
ditioned for homography matrices close to
SO
(3). This is fundamental reasons why
it is preferable to do both image based visual servo control and temporal smooth-
ing directly on the homography group rather than extracting structure variables
explicitly.
The Lie algebra
sl
(3) for
SL
(3) is the set of matrices with trace equal to zero:
sl
(3)=
3
×
3
{
X
∈
R
|
tr(
X
)=0
}
. The adjoint operator is a mapping Ad :
SL
(3)
×
sl
(3)
→
sl
(3) defined by
Ad
H
X
=
HXH
−
1
,
H
∈
SL
(3)
,
X
∈
sl
(3)
.
3
×
3
the Euclidean matrix inner product and Frobenius
For any two matrices
A
,
B
∈
R
norm are defined as
=
= tr(
A
T
B
)
A
,
B
, ||
A
||
A
,
A
.
3
×
3
Let
P
denote the unique orthogonal projection of
R
onto
sl
(3) with respect to
·,·
the inner product
,
(
H
) :=
H
I
tr(
H
)
3
P
−
∈
sl
.
(3)
(8.2)
3
×
3
)is
The projection onto the complementary subspace (the span of
I
in
R
defined by
(
H
)=
tr(
H
)
3
P
⊥
(
H
) :=
H
−
P
I
.
(8.3)
,
P
⊥
(
H
)
Clearly one has
P
(
H
)
= 0.
8.3
Nonlinear Observer on
SL
(3)
Consider the left invariant kinematics defined on
SL
(3)
H
=
HA
(8.4)
where
H
(3). A general framework for nonlinear filtering on the
special linear group is introduced. The theory is developed for the case where
A
is assumed to be unknown and constant. The goal is to provide a set of dynamics
for an estimate
∈
SL
(3) and
A
∈
sl
H
(
t
)
SL
(3) of
H
(
t
) and an estimate
A
(
t
)
∈
∈
sl
(3) of
A
to drive
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