Information Technology Reference
In-Depth Information
⎛
⎞
f
x
su
x
0
f
y
u
y
001
⎝
⎠
,
A
=
(9.2)
f
x
,
f
y
∈
R
being the focal lengths,
u
x
,
u
y
∈
R
the coordinates of the principal
the aspect ratio. Similarly,
q
i
projects onto
F
∗
at the point
point, and
s
∈
R
p
∗
i
=(
p
∗
i
,
1
,
p
∗
i
,
2
,
1)
T
3
∈
R
given by
d
i
p
∗
i
=
AO
∗
T
(
q
i
c
∗
)
−
(9.3)
where
d
i
is the depth of the point with respect to
F
∗
. The camera pose between
F
and
F
∗
is described by the pair
3
(
R
,
t
)
∈
SO
(3)
×
R
(9.4)
where
R
and
t
are the rotational and translational components expressed with respect
to
F
∗
and given by
R
=
O
∗
T
O
O
∗
T
(
c
c
∗
)
(9.5)
−
t
=
O
∗
T
(
c
c
∗
)
−
(
t
is normalized because, by exploiting only the image projections of the points
q
1
,...,
q
N
and the matrix
A
, the translation can be recovered only up to a scale
factor). Let
p
p
∗
∈
R
2
N
,
be the vectors defined as
⎛
⎞
⎛
⎞
p
∗
1
,
1
p
∗
1
,
2
.
p
∗
N
,
1
p
∗
N
,
2
p
1
,
1
p
1
,
2
.
p
N
,
1
p
N
,
2
⎝
⎠
⎝
⎠
p
∗
=
p
=
,
.
The goal condition of an eye-in-hand visual servo system can be expressed as
p
∗
∞
≤
ε
p
−
(9.6)
is a threshold chosen to limit the distance between
p
and
p
∗
(for exam-
ple, via the infinity norm).
This chapter addresses the computation of upper and lower bounds of the worst-
case robot positioning error introduced by image measurement errors through the
goal condition (9.6). In particular, we consider the worst-case rotational error
where
ε
∈
R
p
∗
∞
≤
ε
s
r
(
ε
)=sup
R
t
θ
s.t.
p
−
(9.7)
,
where
θ
∈
[0
,
π
] is the angle in the representation of
R
via exponential coordi-
nates,
i.e.
R
=
e
[
θ
u
]
×
(9.8)
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