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These coefficients represent the variation of the trajectory scalar value induced by
each direction of deformation
E
i
. With these coefficients, (17.15) can be rewritten as
p
i
=1
λ
i
μ
i
.
dV
d
(
τ
j
)=
(17.16)
τ
Thus, if we choose
λ
i
=
−
μ
i
(17.17)
we get a trajectory deformation
.,
τ
j
) that keeps the kinematic constraints satisfied
and that makes the trajectory scalar value decrease. Indeed:
η
(
p
i
=1
μ
dV
d
2
i
(τ
j
)=
−
≤
0
.
τ
0
0
We denote by
λ
this value of vector
λ
. Of course, nothing ensures us that
λ
satisfies the boundary conditions (17.13).
17.3.3.1
Projection over the Subspace of Boundary Conditions
(17.13) states that the set of vectors
λ
satisfying the boundary conditions is a linear
¯
p
. To get such a vector that we denote by
0
subspace of
R
λ
, we project
λ
over this
subspace:
¯
L
+
L
)
0
λ
=(
I
p
−
λ
where
I
p
is the identity matrix of size
p
and
L
+
is the Moore-Penrose pseudo-inverse
of matrix
L
.
It can be easily verified that
¯
λ
satisfies the following properties:
1.
L
¯
λ
= 0;
p
i
=1
¯
2.
η
=
∑
λ
i
E
i
makes the trajectory scalar value decrease.
17.3.3.2
A Better Direction of Deformation
Let us recall that (17.5) is an approximation of order 1 with respect to
τ
. For this
reason,
Δτ
j
η
∞
with
η
∞
max
s
∈
[0
,
S
]
η
(
s
,
τ
j
)
needs to be small.
Δτ
j
is thus
chosen in such a way that
Δ
τ
j
η
∞
is upper bounded by a positive given value
η
max
. The way the
λ
i
's are chosen in (17.17) is not optimal in this respect. Indeed,
the goal we aim at at each iteration is to make the trajectory scalar value
V
decrease
at most for constant
η
∞
. Therefore the optimal value of
λ
realizes the following
minimum:
p
i
=1
μ
i
λ
i
dV
d
min
η
∞
(
τ
j
)=
min
τ
p
i
=1
λ
i
E
i
∞
=1
∑
=1
Unfortunately, this value of vector
λ
is very difficult to determine since
.
∞
is not
a Euclidean norm. Instead, we compute
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