Then the control input of the robot Ri is derived by:
ui = ??’
@Vi
@pi
??’
@Vi
@vi
= ??’
mi
X
j=1
ki(||pi j || ??’ ci j)
pi j
kpi jk
??’ kivvi
= ??’Fi ??’ kivvi,
(4)
where Fi = {Fix, Fiy} is the virtual potential force generated by the one-hop neighbors
of Ri. Here
Fi =
mi
X
j=1
ki(||pi j || ??’ ci j)
pi j
kpi jk
. (5)
in the mobile sensor network, a performance index(candidate For a mobile robotR
Chapter 3 A Scalable Graph Model and Coordination Algorithms 71
If all the other robots but Ri are static, it is easy to prove that the above controller
is globally convergent. Considering the mobile sensor network as a single system,
the performance index in terms of the total kinetic and virtual potential energy of the
system can be described as
V =
n
X
i=1
Vi =
1
2
n
X
i=1
mi
X
j=1
ki(||pij || ??’ ci j)2 +
1
2
n
X
i=1
kvik2. (6)
The control input vector u of the mobile sensor network is derived based on
the total energy function V . The virtual force Fi of robot Ri considering the entire
system is derived by:
@V
@pi
=
@Vi
@pi
+
mi
X
j=1
@Vj
@pi
=
@Vi
@pi
+
mi
X
j=1
kj(||pj i|| ??’ cj i)
??’pj i
kpj ik
=
@Vi
@pi
+
mi
X
j=1
ki(||pi j || ??’ ci j)
pi j
kpi jk
= 2
@Vi
@pi
.
Here we assume ki = kj . The control input for robot Ri considering the entire
system can then be derived by:
ui = ??’
@V
@pi
??’
@V
@vi
= ??’
mi
X
j=1
2ki(||pi j || ??’ ci j)
pi j
kpi jk
+ kivvi.
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