Sensing model and control law

The vehicle's sensor is a quantized variant of bearing-only sensors (see, e.g. [36]). It has a limited angular field-of-view, centered at $\psi_i$ with a span $(-\phi, \phi)$ for some given $\phi \in (0, \pi]$ , which is the same and fixed for all agents in a system (see Fig. 2(b)). By imagining that one sits in the driver's seat, the field-of-view can be considered as a windshield. The sensing range of each agent should be large enough to allow it to track its target until rendezvous occurs, which is certainly guaranteed if the range is unlimited. However, a bounded range is sufficient as a consequence of Proposition 10 and a precise bound can be calculated given the agents' initial configuration. Initially, we assume that the sensor can follow a target in the windshield: An agent cannot occlude another in terms of sensor view. This assumption will be lifted for identical agents in Section 6.

In the previous section we mentioned that agents will try to maintain their targets in the windshield; this appears to require an initial condition in which each agent has its target in the windshield. Assume such an initial condition for the moment; we later show that this requirement is not necessary in Section 6. For an agent $i$ , let its target $j$ initially reside in the $(-\phi, \phi)$ sector of $i$ 's windshield. Let $\epsilon_{\phi}$ be a tiny angle satisfying $0 < \epsilon_{\phi} \ll \phi$ . The introduction of $\epsilon_{\phi}$ provides a way to maintain $j$ in $i$ 's windshield of span $(-\phi, \phi)$ : agent $i$ can notice when $j$ is about to disappear from $i$ 's windshield and start turning to position $j$ towards the center of $i$ 's windshield. As long as $i$ turns fast enough, $j$ will not leave $i$ 's windshield. Assuming $i$ has a single target $j$ , the observation space can be restricted as $Y_i = \{-1, 0, 1\}$ and an observation $y_i$ for agent $i$ is obtained as

$$y_i = \left\{ \begin{array}{rl} 1 & \textrm{agent $j$ appears in... ...ars in $i$'s $(-\phi, -\phi + \epsilon_{\phi})$ sector,} \end{array} \right.$$ (2)

which defines a simple instantaneous mapping $h: X \to Y_i$ . While the mapping given by (2) is not defined on all of $X$ , the part of $X$ on which $h$ is undefined is safe to ignore because of our temporary assumption that agents can keep targets in their windshield span. For each agent $i$ , the sensor does not provide metric information, but instead indicates one of three simple quantized states with respect to some agent $j$ and the windshield. Several possible implementations of the above sensing model are discussed in Appendix A. Our sensor-feedback control law $k_i: Y_i \rightarrow U_i$ is then defined simply as

$$u_i = \omega_i y_i.$$ (3)

Additionally, if an agent has no target, we assume that it does not move. We later remove this extra assumption for identical agents.