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sleeve
funnel
apex
path
start
Fig. 2.
The path, apex, and funnel of the funnel algorithm
define
[
u, v
]=
ʻu
+(1
−
ʻ
)
v,
[
u, v
[=[
u, v
]
\{
v
}
,
]
u, v
[=[
u, v
[
\{
u
}
.
Based on the idea of direct multiple shooting method for solving the optimal
control problems, the proposed method first splits the polygon
PQ
into sub-
polygons
T
i
,for
i
=0
,...,k
, by parallel cutting segments
ʾ
1
,...,ʾ
k
satisfying
following conditions:
ʾ
i
=[
u
i
,v
i
]
ↂD
such that ]
u
i
,v
i
[
ↂ
int
D
,
[
u
i
,v
i
] strictly separates
a
and
b
,
T
i
is bounded by
P
,
Q
, cutting segments
ʾ
i
and
ʾ
i
+1
,
and
SP
(
u
i
,u
i
+1
)
∩
SP
(
v
i
,v
i
+1
)
=
∅
,
∪
i
=0
T
i
where
u
i
∈P
and
v
i
∈Q
, for all
i
=1
,...,k
. We then have that
PQ
=
and int
=
j
.
We initialize a set of shooting points
a
0
T
i
∩
int
T
j
=
∅
,for
i
i
∈
ʾ
i
,for
i
=1
,...,k
. The shooting
point
a
i
∈
∪
i
=0
Z
i
Z
j
:=
ʾ
i
(
j
≥
0) is refined iteratively such that
tends to
SP
(
a, b
), where
a
0
=
a
and
a
k
+1
=
b
and
Z
i
denotes the shortest path between
a
i
and
a
i
+1
in
T
i
(see Fig.
3
). This refinement is finished if the following collinear
condition
SP
(
a
i
,a
i−
1
)
,SP
(
a
i
,a
i
+1
)=
ˀ,
∠
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