Rendezvous without connectivity or distinguishability

In the set of sufficient conditions stated in our earlier theorems that guarantees rendezvous, we have assumed that the assignment graph $G$ has a single connected component initially. We now show that the assumption of single connected assignment graph and agents stopping are not necessary by: 1. Allowing agents to perform reassignment. 2. Allowing agent without targets to merge with other agents within distance $\rho$ . Initially, the assignment graph $G$ has $n$ components: each component contains a single agent. Look at one such component $G_i$ with a single agent $i$ . If there are other agents within distance $\rho$ of $i$ , they will merge with $i$ , therefore, we may assume that there are no other agents within distance $\rho$ of agent $i$ . To rendezvous with other agents, let agent $i$ pick any target in its windshield and if it cannot find any, let it start turning. We will show that with two full turns, it can find agents at least $\rho$ away from it or confirm that there are no such agents to conclude rendezvous is achieved. We call this procedure of finding new targets reassignment. We have the following:

Proposition 17   By allowing merging and reassignment for agents that have no targets, bounded speed pursuit of $n$ Dubins car agents will achieve rendezvous without the initial connectivity requirement, provided that the system will rendezvous if given a connected initial assignment graph and

$$\omega_i \ge \frac{3v_{max}}{\rho}.$$ (41)

Our sensing model (see Appendix A) may do without agents being distinguishable if agents are capable to take continuous videos of their targets in windshield. For agents moving at unit speed, such capability is not necessary provided that when an agent is confused by two possible targets that are colinear with it, it can follow the target that is closer. Such switching is a natural one since a closer target will block a more distant one, effectively removing the assumption requiring no visibility occlusion. Alternatively, if all agents simply send out a beacon signal, a closer agent will have a stronger signal than a more distant one. We have:

Proposition 18   By allowing merging and reassignment for agents that have no targets, as well as the ability for an agent to choose a closer agent as target when two possible targets become colinear with it, unit speed speed pursuit of $n$ Dubins car agents will achieve rendezvous with $\omega_i \ge 3/\rho$ and windshield angle satisfying (27), without initial connectivity or agent distinguishability.

PROOF. Denoting agent $i$ and its two possible targets $t_{near}$ and $t_{far}$ , when such switching of targets happens, $V$ of the system cannot get larger since the affected term in $V$ during a target switching will have $l_{i, t_{near}} \le l_{i, t_{far}}$ and $l_{i, t_{near}}$ is always chosen. At the same time, Theorem 12 guarantees that $\dot{V}$ remains negative.  Proposition 18 also implies that agents will no longer need to stop before they all rendezvous.