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We call this effect the nonholonomic constraint deviation. The goal of this section
is to correct this deviation. If a trajectory is not admissible, the velocity along this
trajectory is not contained in the linear subspace spanned by the
k
control vector
fields and condition (17.1) does not hold.
17.3.4.1
Extended Dynamic System
To take into account this issue, for each configuration
q
,weadd
n
−
k
vector fields
X
k
+1
(
q
)
,···,
X
n
(
q
) to the
k
control vector fields of the system in such a way that
n
. We define the extended system as the system controlled
X
1
(
q
)
,···,
X
n
(
q
) span
R
by all these vector fields:
n
i
=1
u
i
X
i
(
q
)
.
q
=
(17.20)
System (17.20) is not subject to any kinematic constraint. A trajectory
q
(s) of system
(17.20) is admissible for system (17.1) if and only if for any
j
∈{
k
+ 1
,···,
n
}
and
any
s
u
j
(
s
)=0.
In Section 17.2, we deformed a given trajectory, admissible for (17.1) by per-
turbing the input functions
u
1
(
s
∈
[0
,
S
]
,
) of this trajectory in order to avoid
obstacles. In this section, we consider an initial trajectory not necessarily admissi-
ble and we compute input perturbations that make
u
k
+1
(
s
,
τ
)
,···,
u
k
(
s
,
τ
,
τ
)
,···,
u
n
(
s
,
τ
) uniformly
tend toward 0 as
grows.
From now on, we denote by
u
(
s
τ
,
τ
)=(
u
1
(
s
,
τ
)
,···,
u
n
(
s
,
τ
)) the input function
of system (17.20) and by
v
(
s
,
τ
)=(
v
1
(
s
,
τ
)
,···,
v
n
(
s
,
τ
)) the perturbation of these
input functions:
)=
∂
u
i
∂τ
∀
i
∈{
1
,···,
n
},
v
i
(
s
,
τ
(
s
,
τ
)
.
The relation between the input perturbation
v
and the direction of deformation
η
is
similar as in Section 17.2:
η
(
s
)=
A
(
s
)+
B
(
s
,
τ
,
τ
)
η
(
s
,
τ
,
τ
)
v
(
s
,
τ
)
(17.21)
A
(
s
) and
B
(
s
but now,
,
τ
,
τ
) are both
n
×
n
matrices:
n
i
=1
u
i
∂
X
i
A
=
B
=(
BB
⊥
)
q
(
q
) and
(17.22)
∂
where
B
⊥
=(
X
k
+1
(
q
)
X
n
(
q
)) is the matrix the column of which are the additional
vector fields. With this notation, (17.21) can be rewritten as
···
η
(
s
)=
A
(
s
)+
B
⊥
(
s
)
v
⊥
(
s
,
τ
,
τ
)
η
(
s
,
τ
)+
B
(
s
,
τ
)
v
(
s
,
τ
,
τ
,
τ
)
(17.23)
where
v
⊥
(
s
,
τ
)=(
v
k
+1
(
s
,
τ
)
,···,
v
n
(
s
,
τ
)).
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