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allows to consider simultaneously optimality and inequality constraint (visibility).
A numerical method is employed for solving the path-planning problem in the vari-
ational form.
7.2
Preliminaries
This section briefly recalls some mathematical backgrounds on the rotational group,
the standard perspective projection and the associated two views geometry. It also
presents the essential of the path planning method proposed in [23].
7.2.1
Brief Review of
SO
(3)
The group
SO
(3) is the set of all 3
3 real orthogonal matrices with unit determinant
and it has the structure of a Lie group. On a Lie group, the space tangent to the
identity has the structure of a Lie algebra. The Lie algebra of
SO
(3) is denoted by
so
(3). It consists of the 3
×
×
3 skew-symmetric matrices, so that the elements of
so
(3)
are matrices of the form
⎡
⎤
0
−
r
3
r
2
⎣
⎦
.
[
θ]=
r
3
0
−
r
1
−
r
2
r
1
0
One of the main connections between a Lie group and its Lie algebra is the expo-
nential mapping. For every
R
∈
SO
(3), there exists at least one [
θ
]
∈
so
(3) such that
θ
e
[
]
=
R
with (Rodriguez formula)
]
=
I
+
sin
θ
θ
]+
1
−
cos
θ
θ
R
=
e
[
]
2
[
θ
[
θ
,
(7.1)
2
θ
θ
∈
where
is the standard Euclidean norm. Conversely, if
R
SO
(3) such that
−
Trace(
R
)
=
1, then
θ
2sin
R
T
)
[
θ
]=log(
R
)=
(
R
−
,
(7.2)
θ
where
θ
satisfies
= arccos
Trace(
R
)
−
1
.
θ
=
θ
(7.3)
2
If Trace(
R
)=
−
1, log(
R
) can be obtained noticing that
θ
=
±
π
u
where
u
is a unit
length eigenvector of
R
associated with the eigenvalue 1.
Another important connection between
so
(3) and
SO
(3) involves angular veloci-
ties. If
R
(
t
) is a curve in
SO
(3),then
RR
T
and
R
T
R
are skew-symmetric, and hence
element of
so
(3). The element
ω
of
so
(3) such that
]=
R
T
R
[
ω
(7.4)
corresponds to the angular velocity of the rigid body.
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