(7)
Since ki is a control gain parameter, it is seen that the distributed controller eq. (4)
and eq. (7) of robot Ri are essentially the same. We have therefore proved the global
convergence of the mobile sensor network controller based on the virtual potential
field and Delaunay triangulation method. The distributed controller of robot Ri based
on its one-hop neighbors leads to the global convergence of the entire system.
3.2 Convergence Analysis with Topological Events
In a dynamic Delaunay triangulation, the location of the generators varies with respect
to time. The shape of the triangulation is continuous with respect to the motion
of the robots until topological events occur [31]. A topological event for a robot
occurs when it loses a neighbor or gains a new neighbor. Four generators are cocircular
[31] when a topological event occur. As shown in Figure 4, points A,B,C
72 Jindong Tan
and D are co-circular. Sufficiently small continuous changes of point A will lead to
topological changes in the Delaunay triangulation. However, the topological changes
only involve the swap of the triangles ?–?ABD and ?–?BCD to triangles ?–?ABC and
?–?ACD. The perimeter of the polygon ABCD is not changed, and the topological
event does not affect the rest of the topological structure.
According to the definition of the virtual potential function, the continuity of Vi
for a robot vehicle depends on the number of its neighbors.
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