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⎡
⎤
z
1
(
e
1
+
Y
1
)
z
1
1 +(
e
1
+
Y
1
)
2
−
1
/
/
⎣
⎦
.
z
2
(
e
2
+
Y
2
)
/
z
2
1 +(
e
2
+
Y
2
)
2
L
(
z
,
e
)=
−
1
/
(15.7)
z
3
(
e
3
+
Y
3
)
z
3
1 +(
e
3
+
Y
3
)
2
−
1
/
/
The term
∂
e
∂
t
, which represents task variation due to the target motion, can be ex-
pressed by the relation
∂
e
=
B
(
z
,
e
)
ω
,
(15.8)
∂
t
3
×
4
is defined by
where the matrix
B
(
z
,
e
)
∈
ℜ
⎡
⎤
(
e
1
+
Y
1
)
(
e
1
+
Y
1
)
1
/
z
1
−
/
z
1
1
/
z
1
−
/
z
1
⎣
⎦
(
e
2
+
Y
2
)
B
(
z
,
e
)=
1
/
z
2
−
/
z
2
0
0
(15.9)
(
e
3
+
Y
3
)
z
3
(
e
3
+
Y
3
)
1
/
z
3
−
/
z
3
−
1
/
/
z
3
and
⎡
⎣
⎤
⎦
∈
ℜ
−
v
E
cos(
α
)
−
v
E
sin(
α
)
4
ω
=
.
(15.10)
−
l
ω
E
cos(
α
)
−
l
ω
E
sin(
α
)
The target velocity vector
v
E
ω
E
is supposed to be square integrable but un-
known. The vector
e
)
as the associated disturbance matrix. With this notation, the condition C4 can be
specified as follows:
ω
can then be considered as a disturbance vector and
B
(
z
,
ω
∈L
2
and there exists a finite scalar
δ
1
>
0 such that
2
=
∞
=
∞
0
1
δ
1
.
)
ω
(
v
E
(
)+
l
2
E
(
ω
0
ω
(
τ
(
τ
)
d
τ
τ
ω
τ
))
d
τ
≤
(15.11)
To take into account the limits on the actuators dynamics, the statement of condi-
tion C3 can be specified by introducing the following bounds on the camera velocity
and acceleration:
−
u
1
T
u
1
,
(15.12)
T
−
u
0
u
0
.
(15.13)
By considering the extended state vector
x
=
e
T
6
∈
ℜ
,
(15.14)
with the following matrices
x
)=
0
L
(
z
e
)
00
,
6
×
6
;
A
(
z
,
∈
ℜ
B
1
=
0
I
3
x
)=
B
(
z
(15.15)
,
e
)
6
×
3
;
6
×
4
∈
ℜ
B
2
(
z
,
∈
ℜ
0
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