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Table 17.1
Trajectory deformation algorithm: at each step, the direction of deformation
η(
s
,
τ
j
) is computed given the current trajectory
q
(
s
,
τ
j
) and the potential field defined by
obstacles
Algorithm :
Trajectory deformation for nonholonomic systems
/* current trajectory = initial trajectory */
j
= 0; τ
j
= 0
while q
(
s
,
τ
j
) in collision
{
compute
A
(
s
,
τ
j
) and
B
(
s
,
τ
j
) for
s
∈
[0
,
S
]
/* correction of nonholonomic deviation */
for
k
in
{
k
+ 1
,...,
n
}{
compute
u
i
(
s
,
τ
j
)
compute
v
i
(
s
,
τ
j
)=
−
α
u
i
(
s
,
τ
j
)
compute η
1
(
s
,
τ
j
) using (17.24)
}
/* potential gradient in configuration space */
for
i
in
,...,
p
}{
compute
E
i
(
s
{
1
,
τ
j
) by integrating (17.26)
}
compute
∂
U
∂
q
(
q
(
s
,
τ
j
)) for
s
∈
[0
,
S
]
for
i
in
{
1
,...,
p
}{
compute λ
=
−
0
∂
U
i
q
(
q
(
s
,
τ
j
))
T
E
i
(
s
,
τ
j
)
ds
∂
}
/* orthonormalization*/
compute matrix
P
using Gram-Schmidt procedure
/* projection of
λ
over boundary conditions */
¯
λ =
−
P
(
LP
)
+
η
1
(
S
,
τ
j
)+(
I
p
−
P
(
LP
)
+
L
)λ
compute
/* compute and apply deformation */
compute η(
s
,
τ
j
)=
p
i
=1
¯
λ
i
E
i
(
s
,
τ
j
) for
s
∈
[0
,
S
]
q
(
s
,
τ
j
)
←
q
(
s
,
τ
j
)+Δτ η(
s
,
τ
j
) for
s
∈
[0
,
S
]
∑
}
Table 17.1 summarizes an iteration of the trajectory deformation algorithm for
nonholonomic systems.
17.4
Application to Mobile Robot Hilare 2 Towing a Trailer
In this section, we briefly illustrate the developments of the previous section by
applying them to mobile robot Hilare 2 towing a trailer (see Figure 17.2). We refer
the reader to [8] for more details.
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