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p
i
=1
μ
i
λ
i
.
min
p
i
=1
λ
i
E
i
L
2
=1
∑
This is a better approximation than (17.17).
The idea of the computation is to express
in an
L
2
-orthonormal basis in such
a way that the above sum becomes the inner product between two vectors. Let us
build from (
E
1
,···,
η
F
p
) using Gram-Schmidt
orthonormalization procedure. Let
P
be the corresponding
p
E
p
) an orthonormal basis (
F
1
,···,
p
matrix of change
of coordinates (the
j
-th column of P is the vector of coordinates of
F
j
expressed
in (
E
1
,···,
×
E
p
)). If we express
η
in (
F
1
,···,
F
p
) instead of (
E
1
,···,
E
p
), (17.10)
becomes
p
i
=1
λ
i
F
(
s
,
τ
j
)
η(
s
,
τ
j
)=
and (17.16) becomes
i
=1
λ
i
μ
i
=
μ
⊥
|
η
L
2
p
dV
d
(
τ
j
)=
(17.18)
τ
with
S
∂
U
μ
i
j
))
T
F
i
(
s
q
(
q
(
s
,
τ
,
τ
j
)
ds
∂
0
p
i
=1
μ
⊥
=
μ
i
F
i
. The second equality in (17.18) holds since (
F
1
,···,
and
F
p
) is
∑
−
μ
⊥
(
i.e.
−
μ
i
) is the direction
L
2
-orthonormal. At equivalent
L
2
-norm,
η
=
λ
i
=
dV
d
τ
of deformation that minimizes
(
τ
j
). In fact we do not evaluate functions
F
l
's, but
only matrix
P
. The expression of
η
in basis (
E
1
,···,
E
p
) is given by vector
λ
⊥
=
PP
T
0
λ
=
P
λ
.
(17.19)
Using expression of
η
in the orthonormal basis (
F
1
,···,
F
p
), the expression in
(
E
1
,···,
E
p
) of the orthogonal projection of the above
η
over the subspace of vec-
tors satisfying the boundary conditions (17.12) becomes
¯
P
(
LP
)
+
L
)
PP
T
0
λ
=(
I
p
−
λ
Using the above optimal direction of deformation makes the trajectory deformation
algorithm behave much better. It can be explained by the fact that this choice makes
the trajectory scalar value decrease faster and thus is more efficient to get away from
obstacles.
17.3.4
Nonholonomic Constraint Deviation
Approximation (17.5) induces a side effect: after a few iterations, the nonholonomic
constraints are not satisfied anymore and the trajectory becomes non admissible.
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