A drastic change occurs
when it loses or gains a neighbor. The continuity of the virtual force is therefore
affected by the topological events. The virtual potential energy and force have an
abrupt change during the motion, which results in discontinuous motion trajectories.
It can be explained by using Figure 4. Assuming the position of node A??? and A??????
are in a sufficiently small region of node A. The potential energy for node A??? and
A?????? is denoted by VA??? and VA?????? respectively. The link A??????C may significantly change
the value of VA?????? and therefore the virtual force on node A. The stability of node A
should be analyzed when a topological event occurs. In the following analysis, VA???
and VA?????? represent the candidate Lyapunov functions for the left and right topology
structure of Figure 4, which are denoted as topology A??? and A?????? respectively.
Fig. 4. A topological event occurs when a node loses or gains a neighbor.
In the following analysis, the co-circular region is denoted by ?, the region outside
of the circle is denoted by ??? . The two equilibrium points for candidate Lyapunov
functions VA??? and VA?????? are denoted by E??? and E?????? respectively. We have shown that
the point A will converge to an equilibrium if there is no topological event. Node A
will converge to E??? under a controller derived by VA??? ; it will converge to E?????? under
a controller obtained by VA?????? , as shown in Figure 4.
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