Appendix C: Direct cyclic pursuit

If the windshield of an agent is a single point ($\phi=0$ ), every agent moves directly toward its target, and we obtain a version of the classic cyclic pursuit, or $n$ -bug problem. We call this variety the direct cyclic pursuit problem to distinguish it from the Dubins car cyclic pursuit. If we are not sure whether such pursuit is cyclic, we call it direct pursuit. Such cases can be viewed as limiting cases as the windshield goes from an open interval to a point. Here we show that our approach also applies to this special case. Allowing the pursuit formation to be an arbitrary polygon and using the general form of $\theta _i$ defined for self-intersecting polygon, (9) becomes:

$$\dot{V} = \displaystyle\sum _i -v_i(1 + \cos \theta_i),$$ (51)

with $0 < \theta_i \le \pi$ and $\sum \theta_i \le (n-2)\pi$ . Since for at least one $i$ , $\theta_i \le \frac{n-2}{n}\pi$ , $\dot{V}$ is always strictly negative for fixed $n$ and any set of $v_i \ge \epsilon$ for some fixed $\epsilon > 0$ . The intree and cycle plus branch cases also follow. We obtain the following corollary:

Corollary 19   Direct pursuit of $n$ agents with lower bounded speed and arbitrary connected, single-target assignment graph will rendezvous in finite time.

We can say a little more if $v_i = 1$ (or any constant) holds for all agents:

Corollary 20   Unit speed direct pursuit of $n$ agents with connected, single-target assignment graph has the property

$$\dot{V} \le \max\Big\{-1, -n(1 + \cos(\dfrac{n-2}{n}\pi))\Big\} =: -\delta,$$ (52)

and will rendezvous in time no more than $V_0/\delta$ (recall that $V_0$ is the value of $V$ at $t=0$ ).

PROOF. Writing $\dot{V} = h + f$ as that of (13), we readily see that $h = -n$ for such systems. (52) is then easily obtained by substituting in (22).