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17.3.2
Boundary Conditions
We wish the deformation process not to modify the initial and goal configurations
of the trajectory. We thus impose the following boundary conditions:
∀
j
>
0
,
q
(0
,
τ
j
)=
q
(0
,
0)
q
(
S
,
τ
j
)=
q
(
S
,
0)
.
These constraints are equivalent to
∀
j
>
0
,
η
(0
,
τ
j
)=0
(17.11)
η
(
S
,
τ
j
)=0
.
(17.12)
(17.8) and (17.10) ensure us that the first constraint (17.11) is satisfied. The second
constraint (17.12) together with (17.10) becomes a linear constraint over vector
λ
:
L
λ
= 0
(17.13)
where
L
is a
n
×
p
-matrix the columns of which are the
E
i
(
S
,
τ
j
)'s:
L
=
E
1
(
S
.
,
τ
j
)
···
E
p
(
S
,
τ
j
)
Let us notice that in general, the dimension of the subspace of solutions of the above
linear system is equal to
p
−
n
and therefore
p
must be bigger than
n
. The problem is
now to choose a vector
satisfying the above linear constraint and generating a di-
rection of deformation that makes the current trajectory move away from obstacles.
We address this issue in the following section.
λ
17.3.3
Direction of Deformation That Makes Trajectory Scalar
Value Decrease
As explained in Section 17.2.3, a potential field
U
is defined over the configuration
space. This potential field defines by integration a scalar valued function
V
over the
space of trajectories.
Given a vector
p
, the variation of the trajectory scalar value induced by
λ
∈
R
direction of deformation
η
defined by (17.10) is given by
τ
j
)=
S
0
dV
d
∂
U
∂
,
τ
j
))
T
(
q
(
q
(
s
η
(
s
,
τ
j
)
ds
(17.14)
τ
i
=1
λ
i
S
p
∂
U
,
τ
j
))
T
E
i
(
s
=
q
(
q
(
s
,
τ
j
)
ds
.
(17.15)
∂
0
Let us define the coefficients
S
∂
U
∂
,
τ
j
))
T
E
i
(
s
μ
i
q
(
q
(
s
,
τ
j
)
ds
.
0
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