Appendix A: Possible Sensor Implementations

To implement the sensor based strictly on the instantaneous mapping, we may take continuous video of the entire $(-\phi, \phi)$ windshield sector with markers at $-\phi + \epsilon_{\phi}$ and $\phi - \epsilon_{\phi}$ . It is then possible to directly locate agent $j$ in the windshield from each snapshot. This implementation also enables $i$ to track $j$ via $j$ 's continuous motion trajectory since $j$ is always in the video sequences, even if all agents are indistinguishable based on appearance. Moreover, a sensor in reality does not need to decide $\epsilon_{\phi}$ : a physical vehicle has an actual size hence will not disappear from the windshield at a single instant in time. Suppose agent $j$ starts to move outside of $i$ 's field-of-view; $j$ will actually ``stay'' on $i$ 's windshield boundary for some $\Delta t$ time before it disappears from $i$ 's view, which automatically provides $\epsilon_{\phi}$ for agent $i$ . This sensor implementation is illustrated in Fig. 7 (a): arcs $A_l, A_c, A_r$ correspond to the $(-\phi, - \phi + \epsilon_{\phi}), (-\phi + \epsilon_{\phi}, \phi - \epsilon_{\phi}), (\phi - \epsilon_{\phi}, \phi)$ sectors of the windshield, respectively. In fact, the left boundary of $A_l$ and the right boundary of $A_r$ in the sensor implementation are not essential; they can be extended and in the extreme case meet at the back of the vehicle (point $B$ in Fig. 7(b)). In this case, another instantaneous sensor is obtained, with its $2\pi$ angular field-of-view partitioned into three slices.

Figure 7: Three possible implementations of the sensing model.

Alternatively, the sensor may be implemented as simple beam detectors placed at the two windshield boundaries, $-\phi + \epsilon_{\phi}$ and $\phi - \epsilon_{\phi}$ ; we may simply take $\epsilon_{\phi}$ to be a fraction of $\phi$ (see Fig. 7(c)). Except at the two boundaries, the agent is otherwise ``blind''. Such a sensor, even simpler than previous ones, generates time based events and thus needs to have some immediate history of which sector the target is located in to produce the output. It also needs to have some additional mechanism to distinguish the agents that pass the beam detector, since the sensor no longer continuously tracks a target agent.