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e
3
y
3
−
·
V
=
y
3
k
1
e
1
−
k
2
e
2
+
h
1
e
1
e
3
+
h
2
e
2
e
3
+
e
3
f
−
−
φ
.
(12.30)
δ
0and
y
3
−
y
3
δ
Since
φ
(
t
)
>
0,
e
3
(
t
)
<
<
0, the bracketed term in (12.30) is positive,
and (12.30) can be upper bounded as
·
V
k
1
e
1
−
k
2
e
2
+
h
1
e
1
e
3
+
h
2
e
2
e
3
+
e
3
f
≤−
·
V
k
1
e
1
−
k
2
e
2
−
y
2
ω
1
+
y
1
ω
2
+
k
3
)
e
3
.
≤−
(
−
(
y
3
+
y
3
)
b
3
−
By choosing the gain
k
3
(
t
) according to (12.20), the expression for
·
V
(
t
) can be
upper bounded as in (12.28), which can be used to obtain (12.29).
Case 3:
y
3
(
t
)
0: Taking the time derivative of
V
(
e
) and after
utilizing (12.21) and (12.25) yields
<
y
3
and
φ
(
t
)
<
e
3
y
3
−
y
3
·
V
=
k
1
e
1
−
k
2
e
2
+
h
1
e
1
e
3
+
h
2
e
2
e
3
+
e
3
f
−
−
φ
.
(12.31)
δ
0and
y
3
−
y
3
δ
Since
φ
(
t
)
<
0,
e
3
(
t
)
>
<
0, the bracketed term in (12.31) is positive,
and (12.31) can be upper bounded as
·
V
k
1
e
1
−
k
2
e
2
−
y
2
ω
1
+
y
1
ω
2
+
k
3
)
e
3
.
≤−
(
−
(
y
3
+
y
3
)
b
3
−
By choosing the gain
k
3
(
t
) according to (12.20), the expression for
·
V
(
t
) can be
upper bounded as in (12.28), which can be used to obtain (12.29).
The expression in (12.29) indicates that
e
(
t
) is exponentially stable, and the
closed-loop error dynamics can be used to show that all signals remain bounded.
Specifically, since
e
(
t
)
∈L
∞
,and
y
(
t
)
∈L
∞
from Assumption 12.1, then
y
(
t
)
∈L
∞
.
Assumption 12.1-12.2 indicate that
y
(
t
)
,
ω
(
t
)
∈L
∞
, so (12.20) can be used to prove
that the gain
k
3
(
t
)
∈L
∞
. Based on the fact that
e
(
t
),
y
(
t
),
ω
(
t
),
b
(
t
),
k
3
(
t
)
∈L
∞
,
standard linear analysis methods can be used to prove that
·
e
(
t
)
∈L
∞
.Since
y
3
(
t
) is
exponentially estimated, (12.3), (12.4), and (12.6) can be used to recover the struc-
ture
m
(
t
) of the feature points.
12.4.2
Estimation with a Known Linear Velocity
In some scenarios, the linear and angular velocities of the camera may not be com-
pletely known (
e.g.
, the camera is attached to a vehicle that does not contain velocity
sensors, or the sensor feedback becomes temporarily/permanently lost). In this sec-
tion, an estimator is designed for the same perspective dynamic system in (12.11),
yet the angular velocity is considered unknown and only one of the linear velocities
(
i.e.
,
b
3
) is available. Solving the SaM estimation problem is problematic; hence,
some information about the motion of the camera is typically required. To this end,
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