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results such as [6] and [10] and the estimator in this section replace known linear
velocity measurements with an uncertain dynamic model of the linear velocity
b
(
t
)
given by
b
i
(
t
)=
q
(
y
3
b
i
,
t
)
b
i
,∀
i
=
{
1
,
2
,
3
},
(12.32)
where
q
(
y
3
b
i
,
is a known function of unknown states.
To facilitate the design and analysis of the subsequent observer, a new state
u
(
t
)
t
)
∈
R
∈
u
1
(
t
)
u
2
(
t
)
u
3
(
t
)
T
3
U ⊂
R
is defined as
,
,
,
u
1
=
y
3
b
1
u
2
=
y
3
b
2
u
3
=
b
3
(12.33)
where
u
3
(
t
) is a measurable linear velocity. The set
is a closed and bounded
set. After utilizing (12.11) and (12.32), the dynamics for
u
1
(
t
),
u
2
(
t
),
u
3
(
t
) can be
expressed as
U
·
u
1
=
y
3
b
3
u
1
+(
y
2
ω
1
−
y
1
ω
2
)
u
1
+
q
(
u
1
)
u
1
(12.34)
·
u
2
=
y
3
b
3
u
2
+(
y
2
ω
1
−
y
1
ω
2
)
u
2
+
q
(
u
2
)
u
2
(12.35)
·
u
3
=
q
(
y
3
b
3
,
t
)
b
3
.
(12.36)
The observer in this section is developed to estimate the structure and the partial
motion
i.e.
, velocities
b
1
(
t
) and
b
2
(
t
). Using the same strategy developed for the
structure estimator, new estimates
u
i
(
t
)
∀
i
=
{
1
,
2
,
3
}∈
R
are developed for
u
i
(
t
).
To quantify this objective, estimate errors
e
4
(
t
),
e
5
(
t
),
e
6
(
t
)
∈
R
are defined as
u
3
(12.37)
and an augmented error state is defined as
e
(
t
)=
e
T
(
t
)
e
4
(
t
)
e
5
(
t
)
e
6
(
t
)
T
,where
e
1
(
t
),
e
2
(
t
),and
e
6
(
t
) are measurable.
Assumption 12.5.
The functions
q
(
u
i
e
4
=
u
1
−
u
1
,
e
5
=
u
2
−
u
2
,
e
6
=
u
3
−
t
) are piecewise differentiable with respect
to
u
i
(
t
), and the partial derivatives of
q
(
u
i
,
,
t
) with respect to
u
i
(
t
) are bounded,
i.e.
,
q
(
u
i
t
) is Lipschitz continuous [6, 10].
Assumption 12.6.
The function
q
(
y
3
b
3
,
t
) is linear in
y
3
; hence,
q
(
y
3
b
3
,
t
)
−
q
(
y
3
b
3
,
t
)=
c
1
(
y
3
−
y
3
)
,
(12.38)
where
c
1
is a known scalar where
c
1
= 0, which suggests that the known linear
velocity
b
3
(
t
)
= 0.
Assumption 12.7.
The linear velocities
b
1
(
t
)
= 0.
Assumption 12.8.
Since the linear camera velocities
b
(
t
) are upper and lower
bounded by constants, the following inequalities can be determined using Assump-
tion 12.1 and the definition of
u
1
(
t
) and
u
2
(
t
)
= 0and
b
2
(
t
)
b
3
.
u
1
≤
u
1
≤
u
1
,
u
2
≤
u
2
≤
u
2
,
b
3
≤
b
3
=
u
3
≤
(12.39)
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