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1 +
y
3
1 +
y
3
−
y
3
−
y
3
¯
˘
φ
φ
,
φ
φ
.
(12.17)
δ
δ
The projection in (12.15) ensures that the estimate
y
3
∈
Ω
δ
∀
t
≥
0, where
{
|
−
δ
≤
≤
δ
}
Ω
δ
=
y
3
y
3
y
3
y
3
+
δ
>
,
,
∈
R
for some known arbitrary constant
0. In (12.16),
g
(
y
1
y
2
b
)
is defined as
y
1
b
3
)
2
+(
b
2
−
y
2
b
3
)
2
g
(
y
1
,
y
2
,
b
)
(
b
1
−
,
(12.18)
h
1
(
y
1
,
b
1
,
b
3
),
h
2
(
y
2
,
b
2
,
b
3
)
∈
R
are defined as
h
1
(
y
1
,
b
1
,
b
3
)
(
b
1
−
y
1
b
3
)
h
2
(
y
2
,
b
2
,
b
3
)
(
b
2
−
y
2
b
3
)
,
(12.19)
and
k
3
(
t
)
∈
R
is a positive estimator gain defined as
k
3
(
t
)
>|
(2
y
3
+
δ
)
|
b
3
|−
y
2
ω
1
+
y
1
ω
2
|.
(12.20)
Assumption 12.4.
The function
g
(
y
1
,
0. This assumption is an ob-
servability condition for (12.13)-(12.15). This condition indicates that the motion of
the camera should not be parallel to the optical axis of the camera. If the velocities
are parallel to the optical axis of the camera then
y
1
(
t
)
y
2
(
t
)
T
=
b
1
(
t
)
b
3
(
t
)
y
2
,
b
)
= 0
∀
t
≥
b
3
(
t
)
T
b
2
(
t
)
,
∀
= 0, which is called
focus of expansion
[12, 20, 5]. This condition also says
not all linear velocities can be zero at the same instant of time,
i.e.
,
b
1
(
t
)
b
3
(
t
)
,
b
2
(
t
),
b
3
(
t
)
= 0, simultaneously.
12.4.1.2
Error Dynamics
Differentiating (12.12) and using (12.11)-(12.14) yields
·
e
1
=
h
1
e
3
−
·
e
2
=
h
2
e
3
−
k
1
e
1
,
k
2
e
2
.
(12.21)
To facilitate further development, the expressions in (12.18), (12.19) and (12.21) are
used to conclude that
e
3
=
h
1
(
·
e
1
+
k
1
e
1
)+
h
2
(
·
e
2
+
k
2
e
2
)
g
.
(12.22)
Since
·
y
3
(
t
) is generated from a projection law, three possible cases for
·
e
3
(
t
) are
considered.
Case 1:
y
3
≤
0:
After using (12.11), (12.15), (12.16) and (12.22), the time derivative of
e
3
(
t
) in
(12.12), can be determined as
y
3
(
t
)
≤
y
3
or if
y
3
(
t
)
>
y
3
and
φ
(
t
)
≤
0orif
y
3
(
t
)
<
y
3
and
φ
(
t
)
≥
·
e
3
=
f
(12.23)
where
f
(
y
,
y
3
,
b
,
ω
1
,
ω
2
,
e
)
∈
R
is defined as
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