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Remark 12.1.
Assumptions 12.6 and 12.7 form observability conditions for the sub-
sequent estimator,
i.e.
camera must have a velocity in all three directions. This ob-
servability condition is more stringent than for the estimator designed in Section
12.4.1. It is intuitive that if more parameters have to be estimated, more information
is required. The information about
e
3
(
t
) is obtained using known
b
3
(
t
).
Based on Assumption 12.5, the mean-value theorem can be used to conclude that
q
(
u
1
)
−
q
(
u
1
)=
J
1
e
4
,
q
(
u
2
)
−
q
(
u
2
)=
J
2
e
5
,
(12.40)
where
J
1
=
∂
q
(
u
1
)
∂
,
J
2
=
∂
q
(
u
2
)
∂
J
1
,
J
2
.
u
1
∈
R
u
2
∈
R
and are upper bounded as
|
J
1
|≤
|
J
2
|≤
12.4.2.1
Step 1: Angular Velocity Estimation
Solutions are available in literature that can be used to determine the relative angular
velocity between the camera and a target. A brief description of the angular veloc-
ity estimator presented in [8] is provided as a means to facilitate the subsequent
development. The rotation matrices defined in Section 12.2 are related as
R
=
R
(
R
∗
)
T
.
(12.41)
The relationship between angular velocity of the camera
ω
(
t
) and the rotation ma-
trix
R
(
t
) isgivenby[41]
[ω]
×
=
·
R R
T
.
(12.42)
F
c
To quantify the rotation mismatch between
and
F
c
, a rotation error vector
3
is defined by the angle-axis representation of
R
(
t
) as
e
ω
(
t
)
∈
R
e
ω
u
ω
(
t
)
θ
ω
(
t
)
,
(12.43)
3
represents a unit rotation axis, and
where
u
ω
(
t
)
∈
R
θ
ω
(
t
)
∈
R
denotes the rotation
angle about
u
ω
(
t
) thatisassumedtobeconfinedtoregion
−
π
<
θ
ω
(
t
)
<
π
.Taking
time derivative of (12.43), the following expression can be obtained
·
e
ω
=
L
ω
ω
(12.44)
3
×
3
is defined as [8]
∈
R
where the invertible Jacobian matrix
L
ω
(
t
)
⎛
⎞
⎝
⎠
I
3
−
θ
2
θ
ω
(
t
)
2sin
θ
ω
(
t
)
2
sin
[
u
ω
]
2
L
ω
[
u
ω
]
×
+
1
−
×
.
(12.45)
3
A robust integral of the sign of the error (RISE)-based observer
e
ω
(
t
)
∈
R
is gen-
erated in [8] from the following differential equation
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