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12.4.1
Estimation with Known Angular and Linear Velocities
In this section, a nonlinear SfM estimator is presented for the perspective dynamic
system given by (12.11) assuming that all six velocities are known along with the
intrinsic camera calibration matrix
A
. Scenarios where the relative motion
(
t
) and
b
(
t
) are known include a camera attached to the end-effector of a robot or attached to
some vehicle with known motion (acquired from global positioning system (GPS),
inertial measurement unit (IMU), or other sensor data).
ω
12.4.1.1
Observer Design
The Euclidean structure
m
(
t
) can be estimated once
y
3
(
t
) in (12.4) is determined,
since
y
1
(
t
) and
y
2
(
t
) are measurable from (12.3) and (12.6). Since
y
3
(
t
) appears
in the image dynamics for
y
1
(
t
),
y
2
(
t
),and
y
3
(
t
) in (12.11), the subsequent de-
velopment is based on the strategy of constructing the estimates
3
y
(
t
)
∈
R
=
y
1
(
t
)
y
2
(
t
)
y
3
(
t
)
T
. To quantify this objective, an estimate error
e
(
t
)
3
∈
R
e
1
(
t
)
e
2
(
t
)
e
3
(
t
)
T
is defined as
e
1
y
1
−
y
1
,
e
2
y
2
−
y
2
,
e
3
y
3
−
y
3
.
(12.12)
Based on Assumption 12.3,
e
1
(
t
) and
e
2
(
t
) can be determined and used in the esti-
mate design. The estimates
y
1
(
t
) and
y
2
(
t
) are generated according to the update law
·
y
1
=
y
3
b
1
−
y
1
y
2
ω
1
+(1 +
y
1
)
y
1
y
3
b
3
−
ω
2
−
y
2
ω
3
+
k
1
e
1
,
(12.13)
·
y
2
=
y
3
b
2
−
(1 +
y
2
)
y
2
y
3
b
3
−
ω
1
+
y
1
y
2
ω
2
+
y
1
ω
3
+
k
2
e
2
,
(12.14)
where
k
1
,
are strictly positive estimator gains. The estimate
y
3
(
t
) is generated
based on the locally Lipschitz projection defined as [25]
k
2
∈
R
⎧
⎨
if
y
3
≤
y
3
(
t
)
≤
y
3
or
φ
if
y
3
(
t
)
>
y
3
and
φ
(
t
)
≤
0or
·
y
3
(
t
)=
pro j
(
y
3
,
φ
)=
if
y
3
(
t
)
<
y
3
and
φ
(
t
)
≥
0
(12.15)
⎩
¯
φ
if
y
3
(
t
)
>
y
3
and
φ
(
t
)
>
0
˘
φ
if
y
3
(
t
)
<
y
3
and
φ
(
t
)
<
0
·
e
1
,
·
e
2
)
where
φ
(
y
1
,
y
2
,
y
3
,
ω
1
,
ω
2
,
b
3
,
e
1
,
e
2
,
∈
R
is defined as
y
3
b
3
+
y
2
y
3
ω
1
−
φ
y
1
y
3
ω
2
+
h
1
e
1
+
h
2
e
2
+
k
3
h
1
(
k
1
e
1
+
·
e
1
)+
h
2
(
k
2
e
2
+
·
e
2
)
g
,
(12.16)
and
¯
and
˘
φ
(
t
)
∈
R
φ
(
t
)
∈
R
are defined as
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