For the situation shown in Figure 7
(b), the equilibrium on the circle is an unstable equilibrium. The node will converge
to another equilibrium. In summary, the convergence of the controller is guaranteed
even with topological events.
Chapter 3 A Scalable Graph Model and Coordination Algorithms 75
3.3 Obstacles and Constrained Environments
The Delaunay triangulation-based, virtual potential field method can incorporate
with constrained environments and dynamic obstacles. For an environment such as
shown in Figure 8, additional links are added based on the sensor information of the
robots. The additional links are used to define the virtual energy function for robot
Ri. The energy of the system with obstacles can be described as:
Vi =
1
2
mi
X
j=1
kij(kpijk ??’ ci j)2
+
1
2
mo
X
l=1
kil(kpi lk ??’ ci l)2,
where kil is a parameter, mo is the total number of obstacles in the sensing range of
robot Ri, kpi lk and ci l are the actual and the desired distance between the robot
and an obstacle. It is worthing noting that the definition is still distributed. A robot
Ri consider its one-hop neighbors and the obstacles in its own sensing range.
R2
R3
R1
R4
R5
x
1
y
obstacle
Fig. 8. Delaunay Triangulation in a constrained environment.
The virtual potential force of node Ri can then be derived as:
Fi =
@Vi
@pi
=
mi
X
j=1
kij(||pi j || ??’ ci j)
pi j
kpi jk
+
mo
X
j=1
kil(||pi l|| ??’ ci l)
pi l
kpi lk
.
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