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In-Depth Information
∂
u
∂τ
v
(
s
,
τ
)
(
s
,
τ
)
∂
q
∂τ
η
(
s
,
τ
)
(
s
,
τ
)
.
With this notation, the above equation becomes
η
(
s
,
τ
)=
A
(
s
,
τ
)
η
(
s
,
τ
)+
B
(
s
,
τ
)
v
(
s
,
τ
)
(17.3)
where
A
(
s
,
τ
) is the following
n
×
n
matrix:
k
i
=1
u
i
(
s
,
τ)
∂
X
i
A
(
s
,
τ
)=
q
(
q
(
s
,
τ
))
∂
,
τ
and
B
(
s
) is the
n
×
k
matrix the columns of which are the control vector fields:
)=
X
1
(
q
(
s
))
,
τ
,
τ
))
···
X
k
(
q
(
s
,
τ
.
B
(
s
According to (17.3), the derivative with respect to
τ
of the trajectory of parameter
τ
is related to the input perturbation through a linear dynamic system. This system
is in fact the linearized system of (17.1) about the trajectory of parameter
τ
:
s
→
q
(
s
,
τ
). For a given trajectory
q
(
s
,
τ
) of input
u
(
s
,
τ
) and for any input perturbation
v
(
s
) we can integrate (17.3) with respect
to
s
to get the corresponding direction of deformation
,
τ
), and any initial condition
η
0
=
η
(0
,
τ
).
A trajectory deformation process for nonholonomic systems can thus be consid-
ered as a dynamic control system where:
η
(
s
,
τ
•
τ
is the time;
•
s
→
q
(
s
,
τ
) is the state;
•
(
η
0
,
s
→
v
(
s
,
τ
) is the input.
17.2.3
Potential Field and Inner Product
The trajectory deformation method produces at each time
τ
a vector
η
0
and a func-
tion
s
S
] in such a way that the deformation process achieves a
specified goal. This goal is expressed in terms of a scalar value to minimize over the
set of feasible trajectories. The scalar value associated to a trajectory is defined by
integration of a potential field
U
over the configuration space. We denote by
V
(
→
v
(
s
,
τ
) over [0
,
τ
)
→
,
τ
the potential value of trajectory
s
q
(
s
):
S
,
τ
.
V
(
τ
)
U
(
q
(
s
))
ds
0
If the goal to achieve is to avoid obstacles, as in [18, 7, 1], the configuration space
potential field is defined in such a way that the value is high for configurations
close to obstacles and low for configuration far from obstacles. Thus trajectories
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